Introduction to the Mathematical Theory of Compressible Flow (Oxford Lecture Series in Mathematics and Its Applications, 27)

Oxford Lecture Series in Mathematics and its Applications 27 Series editors John Ball Dominic Welsh OXFORD LECTURE SE...

Author: Antonin Novotny | Ivan Straskraba

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Oxford Lecture Series in Mathematics and its Applications 27 Series editors John Ball Dominic Welsh

OXFORD LECTURE SERIES IN MATHEMATICS AND ITS APPLICATIONS 1. J. C. Baez (ed.): Knots and quantum gravity 2. I. Fonseca and W. Gangbo: Degree theory in analysis and applications 3. P.-L. Lions: Mathematical topics in fluid mechanics, Vol. 1: Incompressible models 4. J. E. Beasley (ed.): Advances in linear and integer programming 5. L. W. Beineke and R. J. Wilson (eds): Graph connections: Relationships between graph theory and other areas of mathematics 6. I. Anderson: Combinatorial designs and tournaments 7. G. David and S. W. Semmes: Fractured fractals and broken dreams 8. Oliver Pretzel: Codes and algebraic curves 9. M. Karpinski and W. Rytter: Fast parallel algorithms for graph matching problems 10. P.-L. Lions: Mathematical topics in fluid mechanics, Vol. 2: Compressible models 11. W. T. Tutte: Graph theory as I have known it 12. Andrea Braides and Anneliese Defranceschi: Homogenization of multiple integrals 13. Thierry Cazenave and Alain Haraux: An introduction to semilinear evolution equations 14. J. Y. Chemin: Perfect incompressible fluids 15. Giuseppe Buttazzo, Mariano Giaquinta and Stefan Hildebrandt: One-dimensional variational problems: an introduction 16. Alexander I. Bobenko and Ruedi Seiler: Discrete integrable geometry and physics 17. Doina Cioranescu and Patrizia Donato: An introduction to homogenization 18. E. J. Janse van Rensburg: The statistical mechanics of interacting walks, polygons, animals and vesicles 19. S. Kuksin: Hamiltonian partial differential equations 20. Alberto Bressan: Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem 21. B. Perthame: Kinetic formulation of conservation laws 22. A. Braides: Gamma-convergence for beginners 23. Robert Leese and Stephen Hurley: Methods and algorithms for radio channel assignment 24. Charles Semple and Mike Steel: Phylogenetics 25. Luigi Ambrosio and Paolo Tilli: Topics on analysis in metric spaces 26. Eduard Feireisl: Dynamics of viscous compressible fluids 27. Anton´ın Novotn´ y and Ivan Straˇskraba: Introduction to mathematical theory of compressible flow

Introduction to the Mathematical Theory of Compressible Flow A. Novotn´ y Université du Sud Toulon–Var

I. Straˇskraba Mathematical Institute of the Academy of Sciences of the Czech Republic

1

3 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan South Korea Poland Portugal Singapore Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press 2004 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available ISBN 0-19-853084-6 (Hbk) Typeset by the author in LATEX Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk

For Stania, Filip and Jakub For Jana

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PREFACE The topic of this book is a mathematical description of Newtonian compressible fluids in the steady and unsteady regime. The history of attempts to describe rigorously the flow of a compressible fluid covered a long time beginning from observations of L. Euler in the middle of the 18th century and of C. Navier (Navier, 1827), H. Poisson (Poisson, 1829) and G. Stokes (Stokes, 1845) in the first half of the 19th century, and continue up to now. Despite the fact that the governing equations, called the Euler equations (in the inviscid case) and Navier– Stokes equations (in the viscous case), have been known for a very long time we are far from being satisfied with the completeness of their mathematical analysis (in both cases). Nevertheless the considerable effort of outstanding analysts cited throughout the book brought its fruits and a great number of nontrivial results for compressible fluids has been achieved. This book is an attempt to map the situation in the mathematical theory of compressible flow and present important and up-to-date results in a clear instructive form accessible to a wide audience despite the sophisticated techniques used to overcome modest a priori information derived directly from the equations. We started with the realization of a rather different project including also a numerical treatment of the Euler and Navier–Stokes equations with M. Feistauer and J. Felcman around 1998. It soon appeared that the scope was too large for one monograph and so we accepted with great relief the proposal from Oxford University Press to split the book into two separate monographs, the first of which (Feistauer et al., 2003) has already been published. As already mentioned, the book covers Newtonian compressible fluids, more specifically Euler equations and Navier–Stokes equations in isentropic or barotropic regimes. We do not deal with heat conducting flows except for references to results for small data (that is under the assumption that appropriate norms of the given quantities are small enough). There is currently beeing published a research monograph by E. Feireisl (Feireisl, 2003a) devoted to this subject. There is a vast literature about different kinds of non-Newtonian fluids and we do not go into this business at all. So in this respect the present monograph covers only a part of the available mathematical results for compressible fluids. We have adopted a textbook style. Even well-known basic theorems are recalled in the introductory chapter. This makes the book essentially selfcontained. There are sections called heuristic approach, where we describe the main ideas of proofs. These sections may be sufficient for an experienced reader to understand the subject without wasting time in numerous details. On the other hand, less experienced readers, nonspecialists and students will find in the book even standard technical details. Let us briefly describe the contents. We start with the introductory Chapter 1,

viii

PREFACE

where different models for compressible fluids are derived and some fundamental mathematical results are surveyed. The results are given without proofs but detailed references are given there. Chapter 2 surveys the theoretical aspects of the Euler system for inviscid compressible fluids with the necessary background from the theory of hyperbolic conservation laws. Representative local and global existence results are proved in detail, relying on recent publications, to give the present state of affairs. The generality here is modest, and this is mostly due to the lack of results for the Euler equations. Chapter 3 is preparatory for the subsequent treatment of Navier–Stokes systems. Some specific mathematical tools for these equations, adjusted especially to steady equations, are developed here. The proofs, unlike Chapter 1, are mostly given and only in a few cases are they cited. Chapter 4 is devoted to the theory of weak solutions for steady Navier–Stokes systems for compressible fluids with large data (that is without the restriction described above as the assumption of small data) in the barotropic regime. A complete and detailed proof of the existence of weak solutions is given and modifications for unbounded and exterior domains as well as for different boundary conditions are discussed thoroughly. A survey of known results on this issue is given in the bibliographic remarks. Chapter 5 concerns strong solutions of steady Navier–Stokes equations. The existence of regular solutions is proved, paid for by the assumption of small data. Note that unconditional regularity of solutions both for steady and nonsteady equations is not known. Chapter 6 again collects advanced mathematical tools, now adjusted to nonsteady problems. This includes properties of abstract functions in Bochner spaces, commutators and the study of the (renormalized) equation of continuity. Chapter 7 is mainly devoted to the weak existence theory for nonsteady Navier–Stokes equations in the barotropic regime. Again a discussion of different regions and boundary conditions is included. In Chapter 8, the global behavior of solutions in time is investigated and the related equilibrium problem is described. In the final Chapter 9, the existence of strong solutions for nonsteady Navier– Stokes equations is studied and available existence and uniqueness results are reviewed. Chapter 2 dealing with the Euler equations is essentially self-contained and can be read independently of the other chapters. The same is true for Chapter 4 which deals with weak solutions of steady barotropic Navier–Stokes equations (and which requires only Chapter 3 to be exhaustive) and for Chapter 7 dealing with weak solutions of nonsteady barotropic Navier–Stokes equations (which needs only Chapter 3 and Section 4.4 of Chapter 4 to form a complete treatment). Each of Chapters 5 (about strong solutions for steady equations), 8 (about large time behavior of weak solutions) and 9 (about strong solutions in the nonsteady regime) are essentially self-contained as well. Also, we attempt to treat all

PREFACE

ix

investigated systems in a uniform way taking into account their common nature. We are grateful to E. Feireisl, G. P. Galdi, J. Heywood, P. Krejˇc´ı, V. Lovicar, J. M´ alek, S. Nazarov, J. Neˇcas, J. Neustupa, S. Novo, M. Padula, P. Penel, H. Petzeltová, K. Pileckas, M. Pokorn´ y, M. R˚ uˇziˇcka, R. Salvi, A. Sequeira, A. Valli and A. Zlotnik, who are coauthors with at least one of us of several papers. We enjoyed working with them on more than one problem related to the subject. In particular, we wish to express warm thanks to our friends who helped us to manage the task of the book project by reading the manuscript and with numerous valuable discussions; in alphabetical order: E. Feireisl, J. Heywood, S. Novo, M. Pokorn´ y. Warm thanks also to S. Novotn´ a who typeset, with care, part of the manuscript and compiled the bibliography. Secondly, we would like to acknowledge three stays in Mathematisches Forschungsinstitut in Oberwolfach in the program Research in Pairs which helped us very much in the coordination of our work. We also thank the Grant Agency of the Czech Republic for financial support from projects nos. 201/02/0684, 201/02/0854, and the Czech and French Ministry of Education for support in the frame of project Barrande 99, no. 99004. Our special thanks go, of course, to our families, without whose support we would not have been able to complete this extensive project.

Carqueiranne and Prague, September 2003

A. N. & I. S.


CONTENTS

1

Fundamental concepts and equations 1.1 Some mathematical concepts and notation 1.1.1 Basic notation 1.1.2 Some useful inequalities in IRN 1.1.3 Differential operators 1.1.4 Gronwall’s lemma 1.1.5 Implicit functions 1.1.6 Transformations of Cartesian coordinates 1.1.7 H¨ older-continuous and Lipschitz functions 1.1.8 The symbols “o” and “O” 1.1.9 Partitions of unity 1.1.10 Measure 1.1.11 Description of the boundary 1.1.12 Measure on the boundary of a domain 1.1.13 Classical Green’s theorem 1.1.14 Lebesgue spaces 1.1.15 Lebesgue’s points 1.1.16 Absolutely continuous functions 1.1.17 Absolute continuity of integrals with respect to measurable subsets 1.1.18 Some theorems from integration theory 1.2 Governing equations and relations of gas dynamics 1.2.1 Description of the flow 1.2.2 The transport theorem 1.2.3 The continuity equation 1.2.4 The equations of motion 1.2.5 The law of conservation of the moment of momentum. Symmetry of the stress tensor 1.2.6 Inviscid and viscous fluids 1.2.7 The energy equation 1.2.8 The second law of thermodynamics and the entropy 1.2.9 Principle of material frame indifference 1.2.10 Newtonian fluids 1.2.11 Conservative and dissipation form of the energy equation for Newtonian fluids 1.2.12 Entropy form of the energy equation for Newtonian fluids

1 1 1 3 3 5 5 6 6 7 7 7 8 8 9 10 11 12 12 13 15 16 17 19 19 21 21 22 22 23 24 24 25

xii

CONTENTS

1.3

1.4

2

1.2.13 Some consequences of the Clausius–Duhem inequality 1.2.14 Equations of state 1.2.15 Adiabatic flow of a perfect inviscid gas 1.2.16 Compressible Euler equations 1.2.17 Compressible Navier–Stokes equations for a perfect viscous gas 1.2.18 Barotropic flow of a viscous gas 1.2.19 Speed of sound 1.2.20 Simplified models 1.2.21 Initial and boundary conditions Some advanced mathematical concepts and results 1.3.1 Spaces of H¨ older-continuous and continuously differentiable functions 1.3.2 Young’s functions, Jensen’s inequality 1.3.3 Orlicz spaces 1.3.4 Distributions 1.3.5 Sobolev spaces 1.3.6 Homogeneous Sobolev spaces 1.3.7 Tempered distributions 1.3.8 Radon measure and representation of CB (Ω)∗ 1.3.9 Functions of bounded variation 1.3.10 Functions with values in Banach spaces 1.3.11 Sobolev imbeddings of abstract spaces 1.3.12 Some compactness results Survey of concepts and results from functional analysis 1.4.1 Linear vector spaces 1.4.2 Topological linear spaces 1.4.3 Metric linear space 1.4.4 Normed linear space 1.4.5 Duals to Banach spaces and weak(-∗) topologies 1.4.6 Riesz representation theorem 1.4.7 Operators 1.4.8 Elements of spectral theory 1.4.9 Lax–Milgram lemma 1.4.10 Imbeddings 1.4.11 Solution of nonlinear operator equations

Theoretical results for the Euler equations 2.1 Hyperbolic systems and the Euler equations 2.1.1 Zero-viscosity Burgers equation 2.1.2 One-dimensional Euler equations 2.1.3 Lagrangian mass coordinates 2.1.4 Symmetrizable systems

25 26 27 28 28 29 30 30 31 32 33 33 34 35 40 47 50 52 52 53 57 58 60 60 60 62 62 64 68 68 70 70 71 71 74 74 75 76 76 77

CONTENTS

2.2

2.3

2.1.5 Matrix form of the p-system 2.1.6 The Euler equations of an inviscid gas Existence of smooth solutions 2.2.1 Hyperbolic systems and characteristics 2.2.2 Cauchy problem for system of conservation laws 2.2.3 Linear scalar equation 2.2.4 Solution of a linear system 2.2.5 Nonlinear scalar equation 2.2.6 Piston problem 2.2.7 Complementary Riemann invariants 2.2.8 Solution of the piston problem 2.2.9 Cauchy problem for a symmetric hyperbolic system 2.2.10 Approximations 2.2.11 Existence of approximations 2.2.12 Energy estimate 2.2.13 Convergence of approximations to a generalized solution 2.2.14 Regularity of the generalized solution 2.2.15 Quasilinear system 2.2.16 Local existence for a quasilinear system 2.2.17 Second grade approximations 2.2.18 Higher order energy estimates 2.2.19 Convergence of approximations 2.2.20 Uniqueness 2.2.21 Local existence for equations of an isentropic ideal gas 2.2.22 Existence of global smooth solutions for nonlinear hyperbolic systems 2.2.23 2 × 2 system of conservation laws, Riemann invariants 2.2.24 Plane wave solutions 2.2.25 Plane waves for the Euler system in 2D Weak solutions 2.3.1 Blow up of classical solutions 2.3.2 Generalized formulation for systems of conservation laws 2.3.3 Piecewise smooth solutions 2.3.4 Entropy condition 2.3.5 Physical entropy 2.3.6 General parabolic approximation and the entropy condition 2.3.7 Entropy for a general scalar conservation law

xiii

77 78 79 79 80 81 82 82 84 84 85 89 90 90 91 92 92 94 95 95 95 97 98 99 100 100 103 104 106 107 108 108 110 112 113 115

xiv

CONTENTS

2.4

3

2.3.8 Entropy for a 2 × 2 system of conservation laws in 1D 2.3.9 Entropy function for a p-system 2.3.10 Riemann problem 2.3.11 Riemann problem for 2 × 2 isentropic gas dynamics equations 2.3.12 Existence and uniqueness of admissible weak solution for a scalar conservation law 2.3.13 Plane waves admitting discontinuities 2.3.14 Existence of solutions to the 2 × 2 Euler system for an isentropic gas 2.3.15 Lax–Friedrichs difference approximations 2.3.16 Existence of approximations 2.3.17 Invariant regions for Riemann invariants 2.3.18 Compactness argument 2.3.19 Characterization of the weak limit by Young measure 2.3.20 Div–curl lemma and Tartar’s commutation relation 2.3.21 Existence of weak entropy–entropy flux pairs 2.3.22 Localization of supp ν 2.3.23 Approximative limit is an admissible solution 2.3.24 Global existence for general systems in one dimension Final comments 2.4.1 Local existence results 2.4.2 Global smooth solutions 2.4.3 Blow up and the lifespan of smooth solution 2.4.4 Global weak solutions for multidimensional Euler equations 2.4.5 Riemann problem 2.4.6 Euler equations with source terms 2.4.7 Comments on the 2 × 2 Euler system for an isentropic fluid 2.4.8 Euler equations for a nonisentropic fluid

Some mathematical tools for compressible flows 3.1 Renormalized solutions of the steady continuity equation 3.1.1 Friedrichs’ lemma about commutators 3.1.2 Continuity equation and its prolongation 3.1.3 Renormalized solutions of the continuity equation 3.2 Vector fields with summable divergence

117 118 118 120 125 125 125 128 129 129 130 132 134 135 138 144 145 146 146 147 148 150 151 152 152 154 155 155 155 158 159 163

CONTENTS

3.3

3.4

4

The equation div v = f 3.3.1 Bounded domains 3.3.2 Exterior domains 3.3.3 Domains with noncompact boundaries Some results for monotone and convex operators 3.4.1 Some results from convex analysis 3.4.2 Some results from monotone operators

Weak solutions for steady Navier–Stokes equations of compressible barotropic flow 4.1 Formulation of problems in bounded and exterior domains and main results 4.1.1 Definition of weak solutions 4.1.2 Existence of weak solutions 4.1.3 Exterior domains 4.2 Heuristic approach 4.2.1 Estimates due to the energy inequality and improved estimates of density 4.2.2 Limit passage 4.2.3 Effective viscous flux 4.2.4 Strong convergence of density – Lions’ approach 4.2.5 Strong convergence of density – Feireisl’s approach 4.2.6 Remarks to approximations 4.3 Approximations in bounded domains 4.3.1 First level approximation – artificial pressure 4.3.2 Second level approximation – relaxation in the continuity equation 4.3.3 Third level approximation – relaxed continuity equation with dissipation 4.4 Effective viscous flux 4.4.1 Riesz operators 4.4.2 Div–curl lemma 4.4.3 Commutator lemma 4.4.4 Effective viscous flux 4.5 Neumann problem for the Laplacian 4.5.1 Existence, uniqueness and regularity 4.5.2 Eigenvalue problem 4.6 Relaxed continuity equation with dissipation 4.6.1 Statement of the problem and results 4.6.2 Estimates for the Leray–Schauder fixed points 4.6.3 Homotopy of compact transformations 4.6.4 Nonnegativity of the density 4.7 The Lamé system

xv

165 166 176 178 183 183 186 189 189 190 192 193 194 194 195 196 197 198 199 200 200 202 203 204 205 206 207 208 211 211 212 212 212 213 215 216 216

xvi

CONTENTS

4.8

4.9

4.10

4.11

4.12 4.13

4.14

4.15

4.7.1 Existence, uniqueness and regularity 4.7.2 Eigenvalue problem Complete system with dissipation in the relaxed continuity equation and with artificial pressure 4.8.1 Existence of solutions 4.8.2 Estimates independent of dissipation Complete system with relaxed continuity equation and with artificial pressure 4.9.1 Vanishing dissipation limit 4.9.2 Effective viscous flux 4.9.3 Renormalized continuity equation with powers 4.9.4 Strong convergence of the density 4.9.5 Equation of momentum, energy inequality and estimates independent of the relaxation parameter Complete system with artificial pressure 4.10.1 Vanishing relaxation limit 4.10.2 Effective viscous flux 4.10.3 Renormalized continuity equation with powers 4.10.4 Strong convergence of the density 4.10.5 Momentum equation 4.10.6 Energy inequality and estimates independent of artificial pressure Complete system of a viscous barotropic gas 4.11.1 Vanishing artificial pressure limit 4.11.2 Effective viscous flux 4.11.3 Boundedness of oscillations of density sequence 4.11.4 Renormalized continuity equation 4.11.5 Strong convergence of the density Approximations in an exterior domain 4.12.1 Relaxation on invading domains Complete system with relaxed continuity equation on an exterior domain 4.13.1 Some equivalence inequalities 4.13.2 Bounds due to the energy inequality 4.13.3 Estimates independent of invading domains and relaxation Existence of weak solutions in exterior domains 4.14.1 Vanishing relaxation limit 4.14.2 Effective viscous flux and renormalized continuity equation Existence of weak solutions in bounded and in exterior Lipschitz domains

217 217 218 218 222 223 224 225 226 230

231 231 232 233 234 235 236 236 239 239 241 241 243 244 245 245 247 247 247 248 254 254 255 259

CONTENTS

4.16 Existence of weak solutions in domains with noncompact boundaries 4.16.1 Formulation of the problem, fluxes 4.16.2 Main results 4.16.3 Domains with conical or superconical exits 4.16.4 Domains with cylindrical or subconical exits 4.17 Further results, comments and bibliographic remarks 4.17.1 Weak compactness 4.17.2 Bounded domains 4.17.3 Exterior domains 4.17.4 Domains with noncompact boundaries 4.17.5 Flow of mixtures 5

6

xvii

261 262 264 265 268 268 268 269 274 275 278

Strong solutions for steady Navier–Stokes equations of compressible barotropic flow and small data 5.1 Notation and main results 5.1.1 Formulation of the problem 5.1.2 Existence theorem in a bounded domain 5.1.3 Functional spaces for exterior domains 5.1.4 Existence theorems in exterior domains 5.2 Heuristic approach 5.2.1 Perturbations and linearization of the problem 5.2.2 Helmholtz decomposition and effective viscous flux 5.2.3 Existence theorem for the linearized system 5.3 Auxiliary linear problems 5.3.1 Neumann problem for the Laplacian 5.3.2 Helmholtz decomposition 5.3.3 Dirichlet problem for the Laplacian 5.3.4 Stokes and Oseen problems 5.3.5 Steady transport equation 5.4 The linearized system 5.5 The fully nonlinear system 5.5.1 The case of zero velocity at infinity 5.5.2 The case of nonzero velocity at infinity 5.6 Bibliographic remarks 5.6.1 Bounded domains 5.6.2 Exterior domains

283 285 285 286 286 287 287 289 290 292 292 295 296 296 297

Some mathematical tools for nonsteady equations 6.1 Some auxiliary results from functional analysis 6.1.1 Continuous functions with values in Lqweak 6.1.2 The time and space mollifiers 6.1.3 Local weak compactness in unbounded domains 6.2 Renormalized solutions of the continuity equation

300 300 300 303 304 304

279 279 279 280 280 281 282 282

xviii

CONTENTS

6.2.1 6.2.2 6.2.3 6.2.4 7

Friedrichs’ lemma about commutators Continuity equation and its prolongation Renormalized continuity equation Strong continuity of the density

Weak solutions for nonsteady Navier–Stokes equations of compressible barotropic flow 7.1 Formulation of problems and main results 7.1.1 Definition of weak solutions 7.1.2 Existence in bounded domains 7.1.3 Existence in exterior domains 7.2 Linear momentum and total energy 7.2.1 Linear momentum 7.2.2 Total energy 7.3 Heuristic approach 7.3.1 Compactness of weak solutions 7.3.2 Estimates due to the energy inequality 7.3.3 Improved estimate of the density 7.3.4 Limit passage 7.3.5 Effective viscous flux 7.3.6 Strong convergence of density – Lions’ approach 7.3.7 Strong convergence of density – Feireisl’s approach 7.3.8 Remarks on approximations 7.4 Approximations in bounded domains 7.4.1 First level approximations – artificial pressure 7.4.2 Second level approximation – continuity equation with dissipation 7.4.3 Third level approximation – Galerkin method 7.5 Effective viscous flux 7.6 Continuity equation with dissipation 7.6.1 Regularity for the parabolic Neumann problem 7.6.2 Continuity equation with dissipation 7.6.3 Construction of a solution – Galerkin method 7.6.4 Regularity of solutions 7.6.5 Boundedness from below and from above 7.6.6 L2 -estimates 7.6.7 L2 -estimate of differences 7.6.8 A renormalized inequality with dissipation 7.7 Galerkin approximation of the system with dissipation in the continuity equation and with artificial pressure 7.7.1 Preparatory calculations 7.7.2 Galerkin approximation

304 306 307 310 312 312 313 318 320 321 321 322 324 324 325 325 326 326 327 328 329 330 330 333 335 338 343 343 345 346 348 348 349 350 351 352 352 353

CONTENTS

7.7.3 Local existence of solutions 7.7.4 Existence of maximal solutions 7.7.5 Energy inequalities and estimates 7.8 Complete system with dissipation in the continuity equation and with artificial pressure 7.8.1 Limit in the modified continuity equation 7.8.2 Limit in the momentum equation 7.8.3 Limit in the energy inequality and estimates independent of vanishing dissipation 7.8.4 Improved estimate of density 7.9 Complete system with artificial pressure 7.9.1 Weak limits as dissipation tends to zero 7.9.2 Effective viscous flux 7.9.3 Renormalized equation of continuity and strong convergence of density 7.9.4 Energy inequality and estimates independent of artificial pressure 7.9.5 Improved estimate of density 7.10 Complete system of isentropic Navier–Stokes equations 7.10.1 Weak limits at vanishing artificial pressure 7.10.2 Effective viscous flux 7.10.3 Amplitude of oscillations 7.10.4 Renormalized continuity equation 7.10.5 Strong convergence of the density 7.10.6 Energy inequalities 7.10.7 General initial conditions 7.11 Existence of solutions in exterior domains 7.11.1 Solutions on invading domains 7.11.2 Orlicz spaces Lpq (Ω) 7.11.3 Estimates independent of invading domains 7.11.4 Improved estimates of density 7.11.5 Weak limits at growing invading domains 7.11.6 Effective viscous flux and renormalized continuity equation 7.11.7 Strong convergence of the density 7.11.8 Energy inequality 7.12 Other problems and bibliographic remarks 7.12.1 Bibliographic remarks on basic theorems 7.12.2 Slip boundary conditions 7.12.3 Nonmonotone pressure 7.12.4 Domain dependence 7.12.5 Nonhomogeneous boundary conditions 7.12.6 Unbounded domains and non-zero velocity at infinity

xix

354 357 360 361 362 363 365 366 368 369 372 374 376 376 381 382 386 386 388 390 392 392 393 393 395 396 397 398 400 401 404 404 404 408 409 410 412 424

xx

CONTENTS

7.12.7 Domains with nonsmooth boundaries 8

9

Global behavior of weak solutions 8.1 Formulation of the problem 8.2 Basic assumptions 8.3 Sequential stabilization of the weak solution 8.4 Auxiliary functions 8.5 Existence and estimates of auxiliary functions 8.6 Comparison density and a test function 8.7 Passing to the limit with the regularization parameter 8.8 Comparison density is close to the density as t → ∞. 8.9 Convergence of the density 8.10 Uniqueness of equilibrium 8.11 Global behavior of weak solutions in time in bounded domains – arbitrary forces 8.12 Bounded absorbing sets 8.13 Asymptotically closed trajectories 8.14 Global attractor of short trajectories 8.15 Rapidly oscillating external forces 8.16 Attractors 8.17 Time-periodic solutions 8.18 Uniqueness of equilibrium revisited Strong solutions of nonsteady compressible Navier–Stokes equations 9.1 Problem formulation 9.2 Similarity transformation 9.3 Maximal parabolic regularity 9.4 Resolution of the continuity equation with a given velocity 9.5 Further transcription of the problem 9.6 Fixed point argument and the existence of a local solution 9.7 Uniqueness 9.8 Global a priori estimate 9.9 Global existence 9.10 Bibliographical remarks

429 431 431 432 432 433 435 437 437 438 446 452 456 457 458 459 460 460 461 462 464 464 465 466 467 469 470 473 474 479 480

References

485

Index

499

1 FUNDAMENTAL CONCEPTS AND EQUATIONS This chapter is introductory. In the first section, we introduce the basic mathematical notions and notation used throughout the whole book. The physical background to the governing equations of gas dynamics is presented in Section 1.2. In Section 1.3 we list an important number of advanced mathematical concepts and results needed in the sequel. Finally, in the annexed Section 1.4, a survey of basic general concepts from functional analysis is briefly recalled. Classical results of mathematical analysis are included despite the fact that most readers are familiar with them from basic university courses of real and functional analysis. Nevertheless, we are convinced that the presence of statements of concrete quoted theorems provides considerable comfort for reading the book. Precise references to detailed proofs and arguments are given throughout this chapter, thus making it easier to follow the literature. 1.1

Some mathematical concepts and notation

We assume the reader to be familiar with the elements of linear algebra, mathematical analysis and the theory of the Lebesgue integral – see, e. g., (Rudin, 1974). 1.1.1

Basic notation

By IR and IN we shall denote the set of all real numbers and the set of all positive integers, IR+ := (0, ∞). In the Euclidean space IRN (N ≥ 1) we shall use a Cartesian coordinate system with axes denoted by x1 , . . . , xN . Points in IRN will usually be denoted by x = (x1 , . . . , xN ), y = (y1 , . . . , yN ), etc. In some physical situations we shall call elements of IRN vectors and treat them as columns. To emphasize this fact, the vectors will usually be denoted by bold letters. By e1 , . . . , eN we shall denote the unit vectors in the directions of the coordinate axes. If a is a (column) vector, then aT will denote the (row) vector transposed to a. Let us recall that an open set Ω ⊂ IRN is connected if two arbitrary points can be connected with a piecewise linear curve in Ω. We usually denote by Ω a domain, i.e. an open and connected set in IRN . We say that Ω is a bounded domain if Ω is a domain and if it is bounded. We say that Ω is an unbounded domain if Ω is a domain and if Ω is not bounded. We say that Ω is an exterior domain if IRN \Ω N is a bounded domain. We also denote Br (x) := {y ∈ IRN ; j=1 |xj − yj |2 < r}, a ball in IRN of center x and radius r > 0, and B r (x) = IRN \ Br (x). The ball Br (0) is denoted simply by Br and the complement of its closure by B r . 1

2

FUNDAMENTAL CONCEPTS AND EQUATIONS

For a set M we put Mr = M ∩ Br and M r = M \ Br , and denote by 1M its characteristic function. Thus, e.g., for a function ρ : M → IR, 1{ρ>k} , k ∈ IR, denotes the function equal to 1 on the set {ρ > k} := {x ∈ M ; ρ(x) > k} and equal to 0 otherwise. If P and Q are two sets and f is a mapping (function) defined on P with its values lying in Q, we write f : P → Q. For such a mapping and a subset M ⊂ P the symbol f |M denotes the restriction of f to the set M . In other words, if we write g = f |M , then g : M → Q and g(x) = f (x) for all x ∈ M . Let us note that a one-to-one mapping f of a set P onto a set Q is called a bijection. By f ◦g we denote the composition of functions f and g: (f ◦g) (x) = f (g(x)). For M ⊂ IRN we denote by C(M ) (or C 0 (M )) the linear space of all functions continuous on M . For k ∈ IN and Ω a domain, C k (Ω) will denote the linear space of all functions which have continuous partial derivatives up to order k in Ω. Let ∂Ω and Ω denote the boundary of the set Ω and its closure, respectively. The space C k (Ω) is formed by all functions from C k (Ω) whose all derivatives up to order k can be continuously extended onto Ω. The space Cbk (Ω) is formed by all bounded functions from C k (Ω) whose all derivatives up to order k are bounded. For a bounded domain C k (Ω) = Cbk (Ω). If f : Ω → IRN , i.e. f = (f1 , . . . , fN ), fi : Ω → IR for i = 1, . . . , N , then we write f ∈ [C k (Ω)]N , if fi ∈ C k (Ω) for all i = 1, . . . , N . Similarly we define the space [C k (Ω)]N . By C0k (Ω) is denoted the space of all f ∈ C k (Ω) such that supp f := {x ∈ Ω; f (x) = 0} is compact in Ω. In general, if P is some set, then by the symbol P N we denote the Cartesian product P × P × · · · × P (N times). This means that P N = {(a1 , . . . , aN ); ai ∈ P, i = 1, . . . , N }. For two sets P and Q we put P × Q = {(x, y); x ∈ P, y ∈ Q}. Quantities describing fluid flow are functions of space and time. This means that we write such a quantity as a function f = f (x, t), where t is time and x = (x1 , x2 , x3 ) denotes points of a set Ωt occupied by the fluid at time t. Let (0, T ) (0 < T ) be a time interval during which we follow the fluid motion. Then the domain of definition of the function f is the set M = {(x, t); x ∈ Ωt , t ∈ (0, T )} ⊂ IR4 .

(1.1.1)

Let us assume that the set M is open. (If the domains Ωt = Ω are independent of t, then M = Ω × (0, T ) is open.) If t ∈ (0, T ) is fixed, then f (t) = f (·, t) denotes the function x → f (x, t) whose value at x ∈ Ωt equals f (x, t). Let D be a convex set. We say that a real-valued function η defined on D is convex if η(x + τ (y − x)) ≤ η(x) + τ (η(y) − η(x))

∀ x, y ∈ D, ∀ τ ∈ [0, 1],

(1.1.2)

and strictly convex on D if η(x + τ (y − x)) < η(x) + τ (η(y) − η(x)) ∀x, y ∈ D, x = y, ∀τ ∈ (0, 1).

SOME MATHEMATICAL CONCEPTS AND NOTATION

1.1.2

3

Some useful inequalities in IRN

We recall several elementary inequalities which are very often implicitly used in various calculations.

N

|xi yi | ≤

N

|xi yi | ≤

i=1

N i=1

|xi |p

p1 N

′

i=1

|xi |p

1′ p

,

1 p

+

1 p′

= 1.

(1.1.3)

This inequality is called H¨ older’s inequality in IRN . The next inequality reads i=1

1 p

N

i=1

εp |xi |p +

1 p′

N

′

′

i=1

ε−p |xi |p , ε > 0,

(1.1.4)

and is called Young’s inequality. Finally, we recall

N i=1

|xi + yi |p

p1

≤

N i=1

|xi |p

which is the Minkowski inequality. 1.1.3

p1

+

N i=1

|xi |p

′

1′ p

(1.1.5)

Differential operators

Let us consider a function f = f (x, t) defined on the set M and having partial derivatives (in the classical or generalized sense) ∂f /∂x1 , ∂f /∂x2 , . . . , ∂f /∂xN . Then we put ∂f T ∂f ∇f = ( ∂x ) . , . . . ∂x (1.1.6) 1 N If f = (f1 , f2 , f3 ) : M → IR3 is a vector function with components which have first order partial derivatives (in the classical or generalized sense) with respect to x1 , x2 , x3 in M, then we set div f = curl f =

∂f3 ∂x2

−

3

∂fi i=1 ∂xi ,

∂f2 ∂f1 ∂x3 , ∂x3

−

∂f3 ∂f2 ∂x1 , ∂x1

−

∂f1 ∂x2

T

(1.1.7) .

∂f The derivatives ∂x will also be denoted by ∂f /∂xi or simply, ∂xi f, or ∂i f. i Similar notation is used for the derivative with respect to time: ∂f /∂t = ∂t f and for higher order derivatives: ∂ij , etc. If not stated explicitly otherwise, we shall use the Einstein Nsummation convention over repeated indeces. So, for example, div f = ∂i fi , j=1 aij xj = aij xj , etc. We define the scalar product in IRN by the relation

a·b=

N

i=1

ai bi ,

(1.1.8)

where a = (a1 , . . . , aN ) and b = (b1 , . . . , bN ) ∈ IRN . We use the symbol “ · ” also for multiplication between vectors and matrices, if it is convenient. Thus, for example, for a 3-tensor a and 2-tensor b, the symbol a · b means a 3-tensor with components k aijk bkl . Also, we keep the convention a : b = ij aij bij ,

4

FUNDAMENTAL CONCEPTS AND EQUATIONS 1

for two 2-vectors, in this case. The magnitude of a is the number |a| = (a · a) 2 . For a, b ∈ IR3 we define the vector product a × b = (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 )T

(1.1.9)

and by a ⊗ b we denote the so-called tensor product:

 a1 b1 , a1 b2 , a1 b3 a ⊗ b =  a2 b1 , a2 b2 , a2 b3  . a3 b1 , a3 b2 , a3 b3 

(1.1.10)

If there is no danger of confusion, then we write ab instead of a ⊗ b. In physics one often meets the concept of tensors (of order two) which can be expressed as 3 × 3 matrices. If A = (aij )3i,j=1 is a tensor function depending on x ∈ Ωt , then we define div A =

∂aj1 3 ∂aj2 3 ∂aj3 3 j=1 ∂xj , j=1 ∂xj , j=1 ∂xj

T

.

(1.1.11)

By δij we denote the Kronecker delta: δii = 1, δij = 0 for i = j. Define also the so-called Levi–Civita tensor εijk as follows. If at least two of the numbers i, j, k are equal, then εijk = 0; otherwise ε123 = ε231 = ε312 = 1,

ε132 = ε213 = ε321 = −1.

Let f : M → IRM be a mapping whose components have first order partial derivatives with respect to x1 , . . . , xN . We define the Jacobi matrix of the mapping f (·, t) (t ∈ (0, T ) is fixed) as the matrix 

 ∂f1 (x)   ∂xN     ∂f2   (x)   Df   ∂xN (x) =  .   Dx ..   .       ∂fM ∂fM ∂fM (x), (x), . . ., (x) ∂x1 ∂x2 ∂xN ∂f1 ∂f1 (x), (x), . . ., ∂x1 ∂x2 ∂f2 ∂f2 (x), (x), . . ., ∂x1 ∂x2 . .. , . . ., . , ..

(1.1.12)

In what follows, this matrix will also be denoted simply by ∇f . We also denote ∆f =

N

∂2f i=1 ∂x2i ,

∆f = (∆f1 , . . . , ∆fN ).

The symbol ∆ is called the Laplace operator, or, briefly, the Laplacian.

(1.1.13)


1.1.4

5

Gronwall’s lemma

Gronwall’s lemma in integral form reads: Lemma 1.1 Let us assume that h is continuous on [a, b], r is integrable in (a, b), h, r ≥ 0 a.e. in (a, b). Further assume that the function y is continuous in [a, b] and satisfies the inequality

t (1.1.14) y(t) ≤ h(t) + a r(s)y(s) ds, t ∈ [a, b]. Then

y(t) ≤ h(t) +

t a

h(s)r(s) exp

t s

r(τ ) dτ ds,

t ∈ [a, b].

(1.1.15)

We shall also need the differential version of Gronwall’s lemma, which reads: Lemma 1.2 Let us assume that h, r are integrable on (a, b) and nonnegative a.e. in (a, b). Further assume that y ∈ C([a, b]) and y ′ ∈ L1 ((a, b)) and that the following inequality is satisfied: y ′ (t) ≤ h(t) + r(t)y(t) for a.a. t ∈ (a, b).

(1.1.16)

Then s

t

t y(t) ≤ y(a) + a h(s)exp − a r(τ ) dτ ds exp a r(s) ds , t ∈ [a, b]. (1.1.17)

For the first lemma, see e.g. (Filatov and Sharova, 1976), Theorem 1.1; for the second, e.g. (Evans, 1998), Appendix B, j. 1.1.5

Implicit functions

In this section we formulate two fundamental results which will sometimes be useful. Recall that the Jacobi matrix of a differentiable mapping f from an open subset U of IRN into IRM is the matrix given by (1.1.12). Theorem 1.3 (Implicit function theorem) Let f : U ⊂ IRN × IRM → IRN , where U is an open set in IRN × IRM . Assume that (i) f is continuous in U together with its derivatives ∂f /∂xj , j = 1, . . . , N ; (ii) we have given x0 ∈ IRN , y0 ∈ IRM such that f (x0 , y0 ) = 0; (iii) Df (x0 , y0 ) = 0. J(x0 , y0 ) = det Dx Then (a) there exists a ball Br (y0 ) ⊂ IRM and a unique continuous mapping w : Br (y0 ) → IRN , such that, w(y0 ) = x0 and f (w(y), y) = 0 on Br (y0 );

6


(b) if, in addition, f is continuously differentiable in U with respect to all variables, then w is continuously differentiable in Br (y0 ), Df Dx (w(y), y) invertible therein and ∂w N −1 Df Df j (w(y), y) (w(y), y) ; =− ∂yi i,j=1 Dx Dy

(c) if, moreover, f is continuously differentiable in U up to order k > 1(k ∈ IN ), then w is continuously differentiable in Br (y0 ) up to order k as well.

Theorem 1.4 (Morse–Sard theorem) Let f ∈ [C k (IRN )]M , M, N ∈ IN, where k = 1 + N − M . Then Df (y) = 0 ∩ {y ∈ IRN ; f (y) = z} = ∅ y ∈ IRN ; Dx M for a.a. z ∈ IR (see Section 1.1.14 for a definition of the abreviation a.a.).

For the proof of the first classical statement see, e.g. (Nirenberg, 1974), Theorem 2.7.2. For more details about the second theorem see (Evans, 1992), Section 3.4.2. 1.1.6 Transformations of Cartesian coordinates Let us consider two Cartesian coordinate systems (x1 , . . . , xN ) and (x∗1 , . . . , x∗N ) in IRN . Then the transition from xi to x∗i is realized by the relations N (1.1.18) x∗i = j=1 aij xj + ci , i = 1, ..., N, N

where A = (aij )i,j=1 is an orthonormal matrix. This means that A AT = I,

(1.1.19) N

where I denotes the unit matrix and AT = (aji )i,j=1 is the transpose of A (hence, in view of (1.1.19), A−1 = AT ). Relations (1.1.18) can be written in the vector form x∗ = A x+c, x = (x1 , . . . , xN )T , x∗ = (x∗1 , . . . , x∗N )T , c = (c1 , . . . , cN )T . (1.1.20) The transformation inverse to (1.1.18) is N xk = i=1 aik x∗i + c∗k , k = 1, . . . , N, N where c∗k = − i=1 aik ci , The vector form of (1.1.21) is x = AT x∗ + c∗ ,

c∗ = (c∗1 , . . . , c∗N )T .

(1.1.21)

(1.1.22)

1.1.7 H¨ older-continuous and Lipschitz functions A function f : M → IRN , M ⊂ IRN , is µ-H¨ older-continuous with µ ∈ (0, 1], if there exists a constant L such that |f (x) − f (y)| ≤ L|x − y|µ ,

x, y ∈ M.

(1.1.23)

If µ = 1, we speak about a Lipschitz-continuous (or simply Lipschitz) function. If Ω ⊂ IRN is an open set, then C k,µ (Ω) denotes the set of all functions whose derivatives of order k are µ-Hölder-continuous in Ω.


1.1.8

7

The symbols “o” and “O”

Let f be a function defined in B(0) \ 0, where B(0) is a neighborhood of zero and let α ∈ IR. We write f (h) = o(hα ), if limh→0 f (h)/hα = 0. Further, we say that f (h) = O(hα ), if there is a constant c > 0 such that |f (h)| ≤ chα for all h ∈ B(0) \ {0}. 1.1.9

Partitions of unity

Partitions of unity play an important role in the procedure of “localization” in the theory of function spaces and it is an important tool in many proofs. N Lemma 1.5 (Lemma on the partition of unity) m Let G ⊂ IR be compact, Gi ⊂ N IR , i = 1, . . . , m, be open and let G ⊂ i=1 Gi . Then there exist functions ϕ1 , . . . , ϕm satisfying the following conditions:

(i) ϕi ∈ C0∞ (IRN ), supp ϕi ⊂ Gi , i = 1, . . . , m, (ii) 0 ≤ ϕi ≤ 1 in IRN , i = 1, . . . , m, m ϕi (x) = 1. (iii) x ∈ G =⇒

(1.1.24)

i=1

For the proof, see (Kufner et al., 1977). 1.1.10

Measure

We survey the basics of measure theory, which can be found, e.g., in (Rudin, 1974). (1) A system M of subsets of the space X = IRN is called a σ-algebra on X, if (i) X ∈ M; (ii) A ∈ M =⇒ X \ A ∈ M; ∞ (iii) An ∈ M, n = 1, 2 . . . , A = n=1 An =⇒ A ∈ M. If M is a σ-algebra on X, then X is called a measurable space and the elements of the system M are called measurable sets in X. If X is a measurable space and Y is a metric space, we say that the mapping f : X → Y is measurable, if for any open set V ⊂ Y the set F −1 (V ) := {x ∈ X; f (x) ∈ V } is measurable in X, i.e. f −1 (V ) ∈ M. There exists a smallest σ-algebra B on X containing all open subsets of X. The elements of B are called Borel sets in X. (2) We say that a function µ defined on a σ-algebra M on X is a (positive) measure, if µ : M → [0, ∞] and

∞ ∞ µ i=1 Ai = i=1 µ(Ai )

for every sequence {Ai }∞ i=1 of mutually disjoint sets which are elements of M. (3) If µ is a measure defined on the σ-algebra B of Borel sets, we say that µ is a Borel measure.

8


(4) If a measure µ satisfies the condition µ(X) < ∞, we call it finite. We say that µ is a probability measure if µ(X) = 1. If µ(K) < ∞ for any compact set K ⊂ X, then µ is called a Radon measure. (5) It is known that any Borel measure µ defined on the σ-algebra B is regular in the following sense: (a) µ(E) = inf{µ(V ); E ⊂ V, V open}; (b) µ(E) = sup{µ(K); K ⊂ E, K compact}. (6) The Lebesgue measure µ = meas, defined on the space IRN can be characterized N as a Borel measure such that for any m-dimensional interval I = i=1 (ai , bi ) the measure µ(I) is equal to the volume of I. (7) The support of a Borel measure µ is defined as the set supp µ := {x ∈ X; µ(U ) > 0 for every neighborhood U of x}. Sometimes the Lebesgue measure of a set M ⊂ IRN will be denoted simply by |M |. Hence, |M | is the volume of M or the area of M or the length of M , if N = 3 or N = 2 or N = 1, respectively. 1.1.11

Description of the boundary

Let Ω ⊂ IRN be a bounded domain. Its boundary ∂Ω is called Lipschitz-continuous (or simply Lipschitz) if there exist numbers α > 0, β > 0, and a finite number of local Cartesian coordinate systems xr1 , . . . , xrN and Lipschitz-continuous xr = (xr2 , . . . , xrN ) ∈ IRN −1 ; |ˆ xr | < α} −→ IR (called local functions ar : Mr = {ˆ maps), r = 1, . . . , R (R ∈ IN ), such that R

Λr , r=1 Λr = {(xr1 , x xr ), |ˆ xr | < α} , ˆr ); xr1 = ar (ˆ xr ) + β, |ˆ xr | ˆr ); ar (ˆ xr ) < xr1 < ar (ˆ {(xr1 , x xr ), |ˆ xr | ˆr ); ar (ˆ xr ) − β < xr1 < ar (ˆ {(xr1 , x ∂Ω =

(1.1.25)

< α} ⊂ Ω, r = 1, . . . , R,

< α} ⊂ IRN \ Ω, r = 1, . . . , R.

We often speak of a Lipschitz domain or a domain with Lipschitz-continuous boundary. If ar ∈ C k (Mr ) (resp. C k,µ (Mr ), µ ∈ (0, 1)) for all r = 1, . . . , R, then we write ∂Ω ∈ C k (resp. ∂Ω ∈ C k,µ ). Sometimes, C k is denoted also as C k,0 . The boundary ∂Ω is called smooth, if ∂Ω ∈ C ∞ . 1.1.12

Measure on the boundary of a domain

If ∂Ω is Lipschitz-continuous, then it is possible to define an (N − 1)-dimensional measure on ∂Ω. Let us briefly describe its construction. It is known that the partial derivatives ∂ar /∂xrj , j = 2, . . . , N , of the Lipschitz-continuous function ar : Mr → IR are defined almost everywhere on Mr and are measurable on Mr . For any set M ⊂ Λr let us construct the projection P M of M into Mr . We say that the set M is measurable (on ∂Ω) if P M is measurable with respect to


9

the Lebesgue measure in IRN −1 , and define the measure of M as the Lebesgue integral

N r (ˆxr ) 2 21 r (1.1.26) xr ) 1 + i=2 ∂a∂x dˆ x . µ (M ) = Mr 1P M (ˆ r i

The decomposition (1.1.25) of ∂Ω implies the existence of disjoint measurable sets Mr ⊂ Λr such that R (1.1.27) ∂Ω = r=1 Mr . Now we say that a set M ⊂ ∂Ω is measurable if the sets M ∩ Mr are measurable and we define the measure of M by R (1.1.28) (M ∩ Mr ). µ (M ) = r=1 µ

It is possible to prove that this definition is independent of the decomposition (1.1.27). The function µ is a σ-additive nonnegative measure. We define the surface integral of a function f defined on a measurable subset M of ∂Ω as the integral induced by the measure µ :

f (x) dS = f 1M d µ. (1.1.29) M

If Ω ⊂ IR2 is a plane domain with a Lipschitz-continuous boundary ∂Ω formed by a piecewise smooth curve ϕ, then the integral (1.1.29) with M = ∂Ω is equal to the curvilinear integral. for the In the sequel we shall use the notation meas = µ and measN −1 = µ Lebesgue measure in IRN and the (N − 1)-dimensional measure defined on the boundary ∂Ω of a domain Ω ⊂ IRN , respectively. We also sometimes use the licence |Ω| = meas(Ω) Let us note that the vector n = (n1 , . . . , nN ) with the components n1 = − 1b ; where

ni =

1 ∂ar b ∂xri ,

i = 2, . . . , N,

N ∂ar 2 21 b = 1 + i=2 ∂x , r

(1.1.30) (1.1.31)

i

is the vector of unit outer normal to ∂Ω. For more details see (Kufner et al., 1977), Section 6.3. 1.1.13 Classical Green’s theorem Let Ω ⊂ IRN be a bounded domain with a Lipschitz-continuous boundary and let u, v ∈ C 1 (Ω). Then

∂u

∂v v dx = ∂Ω uvni dS − Ω u ∂x dx, i = 1, . . . , N. (1.1.32) Ω ∂xi i For a proof, see, e.g. (Neˇcas, 1967), Theorem III.1.1. For u ∈ [C 1 (Ω)]N we get from Green’s theorem the identity

div u dx = ∂Ω u · n dS, (1.1.33) Ω

important in fluid dynamics. Later we shall state and use these formulae also for more general functions (see Section 1.3.5.6, Theorem 1.41).

10


1.1.14

Lebesgue spaces

Let M ⊂ IRN be a Lebesgue measurable set. We denote by |M | = meas (M ) the Lebesgue measure of this set. We say that two measurable functions defined on M are equivalent if they differ at most on a set of zero Lebesgue measure. In such a situation we write f1 = f2 a.e. (almost everywhere) in M or f1 (x) = f2 (x) for a.a. (almost all) x ∈ M. For p ∈ [1, ∞) the symbol Lp (M ) will denote the linear space of all (classes of equivalent) functions measurable on M for which

|u|p dx < ∞. (1.1.34) M The space Lp (M ) equipped with the norm

uLp (M ) = u0,p,M = u0,p =

M

1/p |u|p dx

(1.1.35)

is a Banach space. Furthermore, we define the Banach space

L∞ (M ) = {u; u is a measurable function on M and uL∞ (M ) := ess supM |u| (1.1.36) := inf{ sup |u(x)|; Z ⊂ M, meas (Z) = 0} < ∞}. x∈M \Z

Similarly we define Lp (M ) for M ⊂ ∂Ω, where Ω ⊂ IRN is a domain with a Lipschitz-continuous boundary and M is measurable with respect to the (N −1)

. . . dS instead of dimensional measure µ . In (1.1.34) and (1.1.35) we write M

. . . dx and in (1.1.36) we replace meas by meas . N −1 M We say that f ∈ Lploc (M ), if f |M ′ ∈ Lp (M ′ ) for any bounded M ′ ⊂ M such that M ′ ⊂ M . We denote

1 f := |M (1.1.37) | M f dx M and

p

Lp (M ) = {f ∈ Lp (M );

M

f dx = 0},

(1.1.38)

p

the subspace of L (M ) endowed with the norm of L (M ). 1.1.14.1 Basic properties of Lebesgue spaces We shall list several important properties of the spaces Lp (M ): (i) If 1 ≤ p < ∞, Lp (M ) is a separable Banach space and if M is a domain, then C0∞ (M ) is dense in it. On the contrary, the Banach space L∞ (M ) is not separable. (ii) If 1 ≤ p < ∞ then to any f ∈ [Lp (M )]∗ (= dual of Lp (M ) – see Sec′ tion 1.4.5) there exists a unique uf ∈ Lp (M ) which satisfies

f, ϕ = M uf ϕ dx, ϕ ∈ Lp (M ) and f [Lp (M )]∗ = uf Lp′ (M ) . (The symbol f, ϕ denotes the value of the functional f at the point ϕ.) Here and in the sequel, p′ is the so-called conjugate exponent, given by


1 1 + = 1; ′ p p

11

(1.1.39)

of course, if p = ∞, then p′ = 1 and vice versa. For p = p′ = 2 the above ′ result is known as the Riesz theorem. It says that [Lp (M )]∗ and Lp (M ) are ′ isometrically isomorphic Banach spaces (notation [Lp (M )]∗ ≡ Lp (M )). In ′ the sequel, we identify [Lp (M )]∗ with Lp (M ). (iii) We have [L1 (M )]∗ ≡ L∞ (M ) and L1 (M ) ⊂ [L∞ (M )]∗ .

(iv) If 1 < p < ∞, then Lp (M ) is a locally uniformly convex and reflexive Banach space (cf. Section 1.4.5.20.) Spaces Lp (M ) with p = 1 and p = ∞ are neither locally uniformly convex nor reflexive. ′ older inequality (v) Let 1 ≤ p ≤ ∞, f ∈ Lp (M ) and g ∈ Lp (M ). Then the H¨ is valid:

f g dx ≤ f Lp (M ) gLp′ (M ) . (1.1.40) M

If p = 2, then (1.1.40) is called the Cauchy inequality (cf. Section

1.4.4.13). (vi) The space L2 (M ) is a Hilbert space with the scalar product M f g dx. (vii) Let 1 ≤ p < ∞. We denote by Lp (M ) the subspace of Lp (M ) of functions with zero mean. Clearly, if M is a domain, then C0∞ (M ) := {f ∈ C0∞ (M ); M f dx = 0} is dense in Lp (M ).

(viii) Let 1 ≤ p < ∞ and f ∈ Lploc (IRN ). Then limh→0 M |f (x + h) − f (x)|p dx = 0 for any bounded measurable subset M of IRN . The reader can consult, e.g. (Brezis, 1987), Chapter IV, for proofs and more details. 1.1.15

Lebesgue’s points

Lemma 1.6 Let Ω ⊂ IRN be an open set. Then we have: (i) If f ∈ L1loc (Ω) then limdiam(V (a))→0+ (ii) If f ∈ C 0 (Ω), then

limdiam(V (a))→0+

V (a)

(iii) f = 0 in Ω if and only if

f dx = f (a) for a.a. a ∈ Ω.

(1.1.41)

f dx = f (a) for all a ∈ Ω.

(1.1.42)

V (a)

V

f = 0 for any open and bounded set V ⊂ V ⊂ Ω.

In this lemma, both limits are taken over neighborhoods V (a) ⊂ Ω of a, and diam (V) := sup {|x − y|; x, y ∈ V} .

12


Any point a ∈ Ω for

which identity

(1.1.41) is valid is called a Lebesgue point of f. Recall also that M = |M |−1 M . The proof of assertion (iii) is easy and can be left to the reader. The proof of assertion (i) can be found in (Ziemer, 1989), Theorem 1.3.8 and assertion (ii) is a consequence of a more general result for which the reader is referred to, e.g. (Rudin, 1974). 1.1.16

Absolutely continuous functions

1.1.16.1 Definition Let f : I → IR be a (real-valued) function defined on an (bounded or unbounded, open, closed or half-closed) interval I ⊂ IR. Suppose that for any ε > 0 there exists δ > 0 such that k

i=1

|f (dk ) − f (ck )| < ε

for every finite, pairwise disjoint family {(ck , dk )}nk=1 , n ∈ IN of open subintervals of I for which k i=1 (dk − ck ) < δ. Then f is said to be absolutely continuous on I. The set of all absolutely continuous functions on I is denoted AC(I). 1.1.16.2

Some properties of absolutely continuous functions

(i) Let I be a bounded interval of IR. It is well known, and easy to prove, that any absolutely continuous function f on I is continuous on I, and can be written as f1 − f2 where the fi are nondecreasing, and absolutely continuous on I. (ii) The following classical lemma about the characterization of absolutely continuous functions will be used very often whenever we deal with the nonsteady problems.

t Lemma 1.7 (i) If f ∈ L1 (I), then F (t) = 0 f ds belongs to AC(I) (the set of absolutely continuous functions on I) and F ′ (t) = f (t) for a.a. t ∈ I. (ii) Let ψ, ∂t ψ ∈ L1 (I). Then there exists ψ ∈ AC(I) such that ψ = ψ a.e. in I. Moreover

= ψ(0) + t ∂t ψ ds, ∀t ∈ I. ψ(t) 0 If ψ ∈ AC(I) is a function such that ∂t ψ = ∂t ψ a.e. in I, then ψ − ψ = c ∈ IR everywhere in I.

For the proof of the results presented in this section see Theorems 18.13, 18.15, 18.16 and 18.17 in Chapter V of (Hewitt and Stromberg, 1975). 1.1.17

Absolute continuity of integrals with respect to measurable subsets

Lemma 1.8 Let f ∈ L1 (M ). Then we have: (i) For any ε > 0 there exists δ > 0 such that


M′

13

|f | dx ≤ ε for all M ′ ⊂ M, |M | < δ.

(ii) For any ε > 0 there exists n0 > 0 such that

|f | dx ≤ ε, n ≥ n0 (M n = M \ B n ). Mn 1.1.18

Some theorems from integration theory

1.1.18.1 Tonelli’s and Fubini’s theorems Let M ⊂ IRr+s . For arbitrary x ∈ IRr we denote M x,∗ = {y ∈ IRs ; (x, y) ∈ M }. Similarly we define M ∗,y for y ∈ IRs . By Pr and Qs we denote the projection of the set M into the space of the first r variables and into the space of the last s variables, respectively. Theorem 1.9 (Tonelli’s theorem) Suppose that

|f (x, y)| dy < ∞ for a.a. x ∈ Pr M x,∗

and

Pr

Then

|f (x, y)| dy dx < ∞. M x,∗ f ∈ L1 (M ).

See e.g. (Brezis, 1987), Theorem IV.4. Theorem 1.10 (Fubini’s theorem) Suppose that f ∈ L1 (M ). Then:

(i) for a.a. x ∈ Pr , f (x, ·) ∈ L1 (M x,∗ ) and the function x → M x,∗ f (x, y) dy belongs to L1 (Pr );

(ii) for a.a. y ∈ Ps , f (·, y) ∈ L1 (M ∗,y ) and the function y → M ∗,y f (x, y) dx belongs to L1 (Qs ); (iii) it holds

f (x, y) dxdy = Pr M x,∗ f (x, y) dy dx M (1.1.43)

= Qs M ∗,y f (x, y) dx dy.

For this well-known basic result of integral calculus see, e.g. (Brezis, 1987), Theorem IV.5, (Lieb and Loss, 1997), Theorem 1.12. 1.1.18.2 Integrals dependent on a parameter dependent on a parameter reads:

A Basic theorem about integrals

Theorem 1.11 Let M ⊂ IRN be measurable and let U be a neighborhood of a ∈ IRs . (i) Suppose that a function f : U × M → IR has the following properties: a) there exists a set N ⊂ M of measure zero such that for each x ∈ M \ N the function f (·, x) is continuous at a; b) for each t ∈ U , the function f (t, ·) is measurable; c) there exists a function g ∈ L1 (M ) such that |f (t, ·)| ≤ g almost everywhere in M for all t ∈ U .

14


Then for each t ∈ U , f (t, ·) ∈ L1 (M ) and the function

F : t → M f (t, x) dx

is continuous at a. (ii) Suppose that a function f : U × M → IR has the following properties:

a) there is a t0 ∈ U such that f (t0 , ·) ∈ L1 (M ); b) there exists a set N ⊂ M of measure zero such that for each x ∈ M \ N the function f (·, x) is differentiable in U; c) for each t ∈ U , the function f (t, ·) is measurable; d) there exists a function g ∈ L1 (M ) such that | ∂t∂ i f (t, ·)| ≤ g almost everywhere in M for all t ∈ U .

Then for each t ∈ U , f (t, ·) ∈ L1 (M ), the function F is differentiable in U and

∂f ∂F ∂ti (t) = M ∂ti (t, x) dx. For more details see, e.g. (Lukeˇs and Mal´ y, 1995), Theorems 9.1 and 9.2.

1.1.18.3 Substitution theorem Let Ω ⊂ IRN be an open set. We say that the := ϕ(Ω) if the mapping ϕ : Ω → IRN is a diffeomorphism between Ω and Ω following conditions are satisfied: a) ϕ ∈ C 1 (Ω), b) ϕ is one-to-one, c) the Jacobian Dϕ(x) = 0 ∀x ∈ Ω. det Dx = ϕ(Ω) and let f Theorem 1.12 Let ϕ be a diffeomorphism between Ω and Ω be a real function defined (a.e.) in Ω. Then

f (y) dy = Ω

D ϕ(x) f (ϕ(x)) det Dx dx, Ω

(1.1.44)

provided one of the above integrals exists.

For more details see, e.g. (Lukeˇs and Mal´ y, 1995), Theorem 34.18. 1.1.18.4

Convergence in Lp -spaces

Lemma 1.13 Let 1 ≤ p ≤ ∞. Let fk → f strongly in Lp (M ). Then there exists a subsequence such that fk (x) → f (x) for a.a. x ∈ M . Theorem 1.14 (Fatou’s lemma) Let f1 , f2 , . . . be measurable and nonnegative a.e. in M . Then

lim inf k→∞ fk dx ≤ lim inf k→∞ M fk dx. (1.1.45) M

See, e.g. (Lieb and Loss, 1997), Lemma 1.7.

GOVERNING EQUATIONS AND RELATIONS

15

Theorem 1.15 (Monotone convergence). Let f1 , f2 , . . . be a nondecresing sequence of functions belonging to L1 (M ). Then the function f : M → IR ∪ {∞}, f (x) := limk→∞ fk (x) is defined for a.a. x ∈ M and

f dx = limk→∞ M fk dx. (1.1.46) M

See, e.g. (Lieb and Loss, 1997), Theorem 1.6.

Theorem 1.16 (Lebesgue dominated convergence theorem) Let fk , k = 1, 2, . . . , be measurable in M , let ϕ ∈ L1 (M ), |fk (x)| ≤ ϕ(x) for all k = 1, 2, . . . a.e. in M and let f (x) = limk→∞ fk (x) a.e. in M. Then f, fk ∈ L1 (M ), k = 1, 2, . . . and

f dx = limk→∞ M fk dx. M See (Lieb and Loss, 1997), Lemma 1.8.

Theorem 1.17 (Egoroff’s theorem about uniform convergence) Let fn → f a.e. in M, a bounded measurable set in IRN , with f finite a.e. Then for any ε > 0 there exists a measurable subset A ⊂ M such that |M \ A| < ε and fn → f uniformly in A. See, e.g. (Dunford and Schwartz, 1958), Theorem II.6.12. Theorem 1.18 (Vitali’s convergence theorem) Let fn → f a.e. in a measurable set M ⊂ IRN . Then fn → f strongly in L1 (M ) if and only if the following two conditions hold true: (i) for any ε > 0 there exists δ > 0 such that

|f (x)| dx < ε, n = 1, 2, . . . , E ⊂ M, |E| < δ; E n (ii) for any ε > 0 there exists Eε ⊂ M of finite measure such that

|f (x)| dx < ε, n = 1, 2, . . . . M \Eε n

For the proof and more details, see (Dunford and Schwartz, 1963), Theorem II.6.15. 1.2

Governing equations and relations of gas dynamics

In this section we introduce basic facts about physical models of fluid motion. For further details the reader can consult standard monographs as, for example, ˇ (Landau and Lifschitz, 1959), (Rajagopal and Truesdell, 1999), (Silhav´ y, 1997). Let (0, T ) ⊂ IR be a time interval, during which we follow the fluid motion, and let Ωt ⊂ IR3 denote the domain occupied by the fluid at time t ∈ (0, T ). In our considerations we use the fundamental hypothesis that exactly one fluid particle passes through each point x ∈ Ωt at any time t. (We assume that the set M defined by (1.1.1) is open.)

(1.2.1)

16


1.2.1 Description of the flow There are two classical descriptions of the fluid motion. a) In the so-called Lagrangian description of the flow the motion of each individual fluid particle is considered. The trajectories of the particles can be described by the equation x = ϕ(X, t), (1.2.2) where X represents the reference determining the particle under consideration. Usually we assume that X is the initial position of the particle, i.e. X = ϕ(X, 0). The components X1 , X2 , X3 of the reference X are called Lagrangian coordinates. The Lagrangian description is used, for example, if we study the flow of an element of fluid formed by the same particles at each time instant and filling a domain V(t) ⊂ IR3 at time t. The velocity and the acceleration of the fluid particle given by the reference X are defined as (X, t) = ∂t2 ϕ(X, t) (X, t) = ∂t ϕ(X, t) and a v

(1.2.3)

(X, t) = ∂t ϕ (X, t) where x = ϕ(X, t). v (x, t) = v

(1.2.4)

respectively, provided the above derivatives exist. b) The Eulerian description is based on the determination of the velocity v(x, t) of the fluid particle passing through the point x ∈ Ωt at time t. The components x1 , x2 , x3 of the reference x are called Eulerian coordinates. Due to (1.2.2) and (1.2.3) we can write Let us note that in our later considerations the velocity will also be denoted by u. Under the assumption that 3 v ∈ C 1 (M) , (1.2.5)

the acceleration of the particle passing through the point x at time t is expressed as 3 (1.2.6) a(x, t) = ∂t v (x, t) + i=1 vi (x, t)∂i v (x, t). If we omit the variables (x, t), the last formula can be simply written a = ∂t v + (v · ∇) v.

(1.2.7)

Let us also introduce the symbol D = ∂t + v · ∇ (1.2.8) Dt called the material (or total) derivative with respect to time. The partial derivative ∂/∂t is called the local derivative while the term (v · ∇) is referred to as the convective derivative. We see that the acceleration of a fluid particle is expressed in the Eulerian coordinates as the material derivative of the velocity: Dt =

a(x, t) =

Dv (x, t) = ∂t v(x, t) + [(v · ∇) v](x, t). Dt

(1.2.9)


17

1.2.1.1 The transition from the Eulerian description to the Lagrangian description is equivalent to the determination of the paths of fluid particles on the basis of a given velocity field v(x, t). The trajectory of the fluid particle passing through a point X ∈ Ωt0 at time t0 ∈ (0, T ) is given as the solution of the initial value problem dx = v(x, t), x(t0 ) = X. (1.2.10) dt Theorems 10.1.1, 11.1.5 and 13.1.1 from (Kurzweil, 1986) immediately imply the following: Theorem 1.19 Under assumption (1.2.5) the following statements hold: 1. For each (X, t0 ) ∈ M problem (1.2.10) has exactly one maximal solution ϕ(X, t0 ; t) (defined for t from a certain interval (αX,t0 , βX,t0 )). 2. The mapping ϕ has continuous first order partial derivatives with respect to X1 , X2 , X3 , t0 , t and continuous derivatives ∂ 2 ϕ/∂t∂Xi , ∂ 2 ϕ/∂t0 ∂Xi , i = 1, 2, 3, in its domain of definition {(X, t0 , t); (X, t0 ) ∈ M, t ∈ (αX,t0 , βX,t0 )}. Let us recall that a solution of problem (1.2.10) is called maximal if any other solution of the problem is its restriction. By the implicit function theorem 1.3 it is clear that if (X 0 , t0 ) ∈ M is such that (Dϕ/DX)(X 0 , t0 ) = 0 then there exists a neighborhood of (X 0 , t0 ), where the transformation (X, t) → (ϕ(X, t), t) is one-to-one. 1.2.1.2 Derivatives along trajectories If F ∈ C 1 (M) represents some physical quantity transported by moving particles, then the function F(X, t) = F (ϕ(X, t), t) represents the values of this quantity along the trajectory given by the equation x = ϕ(X, t). The derivative of this quantity along the trajectory is expressed with the aid of the chain rule in the form ∂t F(X, t) = ∂t F (ϕ(X, t), t) = ∂t F (ϕ(X, t), t) + ∂t ϕ · ∇F (ϕ(X, t), t) = ∂t F (x, t) + v(x, t) · ∇F (x, t) DF (x, t) . = Dt

(1.2.11)

Therefore, the material derivative is sometimes called the derivative along the trajectory of a fluid particle. 1.2.2

The transport theorem

Let a function F = F (x, t) : M → IR be the Eulerian representation of some physical quantity transported by fluid particles and let us consider a system of fluid particles filling a bounded domain V(t) ⊂ Ωt at time t. Assume that F ∈ C 1 (M), v ∈ [C 1 (M)]3 , ϕ = ϕ(X, t0 ; t) is the mapping from Theorem 1.19. and J(X, t) denotes the Jacobian of the mapping X ∈ V(t0 ) −→ ϕ(X, t0 ; t) ∈ V(t) :

18


Dϕ(X, t0 ; t) . (1.2.12) DX It is possible to prove the following technical result (see, e.g. (Feistauer, 1993)). J(X, t) = det

Lemma 1.20 Let t0 ∈ (0, T ), V(t0 ) be a bounded domain and let V(t0 ) ⊂ Ωt0 . Then there exists an interval (t1 , t2 ) ∋ t0 such that the following conditions are satisfied: a) The mapping t ∈ (t1 , t2 ), X ∈ V(t0 ) −→ x = ϕ(X, t0 ; t) ∈ V(t) has continuous first order derivatives with respect to t, X1 , X2 , X3 and continuous second order derivatives ∂ 2 ϕ/∂t ∂Xi , i = 1, 2, 3. b) The mapping X ∈ V(t0 ) −→ x = ϕ(X, t0 ; t) ∈ V(t) is a continuously differentiable one-to-one mapping of V(t0 ) onto V(t) with the Jacobian (1.2.12) which is continuous and bounded and satisfies the condition J(X, t) > 0 c) The inclusion

∀ X ∈ V(t0 ), ∀ t ∈ (t1 , t2 ).

(x, t); t ∈ [t1 , t2 ], x ∈ V(t) ⊂ M

holds and thus the mapping v has continuous and bounded first order derivatives on {(x, t); t ∈ (t1 , t2 ), x ∈ V(t)}. d) v(ϕ(X, t0 ; t), t) = ∂t ϕ (X, t0 ; t), X ∈ V(t0 ), t ∈ (t1 , t2 ). The following lemma plays an important role in fluid dynamics.

Lemma 1.21 Let conditions a)–d) from Lemma 1.20 be satisfied. Then the function J = J(X, t) has a continuous and bounded partial derivative ∂J/∂t for X ∈ V(t0 ), t ∈ (t1 , t2 ), and ∂t J(X, t) = J(X, t) div v(x, t),

x = ϕ(X, t0 ; t).

(1.2.13)

The proof is an elementary exercise in calculus for functions of several variables and can be found, e.g., in (Chorin and Marsden, 1979). The following theorem, called the transport theorem, holds true: Theorem 1.22 Let conditions a) – d) from Lemma 1.20 be satisfied and let the function F = F (x, t) have continuous and bounded first order derivatives on the set {(x, t); t ∈ (t1 , t2 ), x ∈ V(t)}. Then for each t ∈ (t1 , t2 ) there exists a finite derivative

d (1.2.14) dt V(t) F (x, t) dx = V(t) [∂t F (x, t) + div (F v)(x, t)] dx.

The proof is an elementary exercise in the application of the theorem about integrals dependent on a parameter. In what follows, we shall introduce the mathematical formulation of fundamental physical laws: the law of conservation of mass, the law of conservation of momentum and the law of conservation of energy, called briefly conservation laws, from which we will derive the fundamental differential equations of fluid dynamics: the continuity equation, the equations of motion and the energy equation.


1.2.3

19

The continuity equation

The density of a fluid is a function ρ : M = {(x, t); t ∈ (0, T ), x ∈ Ωt } → (0, ∞) which allows us to determine the mass m(V; t) of the fluid contained in any subdomain V ⊂ Ωt :

m(V; t) = V ρ(x, t) dx. (1.2.15)

Consider at a time t = t0 an arbitrary volume V = V(t0 ) ⊂ V ⊂ Ωt0 , called a control volume and put V(t) := ϕ(V, t), where ϕ is the mapping defined in (1.2.2). Since the domain V(t) is formed by the same particles at each time instant, the conservation of mass can be formulated in the following way: The mass of the element of fluid represented by the domain V(t) does not depend on time t. This means that dm(V(t);t) (1.2.16) = 0, t ∈ (t1 , t2 ), dt

where (t1 , t2 ) is a sufficiently small interval containing t0 . Applying the transport theorem 1.22 to (1.2.16), and taking into account (1.2.15) with V replaced by V(t), we obtain

[∂ ρ(x, t) + div(ρv)(x, t)] dx = 0, t ∈ (t1 , t2 ). V(t) t Now, if we substitute t := t0 and V(t) = V(t0 ) = V, we conclude that

V

[∂t ρ(x, t0 ) + div(ρv)(x, t0 )] dx = 0

(1.2.17)

for an arbitrary t0 ∈ (0, T ) and an arbitrary control volume V in Ωt0 . Using the continuity of the integrand in (1.2.17) and assertion iii) of Lemma 1.6 and writing t instead of t0 , we conclude that ∂t ρ (x, t) + div (ρ(x, t)v(x, t)) = 0,

t ∈ (0, T ), x ∈ Ωt .

(1.2.18)

This equation is the differential form of the law of conservation of mass and is called the continuity equation or equation of continuity. 1.2.4

The equations of motion

Now we apply Newton’s law of conservation of momentum in the following form: The rate of change of the total momentum of an element of fluid formed by the same particles at each time and occupying the domain V(t) at instant t is equal to the force acting on V(t). Let ρ ∈ C 1 (M), v ∈ [C 1 (M)]3 . The total momentum of particles contained in V(t) is given by

20


H(V(t)) =

V(t)

ρ(x, t)v(x, t) dx.

(1.2.19)

Moreover, denoting by F (V(t)) the force acting on the volume V(t), the law of conservation of momentum reads dH(V(t)) = F (V(t)), dt

t ∈ (t1 , t2 ).

(1.2.20)

Using the transport theorem, we get

∂ (ρ(x, t)vi (x, t)) + div (ρ(x, t)vi (x, t)v(x, t)) dx = Fi (V(t)), V(t) t i = 1, 2, 3, t ∈ (t1 , t2 ).

Analogously as in the case of the continuity equation we get

[∂t (ρ(x, t)vi (x, t)) + div (ρ(x, t)vi (x, t)v(x, t))] dx = Fi (V; t), V i = 1, 2, 3, for an arbitrary t ∈ (0, T ) and an arbitrary control volume V in Ωt .

(1.2.21)

The vector F (V; t) with components Fi (V; t) denotes the force acting on the volume V at time t and is to be specified. We distinguish here three types of forces acting in fluids, the so-called volume forces, nonvolume forces and surface forces. a) The volume force (also called the outer or body force) F v (V; t) acting at time t on the particles contained in a control volume V ⊂ V ⊂ Ωt is expressed by its density f ∈ C(M)3 :

F v (V; t) = V ρ(x, t)f (x, t) dx. (1.2.22)

b) The surface force (or inner force) F S , by which the fluid contained outside the domain V acts on a set S ⊂ ∂V, is expressed with the use of the stress tensor T = (τij )3i,j=1 , characterizing the density and direction of the surface force:

F S = S T · n dS. (1.2.23)

Here n(x) is the unit outer normal to ∂V at x. c) Sometimes (usually in the context of magnetohydrodynamics) there might be a necessity to deal with the nonvolume forces g:

F nv (V; t) = V g(x, t) dx. (1.2.24) Let us assume that ρ, vi , τij ∈ C 1 (M) and fi ∈ C(M) (i, j = 1, 2, 3). Inserting (1.2.22) and (1.2.23) into (1.2.21), we obtain

[∂t (ρ(x, t)vi (x, t)) + div (ρ(x, t)vi (x, t)v(x, t))] dx V

3 (1.2.25) = V ρ(x, t)fi (x, t) dx + V g(x, t) dx + ∂V j=1 τij (x, t) nj (x) dS, i = 1, 2, 3, for each t ∈ (0, T ) and an arbitrary control volume V in Ωt .


21

Moreover, applying Green’s theorem from Section 1.1.13 and Lemma 1.6, we get the equations of motion of a fluid in differential conservative form: ∂t (ρvi ) + div (ρvi v) = ρfi + gi + This can be written as

3

j=1

∂j τij ,

i = 1, 2, 3.

∂t (ρv) + div (ρv ⊗ v) = ρf + g + div T . 1.2.5

(1.2.26)

(1.2.27)

The law of conservation of the moment of momentum. Symmetry of the stress tensor

Let us assume that ρ, vi , τij ∈ C 1 (M) and fi , gi ∈ C(M). Similarly as above, consider a control volume V = V(t) formed by the same fluid particles at each time instant t ∈ (t1 , t2 ). The law of conservation of the moment of momentum can be formulated in the following way: The rate of change of the moment of momentum of the element of fluid occupying the volume V(t) at any time t is equal to the sum of the moments of the volume and surface forces acting on this volume. Writing this law in the integral formulation, and then applying the transport theorem we arrive at the following system: ∂t (εjkp ρxk vp ) + div (εjkp ρxk vp v) = ∂i (εjkp ρxk Tpi ) + εjkp ρxk fp + εjkp ρxk gp .

(1.2.28)

It can be derived from (1.2.28) that the stress tensor T has to be symmetric, i.e., τij = τji , i, j = 1, 2, 3. 1.2.6

(1.2.29)

Inviscid and viscous fluids

The relations between the stress tensor and other quantities describing fluid flow, particularly the velocity and its derivatives, represent the so-called rheological equations of the fluid. The simplest rheological equation T = −p I,

I = (δij )3i,j=1 ,

(1.2.30)

characterizes an inviscid fluid. Here p is the pressure. The function p = p(ρ, θ) characterizes the fluid. Its form, which depends on intermolecular interactions, is the subject of studies in kinetic theory and statistical physics. Besides the pressure forces, also the friction shear forces act in real fluids. Such fluids are called viscous. Therefore, in the case of a viscous fluid, we add a contribution T ′ characterizing the shear stress to the term −p I: T = −p I + T ′ .

(1.2.31)

22

1.2.7


The energy equation

The next law to be studied is the energy conservation law. Clearly, the power of the force F acting on the particle passing through the point x at time t equals W (x, t) = F (x, t) · v(x, t).

(1.2.32)

Like in the preceding sections, we consider an element of fluid represented by a control volume V(t) satisfying the assumptions from Section 1.2.3. The energy conservation law can be formulated as follows: The rate of change of the total energy of the fluid particles, occupying the domain V(t) at time t, is equal to the sum of powers of the volume force acting on the volume V(t) and the surface force acting on the surface ∂V(t), and of the amount of heat transmitted to V(t). Denote by E(V(t)) the total energy of the fluid particles contained in the domain V(t) and by Q(V(t)) the amount of heat transmitted to V(t) at time t. Taking into account the character of outer and inner forces acting on the domain V(t), determined by the density f of the volume force and the stress tensor T , we get the identity representing the energy conservation law:

d dt E(V(t)) = V(t) ρ(x, t)f (x, t) · v(x, t) dx + V(t) g(x, t) · v(x, t) dx (1.2.33)

+ ∂V(t) T (x, t) · v(x, t) · n dS + Q(V(t)), where

E = 21 ρ|v|2 + P, P (ρ) = ρe,

Q(V(t)) = V(t) ρ(x, t)q(x, t) dx − ∂V(t) q(x, t) · n(x) dS.

E(V(t)) =

V(t)

E(x, t) dx,

(1.2.34)

Here E is the total energy of the particle at point x at time t, e is the specific internal energy (i.e. the internal energy per unit mass), P is the so-called potential energy of the particle at point x at time t, |v|2 /2 is the density of the kinetic energy, q represents the density of heat sources (related to the unit of mass) and q is the heat flux. The quantities e(ρ, θ) and q(ρ, θ, ∇θ) characterize the fluid. Their form depends on the intermolecular and interatomic forces. In the case of any particular fluid, it is the subject of studies in kinetic theory and in statistical mechanics, or of experiments. Assuming ρ, u, vi , τij , qi ∈ C 1 (M), and fi , q ∈ C(M) (i, j = 1, 2, 3), by virtue of the transport theorem 1.22, Green’s theorem and Lemma 1.6, we find from (1.2.34) the energy equation written in the differential conservative form: ∂t E + div (Ev) = ρf · v + g · v + div(T · v) + ρq − div q. 1.2.8

(1.2.35)

The second law of thermodynamics and the entropy

In order to complete the system of conservation laws, additional equations derived in thermodynamics have to be added.


23

The absolute temperature θ, the density ρ and the pressure p are called the state variables. All these quantities are positive functions. The gas is characterized by the equation of state p = p(ρ, θ)

(1.2.36)

e = e(ρ, θ).

(1.2.37)

and the relation

Any quantity satisfying the relation ds =

1 θ

de + pd

1

(1.2.38)

ρ

is called the specific entropy. Functions p(ρ, θ), e(ρ, θ) and q(ρ, θ, ∇θ) being fixed, we say that a process (ρ, v, θ) ∈ C 1 (M) × C 2 (M)3 × C 2 (M) is an admissible process if there exist f , g ∈ C 0 (M)3 such that (ρ, v, θ) satisfies (1.2.18), (1.2.27) and (1.2.35). The second law of thermodynamics says that d dt

V(t)

ρ(x, t) s(x, t) dx ≥

V(t)

ρ(x,t)q(x,t) θ(x,t)

dx −

∂V(t)

q(x,t)·n(x) θ(x,t)

dS

(1.2.39)

holds for any admissible process. After the application of the transport theorem, (1.2.39) yields ∂t (ρs) + div (ρvs) ≥ −div qθ + ρq. (1.2.40)

The last inequality is known as the Clausius–Duhem inequality. 1.2.9

Principle of material frame indifference

We shall start with a short summary of what we have already said. A thermodynamic process of a viscous fluid is a collection of eight plus three functions of position and time. Their meaning as well as the tensorial nature is as follows: v (velocity vector), θ (absolute temperature – a scalar function), ρ (density – a scalar function), e (specific internal energy – a scalar function), s (specific entropy – a scalar function), f (density of external body forces – a vector function), q (heat sources – a scalar function), q (heat flux – a vector function), T (stress tensor). We already know that the stress tensor is symmetric. Each process satisfies the equations of balance of mass, of momentum, of energy, and the Clausius–Duhem inequality (cf. (1.2.18), (1.2.27), (1.2.35), (1.2.40)). Consider a change of frame xi (t) = Sij (t)xj + ci (t), where S := (Sij )3i,j=1 is an orthonormal matrix. In the last formula, x represents the coordinates of a material point in the old frame and x represents the coordinates of the same material point in the new frame. The principal of material

24


frame indifference postulates that under the above change of frame the quantities θ, ρ, e, s, q, q and T transform as follows: θ = θ, ρ = ρ, e = e, s = s, q = q, q i = Sij qj , τ ij = Sik Sip τkp .

(1.2.41)

In these formulae overlined quantities correspond to the same process described in the new frame. The quantities are evaluated at the same material point; since the spatial description is used, the arguments on the left-hand side are (x, t), while the arguments on the right-hand side are (x, t). 1.2.10 Newtonian fluids In order to identify the viscous part T ′ of the stress tensor (see (1.2.31)), we shall use Stokes’ postulates: 1) The tensor T ′ depends linearly on the gradient of the velocity ∇v. 2) The tensor T ′ satisfies the principle of material frame indifference. Then it is possible to show that Theorem 1.23 Under conditions 1), 2) above, the stress tensor has the form T = (−p + λ div v) I + 2µD(v),

(1.2.42)

where λ, µ are constants or scalar functions of thermodynamical quantities and D(v) = (dij )3i,j=1 , dij = 12 (∂i vj + ∂j vi ).

(1.2.43)

(For the definition of thermodynamical quantites see Section 1.2.9.) The fluid described by (1.2.42) is called Newtonian. 1.2.11

Conservative and dissipation form of the energy equation for Newtonian fluids In virtue of (1.2.35) the conservative form of the energy equation for a Newtonian fluid reads ∂t E + div (Ev) = ρ f · v + g · v − div (pv) + div (λv div v) + div (2µD(v) · v) + ρq − div q.

(1.2.44)

On the basis of the continuity equation (1.2.18), the momentum equations (1.2.27) can be written in the convective form ρ∂t v + ρ(v · ∇)v = ρf + g − ∇p + ∇(λdiv v) + div (2µD(v)). The scalar product of this equation with v yields 2 2 ρ∂t |v2| + ρ(v · ∇) |v2|

(1.2.45)

(1.2.46)

= ρf · v + g · v − v · ∇p + v · ∇(λdiv v) + v · div (2µD(v)).

Recalling that E = 12 ρ|v|2 +ρe, taking into account that div (pv) = v·∇p+pdiv v, div (λvdiv v) = λ(div v)2 + v · ∇(λdiv v), div (2µD(v)v) = v · div (2µD(v)) +


25

2µD(v) : D(v), and the fact that the left-hand side of (1.2.46) can be transformed with the aid of (1.2.18) to the expression ∂(ρ|v|2 )/∂t + div(ρ|v|2 v) /2, we find that the energy equation (1.2.44) can be rewritten in the following form ∂t (ρe) + div (ρev) + p div v = D(v) + ρq − div q.

(1.2.47)

Here the quantity D(v) = λ(div v)2 + 2µD(v) : D(v)

(1.2.48)

is called the dissipation. Equation (1.2.47) is called the energy equation in dissipation form. 1.2.12

Entropy form of the energy equation for Newtonian fluids

By virtue of the continuity equation (1.2.18), the energy equation (1.2.47) is equivalent to ρ De Dt + p div v = D(v) + ρq − div q. Since (1.2.38) can also be written in the form Ds Dt

=

1 De θ Dt

+ p div v,

(1.2.49)

we can rewrite (1.2.47) in the form ρ Ds Dt = (D(v) + ρq − div q)/θ

(1.2.50)

∂t (ρs) + div (ρsv) = (D(v) + ρq − div q)/θ.

(1.2.51)

which is equivalent to

This is the so-called entropy form of the energy equation. 1.2.13

Some consequences of the Clausius–Duhem inequality

If we apply identities (1.2.49) and (1.2.50) in the Clausius–Duhem inequality (1.2.40), we deduce that D(v) − qθ · ∇θ ≥ 0 for any admissible process. This means that we must have at least −q · ∇θ ≥ 0

(1.2.52)

D(v) ≥ 0.

(1.2.53)

and also, To see the first inequality it suffices to take all admissible processes with v = 0 and to see the second inequality, all admissible processes with θ = const > 0.

26


By an elementary argument of linear analysis formula (1.2.53) yields µ ≥ 0,

2µ + 3λ ≥ 0.

(1.2.54)

If we suppose that q depends only on ρ, θ and ∇θ, from the principle of material frame indifference we derive that it must obey the Fourier law q = −k(θ, |∇θ|)∇θ.

(1.2.55)

k(θ, |∇θ|) ≥ 0.

(1.2.56)

Moreover, due to (1.2.52) 1.2.14

Equations of state

1.2.14.1 Relations between pressure and internal energy According to classical thermodynamical theory, from the quantities e, p, s, θ, ρ only two are independent thermodynamical variables. Usually, we consider ρ and θ independent, and we characterize the fluid by prescribing the specific internal energy e(ρ, θ) and the pressure p(ρ, θ). In this case, the specific entropy s(ρ, θ) is explicitly given by (1.2.38). The laws p(ρ, θ) and e(ρ, θ) are specific for a given gas and are the 2 ), then subject of studies in kinetic theory and statistical physics. If s ∈ C 1 (IR+ (1.2.38) implies

∂ρ s = θ1 ∂ρ e − ρp2 , ∂θ s = θ1 ∂θ e (1.2.57) 2 and if s is C 2 (IR+ ), then we necessarily have

1 ρ2 p − θ∂θ p = ∂ρ e.

(1.2.58)

1.2.14.2 Perfect gas If the fluid obeys the Mariotte and Joule laws, i.e., if p(ρ, θ) takes the form ρf (θ) and e(ρ, θ) takes the form e(θ), then formula (1.2.58) yields p(ρ, θ) = Rρθ, e(ρ, θ) = e(θ), (1.2.59) where R is called the universal gas constant. Physical considerations usually suggest for a standard gas e(θ) = cv θ,

cv > 0,

(1.2.60)

where the constant cv is called the specific heat at constant volume. If a gas obeys state equations (1.2.59) and (1.2.60), then it is called a perfect gas. From definition (1.2.38), it is easy to calculate explicitly the state equation for the specific entropy: Exercise 1.24 For a perfect gas we have θ/θ0 p/p0 s = cv ln (ρ/ρ γ + const = cv ln (ρ/ρ )γ−1 + const, 0) 0

(1.2.61)

where p0 and ρ0 are fixed (reference) values of pressure and density, respectively, γ = 1 + cRv and θ0 = p0 /(Rρ0 ).


27

1.2.14.3 Heat flux For most gases for the coefficient k in the Fourier law (1.2.55) there holds k = const > 0. (1.2.62) This constant is called the coefficient of heat conduction. For inviscid gases, one usually puts k = 0. (1.2.63) 1.2.15

Adiabatic flow of a perfect inviscid gas

If there is no heat transfer and heat exchange between fluid volumes, we speak about adiabatic flow. Hence, in adiabatic flow the heat sources and heat flux are zero, so that q = 0, q = 0 and, with respect to (1.2.55), also k = 0. It is known that heat conductivity and internal friction represent two faces of molecular transmission. Heat conductivity is related to the transmission of molecular kinetic energy and internal friction is conditioned by the transmission of molecular momentum. Therefore, it makes sense to speak about adiabatic flow particularly in the case of an inviscid gas. From (1.2.50) it follows that for adiabatic inviscid flow Ds Dt

= ∂t s + v · ∇s = 0.

(1.2.64)

If x = ϕ(X, t) is the trajectory of a fluid particle and, hence, v(x, t) = ∂t ϕ(X, t), in view of (1.2.11) we have ∂t s(ϕ(X, t), t) = 0. Hence, s(ϕ(X, t), t) = const.

(1.2.65)

From this and (1.2.61) we see that we have proved Theorem 1.25 In adiabatic flow of an inviscid perfect gas s = const along the trajectory of any fluid particle, p = Cργ along the trajectory of any fluid particle,

(1.2.66) (1.2.67)

where C is a constant dependent on the trajectory considered. If condition (1.2.66) is satisfied, then we speak about isentropic flow. If s = const in the whole flow field, then the flow is called homoentropic. Remark 1.26 It is necessary to say that the statement p = Cργ is not quite correct, although it is commonly used in physics. To be correct, we should write p = cp0 (ρ/ρ0 )γ where p0 and ρ0 are suitable reference values of the pressure and density, respectively, and c is a dimensionless constant. If it makes sense to substitute p := p0 and ρ := ρ0 , we see that c = 1. For the sake of simplicity we, however, often use the incorrect statement with C := p0 /ργ0 mentioned above.

28


Summarizing the above results, we see that the flow of a perfect gas, described by sufficiently smooth functions, satisfies the second law of thermodynamics. If the flow is moreover inviscid and adiabatic, then it is isentropic. 1.2.16

Compressible Euler equations

Compressible Euler equations are equations governing the motion of an inviscid perfect gas. By virtue of (1.2.18), (1.2.27), (1.2.35), (1.2.59), (1.2.63), (1.2.18), they read: ∂t ρ + div (ρv) = 0, ∂t (ρv) + div (ρv ⊗ v) + ∇p = ρf + g,

∂t E + div ((E + p)v) = ρq + ρf · v + g · v, 1 E = |v|2 + cv θ, p = p(ρ, θ) = Rρθ. 2

(1.2.68) (1.2.69) (1.2.70) (1.2.71)

This system is simply called the compressible Euler equations. If the temperature of the gas is kept constant (θ = θ0 > 0) during the process, we deal with isothermal flow. Isothermal flow of a perfect inviscid gas is governed by the following system of equations: ∂t ρ + div (ρv) = 0, ∂t (ρv) + div (ρv ⊗ v) + Rθ0 ∇ρ = ρf + g.

(1.2.72) (1.2.73)

This system is called the compressible Euler equations for isothermal flow of a perfect gas. If the flow is adiabatic (i.e. the entropy is kept constant during the process) the governing equations are the compressible isentropic Euler equations for a perfect gas. They read: ∂t ρ + div (ρv) = 0, ∂t (ρv) + div (ρv ⊗ v) + ∇p(ρ) = ρf + g, p(ρ) = cργ . 1.2.17

(1.2.74) (1.2.75) (1.2.76)

Compressible Navier–Stokes equations for a perfect viscous gas

A Newtonian fluid satisfying (1.2.59)–(1.2.62) is called a perfect viscous gas. It is governed by the following system of equations: ∂t ρ + div (ρv) = 0,

(1.2.77) ∂t (ρv) + div (ρv ⊗ v) = ρf + g − ∇p + ∇(λdiv v) + div (2µD(v)), (1.2.78) ∂t E + div ((E + p)v)) + div q (1.2.79) = ρq + ρf · v + g · v + div (λv div v) + div (2µD(v) · v), 1 E = ρ|v|2 + cv θ, q = k∇θ, p(ρ, θ) = Rρθ. 2

(1.2.80)


1.2.18

29

Barotropic flow of a viscous gas

Suppose that during the thermodynamic process of a viscous compressible perfect gas, the entropy is kept constant. In this case, the energy equation in the entropy form (cf. (1.2.50)) yields div q − ρq = D(v), (1.2.81) and from (1.2.61), (1.2.60), we obtain P (ρ) := ρe(θ) =

C γ γ−1 ρ ,

p(ρ) = Cργ , C =

p0 s0 /cv e . ργ 0

(1.2.82)

Here s0 is the constant value of the specific entropy. Now, let us investigate the thermodynamic processes with constant entropy s = s0 = const of a general viscous gas which is characterized by the state equations (1.2.83) e = cv θ, p = p(ρ). The energy equation (1.2.50) guarantees that equation (1.2.81) holds again. By virtue of (1.2.38), the state equations (1.2.83) yield the following formula for the potential energy P = ρe of the gas: ρP ′ (ρ) − P (ρ) = p(ρ) or equivalently P (ρ) = ρ

ρ 1

p(τ ) τ 2 dτ

(1.2.84) (1.2.85)

up to a constant. Such a thermodynamic process is called barotropic. We see that an isentropic process of a viscous gas can be viewed as one of its particular cases with p(ρ) = Cργ . To resume, the barotropic process of a viscous gas is governed by the following system of equations: ∂t ρ + div (ρv) = 0, (1.2.86) ∂t (ρv) + div (ρv ⊗ v) − div (λ∇v) − div (2µD(v)) + ∇p = ρf + g, (1.2.87) p = p(ρ). (1.2.88) Of course, the energy equation (1.2.44) must hold as well. By virtue of (1.2.81), it takes the form ∂t E + div ((E + p)v) + D(v) = div (λvdiv v) + div (2µD(v) · v) + ρf · v + g · v,

(1.2.89)

E = 12 ρ|v|2 + P (ρ) with ρP ′ (ρ) − P (ρ) = p(ρ).

(1.2.90)

where In contrast to energy equation (1.2.44), energy equation (1.2.87) is not independent of (1.2.86)–(1.2.88). It can be derived formally from (1.2.86)–(1.2.88) by scalar multiplying (1.2.86) by v applying similar arguments as those used in

30


Section 1.2.11. Vice versa, if we suppose that (1.2.86)–(1.2.89) hold, and that s = const, then the energy E in the energy equation (1.2.89) must be given by (1.2.90). Of course, it is a difficult task to keep condition (1.2.81) satisfied in physical experiments. Nevertheless, from the point of view of the mathematical theory, isentropic processes of a viscous gas are possible. If the viscosity of the gas is small, we can consider that condition (1.2.81) is satisfied approximately under the same physical conditions evoked in Section 1.2.11. This justifies the use of barotropic models for viscous gases. 1.2.19 Speed of sound A more general model than barotropic flow is obtained in thermodynamics under the assumption that the pressure is a function of the density and entropy: p = p(ρ, s) where p is a continuously differentiable function and ∂p/∂ρ > 0. For example, for perfect gas, in view of Exercise 1.24, we have p = f (ρ, s) = Cργ exp(s/cv ),

C = const > 0.

(1.2.91)

(The adiabatic barotropic flow of an ideal perfect gas with s = const is obviously a special case of this model.) Let us introduce the quantity a = ∂ρ f (1.2.92)

which has the dimension m s−1 of velocity and is called the speed of sound. This terminology is based on the fact that a represents the speed of the propagation of pressure waves of small intensity. 1.2.20 Simplified models If the quantities describing the flow are independent of time and, hence, all their time-derivatives are zero, then we speak about stationary (or steady) flow. Sometimes the geometry of the domain occupied by the fluid and the character of the flow allows us to introduce a Cartesian coordinate system x = (x1 , x2 , x3 ) in IR3 in such a way that the quantities describing the flow are independent of x3 and the velocity component v3 and the component f3 of the volume force vanish. Then we can assume that Ω ⊂ IR2 , x = (x1 , x2 ), v = (v1 , v2 ), f = (f1 , f2 ), ∇ = (∂/∂x1 , ∂/∂x2 )T ,

D(v) = (dij (v))2i,j=1 , dij (v) = (∂j vi + ∂i vj )/2,

etc. We easily find that the governing system of equations can again be written as (1.2.18) (the continuity equation), (1.2.45) with N = 2 (the Navier–Stokes equations) and (1.2.44) (the energy equation). Also relations from Sections 1.2.8– 1.2.19 remain formally unchanged. In this way we get a two-dimensional model of the flow. We also speak of plane flow. Similarly, we arrive at a 1D model, when the flow is described by quantities ρ, v = v1 , p, θ, etc., depending on x = x1 ∈ Ω ⊂ IR only. Hence, the flow problems can be formulated in IRN , where N = 1, 2 or 3.


1.2.21

31

Initial and boundary conditions

The system of governing equations and relations has to be equipped with initial conditions determining the state at the initial time t = 0 and with boundary conditions which characterize the behavior of the flow on the boundary ∂Ω. The initial conditions can be formulated, e.g., as v(x, 0) = v 0 (x), ρ(x, 0) = ρ0 (x), θ(x, 0) = θ0 (x),

x ∈ Ω,

(1.2.93)

with given initial data v 0 , ρ0 , θ0 . The choice of the boundary conditions is more complex. First, let us assume that Ω is a fixed domain. For the velocity, the simpliest choice is the Dirichlet (also called essential) condition v|∂Ω = v D ,

(1.2.94)

with a given function v D . On a fixed impermeable wall Γ ⊂ ∂Ω we assume the so-called no-slip condition (1.2.95) v|Γ = 0 (i.e., v D |Γ = 0), expressing the physical fact that a real fluid adheres to Γ. If condition (1.2.94) is used, then on ∂Ω we distinguish inflow through an inlet (1.2.96) ΓI (t) = {x ∈ ∂Ω; v(x, t) · n(x) < 0} and outflow through an outlet ΓO (t) = {x ∈ ∂Ω; v(x, t) · n(x) > 0},

(1.2.97)

where n denotes the unit outer normal to ∂Ω. If ΓI (t) = ∅, then it is necessary to prescribe the density on ΓI (t): ρ(x, t) = ρD (x, t),

x ∈ ΓI (t), t ∈ (0, T ).

(1.2.98)

In the case of an inviscid model, when µ = λ = 0 (and eventually k = 0), the boundary conditions (1.2.94) and (1.2.95) must be relaxed. There is no reason for the fluid to adhere to an impermeable wall Γ now and we consider the condition v · n|Γ = 0.

(1.2.99)

On the whole boundary, possibly with an inlet or an outlet, we use the boundary condition (1.2.100) v · n|∂Ω = vnD with a given scalar function vnD . On the inlet ΓI (t) = ∅ we again prescribe the density, i.e., we use condition (1.2.98).

32


There is also a model of viscous flow with assumption that the fluid slips on the boundary. Then we use the no-stick conditions, also called the slip conditions N (1.2.101) v · n|∂Ω = 0, i,j=1 τi (∂i vj + ∂j vi )nj |∂Ω = 0,

for any vector τ = (τ1 , . . . τN ), tangent to the boundary. Another possibility is that on the boundary ∂Ω the normal component of the stress tensor is prescribed: Tn |∂Ω ≡ T n|∂Ω ≡ ((−p + λ div v)n + 2µD(v)n) |∂Ω = H

(1.2.102)

with a given function H. These conditions may appear in problems with a free boundary. For the definition of the boundary we use the so-called kinematic condition ∂Ω(t) = {x; η(x, t) = 0}, (1.2.103) where η satisfies

∂t η + (v · ∇)η = 0.

(1.2.104)

This equation means that the particle being on the free boundary can move only along this boundary. Indeed, if ξ 0 ∈ ∂Ω(0) and

dξ dt

= v(ξ, t),

t > 0,

then d dt η(ξ(t), t)

which implies

= ∂t η(ξ(t), t) + v(ξ(t), t) · ∇η(ξ(t), t) = 0,

η(ξ(t), t) = η(ξ 0 , 0) = 0,

t > 0.

If heat conduction is included in the model, then it is necessary to add boundary conditions characterizing the heat processes at the boundary. For example, we can assume that (1.2.105) k∂n θ|∂Ω = κ(θ − χ)|∂Ω ,

where k is the heat conduction coefficient and κ and χ are given functions defined on ∂Ω. There are also other possibilities: Either θ|∂Ω = χ (given temperature on the boundary)

(1.2.106)

or k∂n θ = q

(given heat flux through the boundary).

(1.2.107)

If the regularity of a solution up to the boundary is required, then initial and boundary data must satisfy certain compatibility conditions to be specified accordingly. 1.3

Some advanced mathematical concepts and results

Here we summarize all important auxiliary results which will be used in our further considerations. Let us note that the basic concepts from functional analysis are explained in Section 1.4.

SOME ADVANCED MATHEMATICAL CONCEPTS AND RESULTS

33

1.3.1 Spaces of H¨ older-continuous and continuously differentiable functions Let Ω ⊂ IRN be a domain. We shall work with the linear spaces C k (Ω), C k (Ω) and Cbk (Ω) defined in Section 1.1.1 and the linear spaces C k,µ (Ω) introduced in Section 1.1.7. Let us put ∞ ∞ (1.3.1) C ∞ (Ω) = k=1 C k (Ω) and C ∞ (Ω) = k=1 C k (Ω).

A vector α = (α1 , . . . , αN ) with integer components αi ≥ 0 is called a multiindex N of dimension N . The number |α| = i=1 αi is called the length of the multiindex α. In the following we shall often use the simplified notation Dα v :=

∂ |α| v α α ∂x1 1 ...∂xNN

(1.3.2)

for partial derivatives of a function v = v(x1 , . . . , xN ) of N real variables. We also denote αN 1 xα := xα 1 . . . xN . The linear space Cbk (Ω), k = 0, 1, . . . endowed with norm uC k (Ω) := |α|≤k supx∈Ω |Dα u(x)| b

is a Banach space. Sometimes if there is not a danger of confusion we write simply uC k (Ω) = uC k (Ω) = uC k . b

If Ω is a bounded domain, then Cbk (Ω) = C k (Ω). Spaces Cbk (Ω) are separable and nonreflexive if Ω is a bounded domain and they are neither separable nor reflexive if Ω is an unbounded domain. The linear space C k,µ (Ω), k = 0, 1, . . ., µ ∈ (0, 1] endowed with norm α α u)(y)| uC k,µ (Ω) := uC k−1,µ (Ω) + |α|=k supx,y∈Ω, x=y |(D u)(x)−(D |x−y|µ

is a Banach space. It is called the H¨ older space. It is well known that this space is not separable and that it is not reflexive. For proofs and details see Chapter I in (Kufner et al., 1977). 1.3.2 Young’s functions, Jensen’s inequality (a) We say that Φ is a Young’s function if

t Φ(t) = 0 φ(s) ds, t ≥ 0,

(1.3.3)

where the real-valued function φ defined on [0, ∞) has the following properties φ(0) = 0, φ(s) > 0, s > 0, lims→∞ φ(s) = ∞,

We define

φ is right continuous and nondecreasing on [0, ∞).

(1.3.4)

t (1.3.5) ψ(t) = sup{φ(s)≤t} s, t ≥ 0, Ψ(t) = 0 ψ(s) ds. Then Ψ is a Young’s function as well. It is called the complementary Young’s function to Φ. If Φ is complementary to Ψ, then Ψ is complementary to Φ.

34


(b) If Φ and Ψ are complementary Young’s functions then the following generalization to Young’s inequality in IR holds: st ≤ Ψ(s) + Φ(t), s, t ≥ 0.

(1.3.6)

(c) We say that Young’s function Φ satisfies the ∆2 condition if there exists c > 0 such that Φ(2t) ≤ c Φ(t), t ≥ 0.

(d) Young’s functions Φ1 and Φ2 are equivalent if there exist c1 , c2 > 0 such that Φ1 (c1 t) ≤ Φ2 (t) ≤ Φ1 (c2 t), t ≥ 0.

(e) Concluding this section, let us mention the so-called Jensen’s inequalities: Let Φ be a convex function on IR. (i) Let u1 , . . . , un ∈ IR and let α1 , . . . , αn be positive numbers. Then

α1 Φ(u1 )+α2 Φ(u2 )+···+αn Φ(un ) 2 u2 +···+αn un (1.3.7) ≤ . Φ α1 uα1 +α α1 +α2 +···+αn 1 +α2 +···+αn

This inequality is called Jensen’s inequality. (ii) Let α = α(x) be defined and positive almost everywhere on a measurable set Ω ⊂ IRN . Then u(x)α(x) dx Φ(u(x))α(x) dx (1.3.8) Φ Ω α(x) dx ≤ Ω α(x) dx Ω

Ω

for every nonnegative function u provided that all integrals in the inequality are convergent. This inequality is sometimes called Jensen’s integral inequality. 1.3.3 Orlicz spaces Theorem 1.27 Let Ω be a domain in IRN and Φ a Young’s function. The subset of L1loc (Ω) given by

LΦ (Ω) := {u ∈ L1loc (Ω); uΦ := sup{ Ψ(|v|)≤1} Ω uv dx < ∞}, Ω

where Ψ is the complementary Young’s function to Φ, is a Banach space with norm uΦ .

The Banach space LΦ (Ω) is called the Orlicz space corresponding to Φ and ·Φ is called the Orlicz norm. For more details see Theorems 3.6.4 and 3.9.1 in (Kufner et al., 1977). The basic properties of Orlicz spaces are recalled in the following four theorems.

Theorem 1.28 (i) If u ∈ LΦ (Ω), then Ω Φ(|u|) < ∞.

(ii) If Φ satisfies ∆2 condition, then u ∈ LΦ (Ω) if and only if Ω Φ(|u|) < ∞. (iii) If u ∈ LΦ (Ω) and v ∈ LΨ (Ω) where Φ and Ψ are the complementary Young’s older’s inequality in the form functions, then uv ∈ L1 (Ω) and H¨

uv dx ≤ uΦ vΨ Ω holds.


35

Theorem 1.29 LΦ1 (Ω) = LΦ2 (Ω) with · Φ1 , · Φ2 equivalent norms if and only if Φ1 and Φ2 are equivalent Young’s functions. Theorem 1.30 Let the Young’s function Φ satisfy the ∆2 condition and F ∈ LΦ (Ω)∗ . Then there exists vF ∈ LΨ (Ω) such that

F (u) = Ω vF u dx and the norms F (LΦ (Ω))∗ and vF Ψ are equivalent.

Theorem 1.31 (i) If Φ satisfies the ∆2 condition, then the Orlicz space LΦ (Ω) is separable and C0∞ (Ω) is dense in it. (ii) Let Φ and Ψ be two complementary Young’s functions. Then LΦ (Ω) is reflexive if and only if both Φ and Ψ satisfy the ∆2 condition. For details about these results see Chapter 3 in (Kufner et al., 1977), more specifically Theorems 3.7.3, 3.7.5 and 3.13.1. 1.3.4 Distributions In this section, if not stated explicitly otherwise, Ω is a domain in IRN . Let C0∞ (Ω) denote the linear space of all functions v ∈ C ∞ (Ω) whose support supp v = {x; v(x) = 0}

(1.3.9) C0∞ (Ω)

is a compact (i.e. bounded and closed) subset of the domain Ω. In we introduce the topology of locally uniform convergence: we say that vn → v in C0∞ (Ω) as n → ∞, if there exists a compact set K ⊂ Ω such that a) supp vn ⊂ K ∀ n = 1, 2, . . . , and b) Dα vn −→ Dα v uniformly in K

(1.3.10)

as n −→ ∞ for all multiindices α = (α1 , . . . , αn ).

The space C0∞ (Ω) endowed with this topology is usually denoted D(Ω). By D′ (Ω), we denote the space of all continuous linear functionals defined on D(Ω), i.e. D′ (Ω) = (C0∞ (Ω))∗ . This means that f : D(Ω) → IR is an element of D′ (Ω) if the following conditions are satisfied (the symbol f, v denotes the value of the functional f at a point v ∈ D(Ω)): a) f, α1 v1 + α2 v2 = α1 f, v1 + α2 f, v2 ∀ α1 , α2 ∈ IR, ∀ v1 , v2 ∈ D(Ω), b) v, vn ∈ D(Ω), vn → v in D(Ω) =⇒ f, vn → f, v.

(1.3.11)

The space D′ (Ω) is endowed with the topology induced by the convergence fn → f in D′ (Ω) if fn , v → f, v, v ∈ D(Ω).

The elements of the space D′ (Ω) are called generalized functions or distributions.

36


Example 1.32 a) Let f be a locally integrable function in Ω, i.e. f ∈ L1loc (Ω) (see Section 1.1.14). It can be shown that the mapping

v ∈ C0∞ (Ω) → Ω f v dx (1.3.12)

is an element of D′ (Ω). Hence, the space L1loc (Ω) can be identified with a certain subspace in D′ (Ω). The distributions associated with a function f by (1.3.12) are called regular distributions and they will again be denoted by the symbol f . Therefore, we write

f, v = Ω f v dx, v ∈ D(Ω). (1.3.13)

b) For a ∈ Ω we define the functional δa : C0∞ (Ω) → IR attaining the value v(a) for any v ∈ C0∞ (Ω). It is easy to see that δa ∈ D′ (Ω). Hence, we write δa , v = v(a),

v ∈ C0∞ (Ω).

(1.3.14)

The distribution δa is called the Dirac distribution. The following assertion is sometimes useful. Lemma 1.33 Let f1 , f2 ∈ L1loc (Ω) and f1 , v = f2 , v for all v ∈ C0∞ (Ω). Then f1 = f2 a.e. in Ω. For the proof see (Brezis, 1987), Lemma IV.2. 1.3.4.1 Derivatives of distributions Let α be a multiindex. A distribution fα ∈ D′ (Ω) is called the α-th distributional derivative (or generalized derivative) of a distribution f ∈ D′ (Ω) if fα , v = (−1)|α| f, Dα v

∀ v ∈ D(Ω).

(1.3.15)

It is obvious that every distribution has generalized derivatives of all orders. Moreover, Green’s theorem implies that the distributional derivatives of any function f ∈ C k (Ω) (considered as a distribution in the sense of (1.3.13)) up to order k are equal to the classical derivatives of f . Therefore, we use the notation Dα f also for distributional derivatives and write Dα f, v = (−1)|α| f, Dα v for f ∈ D′ (Ω) and v ∈ C0∞ (Ω).

(1.3.16)

1.3.4.2 Multiplication of distributions If f ∈ D′ (Ω) and a ∈ C ∞ (Ω), we define af ∈ D′ (Ω) by setting af, v = f, av, v ∈ D(Ω). 1.3.4.3 where

(1.3.17)

Convolutions Let f, g ∈ L1loc (IRN ) be such that the map x → h(x), h(x) = L1loc (IRN ).

IRN

|g(y)f (x − y)| dy,

Then the function

x → [f ∗ g](x), [f ∗ g](·) = IRN f (y)g(· − y) dy ∈ L1loc (IRN )

belongs to

is called the convolution of f and g.


37

Convolution f ∗ g is well defined, e.g., in the following cases: (i) if one of the functions f , g has a compact support; (ii) if f ∈ Lp (IRN ), g ∈ Lq (IRN ), 1 ≤ p, q ≤ ∞,

1 r

:=

1 p

+

1 q

− 1 ≥ 0.

In this case f ∗ g ∈ Lr (IRN ) and f ∗ g0,r ≤ f 0,p g0,q .

(1.3.18)

This inequality is known as Young’s inequality for convolutions. 1 Property (i) is easy to prove, see e.g. (Vladimirov, 1967), Section II.7.4. Statement (ii) can be found e.g. in (Brezis, 1987), IV.30 or in (Bergh and L¨ ofstr¨ om, 1976), Theorem 1.2.2. We shall also need convolutions of f ∈ D′ (IRN ) with g ∈ D(IRN ). We can define it by

f ∗ g, v := f, IRN v(· + y)g(y) dy (1.3.19) = f, g ∗ v, g(x) = g(−x). For the definition of convolution for more general distributions f and g, see, e.g., (Vladimirov, 1967), Section II.7 or (Yosida, 1974), Section VI.3. We shall need the following properties of convolutions: (i) If f ∗ g exists, then g ∗ f exists and f ∗ g = g ∗ f.

(1.3.20)

(ii) If f ∗ g exists, then Dα f ∗ g and f ∗ Dα g exist and Dα [f ∗ g] = Dα f ∗ g = f ∗ Dα g.

(1.3.21)

For proofs, see, e.g. (Vladimirov, 1967), Section II.7.5. 1.3.4.4

Mollifiers Let ω0 ∈ D(IRN ), ω0 (x) ≥ 0, x ∈ IRN ,

ω dx = 1, supp ω0 ⊂ B1 (0). IRN 0

(1.3.22)

One example of such a function is

p say that f ∈ L weak (Ω), 1 ≤ p < ∞ if there exists A > 0 such that |{x ∈ Ω; |f (x)| > 1 −p p t}| ≤ At , t > 0. The space L weak is a Banach space endowed with norm f p,weak = inf A, where the infimum is taken over all A satisfying the previous estimate. (Do not mix up the p p p spaces L weak and Lweak ; the latter space means L considered as a topological space endowed with the weak topology, see Section 1.3.10.2!) Young’s inequality for convolutions holds even 1 N p if f belongs only to L weak (IR ): It reads f ∗ g0,r ≤ f 0,p,weak g0,q , 1 ≤ p < ∞, r := 1 1 + − 1 ≥ 0. See Definition II.13 and Proposition II.27 in (Bourdaud, 1988) for more details. p q 1 We

38


ω0 (x) = For ε > 0, we set



 

exp |x|21−1 dx B1 (0)

0

ωε (x) =

−1

exp |x|21−1 if |x| < 1,

(1.3.23)

if |x| ≥ 1.

x

1 , i.e. IRN ϕε = 1. ω0 N ε ε

(1.3.24)

For any f ∈ D′ (IRN ), we define the function

Sε (f ) = ωε ∗ f.

(1.3.25)

called the regularization of f. The operator Sε is called a mollifier or a regularizing operator. The following result is well known. Lemma 1.34 We have Sε (f ) ∈ C ∞ (IRN ), Sε (f ), v = f, Sε (f ),

(1.3.26)

ε ∗ f , cf. (1.3.19). Moreover, where Sε (f ) = ω

(i) If f ∈ D′ (IRN ), then

Sε (f ) → f in D′ (IRN ).

(1.3.27)

(ii) If f ∈ Lploc (IRN ), 1 ≤ p < ∞, then Sε (f ) ∈ Lploc (IRN ) and Sε (f ) → f strongly in Lploc (IRN ).

(1.3.28)

(iii) If f ∈ Lp (IRN ), 1 ≤ p < ∞, then Sε (f ) ∈ Lp (IRN ), Sε (f )0,p ≤ f 0,p (also for p = ∞) and Sε (f ) → f strongly in Lp (IRN ).

(1.3.29)

(iv) Let Ω be a domain of IRN and f ∈ L1loc (Ω) such that suppf ⊂ Ω. Then (1.3.30) Sε (f ) ∈ D(Ω), 0 < ε < dist (suppf, Ω) p p and Sε (f ) → f in L (Ω) whenever f ∈ L (Ω) (1 ≤ p < ∞). If moreover f ∈ C00 (Ω), then Sε (f ) → f in C 0 (Ω).

(1.3.31) 2

For details and proofs, see, e.g. (Kufner et al., 1977), Section 2.5.


39

1.3.4.5 Weakly singular integrals We shall recall some basic properties of integral transforms

[T f ](x) = IRN K(x, x − y)f (y) dy, (1.3.32) where

K(x, z) =

z θ(x, |z| )

|z|λ

, 0 < λ < N, θ ∈ L∞ (IRN × ∂B1 ).

Theorem 1.35 Let K be a weakly singular kernel and 1 < q < ∞. If 0, then λ + 1q − 1 T f ∈ Lr (IRN ), f ∈ C0∞ (IRN ), 1r = N

(1.3.33) λ N

+ 1q −1 > (1.3.34)

and there exists a constant c(N, λ, q) such that T f 0,r,IRN ≤ cf 0,q,IRN .

(1.3.35)

For the proof, see Chapter V in (Stein, 1970). Remark 1.36 According to Theorem 1.35, T : Lq → Lr is a densely defined bounded linear operator. Its closure T is a continuous linear operator from Lq (IRN ) → Lr (IRN ). As usual we shall denote it simply by T and we shall say that (1.3.32) is defined on Lq (IRN ). In a sense, a complementary situation is treated in the next theorem. Let Ω be a bounded domain of IRN . We set 1 , |x−y|λ

(x, y) ∈ Ω × Ω, x = y 0, (x, y) ∈ / Ω × Ω.

K(x, x − y) =

(1.3.36)

The following theorem holds: Theorem 1.37 Suppose that K is a weakly singular kernel (1.3.36), 1 < q < ∞ and 0 < λ < N . λ (a) If N + 1q − 1 = 0, then T : Lq (Ω) → Lr (Ω). Moreover, T f 0,r ≤ |Ω| 1 ≤ r ≤ r0 , and T f 0,q ≤ |Ω| (b) If

λ N

+

1 q

N −λ N

1 s

+

1 q

r0 −r rr0

K0,s f 0,q ,

−1=

1 r0 ,

1≤s
0 such that cuf k,p ≤ f −k,p ≤ uf k,p .

(1.3.53)

Theorem 1.40 Let 1 < p < ∞ and f ∈ W −k,p (Ω). Then there exists a family {fα }|α|≤k , fα ∈ Lp (Ω) such that f= Moreover,

k

|α| α |α|=0 (−1) D fα

f −k,p ≤ inf

|α|≤k

in D′ (Ω). fα 0,p ,

(1.3.54)

(1.3.55)

where the infimum is taken over all families {fα }|α|≤k such that (1.3.54) holds. For proofs of these statements, see, e.g. (Kufner et al., 1977), Theorems 5.9.3 and 5.9.2. An easy consequence of Theorem 1.40 is D(Ω) is dense in W −k,p (Ω), 1 < p < ∞.

(1.3.56)

1.3.5.4 Derivatives almost everywhere and distributional derivatives Suppose 1,1 that f ∈ Wloc (Ω). Then any of its partial derivatives ∂i f exists almost everywhere in Ω and is equal almost everywhere to the corresponding distributional derivative (see, e.g. (Kufner et al., 1977), Theorem 5.6.3).


1.3.5.5

43

Composed functions, positive (negative) parts of functions We set ρ+ =

ρ a.e. in {ρ > 0} , ρ− = 0 a.e. in {ρ ≤ 0}

−ρ a.e. in {ρ < 0} . 0 a.e. in {ρ ≥ 0}

We recall two well known properties of W 1,p -spaces, 1 0} , ∂j ρ− = 0 a.e. in {ρ ≤ 0}

−∂j ρ a.e. in {ρ < 0} . 0 a.e. in {ρ ≥ 0}

(ii) If F ∈ C 1 (IR) is such that F ′ is bounded and if ρ ∈ W 1,p (Ω) then F (ρ) ∈ W 1,p (Ω), and ∂j F (ρ) = F ′ (ρ)∂j ρ. For the proof, see, e.g., Exercises 5.10.16 and 5.10.17 in (Evans, 1998). 1.3.5.6

Theorem on traces

Theorem 1.41 Let 1 ≤ p < ∞ and Ω be a Lipschitz domain.

(i) Then there exists a uniquely determined continuous linear mapping γ0Ω : W 1,p (Ω) → Lp (∂Ω) such that γ0Ω (u) = u|∂Ω

for all u ∈ C ∞ (Ω).

(ii) If 1 < p < ∞, then Green’s formula

(u∂i v + v∂i u) dx = ∂Ω γ0Ω (u)γ0Ω (v)ni dS, Ω ′

u ∈ W 1,p (Ω), v ∈ W 1,p (Ω)

(1.3.57)

(1.3.58)

holds. For the proof, see, e.g. (Kufner et al., 1977), Section 6.4 and (Neˇcas, 1967), Theorem III.1.1. The function γ0Ω (u) ∈ Lp (∂Ω) is called the trace of the function u ∈ W 1,p (Ω) on the boundary ∂Ω. For simplicity, when it is not confusing, the notation u|∂Ω = γ0Ω (u) is used not only for u ∈ C ∞ (Ω) but also for u ∈ W 1,p (Ω). In the sequel, whenever it is not confusing, we omit the exponent Ω at γ0 and write simply γ0 instead of γ0Ω . The continuity of the mapping γ0 is equivalent to the existence of a constant c > 0 such that u|∂Ω Lp (∂Ω) = γ0 (u)Lp (∂Ω) ≤ cu1,p,Ω ,

u ∈ W 1,p (Ω).

(1.3.59)

Let us note that in the sequel the symbol c will often denote a positive generic constant, attaining, in general, different values at different places.

44


1.3.5.7

The Friedrichs–Poincaré inequalities

Lemma 1.42 (Friedrichs inequality) Let 1 ≤ p < ∞ and Ω be a bounded Lipschitz domain. Let a set Γ ⊂ ∂Ω be measurable with respect to the (N − 1)dimensional measure µ := measN −1 defined on ∂Ω and let measN −1 (Γ) > 0. Then there exists a constant c(p, n, Ω, Γ) > 0 such that u1,p,Ω ≤ c∇u0,p,Ω

for all u ∈ W 1,p (Ω) with γ0 (u) = 0 µ -a.e. on Γ. (1.3.60)

Lemma 1.43 (Poincaré inequality) Let 1 ≤ p < ∞ and Ω be a bounded Lipschitz domain. There exists a constant c(p, N, Ω) > 0 such that p

(1.3.61) |u|p dx ≤ c Ω |∇u|p dx + Ω u dx , u ∈ W 1,p (Ω). Ω

Both classical lemmas are consequences of a slightly more general Theorem 5.11.2 in (Kufner et al., 1977) or of Theorem 4.1.1 and consequent remarks in (Ziemer, 1989). 1.3.5.8 Imbedding theorems a) Let k ≥ 0 and 1 ≤ p ≤ ∞ and Ω be a bounded Lipschitz domain. Then W k,p (Ω) ֒→ Lq (Ω) where

1 q

=

1 p

−

k N,

W k,p (Ω) ֒→ Lq (Ω) for all q ∈ [1, ∞), if k = W k,p (Ω) ֒→ C 0,k−N/p (Ω), W

k,p

(Ω) ֒→ C

0,α

if

N p

NN−1 (1.3.64) p := p+N arbitrary > 1 if p ≤ NN−1 , p∗ :=

1.3.5.9

Np N −p

∞

if p < N (the Sobolev conjugate to p) if p ≥ N,

 Np if p < N  N −p p := arbitrary ≥ 1 if p = N  ∞ if p > N .

Some properties of W01,p

Lemma 1.44 As follows from Theorem 1.41, we have: W01,p (Ω) with 1 ≤ p < ∞, defined in (1.3.48), can be characterized as " # (1.3.65) W01,p (Ω) = v ∈ W 1,p (Ω); γ0 (v) = 0 .

We shall also need the following Hardy-type imbedding.

Lemma 1.45 Let 1 ≤ p < ∞ and Ω be a bounded Lipschitz domain. Then the function (x) , f ∈ W01,p (Ω) x → distf(x,∂Ω) belongs to the space Lp (Ω) and there exists c(p, N, Ω) such that f (·) ≤ cf 1,p dist (·,∂Ω) 0,p

For more details see (Kufner et al., 1977), Section 8.10.4.

1.3.5.10 Sobolev–Slobodetskii spaces of functions with “fractional derivatives” If k ≥ 0 is an integer, ε ∈ (0, 1) and p ∈ [1, ∞), then W k+ε,p (Ω) denotes the space of all functions u ∈ W k,p (Ω) such that

α α u(y)|p dx dy < ∞ for |α| = k. Iα,ε,p,Ω (u) = Ω Ω |D u(x)−D (1.3.66) |x−y|N +pε

46


The space W k+ε,p (Ω) equipped with the norm 1/p uk+ε,p,Ω = upk,p,Ω + |α|=k Iα,ε,p,Ω (u)

(1.3.67)

is a Banach space, see, e.g. (Triebel, 1978), Remark 2.5.1/4. The extension of Sobolev spaces to negative exponents is possible and we refer the reader to (Triebel, 1995), Definition 2.3.1/1 and Definition 4.2.1/1. 1.3.5.11 Trace theorem revisited The Sobolev spaces can also be defined on the Lipschitz-continuous boundary ∂Ω of a domain Ω. Let ∂Ω be characterized with the use of local (sufficiently regular) maps x′1 = ar (x′ ), x′ ∈ Mr , ar ∈ C k−1,1 (Mr ), r = 1, . . . , R, from Section 1.1.12. We say that a function u : k+ε,p (∂Ω) where ε ∈ [0, 1) and p ≥ 1, if ∂Ω → IR is an space relement of the

W r r the function x → u(ar (x ), x ) belongs to W k+ε,p (Mr ) for all r = 1, . . .,R. The space of the traces of all functions u ∈ W 1,p (Ω) can be identified with the space W 1−1/p,p (∂Ω). More precisely, the following theorem holds. Lemma 1.46 Let 1 < p < ∞ and Ω be a bounded Lipschitz domain. Then the operator γ0 defined in Section 1.3.5.6 is continuous from 1

W 1,p (Ω) → W 1− p ,p (∂Ω). The proof can be found in (Neˇcas, 1967), Theorem 5.5. We finish this section by recalling the so-called inverse trace theorem. Lemma 1.47 Under the assumptions of Lemma 1.46, there exists a continuous linear opertator 1 L : W 1− p ,p (∂Ω) → W 1,p (Ω) such that

1

-a.e. on ∂Ω, u ∈ W 1− p ,p (∂Ω). γ0 (L(u)) = u µ

See (Neˇcas, 1967), Theorem 5.7. There is a sort of Poincaré inequality. Let S ⊂ ∂Ω, µ (S) = 0. We set 1 1− p ,p

WS

1 1− p ,p

On WS

1

-a.e. in S}. (∂Ω) := {u ∈ W 1− p ,p (∂Ω); u = 0 µ

p1 (∂Ω), the norms · 1− p1 ,p,∂Ω and I0, p−1 ,p,∂Ω (·) are equivalent. p

1.3.5.12 Interpolation of Lebesgue and Sobolev spaces For Sobolev–Slobodetskii spaces, i.e. in particular for Sobolev spaces, the following result holds true: Theorem 1.48 Let Ω ⊂ IRN be a domain and 0 ≤ sj < ∞, 1 ≤ pj < ∞, θ j = 0, 1, 0 ≤ θ ≤ 1, s = (1 − θ)s0 + θs1 , p1 = 1−θ p0 + p1 . Then there is a constant C > 0 such that θ s0 ,p0 f s,p ≤ C f 1−θ (Ω) ∩ W s1 ,p1 (Ω). s0 ,p0 f s1 ,p1 , f ∈ W


47

This statement in a more general situation can be found, e.g., in (Triebel, 1978), Assertion (f) in Section 3.4.2. The extension of Theorem 1.48 to sj ∈ IR is given in (Triebel, 1995), Theorem 2.4.1 and Theorem 4.3.1/1. If we limit ourselves to Lebesgue spaces, we have a slightly more general result. Theorem 1.49 Let M ⊂ IRN be a measurable set, 1 ≤ p ≤ r ≤ q < ∞, p q−r 0 ≤ θ ≤ 1 and 1r = pθ + 1−θ q (i.e. θ = r q−p ). Then q p θ f 0,r ≤ f 1−θ 0,q f 0,p , f ∈ L (Ω) ∩ L (Ω).

1.3.5.13 Extension operators for Sobolev spaces Theorem 1.50 Let 1 ≤ p < ∞ and Ω ⊂ IRN be a bounded domain of class C k−1,1 (k ∈ IN ). Then a continuous linear operator E from W k,p (Ω) into W k,p (IRN ) exists such that [E(u)]|Ω = u, u ∈ W k,p (Ω). Moreover, E(u) has compact support in IRN . This theorem is known as the extension result of Nikolskii. For more details and references for the proof, see (Kufner et al., 1977), Section 6.5. 1.3.6

Homogeneous Sobolev spaces

If not stated explicitly otherwise, throughout this section Ω is an unbounded domain in IRN . 1.3.6.1 Definition with the norm

It is easy to see that the sets C0∞ (Ω) and C0∞ (Ω) endowed3 |u|1,q := ∇u0,q,Ω

(1.3.68)

are normed linear spaces. We define the homogeneous Sobolev spaces as |·|1,q

D01,q (Ω) = C0∞ (Ω) D1,q (Ω) = C0∞ (Ω)

|·|1,q

(1.3.69) ,

where the sign “overline with norm” means completion with respect to that norm. Remark 1.51 The definition of D01,q (Ω) makes sense also for Ω a bounded domain: in this case the spaces W01,q (Ω) and D01,q (Ω) are isometrically isomorphic. The definition of D1,q (Ω) as a Banach space does not make sense for Ω a bounded domain due to the fact that (1.3.68) is only a seminorm on C0∞ (Ω). The spaces D01,q (IRN ) and D1,q (IRN ) coincide. 3 The latter space is a linear space of all functions from C ∞ (Ω) whose support is compact in Ω.

48


1.3.6.2 Some useful formulas a) Let Ω be an exterior domain or Ω = IRN and suppose that u ∈ Lqloc (Ω), ∇u ∈ Lq (Ω). Then there exists u∞ ∈ IR such that for any R0 which satisfies IRN \ Ω ⊂ BR0 , we have:

|u − u∞ |q dS ≤ c(q, N, R0 )Rq−N B R |∇u|q dx, 1 ≤ q < N, R > R0 ∂BR (1.3.70) and

(ln R)N −1 , q = N |u|q dS = c(q, N, R0 )o(1) × (1.3.71) R > R0 . ∂BR Rq−N , q > N, b) Under the same assumptions as in a), with the same u∞ and R0 the following Hardy-type inequality u−u∞ ≤ c(q, N, R0 )∇u0,q,B R , 1 ≤ q < N, R > R0 |x| (1.3.72) 0,q,B R

holds.

c) Under the same assumptions as in a), with the same u∞ and R0 the following Sobolev-type inequality u − u∞ 0,

Nq N −q ,Ω

≤ c(q, N )∇u0,q,Ω , 1 ≤ q < N

(1.3.73)

holds. 1.3.6.3 Cut-off functions for unbounded domains a) For R > 0 we introduce the function ΦR ∈ C0∞ (IRN ) as follows: x , Φ ∈ C0∞ (IRN ), 0 ≤ Φ ≤ 1, Φ(x) = ΦR (x) = Φ R

1 if x ∈ B1 0 if x ∈ B 2 .

(1.3.74)

Functions ΦR are usually called classical cut-off functions. We recall two of their properties needed in the sequel in the form of two exercises. Exercise 1.52 2R supp (Dα ΦR ) ⊂ BR , |Dα ΦR (x)| ≤ cR−|α| , x ∈ IRN

for any multiindex α = 0.

Exercise 1.53 Let 1 ≤ q < ∞. (i) If kq > N , then we have Dα ΦR 0,q,IRN → 0 as R → ∞, |α| = k .

(ii) If u ∈ Lqloc (IRN ), ∇u ∈ Lq (IRN ) then

(u − u∞ ) · ∇ΦR 0,q,IRN → 0 as R → ∞, 1 ≤ q < N , u · ∇ΦR 0,q,IRN → 0 as R → ∞, q > N

where u∞ is defined in Section 1.3.6.2.


49

1.3.6.4 Fundamental properties of homogenous Sobolev spaces a) The spaces D01,q (Ω) and D1,q (Ω) are separable if 1 ≤ q < ∞ and reflexive if 1 < q < ∞. b) u0,

Nq N −q ,Ω

≤ c(q, N )∇u0,q,Ω , 1 ≤ q < N, u ∈ D1,q (Ω).

(1.3.75)

We emphasize that the constant in (1.3.75) does not depend on Ω. This inequality is known as the Sobolev inequality. c) Homogenous Sobolev spaces can be characterized as follows: D01,q (Ω) = Nq

{u ∈ D′ (Ω); u ∈ L N −q (Ω), ∇u ∈ Lq (Ω), u|∂Ω = 0}, if 1 ≤ q < N {u ∈ D′ (Ω); u ∈ Lqloc (Ω), ∇u ∈ Lq (Ω), u|∂Ω = 0}, if q ≥ N , Ω = IRN (1.3.76) Nq

D1,q (Ω) = {u ∈ D′ (Ω); u ∈ L N −q (Ω), ∇u ∈ Lq (Ω)}, if 1 ≤ q < N u ∈ Lq (Ω)} if q ≥ N . D1,q (Ω) = {u = { u + c}c∈IR ; u ∈ Lqloc (Ω), ∇

In the last formula u|∂Ω = 0 means γ0Ω∩B (uφ) = 0 for any ball B such that Ω ∩ B = ∅ and for any φ ∈ C0∞ (B). Notice that elements of D1,q (Ω) and D01,q (IRN ) = D1,q (IRN ) for q ≥ N can be viewed only as the sets of equivalent classes (they are not distributions). 1.3.6.5 Dual spaces to homogeneous Sobolev spaces If 1 < q < ∞, it is usual ′ to denote by D−1,q (Ω) the dual space to D01,q (Ω). The corresponding dual norm is denoted by | · |−1,q′ . The following representation theorem is valid: Theorem 1.54 Let 1 < q < ∞ and f ∈ D−1,q (Ω). Then there exists a family {f1 , . . . , fn }, fi ∈ Lq (Ω) such that f, u = Moreover,

N

i=1 Ω

′

fi ∂i u dx, u ∈ D01,q (Ω).

|f |−1,q ≤ inf

N

i=1

fi 0,q,Ω ,

(1.3.77)

(1.3.78)

where the infimum is taken over all families {f1 , . . . , fN } such that (1.3.77) holds. An easy consequence of Theorem 1.40 is D(Ω) is dense in D−1,q (Ω), 1 < q < ∞.

(1.3.79)

For proofs and more details about the homogeneous Sobolev spaces in unbounded domains (especially in exterior domains) see e.g. (Galdi, 1994a), Chapter II.

50


1.3.7 Tempered distributions We denote S(IRN ) := {f ∈ C ∞ (IRN ); sup |xβ Dα f (x)| ≤ c(α, β)},

(1.3.80)

x∈IRN

for all multiindices α, β of N with some c(α, β) > 0. In S(IRN ), we introduce the topology of locally uniform convergence: Let N N {fn }∞ n=1 , fn ∈ S(IR ), f ∈ S(IR ). We say fn → f in S(IRN )

(1.3.81)

if xβ Dα fn → xβ Dα f uniformly in IRN for all N -dimensional multiindices α, β. (1.3.82) The space of all continuous linear functionals on S(IRN ) is denoted by S ′ (IRN ) and called the space of tempered distributions. More precisely, f ∈ S ′ (IRN ), if a) f, α1 v1 + α2 v2 = α1 f, v1 + α2 f, v2 ∀ α1 , α2 ∈ IR, ∀ v1 , v2 ∈ S(IRN ), b) v, vn ∈ S(IRN ), vn → v in S(IRN ) =⇒ f, vn → f, v.

(1.3.83)

The space S ′ (IRN ) is endowed with the topology induced by the convergence fn → f in S ′ (IRN ) if fn , v → f, v, v ∈ S(IRN ).

The elements of the space S ′ (IRN ) are called tempered distributions. 1.3.7.1 Fourier transform The Fourier transform F is a continuous linear operator of S(IRN ) into S(IRN ) defined by [F(f )](ξ) =

1 (2π)N/2

IRN

f (x) exp(−ix · ξ) dx, f ∈ S(IRN ).

(1.3.84)

It can be proved that F : S(IRN ) → S(IRN ) is a bijection and that the operator

[F −1 (f )](ξ) = (2π)1N/2 IRN f (x) exp(ix · ξ) dx, f ∈ S(IRN ) (1.3.85)

is the inverse operator to F, i.e.

F(F −1 (f )) = F −1 (F(f )) = f, f ∈ S(IRN ).

(1.3.86)

We define the Fourier transform on S ′ (IRN )

F : S ′ (IRN ) → S ′ (IRN )

by putting F(f ), v = f, F(v), v ∈ S(IRN ).

(1.3.87)

Then F is bijective and the operator

F −1 : S ′ (IRN ) → S ′ (IRN ), F −1 (f ), v = f, F −1 (v), v ∈ S(IRN )

is the inverse operator to F called the inverse Fourier transform.

(1.3.88)


51

Basic properties of the Fourier transform needed in the book are gathered in the following theorem. Theorem 1.55 (i) The operator F defined by (1.3.84) can be extended by continuity so that we have F ∈ L(L2 (IRN ), L2 (IRN )) and

f ∗ g dx = IRN [F(f )]∗ F(g) dx, f, g ∈ L2 (IRN ). IRN (Here a∗ means the complex conjugate of a.)

(ii) Dα [F(f )] = F[(−ix)α f ], F(Dα f ) = (iξ)α Ff, f ∈ S ′ (IRN ), α an N -dimensional multiindex. (iii) F(f ∗ g) = F(f ) F(g) f ∈ S ′ (IRN ), g ∈ S(IRN ).

For more details, see, e.g. (Lions and Magenes, 1968), Section I.1.2, (Vladimirov, 1967), Section II.9, (Yosida, 1974), Section VI.2. 1.3.7.2 Fourier multipliers A bounded measurable function m : IRN → IR is called a Fourier multiplier of type (p, q), 1 ≤ q, p < ∞, if there exists a positive constant c(p, q) such that

F −1 mF(f ) 0,q,IRN ≤ c(p, q)f 0,p,IRN , f ∈ S(IRN ). Hence, if m is a Fourier multiplier of type (p, q), then a linear operator

(1.3.89) T : S(IRN ) ⊂ Lp (IRN ) → Lq (IRN ), T f = F −1 mF(f )

with domain D(T ) = S(IRN ) is a densely defined continuous linear operator from Lp (IRN ) to Lq (IRN ). Therefore, its closure (denoted again by T ) is a continuous p N linear operator from L L (IR ), Lq (IRN ) (see Section 1.4.7.4 and 1.4.7.6), i.e. F −1 mF(f ) 0,q,IRN ≤ c(p, q)f 0,p,IRN , f ∈ Lp (IRN ).

(1.3.90)

Sufficient conditions for m to be a Fourier multiplier of type (p, p) are given in the following theorem. Theorem 1.56 Let 1 0 such that for all R > 0 and with any multiindex α with |α| ≤ [N/2] + 1, we have either

R2|α|−N B 2R |Dα m| dx ≤ B (1.3.91) R/2

or

|Dα m(ξ)| ≤ B|ξ|−|α| , ξ ∈ IRN \ {0}.

(1.3.92)

Then m is a Fourier multiplier of type (p, p). This means that the

operator T defined in (1.3.89) is an element of the space L Lp (IRN ), Lp (IRN ) .

52


This result is known as Mikhlin’s and H¨ ormander’s (with the condition (1.3.91)) or Marcinkiewicz’ (with the condition (1.3.92)) theorem about multipliers. See (Triebel, 1978), Section 2.2.4 or (Torchinsky, 1986), Section XII.5 for more details. The generalization of this theorem to multipliers of type (p, q) is due to (Lizorkin, 1952), (Lizorkin and Nikol’skii, 1965), (Lizorkin, 1969). It reads: Theorem 1.57 Let 1 0 such that αN m(ξ)| ≤ B, ξ ∈ IRN \ {0}, |ξ1 |α1 +β . . . |ξN |αN +β |∂1α1 . . . ∂N

(1.3.93)

where αi is 0 or 1. Then m is a Fourier multiplier of type (p, q) with 1q = p1 − β. This means that the operator T defined in (1.3.89) is an element of the space L Lp (IRN ), Lq (IRN ) .

If β = 0, Theorem 1.57 gives another condition for m to be a multiplier of type (p, p). 1.3.8

Radon measure and representation of CB (Ω)∗

Let Ω ⊂ IRN be an open set and (1.3.94) CB (Ω) = {w : Ω → IRm ; w is bounded and continuous on Ω and w|∂Ω = 0, w tends to 0 as |x| → ∞, if Ω is unbounded}. Further, denote by M (Ω) the space of m-vectors whose components are bounded Radon measures (see Section 1.1.10 for definition of the Radon measure) on Ω which is isometrically isomorphic to the dual space of the space CB (Ω). The correspondence is realized by the mapping J : M (Ω) → CB (Ω)∗ as follows: For µ ∈ M (Ω) we have Φ = Jµ if and only if Φ, w = Ω w dµ(x) for any w ∈ CB (Ω), where ·, · denotes duality pairing between CB (Ω) and CB (Ω)∗ . The assertion about the representation of CB (Ω)∗ is proved in (Dunford and Schwartz, 1963), Theorem IV.5.1. 1.3.9

Functions of bounded variation

Definition 1.58 We say that a function w : Ω ⊂ IRN → IRm with Ω open is of locally bounded variation if w ∈ L1loc (Ω)m and for any j = 1, . . . , m the distributional derivative ∂xj w is a locally finite (vector-valued) Radon measure µj on Ω, i.e.

∂ ϕ(x)w(x) dx = − Ω ϕ(x) dµj (x), for j = 1, . . . , m and ϕ ∈ C0∞ (Ω)m . Ω j (1.3.95)


53

If in addition, w ∈ L1 (Ω)m and µj are finite, then w is called a function of bounded variation, with total variation m T VΩ w = j=1 |µj |(Ω), (1.3.96)

where |µ| denotes the total variation of the measure µ ∈ M (Ω), i.e.

|µ|(Ω) = sup{ Ω ϕ(x) dµ(x); ϕ ∈ CB (Ω), supx∈Ω |ϕ(x)| ≤ 1}, µ ∈ M (Ω). (1.3.97) We denote by BVloc (Ω) the space of functions of locally bounded variation and analogously, by BV (Ω) the space of functions of bounded variation in Ω. The space BV (Ω) is a Banach space under the norm wBV (Ω) = wL1 (Ω) + T VΩ w,

w ∈ BV (Ω)

(1.3.98)

with T VΩ w given by (1.3.96). It is immediate that the following imbeddings hold true: W 1,1 (Ω)m ֒→ BV (Ω),

1,1 Wloc (Ω)m ֒→ BVloc (Ω).

The latter imbedding is understood with respect to topologies defined by corresponding seminorms. If N = m = 1, then BV (Ω) is the space of classical functions of bounded variation in one real variable. Then Ω = (a, b) ⊂ IR and given w ∈ BV (Ω), such a representant of w (denoted again by w) can be chosen so that there exist limits w(x±) for any x ∈ (a, b) and are equal to w(x) with the exception of at most a countable set of points from (a, b). Also we recall that "k−1 # (1.3.99) T V(a,b) w = sup j=1 |w(xj+1 ) − w(xj )|; a < x1 < . . . < xk < b . From the theory of functions with bounded variation in IR the classical Helly compactness theorem is known. Here is its multidimensional generalization. Theorem 1.59 Let {wj }∞ j=1 ⊂ BVloc (Ω) be such that

sup wj L1 (Ω) + T VG wj < ∞ j=1,2,...

for any bounded set G satisfying G ⊂ Ω. Then {wj }∞ j=1 contains a subsequence ∞ denoted again by {wj }j=1 such that wj → w in L1loc (Ω) and a.e. in Ω with w ∈ BVloc (Ω) and T VG (w) ≤ lim inf j→∞ T VG wj .

The proof can be found in (Dafermos, 2000), Theorem 1.7.2. 1.3.10

Functions with values in Banach spaces

In the investigation of nonstationary problems we will work with functions which depend on time and have values in Banach spaces. If u(x, t) is a function of the

54


space variable x and time t, then it is sometimes suitable to separate these variables and consider u as a function u(t) = u(·, t) which for each t in consideration attains a value u(t) that is a function of x and belongs to a suitable space of functions depending on x. This means that u(t) represents the mapping x → [u(t)] (x) = u(x, t). Let a, b ∈ IR, a < b, and let X be a Banach space with norm ·. By a function defined on the interval [a, b] with its values in the space X we understand any mapping u : [a, b] → X. We say that a function u : [a, b] → X is continuous at a point t0 ∈ [a, b] if lim u(t) − u(t0 ) = 0.

t→t0 t∈[a,b]

(1.3.100)

By the symbol C([a, b], X) we shall denote the space of all functions continuous on the interval [a, b] (i.e. continuous at each t ∈ [a, b]) with values in X. The space C([a, b], X) equipped with the norm uC([a,b], X) = max u(t) t∈[a,b]

(1.3.101)

is a Banach space. 1.3.10.1 Bochner integral An important concept is the Bochner integral of a function u : (a, b) → X:

b (1.3.102) u(t) dt a

which can be introduced similarly as the Lebesgue integral of real functions. Let us summarize some important definitions and results concerning the Bochner integral. The details can be found, e.g., in (Kufner et al., 1977) and (Gajewski et al., 1974). A mapping f : (a, b) → X is called a simple function if there exist measurable sets Bi ⊂ [a, b] and elements ci ∈ X, i = 1, . . . , n, such that Bi ∩ Bj = ∅ for n i = j, [a, b] = i=1 Bi and n f (t) = i=1 1Bi (t) ci , t ∈ (a, b). (1.3.103)

A function f : (a, b) → X is called strongly measurable if there exists a sequence {fn } of simple functions such that lim fn (t) − f (t) = 0 for a.a. t ∈ (a, b).

n→∞

(1.3.104)

Lemma 1.60 Let f : (a, b) → X be strongly measurable. Then the function t → f (t) is measurable. 2 The Bochner integral of a simple function (1.3.103) is defined by the relation

b a

f (t) dt =

n

i=1

meas (Bi ) ci .

(1.3.105)


55

Definition 1.61 We say that f : (a, b) → X is Bochner integrable if there exists a sequence {fn } of simple functions satisfying (1.3.104) and limn→∞

b a

fn (t) − f (t) dt = 0.

(1.3.106)

For a measurable B ⊂ (a, b) and a Bochner integrable f we define the Bochner

integral B f (t) dt by

B

f (t) dt = limn→∞

b a

χB (t)fn (t) dt.

(1.3.107)

The following result plays an important role. Theorem 1.62 (Bochner’s theorem) A strongly measurable function f : (a, b) → X is Bochner integrable if and only if the real function f (t) has the finite

b Lebesgue integral a f (t) dt.

By X ∗ we shall denote the dual space to X and ·, · will denote the duality between X ∗ and X. This means that η, f is the value of a functional η ∈ X ∗ at the point f ∈ X. (See Section 1.4.5.2.) Lemma 1.63 If f is Bochner integrable in (a, b), then

b f (t) dt ≤ b f (t) dt. a

a

(1.3.108)

If moreover η ∈ X ∗ , then η, f (t) is integrable and

b a

$ b % η, f (t) dt = η, a f (t) dt .

(1.3.109)

Definition 1.64 We say that a function f : (a, b) → X is differentiable at a point t0 ∈ (a, b), if there exists w ∈ X such that (t0 ) − w = 0. limh→0 f (t0 +h)−f h

(1.3.110)

Then f ′ (t0 ) = (df /dt)(t0 ) := w is called the strong derivative of f at t0 . Lemma 1.65 If u is Bochner integrable in (a, b), t0 ∈ [a, b] and ξ ∈ X, then the function

t (1.3.111) v(t) = ξ + t0 u(s)ds, t ∈ [a, b], is continuous in [a, b], differentiable at a. e. t ∈ (a, b) and dv dt (t)

= u(t)

for a. e. t ∈ (a, b).

(1.3.112)

If u : (a, b) → X is Bochner integrable and ϕ ∈ C0∞ (a, b), then obviously u(t)ϕ(t) is also Bochner integrable in (a, b).

56


Lemma 1.66 Let u, v : (a, b) → X be Bochner integrable. Then (1.3.111) is equivalent to each of the following conditions:

b

b (1.3.113) u(t)ϕ(t) dt = − a v(t)ϕ′ (t) dt ∀ ϕ ∈ C0∞ (a, b) a and

d dt η, v

∀ η ∈ X ∗,

= η, u

(1.3.114)

where the derivative d/dt is considered in the sense of scalar distributions on (a, b). For details of the theory of the Bochner integral see (Yosida, 1974), Section V.5 and (Brezis, 1973), Appendix. 1.3.10.2 Examples of spaces of functions with values in a Banach space Let X be a Banach space. For p ∈ [1, ∞] we denote by Lp ((a, b), X) the space of (equivalent classes of) strongly measurable functions u : (a, b) → X such that uLp ((a,b),X) := and

b a

u(t)pX dt

1/p

< ∞,

if 1 ≤ p < ∞,

uL∞ ((a,b),X) := ess sup u(t)X

(1.3.115)

(1.3.116)

t∈(a,b)

=

inf

sup

meas(N )=0 t∈(a,b)\N

u(t)X < ∞,

if p = ∞.

We speak of Bochner spaces. It can be proved that Lp ((a, b), X) is a Banach space. If the space X is reflexive, so is Lp ((a, b), X) for p ∈ (1, ∞). Let 1 ≤ p < ∞. Then the dual of Lp ((a, b), X) is isometrically isomorphic to the space Lq ((a, b), X ∗ ), where 1/p + 1/q = 1 and X ∗ is the dual of X (for p = 1 we set q = ∞). The duality between Lq ((a, b), X ∗ ) and Lp ((a, b), X) becomes f, v =

b a

f (t), v(t)X ∗ ,X dt, f ∈ Lq ((a, b), X ∗ ), v ∈ Lp ((a, b), X). (1.3.117)

f (t), v(t)X ∗ ,X denotes the value of the functional f (t) ∈ X ∗ at v(t) ∈ X. The proof can be found in (Gajewski et al., 1974). If X is a separable Banach space, then Lp ((a, b), X) is also separable, provided p ∈ [1, ∞). (See, e. g., (Edwards, 1965), Section 8.18.1.) We define analogously Sobolev spaces of functions with values in X: First, we say that a function f ∈ L1 ((a, b), X) has in (a, b) a generalized derivative if there exists a function g ∈ L1 ((a, b), X) such that

b

dϕ (t)f (t) dt a dt

=−

b a

ϕ(t)g(t) dt

for all ϕ ∈ D(a, b).

df dt

:= g. Now we define

If the condition is satisfied, then we denote


57

j

W k,p ((a, b), X) = {f ∈ Lp ((a, b), X); ddtfj ∈ Lp ((a, b), X), j = 1, . . . , k}, for k = 1, 2, . . . and p ∈ [1, ∞]. The norm of f ∈ W k,p ((a, b), X) is defined by f k,p := f W k,p ((a,b),X) =

k

j=1

dj f p j p dt

L ((a,b),X)

(1.3.118)

1/p

.

(1.3.119)

These spaces have exactly the same properties as listed above for Lp -spaces. We also define respectively spaces of continuous and differentiable functions on the interval I with values in X: C(I, X) = C 0 (I, X)

(1.3.120) = {f : I → X; f is bounded and continuous at each point of I} dj f C k (I, X) = {f ∈ C(I, X); ∈ C(I, X) for all j = 1, . . . , k}. dtj The norm of f ∈ C k (I, X), k = 0, 1, . . . is defined by j f C k (I,X) = sup ddtfj (t) , j = 0, . . . , k; t ∈ I . C(I,X)

(1.3.121)

These spaces are nonreflexive Banach spaces, separable, if so is X. Finally, we define the spaces with weak and weak-∗ topology: C(I, Xweak ) = {f : I → X; f, vX ∗ ,X ∈ C(I) for any v ∈ X};

(1.3.122)

and if X = B ∗ , where B is a Banach space, then C(I, X∗−weak ) = {f : I → X; f, vB ∗ ,B ∈ C(I) for any v ∈ B}.

(1.3.123)

The topologies in these spaces are respectively induced by the weak and weak-∗ topology in X. We add the following theorem on integration by parts (Gajewski et al., 1974): Theorem 1.67 Let H be a Hilbert space and V ֒→ H be dense in H. If u, v ∈ Lp ((a, b), V ) with a, b ∈ IR, a < b, 1 < p < ∞, and u′ , v ′ ∈ Lq ((a, b), V ∗ ), p1 + 1q = 1, then u, v ∈ C([a, b], H) and

t

(1.3.124) u(t), v(t) − u(s), v(s) = s u′ (τ ), v(τ ) + v ′ (τ ), u(τ ) dτ. (Here ·, · is the duality between V and V ∗ .)

1.3.11

Sobolev imbeddings of abstract spaces

A frequently used consequence of Theorem 1.67 is the following Theorem 1.68 Suppose u ∈ L2 ((0, T ), H01 (Ω))∩W 1,2 ((0, T ), H −1 (Ω)), (T > 0). Then

58


(i) u ∈ C([0, T ]), L2 (Ω)) (after possibly being redefined on a set of measure zero); (ii) the mapping t −→ u(t)2L2 (Ω) is absolutely continuous, with d 2 dt u(t)L2 (Ω)

= 2u′ (t), u(t)

for a.e. 0 ≤ t ≤ T ; (iii) the estimate max u(t)2L2 (Ω) ≤ C(uL2 ((0,T ),H01 (Ω)) + u′ L2 ((0,T ),H −1 (Ω)) ),

0≤t≤T

holds with the constant C independent of u. 1.3.12

Some compactness results

In the sequel we shall need some fundamental results concerning the imbedding of the spaces Lp ((a, b), X). Let us consider three Banach spaces X0 , X and X1 such that (1.3.125) a) X0 ⊂ X ⊂ X1 , b) the imbeddings of X into X1 and of X0 into X are continuous, c) the imbedding of X0 into X is compact. Lemma 1.69 Let the spaces X0 , X1 , X satisfy assumptions (1.3.125), a)–c). Then for any δ > 0 there exists a constant cδ such that vX ≤ δvX0 + cδ vX1

∀ v ∈ X0 .

(1.3.126)

The proof can be found in (Neˇcas, 1967), Chapter 2, Lemma 6.1. The following assertion is an abstract version of the Arzel` a–Ascoli theorem. Theorem 1.70 Let B and X be Banach spaces such that B ֒→֒→ X is compact4 . Let fn be a sequence of functions I → B uniformly bounded in B and uniformly continuous in X. Then there exists f ∈ C 0 (I, B) such that fn → f strongly in C 0 (I, X) at least for a chosen subsequence. Theorem 1.70 can be deduced from (Kufner et al., 1977), Theorem 1.6.9. The following theorem provides us with a sufficient condition for a family of functions from the space L1 (I, X) to be compact in the space Lp (I, X) for some p ≥ 1. 4 Of

course, if X has finite dimension, then one can take B = X.


59

Theorem 1.71 (Aubin–Lions) Let X ֒→֒→ B ֒→ Y be Banach spaces and {fn } a sequence bounded in Lq (I, B) ∩ L1 (I, X), (1 < q ≤ ∞) and {dfn /dt} bounded in L1 (I, Y ). Then {fn } is relatively compact in Lp (I, B) for any 1 ≤ p < q. The above theorem is a special case of the following more general result from (Simon, 1987), Corollary 9: Theorem 1.72 Let X ֒→ B ֒→ Y be three Banach spaces with compact imbedding X ֒→֒→ Y. Further, let there exist 0 < θ < 1 and M > 0 such that θ vB ≤ M v1−θ X vY

for all v ∈ X ∩ Y.

(1.3.127)

Denote for T > 0, W (0, T ) := W s0 ,r0 ((0, T ), X) ∩ W s1 ,r1 ((0, T ), Y )

(1.3.128)

with s0 , s1 ∈ IR; r0 , r1 ∈ [1, ∞], sθ := (1 − θ)s0 + θs1 ,

1 rθ

:=

1−θ r0

+

θ r1 ,

s∗ := sθ −

1 rθ .

(1.3.129)

Assume that sθ > 0 and F is a bounded set in W (0, T ). If s∗ ≤ 0, then F is relatively compact in Lp ((0, T ), B) for all 1 ≤ p < p∗ := −1/s∗ . If s∗ > 0, then F is relatively compact in C((0, T ), B). In view of future applications we apply the above theorem to three Sobolev spaces. Theorem 1.73 Let Ω be a bounded open subset of IRN satisfying the cone property see (Adams, 1975), Section 4.3. Put X := W α0 ,ζ0 (Ω), B := W α,ζ (Ω) and Y := W α1 ,ζ1 (Ω).

(1.3.130)

Then X ֒→ B ֒→ Y and X ֒→֒→ Y if and only if α0 ≥ α ≥ α1 , α0 > α1 and β0 ≥ β ≥ β1 , β0 > β1 , where β := α −

N ζ ,

βi := αi −

N ζi ,

i = 0, 1,

(1.3.131) (1.3.132)

and (1.3.127) holds for all " −α β0 −β # θ < θ∗ := min αα00−α . , 1 β0 −β1

(1.3.133)

The value of θ∗ is optimal.

The proof can be found in (Simon, 1987), Lemma 12. Let us note that domains with Lipschitz boundary have the cone property (see (Adams, 1975)).

60

1.4


Survey of concepts and results from functional analysis

In this section we give a survey of basic concepts and results of functional analysis which are frequently used. All results are stated without proofs, which can be found in standard monographs ((Aze, 1997), (Brezis, 1987), (Edwards, 1965), (Fonseca and Gangbo, 1995), (Kolmogorov and Fomin, 1972), (Ljusternik and Sobolev, 1974), (Taylor, 1967), (Yosida, 1974), (Zeidler, 89)). 1.4.1

Linear vector spaces

The set X is called a (real) linear vector space (briefly, a linear space) if addition of elements of X and multiplication by scalars is defined: u, v ∈ X −→ u + v ∈ X, u ∈ X, λ ∈ IR −→ λu ∈ X so that for any u, v, w ∈ X and λ, µ ∈ IR the following axioms are satisfied: (i) u + v = v + u; (ii) u + (v + w) = (u + v) + w; (iii) in X there exists a uniquely determined element denoted by 0 and called the zero element such that u + 0 = u; (iv) to each u ∈ X there exists a uniquely determined element (−u) such that u + (−u) = 0; (v) λ(u + v) = λu + λv; (vi) (λ + µ)u = λu + µu; (vii) (λµ)u = λ(µu); (viii) 1u = u; (ix) 0u = 0. (Zero in IR is denoted by the same symbol as the zero element in X without the danger of misunderstanding.) In X we also define subtraction: u − v = u + (−v). 1.4.1.1

Convex sets A subset M of a linear vector space X is called convex if u, v ∈ M, λ ∈ [0, 1] =⇒ λu + (1 − λ) v ∈ M.

1.4.2

Topological linear spaces

1.4.2.1 Definition Let X be a linear vector space. A family τ composed of subsets M of X is called a topology on X, if it is closed with respect to (i) the intersection of any finite number of its elements; (ii) the union of any arbitrary number of its elements. The couple (X, τ ) is called a topological linear space. Elements of τ are called open sets of X with respect to the topology τ . An open set U ∈ τ is called a

SURVEY OF CONCEPTS AND RESULTS FROM FUNCTIONAL ANALYSIS 61

neighborhood of a point a ∈ X, if a ∈ U . A generic neighborhood of a point a ∈ X is denoted by U (a). Any topology is uniquely determined by families of neighborhoods of all points belonging to X as follows: M ∈ τ if and only if M can be written as an at most finite intersection of possibly infinite unions of neighborhoods of points of X. 1.4.2.2 Sequences and limits Let {un }∞ n=1 be a sequence of elements of X. We say that limn→∞ un = u ∈ X (or briefly un → u) if for every U (u) ∈ τ there exists n0 ∈ IN such that un ∈ U (u), n ≥ n0 . In what follows we shall sometimes use a simplified notation {un } instead ∞ of {un }∞ n=1 , the fact that {un }n=1 is a sequence with elements un ∈ M will be written as {un } ⊂ M. 1.4.2.3 Interior points and closure A point a ∈ M ⊂ X is called an interior point of M , if there exists a neighborhood U (a) ∈ τ such that U (a) ⊂ M . The set of all interior points of M is denoted intτ (M ). τ We define the closure M of a subset M ⊂ X (with respect to the topology τ ) as the set of all points a ∈ X such that U (a) ∩ M = ∅ for any neighborhood U (a) ∈ τ . If there is no danger of confusion, we write int (M ) instead of intτ (M ) and τ M instead of M . We have int (M ) ⊂ M ⊂ M . 1.4.2.4 Closed sets (i) The set M is called closed if M = M . If M is closed, then X \ M is open. (ii) The set M is sequentially closed if the following implication holds: un ∈ M, un −→ u in X =⇒ u ∈ M. Consequently any closed set is sequentially closed, as well. 1.4.2.5

Boundary The set ∂M = M ∩ (X \ M )

is called the boundary of the set M. 1.4.2.6 Continuity Let f be a mapping from (X, τ ) to (Y, σ) with the domain of definition D(f ). We say that f is continuous at the point a ∈ D(f ) if for any U (f (a)) ∈ σ there exists V (a) ∈ τ such that f (x) ∈ U ((f (a)), x ∈ V (a) ∩ D(f ).

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1.4.2.7 Lower semicontinuity Let f be a mapping of (X, τ ) to IR ∪ {∞}. (i) We say that f is lower semicontinuous in a ∈ X if f (a) ≤ sup inf f (u) := lim inf f (u), V (a)∈τ u∈V (a)

u→a

where the infimum is taken over all neighborhoods V (a) ∈ τ of a. (ii) We say that f is sequentially lower semicontinuous in a ∈ X if f (a) ≤ lim inf f (un ) whenever un → a in X. n→∞

(iii) The mapping f is said to be lower semicontinuous (resp. sequentially lower semicontinuous) in X, if it is lower semicontinuous (resp. sequentially lower semicontinuous) in any point a ∈ X. (iv) The following result is well known: f is lower semicontinuous (resp. sequentially lower semicontinuous) in X if and only if the sets {x ∈ X; f (x) ≤ λ} are closed (resp. sequentially closed) for any λ ∈ IR. Consequently, any lower semicontinuous mapping is sequentially lower semicontinuous, as well. 1.4.3

Metric linear space

Let X be a linear vector space. A function d(·, ·) : X × X → [0, ∞) is called a distance or metric, if it satisfies the following axioms: (i) u, v ∈ X, d(u, v) = 0 ⇔ u = v; (ii) d(u, v) = d(v, u), u, v ∈ X;

(iii) d(u, v) ≤ d(u, z) + d(z, v), u, v, z ∈ X

(triangle inequality).

A linear space X equipped with a distance d is called a metric linear space. The distance d induces a topology defined by neighborhoods of any point a U (a) := {x ∈ X; d(x, a) < ε}, ε ∈ (0, ∞). These neighborhoods are also called balls centered at a with radius ε and denoted by Bε (a). The distance of two subsets M1 , M2 of X is defined by dist (M1 , M2 ) = inf{d(x1 , x2 ); x1 ∈ M1 , x2 ∈ M2 }. 1.4.4

Normed linear space

1.4.4.1 Definitions Let X be a linear vector space. A function · : X → [0, ∞) is called a norm on X if it satisfies the following axioms: (i) u ∈ X, u = 0 ⇔ u = 0; (ii) λu = |λ| u, λ ∈ IR, u ∈ X; (iii) u + v ≤ u + v, u, v ∈ X (triangle inequality). A linear space X equipped with a norm · is called a normed linear space, and it is a metric space under the distance d(u, v) = u − v.


1.4.4.2 Balls Every normed linear space X is a metric space with the metric (i.e. distance) defined as u − v for u, v ∈ X. Therefore, the topology induced by a norm is determined in the same way as described in Section 1.4.3. The ball of center a ∈ X with radius ε > 0 is defined as the set Bε (a) := {u ∈ X; u − a < ε}. 1.4.4.3 Bounded sets A set M ⊂ X (sequence {un }∞ n=1 with un ∈ X) is called bounded in X if there exists a constant K ≥ 0 such that u ≤ K for all u ∈ M (un ≤ K for all n = 1, 2, . . .). 1.4.4.4 Topology induced by a norm For the topology induced by a norm, the general results for topological spaces presented in Sections 1.4.2.1–1.4.2.7 can be rephrased in a more convenient form. We announce these results in the following Sections 1.4.4.5–1.4.4.10. 1.4.4.5 Open sets We say that a set M ⊂ X is open if for every a ∈ M there exists a ball centered at a contained in M . 1.4.4.6

Sequences and limits Let u, un ∈ X for n = 1, 2, . . . . We say that lim un = u (in X) or, more simply, un → u,

n→∞

if limn→∞ un − u = 0. In this situation, we speak about strong convergence of un to u in X. 1.4.4.7 Closure The closure of a set M ⊂ X is defined as the set M = {u ∈ X; there exists a sequence {un } ⊂ M such that un → u}. A set M is closed if and only if M = M . 1.4.4.8 Closed sets In the normed linear space the notions of (strong) closure and sequentially (strong) closure coincide. 1.4.4.9 Continuity Let f be a mapping from X into Y with the domain of definition D(f ). We say that f is continuous at a point a ∈ D(f ), if for any ε > 0 there exists a δ > 0 such that for any x ∈ D(f ) : x − aX < δ we have f (x) − f (a)Y < ε. 1.4.4.10 Compactness A set M ⊂ X is called compact (precompact or relatively compact), if for every bounded sequence {un }∞ n=1 ⊂ M there exist a subsequence and an element u ∈ M (u ∈ X) such that unk → u in X. {unk }∞ k=1 1.4.4.11 Connected sets implication holds:

A set M ⊂ X is called connected if the following

M = A ∪ B, A ∩ B = ∅ = A ∩ B =⇒ either A or B is empty.

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1.4.4.12 Banach space Let X be a normed linear space. A sequence {un } ⊂ X is called a Cauchy (or fundamental) sequence if for any ε > 0 there exists n0 = n0 (ε) such that um − un < ε for all integers m, n > n0 . We say that X is a complete space if every Cauchy sequence {un } ⊂ X is convergent: there exists a u ∈ X such that un → u as n → ∞. A complete normed linear space is called a Banach space. 1.4.4.13 Hilbert space A real scalar function (·, ·) defined on X × X, where X is a linear vector space, is called a scalar (or inner) product on X if for any u, v, w ∈ X and λ ∈ IR the following relations hold: (i) (u + v, w) = (u, w) + (v, w); (ii) (λu, v) = λ(u, v); (iii) (u, v) = (v, u); (iv) (u, u) > 0, provided u = 0. 1

A scalar product induces the norm in X defined as u = (u, u) 2 for u ∈ X. The so-called Cauchy inequality |(u, v)| ≤ u v for u, v ∈ X can be proved. A linear vector space with a scalar product which is complete with respect to the induced norm is called a Hilbert space. 1.4.4.14 Density, separability A subset M ⊂ X of a normed linear space X is called dense in X if M = X. The space X is called separable if there exists a countable set M ⊂ X, dense in X. (M is countable if all its elements can be ordered into a sequence.) 1.4.4.15 Subspaces Let X be a normed linear space with a norm ·X and let M ⊂ X be a linear vector space. Then M endowed with the norm ·X is called a subspace of the space X. If M is closed in X, we speak about a closed subspace of X. 1.4.4.16 Subspaces and separability A closed subspace of a separable normed linear space is also a separable space. 1.4.4.17 Subspaces of Banach spaces A closed subspace of a Banach (Hilbert) space is also a Banach (Hilbert) space. 1.4.5

Duals to Banach spaces and weak(-∗) topologies

1.4.5.1 Continuous linear functionals on normed linear spaces A continuous linear functional defined on a normed linear space X is a continuous linear mapping f : X → IR. For the value of f at a point u ∈ X, we use the notation f (u) or f, u.


1.4.5.2 Dual spaces The set of all continuous linear functionals on X forms a linear vector space, which is denoted by X ∗ (or X ′ ) and is called the dual of X. The space X ∗ together with the norm f X ∗ = sup u∈X u=0

|f, u| , uX

f ∈ X∗

is a normed linear space. The mapping ·, · : X ∗ × X → IR is called the duality (mapping) between X and X ∗ . It is known that X ∗ endowed with the norm · X ∗ is a Banach space. 1.4.5.3 Hahn–Banach theorem Let the following assumptions be satisfied: X is a normed linear space, M ⊂ X is a linear subspace, ϕ : M → IR is a linear mapping continuous on M in the norm ·X (which means that un , u ∈ M , un − uX → 0 =⇒ ϕ(un ) → ϕ(u)). Then there exists φ ∈ X ∗ such that φ(u) = ϕ(u) for u ∈ M and φX ∗ = sup u∈M u=0

1.4.5.4 Another calculation of the norm in X Banach theorem is the following formula xX = sup

f ∈X ∗

|ϕ(u)| . uX

One consequence of the Hahn–

f, x . f X ∗

1.4.5.5 Weak topology on X We denote by σ(X, X ∗ ) the so-called weak topology on X. A ball at the point a ∈ X is the set B(a) = {x ∈ X; |f1 , x − a| < ε1 , . . . , |fk , x − a| < εk }, where k ∈ IN , εi > 0 and fi ∈ X ∗ . Any neighborhood of a is then an arbitrary union of balls of the above type such that at least one of them is centered at the point a. 1.4.5.6 Weak convergence on X ⊂ X. We say that

Let X be a normed linear space and {un }

un → u ∈ X weakly in X as n → ∞, if un → u with respect to the weak topology described in Section 1.4.5.5. In agreement with Section 1.4.5.5, it is equivalent to the condition lim f (un ) = f (u),

n→∞

f ∈ X ∗. w

For weak convergence, sometimes the notation un −→ u (or un ⇀ u) is used.

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1.4.5.7 Strong convergence implies weak convergence X, then un → u weakly in X.

If un → u (strongly) in

1.4.5.8 Weakly convergent sequences are bounded If X is a Banach space, {un } ⊂ X, which is weakly convergent in X to u ∈ X, then it is bounded and uX ≤ lim inf n→∞ un X . 1.4.5.9 Weakly closed sets A subset M ⊂ X of a normed linear space X is weakly closed if it is closed with respect to the topology σ(X, X ∗ ). A subset M ⊂ X of a normed linear space X is called sequentially weakly closed in X if the following implication holds: un ∈ M, un −→ u weakly in X =⇒ u ∈ M. In general, for a set M , the weak closure is larger than the sequential weak closure which is larger than the strong closure. Similarly, a weakly closed set is also a sequentially weakly closed set, which is strongly closed, as well. 1.4.5.10 A version of Mazur’s theorem Let M be a convex subset of a Banach space X. Then M is weakly closed if and only if it is strongly closed. 1.4.5.11 A version of Mazur’s theorem for lower semicontinuous functions Let X be a Banach space and f : X → IR ∪ {∞} a mapping. We have: (i) If f is a weakly lower semicontinuous function, then f is also sequentially weakly lower semicontinuous and (strongly) lower semicontinuous (an evident statement following directly from the definition of the lower semicontinuity, see 1.4.2.7). (ii) If f is convex and (strongly) lower semicontinuous then it is also weakly lower semicontinuous (a deeper statement, a consequence of 1.4.2.7 and 1.4.5.11). 1.4.5.12 Weak-∗ topology on X ∗ We denote by σ(X ∗ , X) the so-called weak topology on X ∗ . A ball centered at a point a ∈ X ∗ is the set B(a) = {f ∈ X ∗ ; |f − a, x1 | < ε1 , . . . , | < f − a, xk > | < εk }, where k ∈ IN , εi > 0 and xi ∈ X. Any neighborhood of a is then an arbitrary union of balls of the above type such that at least one of them contains a. 1.4.5.13 Weak-∗ convergence in X ∗ Let X be a normed linear space, X ∗ be ∗ the dual of X and {fn }∞ n=1 ⊂ X . We say that fn → f weak-∗ (or weakly-∗) in X ∗ as n → ∞, if lim fn (u) = f (u),

n→∞

u ∈ X.

This definition is in agreement with Section 1.4.5.12.


1.4.5.14 Strong convergence implies weak-∗ convergence fn → f weak-∗ in X ∗ .

If fn → f in X ∗ , then

1.4.5.15 A weakly-∗ convergent sequence is bounded If X is a Banach space, then any sequence {fn } ⊂ X ∗ weakly-∗ convergent in X ∗ to f ∈ X ∗ is bounded and f X ∗ ≤ lim inf fn X ∗ . 1.4.5.16 Continuity of the duality pairing If X is a Banach space and xn → x strongly in X and fn → f weakly-∗ in X ∗ , then fn , xn → f, x, where ·, · is the duality pairing between X ∗ and X. If xn → x weakly in X and fn → f strongly in X ∗ , then fn , xn → f, x. 1.4.5.17 Reflexivity If X is a Banach space and X ∗ its dual, then we can consider the dual of X ∗ : X ∗∗ = (X ∗ )∗ . Let u ∈ X. It is obvious that the mapping ϕ ∈ X ∗ → ϕ, u ∈ IR defines an element of X ∗∗ . This means that for each u ∈ X there exists Ju ∈ X ∗∗ such that (Ju) (ϕ) := ϕ, u,

ϕ ∈ X ∗.

J : X → X ∗∗ is called the canonical mapping of the space X into X ∗∗ . 1.4.5.18 Basic properties of the canonical mapping The mapping J is linear, continuous and one-to-one. Its inverse J −1 is also continuous. (This means that J is an isomorphism between X and J(X).) Further, uX = JuX ∗∗ . (J is an isometric mapping.) 1.4.5.19 Reflexive Banach spaces We say that a Banach space is reflexive if J(X) = X ∗∗ . This means that for each g ∈ X ∗∗ there exists a uniquely determined element ug ∈ X such that g(ϕ) = ϕ, ug for all ϕ ∈ X ∗ and gX ∗∗ = ug X . For a reflexive Banach space X we simply write X = X ∗∗ . 1.4.5.20 Uniformly convex spaces A Banach space X is uniformly convex if for any ε > 0 there exists δ > 0 such that 1 2 x

+ yX < 1 − δ,

for all x, y ∈ X such that xX ≤ 1, yX ≤ 1, x − yX > ε. 1.4.5.21 flexive.

Milman–Pettis theorem Any uniformly convex Banach space is re-

1.4.5.22 Strong convergence versus weak or weak-∗ convergence (i) Let X be a uniformly convex Banach space. Then un → u strongly in X if and only if un → u weakly in X and un X → uX . (ii) Let X be a Banach space such that X ∗ is uniformly convex. Then fn → f strongly in X ∗ if and only if fn → f weakly-∗ in X ∗ and fn X ∗ → f X ∗ .

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1.4.5.23 Subspaces of reflexive Banach spaces Any closed subspace of a reflexive Banach space is also a reflexive Banach space. 1.4.5.24 Let X be a separable reflexive Banach space. Then X ∗ is also a reflexive and separable Banach space. 1.4.5.25 A version of the Banach–Alaoglu theorem Let X be a reflexive Banach space and let {un } ⊂ X be a bounded sequence. Then there exists a subsequence {unk }∞ k=1 weakly convergent in X. 1.4.5.26 Another version of the Banach–Alaoglu theorem Let X be a separable Banach space and let {fn } ⊂ X ∗ be a bounded sequence. Then there exists ∗ a subsequence {fnk }∞ k=1 weakly-∗ convergent in X . 1.4.6

Riesz representation theorem

Let X be a Hilbert space. Then for each ϕ ∈ X ∗ there exists a uniquely determined element uϕ ∈ X such that ϕ, v = (uϕ , v) ∀v ∈ X. Moreover, uϕ X = ϕX ∗ . 1.4.6.1 A corollary of the Riesz representation theorem Any Hilbert space is a reflexive Banach space. 1.4.7

Operators

1.4.7.1 Linear operators Let X and Y be Banach spaces with norms ·X and ·Y . A mapping A : D(A) ⊂ X → Y, where D(A) is a subspace of X, is called a linear operator if it satisfies the conditions (i) A(u + v) = A(u) + A(v), u, v ∈ X

(ii) A(λu) = λA(u), λ ∈ IR, u ∈ X. We call D(A) the domain of A.

R(A) = A(D(A)) = {y ∈ Y ; y = Ax, x ∈ D(A)} is called the range of A, N (A) = {x ∈ X; Ax = 0} the null space (or kernel) of A, and G(A) = {(x, Ax); x ∈ D(A)} ⊂ X × Y the graph of A.


1.4.7.2 Bounded linear operators We say that a linear operator A defined in Section 1.4.7.1 is bounded, if there exists c ≥ 0 such that AuY ≤ cuX , u ∈ D(A).

1.4.7.3 Continuous operators We say that a (not necessarily linear) mapping A : X → Y is continuous if the implication un → u in X =⇒ A(un ) → A(u) in Y

is valid. 1.4.7.4 Continuous linear operators A linear operator A : X → Y is called a continuous linear operator if it is linear and continuous on X. The set L(X, Y ) of all continuous linear operators A : X → Y equipped with the norm AuY A := sup 0 for all u = 0, then λk > 0, k ∈ IN. Moreover, Au =

k∈S (Au, vk )H

vk =

k∈S

λk (u, vk )H vk

holds for all u ∈ H in the sense of the norm induced by the scalar product (·, ·)H in the space H. For any j ∈ S the dimension of the space generated by all linear combinations of eigenvectors corresponding to λk is finite and equal to the number of appearances of λj in the sequence {λk }k∈S . Further, if M is the closure of the space of all linear combinations of all vk , k ∈ S, then the null space of A is complementary to M in H, i.e. N (A) = {u ∈ H; (u, v)H = 0 ∀v ∈ M }. The proof of the above results can be found in (Taylor, 1967), Theorem 6.4-B and Theorem 6.4-D. 1.4.9

Lax–Milgram lemma

Let us assume that H is a Hilbert space with scalar product (·, ·)H and let a : H × H → IR be a bilinear form having the following properties: (i) a is continuous, i.e. |a(z, v)| ≤ M zH vH , z, v ∈ H; (ii) a is H-elliptic, i.e. a(z, z) ≥ αz2H , z ∈ H, where M, α > 0 are constants idependent of z, v.


Let ϕ ∈ H ∗ . Then there exists exactly one solution z ∈ H of the equation a(z, v) = ϕ, v

∀v ∈ H.

The mapping ϕ → z assigning to ϕ ∈ H ∗ the solution u of the above equation is an invertible continuous linear operator of H ∗ onto H. 1.4.10

Imbeddings

1.4.10.1 Identity operator Let X ⊂ Y (in the sense of sets) be normed linear spaces with norms ·X and ·Y , respectively. Let us define the identity operator I from X into Y with the domain of definition X and the range I(X) = X by the relation Iu = u for u ∈ X. 1.4.10.2 Continuous imbedding The identity operator I is linear. If I is continuous, we speak of the continuous imbedding of X into Y . 1.4.10.3 Equivalence The continuity of the imbedding I is equivalent to the existence of a constant c > 0 such that uY ≤ c uX

∀u ∈ X.

1.4.10.4 Notation The fact of the continuous imbedding of X into Y is written in the form X ֒→ Y . 1.4.10.5 Compact imbedding In case when the imbedding operator I is completely continuous we speak of a compact imbedding and write X ֒→֒→ Y . 1.4.10.6 Consequence of compact imbedding a) If X ֒→֒→ Y, then from any bounded sequence un ∈ X it is possible to extract a subsequence strongly convergent in Y. b) If X is reflexive Banach space, then X ֒→֒→ Y if and only if (un → u weakly in X =⇒ un → u strongly in Y ). 1.4.10.7

Dual imbedding Let X and Y be normed linear spaces. Then we have:

a) X ֒→ Y =⇒ Y ∗ ֒→ X ∗ ; 1.4.11

b) X ֒→֒→ Y =⇒ Y ∗ ֒→֒→ X ∗ .

Solution of nonlinear operator equations

A series of problems can be transformed to an equation of the form u = F (u), for an unknown u ∈ X, where X is a Banach space and F : X → X. Every solution of this equation is called a fixed point of the mapping F . Let us quote some fundamental theorems on the existence of a fixed point.

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1.4.11.1 The method of contractive mapping A mapping F : X → X is called contractive if there exists a constant q ∈ [0, 1) such that F (u) − F (v) ≤ q u − v

∀u, v ∈ X.

1.4.11.2 Banach theorem Let X be a Banach space and F : X → X be a contractive mapping. Then there exists exactly one fixed point u ∈ X of the mapping F . The fixed point u can be obtained as u = lim uk , k→∞

where u0 ∈ X is arbitrary and uk+1 = F (uk ) for k ≥ 0. This result remains true if M is a closed convex subset of X, F maps M into itself and if it is contractive on M. 1.4.11.3 Brouwer theorem Let M ⊂ IRN be a nonempty, bounded, closed, convex set and let F : M → M be a continuous mapping. Then there exists at least one fixed point u ∈ M of the mapping F . Extensions of the Brouwer theorem to general Banach spaces are the Schauder and the Tikhonov theorems. 1.4.11.4 Completely continuous (nonlinear) operator Let X be a Banach space, M ⊂ X and F : M → X. F is completely continuous (or compact) in M if & ⊂ M onto a precompact set. F is continuous and maps any bounded set M

1.4.11.5 Schauder theorem Let X be a Banach space, M ⊂ X be a nonempty, bounded, closed, convex set and let F : M → M be a completely continuous mapping. Then there exists at least one fixed point u ∈ M of the mapping F .

1.4.11.6 Tikhonov theorem Let M be a nonempty bounded closed convex subset of a separable reflexive Banach space X and let F : M → M be a weakly continuous mapping (i.e. if xn ∈ M, xn → x weakly in X, then F (xn ) → F (x) weakly in X as well). Then F has at least one fixed point in M. Finally, we mention the well-known Leray–Schauder theorem. 1.4.11.7 Definition of homotopy Let (X, ·X ) be a Banach space and D ⊂ X a bounded open set. We say that H : D × [0, 1] → X is a homotopy of compact transformations on D if (i) H(·, t) : D → X is a compact operator for any t ∈ [0, 1]; (ii) for any κ1 > 0 and for any B ⊂ D there exists κ2 > 0 such that H(x, t) − H(x, s) < κ1 , ∀x ∈ B, ∀s, t ∈ [0, 1], |s − t| < κ2 .


1.4.11.8 Leray–Schauder theorem Let X be a Banach space, D ⊂ X a bounded open set, and let H : D × [0, 1] → X be a homotopy of compact transformations on D such that 0 ∈ (I − H(·, t))(∂D), t ∈ [0, 1]. If there exists at least one u0 ∈ D such that H(u0 , 0) = u0 , then for any t ∈ [0, 1] the problem to find ut ∈ D such that H(ut , t) = ut , admits at least one solution as well. This result is a version of the Leray–Schauder theorem and it follows, e.g., from Theorems 7.8 and 7.10 in (Fonseca and Gangbo, 1995).

2 THEORETICAL RESULTS FOR THE EULER EQUATIONS This chapter is devoted to the study of qualitative properties of the Euler equations of compressible flow. In what follows basic properties of the Euler equations as a nonlinear hyperbolic system will be pointed out and the relevance of boundary conditions discussed. The method of characteristics is demonstrated on representative examples as is, for example, the piston problem. Then the existence of smooth solutions, loss of smoothness and lifespan of the smooth solution of the general hyperbolic system in connection with the Euler system is treated thoroughly. Special solutions as plane waves are also studied. This is followed by an introduction to weak solutions theory with such concepts as shock and rarefaction waves, Rankine–Hugoniot conditions and entropy admissible solutions. The method of parabolic approximation (widely known as the method of vanishing viscosity) and entropy construction is described. The solution of the Riemann problem for gas dynamics equations is established. Existence results for one, two and m scalar conservation laws are given, and the Riemann problem in one space dimension is studied. Particular attention is paid to a rigorous proof of the existence of a global solution to the 2 × 2 Euler system in one space variable. In the final comments further results on Euler equations, and conservation laws in general, are surveyed and a representative base of references is given. 2.1

Hyperbolic systems and the Euler equations

The Euler equations of (adiabatic) compressible fluid flow written in conservation form are as follows (see (1.2.68)–(1.2.70))

Here

∂t ρ + div (ρv) = 0, ∂t (ρv) + div (ρv ⊗ v) + ∇p = 0,

∂t E + div v(E + p) = 0. 2

E = ρ |v2| + e

(2.1.1)

(2.1.2)

is the total energy, ρ(x, t) the density, v the velocity and e is the internal energy, which is to be specified by an appropriate constitutive equation, for example e = e(ρ, p), obtained from thermodynamical laws (cf. Section 1.2.16). 74

(2.1.3)

HYPERBOLIC SYSTEMS AND THE EULER EQUATIONS

75

The nonlinear system (2.1.1) can be written in the form ∂t w +

N

∂j f j (w) = 0.

(2.1.4)

j=1

We assume that f j = (fj1 , . . . , fjm )T : D → IRm , j = 1, . . . , N (m, N ∈ IN ), are continuously differentiable functions and D ⊂ IRm is an open set. We consider (2.1.4) in a space-time cylinder QT = Ω × (0, T ), where Ω ⊂ IRN is a domain occupied by gas and T > 0. The vector functions f j are called the fluxes of the quantity w = (w1 , . . . , wm )T in the directions xj . System (2.1.4) is called a system of conservation laws. Assuming that w ∈ C 1 (QT )m , it can be written as a quasilinear system of the type A0 (w)∂t w +

N

Aj (w)∂j w = 0

(2.1.5)

j=1

with m×m matrices Aj (w), j = 0, . . . , N, which depend on the unknown function w in a generally nonlinear way. Namely, in case of (2.1.4), A0 (w) = I is the unit m × m matrix and Aj = Df j (w)/Dw is the Jacobi matrix of f j , j = 1, . . . , N. If the matrices Aj (w) are symmetric for all w ∈ D, then system (2.1.5) is called symmetric. It is a well known fact that even the simplest equations of the type (2.1.5) exhibit such nonlinear phenomena as nonexistence of global smooth solutions on a massive set of initial and/or boundary data. Perhaps the simplest example used in the literature shedding light on the above mentioned facts follows. 2.1.1

Zero-viscosity Burgers equation

Example 2.1 Take the so-called (zero-viscosity) Burgers equation ∂t u + u∂x u = 0 with the initial condition

or ∂t u + ∂x

u(x, 0) = s x,

u2 2

=0 ,

x ∈ IR,

x ∈ IR

(2.1.6)

(2.1.7)

where s = ±1. Using the classical method of characteristics which will be explained later in Section 2.2.1 we can obtain for the solution formula u(x, t) =

sx . 1 + st

(2.1.8)

x If s = 1, then (2.1.8) defines an infinitely differentiable solution u(x, t) = t+1 of (2.1.6), (2.1.7) for all t ∈ [0, ∞). On the other hand, if s = −1, then the initially x breaks down at time t = 1. smooth solution u(x, t) = 1−t

76

THEORETICAL RESULTS FOR THE EULER EQUATIONS

As for the space dimension, of course, the only fully realistic view-point is to consider three-dimensional Euler equations. But there are many practical situations where a two-dimensional or even a one-dimensional model is sufficient to describe substantial features of a particular physical configuration. So to start with let us consider the one-dimensional Euler equations of compressible isentropic flow. 2.1.2

One-dimensional Euler equations

One-dimensional compressible isentropic inviscid flow in IR is governed by the equations (cf. (1.2.68), (1.2.69)) ρ(∂t v + v∂x v) + ∂x p(ρ) = 0, ∂t ρ + ∂x (ρv) = 0, x ∈ IR, t > 0

(2.1.9)

with a given constitutive barotropic relation p = p(ρ). These equations can more or less precisely describe the motion of a compressible fluid in a long tube which is approximated by an infinitely long one (a nonphysical idealization, of course). Then x runs over all of IR and if initially ρ(x, 0) = ρ0 (x),

v(x, 0) = v0 (x),

x ∈ IR

(2.1.10)

with given initial distribution of the density and velocity, respectively, then we can ask about values of ρ and v at any t > 0 and x ∈ IR. Notice that if we put p(ρ) = const and assume ρ > 0, then the first equation in (2.1.9) is the Burgers equation (2.1.6) for u = v and system (2.1.9) separates. 2.1.3

Lagrangian mass coordinates

∞ Assume ρ0 ∈ L1 (IR), ρ0 > 0, −∞ ρ0 (x) dx = m0 < ∞. In one space dimension we enjoy the advantage of a special transform into the so-called Lagrangian mass coordinates which simplifies system (2.1.9) considerably. We define the transformation (x, t, v, ρ) → (y, t, u, V ) by (formal) relations

x 1 y = ϕ(x, t) := −∞ ρ(ξ, t) dξ, u(y, t) = v(x, t), V (y, t) = ρ(x,t) . (2.1.11)

In this stage of the investigation we take such candidates for solutions for which (2.1.11) makes sense. It is an easy exercise in differentiation to show that, for smooth v and ρ, problem (2.1.9), (2.1.10) assumes the form ∂t u + ∂y p(1/V ) = 0, ∂t V − ∂y u = 0, u(y, 0) = u0 (y), where

y ∈ IR, t > 0, V (y, 0) = V0 (y),

u0 (y) = u0 ϕ−1 (y, 0) , V0 (y) =

y ∈ IR,

1 ρ0 (ϕ−1 (y,0)) .

Here ϕ−1 (·, 0) means the function inverse to ϕ(·, 0), i.e. we have

(2.1.12)

HYPERBOLIC SYSTEMS AND THE EULER EQUATIONS

y=

ϕ−1 (y,0) −∞

77

ρ0 (ξ) dξ.

To keep consistent notation let us write x instead of y, and p(V ) instead of p(ρ) = p( V1 ). With this convention in mind problem (2.1.9), (2.1.10) now reads ∂t u + ∂x p(V ) = 0, ∂t V − ∂x u = 0, x ∈ IR, t > 0, u(x, 0) = u0 (x), V (x, 0) = V0 (x),

(2.1.13) x ∈ IR.

System (2.1.13) is known in the literature as the p-system. It may be shown that system (2.1.13) exhibits analogous features to the single equation (2.1.6). Even if the data in (2.1.12) are smooth, the solution which is initially smooth, may break down in derivatives at some finite time t∗ in the sense u(x, t)|+|∂ u(x, t)|+|∂ V (x, t)|+|∂ V (x, t)| = ∞ for some x ∈ IR. lim inf |∂ t x t x ∗ t↑t

(2.1.14)

2.1.4

Symmetrizable systems

To understand the main features of the Euler equations it is desirable to present some facts from the theory of quasilinear systems of partial differential equations, that is systems of the form (2.1.5). Clearly, the Euler system belongs to this class of equations. Let us begin with the following definition. Definition 2.2 Let D be an open set in IRm , a definition region of matrices Aj (w), j = 1, . . . , N. We say that system (2.1.5) with A0 (w) = I is symmetriz 0 (w) able (in D) if for any w ∈ D there is a positive definite symmetric matrix A such that (i) for any subregion D1 satisfying D1 ⊂ D we have 0 (w)z, z)IRN ≤ cz2 N , c−1 z2IRN ≤ (A IR

0 (w) = A 0 (w)T A

with a positive constant c independent of w ∈ D1 and z ∈ IRN ; (ii) the following relation holds: j (w), where A j (w) = A j (w)T , j = 1, . . . , N. 0 (w)Aj (w) = A A

2.1.5

(2.1.15)

(2.1.16)

Matrix form of the p-system

Example 2.3 The system given by the first two equations in (2.1.13) can be written in the form (2.1.5) with w = (u, V ), N = 1, A0 (w) := I, and ( ' 0 p′ (w2 ) A1 (w) := . −1 0 Multiplying this system by

78


0 (w) := A

'

10 0 −p′ (w2 )

(

0 (w), yields the quasilinear symmetric system with A ' ( 0 p′ (w2 ) A1 (w) := . p′ (w2 ) 0

Note that this symmetrization corresponds to the multiplication of the second equation in (2.1.13) by (−p′ (V )).

2.1.6 The Euler equations of an inviscid gas Example 2.4 The Euler equations for an adiabatic inviscid gas in terms of the variables p (pressure), v (velocity) and S (entropy) can be written in the form (cf. (1.2.68)–(1.2.70), (1.2.91), (1.2.64)) ∂t p + (v · ∇)p + γp div v = 0, ρ(∂t v + (v · ∇)v) + ∇p = 0, ∂t S + (v · ∇)S = 0,

(2.1.17)

where the thermodynamical relation confining p, S and the density ρ is assumed in the normalized form eS = pρ−γ

with a constant γ > 1.

(2.1.18)

Indeed, if we take equations (1.2.68), (1.2.69), (1.2.64), (2.1.18), compute Dt p = ∂t p + (v · ∇)p and use (1.2.68), (1.2.64) and (2.1.18), we obtain the equation of continuity in the form of the first equation in (2.1.17). Evaluating from (2.1.18) 1 ρ = ρ(p, S) = p γ e−S/γ we can verify that system (2.1.17) is symmetrizable by the 5 × 5 matrix   (γp)−1 ∅ 0 0 (p, v, S) =  ∅ ρ(p, S)I 0  . (2.1.19) A 0 ∅ 1 Example 2.5 Using the normalized thermodynamic relations e(ρ, p) =

p ρ(γ−1)

=

θ γ−1

(2.1.20)

with θ as the temperature, we can write the inviscid flow equations in variables ρ, v and θ as follows: ∂t ρ + (v · ∇)ρ + ρ div v = 0, ρ(∂t v + (v · ∇)v) + ρ∇θ + θ∇ρ = 0, ∂t θ + (v · ∇)θ + (γ − 1)θ div v = 0. The corresponding (not unique!) symmetrizer can be chosen as θ  0 ρ ∅ 0 (ρ, v, θ) =  0 ρI 0  . A ρ 0 0 (γ−1)θ

(2.1.21)

(2.1.22)

EXISTENCE OF SMOOTH SOLUTIONS

79

The possibility to symmetrize important systems of inviscid fluid equations allows us to take advantage of the theory of linear symmetric hyperbolic systems to treat auxiliary systems arising when we define successive approximations in order to solve nonlinear systems of the type (2.1.5). This will be the main objective of Section 2.2. 2.2 Existence of smooth solutions In this section we are going to introduce the method of characteristics, local existence results for quasilinear hyperbolic systems and its consequences for Euler equations, breakdown of solutions phenomena and some particular cases of existence of global smooth solutions. 2.2.1 Hyperbolic systems and characteristics First, consider system (2.1.5) and introduce the notion of hyperbolicity. Definition 2.6 System (2.1.5) is called hyperbolic in the region D ⊂ IRm if all solutions λj = λj (w, n), j = 1, . . . , m of the m-th order algebraic equation

N (2.2.1) det λA0 (w) − j=1 nj Aj (w) = 0

are real for any n ∈ IRN and w ∈ D. We call λj generalized eigenvalues of system (2.1.5). If moreover the generalized eigenvalues λj are simple, then the system is called strictly hyperbolic. We will say that system (2.1.5) is diagonally (strictly) hyperbolic if in adN dition the matrix P := j=1 nj Aj (w) is diagonalizable. This means that there exists a nonsingular matrix T = T(w, n) such that   ∅ λ1   .. T−1 P T = Λ\(w, n) = diag (λ1 , . . . , λm ) =  (2.2.2) . . ∅

λm

Example 2.7 Write the equations of isentropic gas dynamics in the form 1 ∂t v + (v · ∇)v + ∇p(ρ) = f , ρ ∂t ρ + div (ρv) = 0

v = (v1 , v2 , v3 ),

(2.2.3)

with the equation of state p = p(ρ) (ρ > 0) assuming p′ (ρ) > 0. Then the corresponding determinant in (2.2.1) is equal to

2

2 3 3 3 (2.2.4) λ − k=1 nk vk − p′ (ρ) k=1 n2k λ − k=1 nk vk

and the corresponding equation (2.2.1) has four real roots (one of them multiple) 1/2 3 3 3 (2.2.5) . λ1 = λ2 = k=1 nk vk , λ3,4 = k=1 nk vk ± p′ (ρ) k=1 n2k

Hence system (2.2.3) is hyperbolic in any region (v, ρ) ∈ D ⊂ IR × (0, ∞).

80


Definition 2.8 If N = 1, A0 = I, A1 (w) = A(w) and system (2.1.5) is hyperbolic, then the real roots λj , j = 1, . . . , m, of the equation

det λI − A(w) = 0 (2.2.6) are called eigenvalues and for a given smooth function w satisfying (2.1.5) with the above choice of the matrices Aj , j = 0, 1, the curves in the (x, t)-plane parametrized by x := ξj (s), t = t(s) = s, s ∈ J, where dξj ds

= λj (ξj (s), s),

s ∈ J, j = 1, . . . , m,

(2.2.7)

J is an open interval, are called characteristics. If in addition the matrices Aj (w), j = 0, 1, . . . , N in (2.1.5) are symmetric for all w ∈ D ⊂ IRm , then system (2.1.5) is called a symmetric hyperbolic system (in D). For example, for equation (2.1.6) (m := 1) we have λ = λ1 (u), and ξ = ξ1 (s) is given by dξ (2.2.8) ds = u(ξ(s), s), whenever u is an appropriate solution of (2.1.6). Notice that the trajectory {(x, t); x = ξ(s), t = s} in the (x, t)-plane is not determined uniquely unless we fix “the initial” value of ξ at some point s = s0 . The concept of characteristics in some cases helps us to say something about possible solutions to one-dimensional equations. We illustrate this method later in Sections 2.2.3 and 2.2.5. 2.2.2 Cauchy problem for system of conservation laws The nonstationary system (2.1.4), if considered in Ω = IRN , should naturally be equipped with the initial condition w(x, 0) = w0 (x), 0

x ∈ Ω,

(2.2.9)

where w is a given vector-valued function. In what follows, we shall be concerned with the Cauchy problem given by N ∂t w + j=1 ∂j f j (w) = 0 in QT := IRN × (0, T ), (2.2.10) w(x, 0) = w0 (x), x ∈ IRN . Definition 2.9 Let D ⊂ IRN be the domain of definition of the functions f j , j = 1, . . . , N . We say that a vector-valued function w is a classical solution of the Cauchy problem (2.2.10) if a) w ∈ C 1 (IRN × (0, T ))m ∩ C(IRN × [0, T ))m , b) w(x, t) ∈ D for all (x, t) ∈ QT , c) w satisfies (2.2.10) for all (x, t) ∈ IRN × (0, T ) and x ∈ IRN , t = 0, respectively. We say that w is a classical solution of the first equation in (2.2.10) if w ∈ C 1 (IRN × (0, T ))m and the equation is satisfied pointwise. In what follows, a few simple examples illustrate some specific properties of hyperbolic problems and possibilities of the so-called method of characteristics.


81

2.2.3 Linear scalar equation Consider problem (2.2.10) with N = 1, m = 1, f1 (w) = aw, where a is a constant. Then the differential equation consists of one linear partial differential equation for a scalar function w with a scalar initial condition w0 . To show the possibilities of the method we are going to use, let us allow the constant a be a function of x and t, i.e. a = a(x, t). Consider the Cauchy problem ∂t w + a(x, t) ∂x w = 0 in IR × (0, T ), w(x, 0) = w0 (x),

x ∈ IR,

(2.2.11)

for w = w(x, t) : IR × [0, T ) → IR. This problem can be solved by the method of characteristics as is illustrated by the following theorem. Theorem 2.10 Let w0 ∈ C 1 (IR) and a, ∂x a ∈ C(QT ). Then for any (x, t) ∈ QT there exists a unique solution ξ = ξ(s; x, t), ξ ∈ C 1 ({(s; x, t); (x, t) ∈ QT , s ∈ (0, t)}) of the problem ∂s ξ = a(ξ, s),

s ∈ (0, t),

ξ(t; x, t) = x

(2.2.12)

and the function w(x, t) := w0 (ξ(0; x, t))

(2.2.13)

is a unique classical solution of problem (2.2.11). (The definition of a classical solution to (2.2.11) is quite analogous to that for (2.2.10).) Proof The existence of a unique solution ξ = ξ(s; x, t) of problem (2.2.12) is a consequence of the classical existence result for ordinary differential equations (see, e.g. (Kurzweil, 1986)). For a fixed (x, t) ∈ QT , the curve in the (x, t)-plane given by {(ξ(s; x, t), s); s ∈ (0, t)} is the characteristic (issuing from the point (x, t)). If w is a classical solution of (2.2.11), then, writing briefly ξ = ξ(s), from (2.2.11), (2.2.12) we find (d/ds)w(ξ(s), s) = ∂t w(ξ(s), s) + ∂s ξ(s)∂x w(ξ(s), s) = ∂t w(ξ(s), s) + a(ξ(s), s)∂x w(ξ(s), s) = 0. Thus w is constant along characteristics. This fact leads to formula (2.2.13). The rigorous proof of the existence and uniqueness is now based on the relation ∂t ξ(s; x, t) + a(x, t)∂x ξ(s; x, t) = 0,

(2.2.14)

which can be obtained by differentiation of (2.2.12) with respect to x and t, respectively. It appears that ∂x ξ and ∂t ξ are solutions of the same first-order homogeneous linear differential equation and so one is a multiple of the other. The coefficient of proportionality is obtained by differentiation of the relation ξ(t; x, t) = x with respect to t. Here again, classical results on the differentiation of solutions to ordinary differential equations with respect to parameters are applied. Finally, assuming (2.2.13) and taking s = 0 in (2.2.14) we find

∂t w(x, t)+a(x, t)∂x w(x, t) = (w0 )′ (ξ(0; x, t)) ∂t ξ(0; x, t)+a(x, t)∂x ξ(0; x, t) = 0. 2

82


From Theorem 2.10 we easily conclude that for a = const and w0 ∈ C 1 (IR), there exists a unique classical solution of problem (2.2.11) and it is given by w(x, t) = w0 (x − at),

x ∈ IR, t ∈ (0, ∞).

(2.2.15)

This solution is called a travelling wave. 2.2.4

Solution of a linear system

The result of Theorem 2.10 can also be used for the resolution of the Cauchy problem for the linear hyperbolic system with constant coefficients. Consider the problem ∂t w + A ∂x w = 0 in IR × (0, ∞) (2.2.16) w(x, 0) = w0 (x), x ∈ IR. Theorem 2.11 Assume that A is an m × m matrix which is diagonalizable and has real eigenvalues λ1 , . . . , λm . Let r j , j = 1, . . . m, be eigenvectors of A and w0 ∈ C 1 (IR)m be expanded into the basis formed by {r j }, i.e. m w0 (x) = j=1 wj0 (x) r j , x ∈ IR, t ∈ [0, ∞). (2.2.17)

Then there exists a unique classical solution w ∈ C 1 (IR × (0, ∞))m of problem (2.2.16) and it is given by m w(x, t) = j=1 wj0 (x − λj t) r j , (2.2.18) where λj are respective eigenvalues corresponding to the eigenvectors r j .

Proof Inserting (2.2.18) into (2.2.16), using Theorem 2.10 and (2.2.17), we get

m ∂t w + A ∂x w = j=1 −λj (wj0 )′ (x − λj t) + (wj0 )′ (x − λj t)Ar j = 0, m w(x, 0) = j=1 wj0 (x)r j = w0 (x), (x, t) ∈ IR × (0, ∞).

For uniqueness it suffices to prove that if w0 (x) ≡ 0, then the corresponding solution of (2.2.16) is identically zero. This is trivially done by expanding a solution with w(x, 0) ≡ 0 into r j , inserting the expansion into the system and using the linear independence of r j to obtain m scalar equations for coefficients with zero initial conditions, which yields that all coefficients must be zero. 2 Notice that the coefficients in expansion (2.2.18) are of the form (2.2.15) and consequently the solution of (2.2.16) is a composition of travelling waves in the directions of the eigenvectors r j with respective speeds λj . 2.2.5

Nonlinear scalar equation

The method of characteristics can be also be applied to the nonlinear problem ∂t w + ∂x f (w) = 0 in IR × (0, T ) (T > 0), w(x, 0) = w0 (x),

x ∈ IR.

(2.2.19) (2.2.20)


83

Theorem 2.12 Let f ∈ C 2 (IR) and w0 ∈ C 1 (IR) ∩ W 1,∞ (IR). Then, denoting D = {w0 (x); x ∈ IR}, there exists a positive number −1 T < (w0 )′ L∞ (IR) sup |f ′′ (z)|

(2.2.21)

z∈D

such that there is a unique classical solution of (2.2.19), (2.2.20) in IR × (0, T ), and it can be obtained as a solution w = w(x, t) of the equation w = w0 (x − f ′ (w)t),

x ∈ IR, t ∈ (0, T ).

(2.2.22)

Proof Denote G(w, x, t) := w − w0 (x − f ′ (w)t). Then ∂w G(w, x, t) = 1 + tf ′′ (w)(w0 )(x − f ′ (w)t). If (2.2.21) is satisfied, then ∂w G(w, x, t) > 0 for all x ∈ IR, t ∈ [0, T ) and w ∈ D. By the implicit function theorem (Theorem 1.3), given a fixed x ∈ IR, there is a T (x) > 0 such that equation (2.2.22) has a unique solution w(x, t) for t ∈ (0, T (x)). Let Tmax (x) be the maximum of τ > 0, for which w(x, t) exists on (0, τ ). If Tmax (x) ≤ T, then still ∂G (w(x, Tmax (x)), x, Tmax (x)) > 0, ∂w and again, by the implicit function theorem there is a T1 > Tmax (x) such that w(x, t) exists on [Tmax , T1 ), which is a contradiction to the maximality of Tmax (x). Thus, independently of x, the solution w(x, t) of (2.2.22) exists on (0, T ). It is a routine matter to show with the help of the implicit function theorem that w satisfies (2.2.19), (2.2.20) pointwise. Uniqueness of the solution w can be proved so that given another solution z of the same problem and a characteristic given by ∂s ξ(s; x, t) = f ′ (z(ξ(s; x, t), s)),

ξ(t; x, t) = x,

from this equation we deduce that z is constant along the characteristic, which leads to equation (2.2.22). But this equation is uniquely solvable on (0, T ) and consequently z must be equal to w. 2 Let us note that if f ′′ (z)(w0 )′ (y) ≥ 0 for all z ∈ D and y ∈ IR, then

∂G ∂w

> 0 in any case and T can be taken to be infinite.

84

2.2.6


Piston problem

The method of characteristics exemplified above can be used also for systems, but is not so straightforward as for one equation. For example, the so-called piston problem can be analyzed in detail (see (Chorin and Marsden, 1979), §3.1). The problem consists in finding the evolution of compressible fluid flow in a tube where initially the fluid of constant density is in the rest state and is bounded by a piston which at time t = 0 starts to move with a given velocity u1 (t), a function of time. Model equations for this problem are given by the system ∂t u + ∂x p(V ) = 0,

∂t V − ∂x u = 0,

x > 0, t ∈ (0, T ) (T > 0), x ≥ 0,

u(x, 0) = u0 = 0, V (x, 0) = V0 = const > 0,

t > 0, (2.2.23) where p ∈ C 1 (0, ∞), p′ (V ) < 0 for V > 0. As we know, system (2.2.23)1 can be written in the form ∂t w + A(w)∂x w = 0, (2.2.24) where T

w = (u, V ) , A(ρ, w) =

'

u(0, t) = u1 (t),

0 p′ (V ) −1 0

(

.

In contrast with the scalar case, here we cannot expect w to be constant along characteristics but we might seek functions ψj = ψj (w), j = 1, 2 that are constant along the characteristics associated with the eigenvalues λ1 , λ2 which are clearly )

(2.2.25) λ1,2 (u, V ) = ± −p′ (V ) . 2.2.7

Complementary Riemann invariants

To construct functions ψj for the general m × m hyperbolic system (2.2.24) assume first that for a given eigenvalue λ we can find a scalar function ψ = ψ(w) such that (2.2.26) A(w)T ∇w ψ = λ(w)∇w ψ. Definition 2.13 A scalar-valued function w → ψ(w) is called a complementary Riemann invariant if it satisfies (2.2.26) for some λ = λ(w). If λ = λk (w) is the k-th eigenvalue in the field of all eigenvalues of (2.2.26), then we speak about the k-complementary Riemann invariant (corresponding to the k-th eigenvector r k (w) and the eigenvalue λk (w), w ∈ D). If (2.2.26) is true, then eigenvalue λ(w).

Dψ Dw

is an eigenvector of A(w)T with the corresponding

Theorem 2.14 Assume that w is a classical solution of the strictly hyperbolic system (2.2.24) with values in a region D ⊂ IRm such that the matrix-valued function w → A(w) is continuous from D into IRm×m . Then the complementary


85

Riemann invariant corresponding to the eigenvalue λ(w) is constant along any characteristic given in the (x, t)-plane by the curve {(ξ(s), s); s ∈ J}, where dξ ds

= λ(w(ξ(s), s)),

(2.2.27)

s ∈ J,

and J is some time interval. Proof Assume that w = (w1 , . . . , wm )T is a classical solution to (2.2.24) and let ξ = ξ(t) be a characteristic corresponding to λ, given by (2.2.27). Now differentiate ψ along the characteristic satisfying (2.2.27):

m ∂ψ dwl m ∂ψ ∂wl ∂wl dξ d l=1 ∂wl ds = l=1 ∂wl ∂s + ∂x ds ds ψ(w(ξ(s), s)) =

m ∂ψ m ∂wl i − i=1 Ali ∂w = l=1 ∂w (2.2.28) ∂x + λ ∂x l

m m m ∂ψ ∂wl ∂ψ ∂wl = 0. = l=1 − i=1 Ail ∂w + λ l=1 ∂w i ∂x l ∂x

So we see that ψ(w) is constant along the curve {(ξ(s), s); s ∈ J}.

2

Note that in general complementary Riemann invariants may not exist. Especially for m > 2 only systems of special structure enjoy the property of existence of complementary Riemann invariant(s). 2.2.8

Solution of the piston problem

Let us return to the piston problem defined by (2.2.23). In this case m = 2 and two complementary Riemann invariants exist under reasonable conditions, in particular for system (2.2.23). Indeed, assuming p′ (r) < 0 for r> 0, we find eigenvectors of AT corresponding to eigenvalues ±a, where a = −p′ (V ), i.e. vectors (h± , k± )T such that ( ' ( ( ' ' h± h± 0 −1 . . (2.2.29) = ±a k± k± −a2 0 We find easily that h± = 1, k± = ∓a(V ) modulo a multiplicative constant. From ∂ψ ∂ψ the requirements ∂u1,2 = h± , ∂V1,2 = k± we obtain by integration ψ1 (u, V ) = u −

V

a(η) dη,

ψ2 (u, V ) = u +

V

a(η) dη.

(2.2.30)

The functions given by (2.2.30) are the complementary Riemann invariants of system (2.1.10) (see Definition 2.13). Let us now use the properties of complementary Riemann invariants

to solve at least locally the piston problem. First, assume the primitive a(η) dη in

V (2.2.30) fixed by b a(η) dη with a constant b > 0. Second, we assume u1 continuously differentiable on [0, ∞) together with the compatibility condition u1 (0) = u0 (0) = 0. Now we proceed by heuristic considerations to find the formula for the solution of (2.2.23). This will consist in assuming that we have a smooth solution (u, V ) of the problem and proving that under this assumption u

86


and V must have a certain form. Then using this formal representation we insert it into the equations and verify that they are satisfied with this choice of u and V. Following this scenario, let u, V exist as we have assumed. Then given x > 0 and t small, from the continuity, both characteristics, given respectively by dξ dt

= ±a(V (ξ, t))

and issuing from (x, t), fall to the halfline {x ≥ 0, t = 0}. Since ψ1 , ψ2 are constant

V along corresponding characteristics and by (2.2.23), ψ1 |t=0 = − b 0 a(η) dη =

V const, ψ2 |t=0 = b 0 a(η) dη = const we have ψ1 (x, t) = −

V0 b

a(η) dη,

ψ2 (x, t) =

V0 b

a(η) dη.

(2.2.31)

Consider the first characteristic issuing from x = 0, t = 0. Then in any right neighborhood of it the characteristics of the same family must be straight lines at least for short times since they are contained in Ω0 := {x ≥ a(V0 )t}, where u and V are constant as a consequence of the constancy of ψ1 , ψ2 and the one-to-one correspondence in the transformation (u, V ) → (ψ1 , ψ2 ). So this characteristic is also straight line. Then, in Ω0 , u = 0 and V = V0 is a solution of the problem. Let Ω1 := {0 ≤ x < a(V0 )t}. Take some (x, t) ∈ Ω1 and issue the second characteristic backwards with respect to time. Assuming that it reaches the line x = a(V0 )t we see that

V ψ2 (x, t) = b 0 a(η) dη. This yields

u=−

V

V0

a(η) dη := ϕ(V ) in Ω1 .

(2.2.32)

Denote by h the function ϕ−1 . Now issuing backwards from (x, t) the characteristic ξ1 = ξ1 (s) of the first family we get ψ1 (x, t) = ψ1 (0, τ ) = ψ1 (ξ1 (s), s),

s ∈ (τ, t),

(2.2.33)

where dξ1 ds

= a(ξ1 , s),

ξ1 (t) = x,

s ∈ [τ, t]

(2.2.34)

and τ is such that ξ1 (τ ) = 0. But (2.2.32), (2.2.33) imply that ψ1 (ξ1 (s), s), ψ2 (ξ1 (s), s) are independent of s. This in particular means that a(ξ1 (s), s) is independent of s as well. So by (2.2.34), (2.2.32)

s ξ1 (s) = x + t a(ξ1 (σ), σ) dσ = x + (s − t)a(ϕ−1 (u1 (τ ))). (2.2.35) In particular

0 = ξ1 (τ ) = x + (τ − t)a(h(u1 (τ )))

(2.2.36)

and if τ = τ (x, t) were a solution of (2.2.36), then our candidate solution in Ω1 is a couple of functions (u, V ) satisfying


u(x, t) −

V (x,t)

a(η) dη = u1 (τ (x, t)) −

V (0,τ (x,t))

a(η) dη,

V (x,t) V (0, τ (x, t)) = ϕ−1 (u1 (τ (x, t))), u(x, t) = − V0 a(η) dη. b

b

87

(2.2.37)

Let us show that equation (2.2.36) has at least one solution τ = τ (x, t) ∈ [0, t]. Indeed, the function ξ1 is continuous and ξ1 (0) = x − a(V0 )t < 0, ξ1 (t) = x > 0. By the Darboux property of continuous functions there must be at least one τ satisfying (2.2.36). Let us investigate the uniqueness of the solution to (2.2.36). Assume d2 p (2.2.38) dV 2 (η) > 0 for η > 0. If in addition to (2.2.38) du1 ds (s)

≤ 0 for s ∈ (0, t0 ) (t0 > 0),

(2.2.39)

then from (2.2.36) dξ1 dτ (s)

da = a ◦ h ◦ u1 (s) + (s − t) dV ◦ h ◦ u1 (s) · h′ (u1 (s)) · u′1 (s), ′′

p ◦h◦u1 (s) ′ = a ◦ h ◦ u1 (s) + (s − t) 2(a◦h◦u 2 · u1 (s) > 0, s ∈ (0, t0 ) 1 (s))

(2.2.40)

and uniqueness immediately follows. Now, in contrast to (2.2.39), let there be an s0 ∈ (0, t) such that du1 ds (s0 )

(2.2.41)

> 0.

Then putting 1 s1 := sup{s; dξ ds (σ) > 0 for σ ∈ (0, s)},

(2.2.42)

dξ1 ds (0)

= a ◦ h ◦ u1 (0) = a(V0 ) > 0) and if t ≤ s1 , we clearly have s1 > 0 (since then (2.2.40) continues to hold for s ∈ (0, t), and τ (x, t) is determined uniquely. But in this case we pay for uniqueness by the assumption t ≤ s1 while s1 given by (2.2.33) can be small. Nevertheless, in both cases we have a uniquely determined τ = τ (x, t) satisfying (2.2.36). Assume that (2.2.39) or t0 ≤ s1 holds with some t0 > 0. Then writing ξ1 = ξ1 (τ ; x, t) for 0 ≤ x ≤ c0 t, t ∈ (0, t0 ), τ > 0 by the implicit function theorem (Theorem 1.3) we have 1 ∂x τ (x, t) = − (ξ1 )τ (τ (x,t);x,t) ,

∂t τ (x, t) =

a◦h(u1 (τ (x,t))) (ξ1 )τ (τ (x,t);x,t)

and in particular ∂t τ = −a ◦ u1 ◦ τ · ∂x τ.

(2.2.43)

From (2.2.37) we easily get u(x, t) = u1 (τ (x, t)), V (x, t) = h(u1 (τ (x, t))) for 0 ≤ x ≤ a0 t, t ∈ (0, t0 ). (2.2.44) Show that (u, V ) defined by (2.2.44) satisfies the first two equations in (2.1.13). Indeed, by (2.2.43), (2.2.32) we have

88


∂t u = u′1 (τ )∂t τ = −u′1 (τ )a(V )∂x τ = a2 ∂x V = −p′ (V )∂x V = −∂x p(V ), 1 ′ ′ ∂t V = h′ (u1 (τ ))u′1 (τ )∂t τ = − a(V ) (−a∂x τ )u1 (τ ) = ∂x τ u1 (τ ) = ∂x u

in the region {(x, t); 0 < x < a0 t, t ∈ (0, t0 )}. Also, u(0, t) = u1 (τ (0, t)) = u1 (t) by (2.2.36). We further define u(x, t) = 0,

V (x, t) = V0

for x ≥ a(V0 )t, t > 0.

(2.2.45)

This couple satisfies trivially the first two equations in (2.2.23) for x > a(V0 )t, t > 0 and initial data in (2.2.23) as well. So at least for small t the above constructed pair is the classical solution of problem (2.2.23). The compatibility condition u1 (0) = 0 guarantees the continuity of the solution across the line x = a(V0 )t. If, in addition, we also have (2.2.46) u′1 (0) = 0, then the solution is also continuously differentiable across x = a(V0 )t. Let us summarize our result in the following theorem. Theorem 2.15 Let 1 ([0, ∞)), u1 (0) = 0, u0 = 0, V0 = const > 0, u1 ∈ Cloc 2 p(·) ∈ Cloc ([0, ∞)) ∩ Cloc ((0, ∞)), p′ (η) < 0, p′′ (η) > 0 for η > 0,

and (2.2.39) or (2.2.41) hold respectively for t ∈ (0, t0 ) with some t0 > 0 satisfying (2.2.39), or t0 ≤ s1 with s1 defined by (2.2.42), respectively. Then the piston problem given by (2.2.23) has exactly one classical solution (u, V ) in IR × (0, t0 ) which is given by (2.2.44), (2.2.45). Note that similar result in a more general setting can be found in (Straˇskraba, 1984). We have seen above that even in the simple situation of one-dimensional isentropic Euler equations with special data the resolution is not straightforward. We cannot simply use tools like the method of characteristics in more realistic 2D or 3D models and rather employ the theory of symmetric hyperbolic systems. We already know that Euler equations can be symmetrized, which means they can be transformed into the system of the type (2.1.5) with Aj , j = 1, . . . , m symmetric. So the next section will be devoted to the local theory of symmetric hyperbolic systems.


2.2.9

89

Cauchy problem for a symmetric hyperbolic system

In what follows we are going to prove a local existence theorem for the first order symmetric hyperbolic system (2.1.5) (see Definition 2.8) to be applied later to Euler equations in several space variables. Since the solution to (2.1.5) is constructed via successive approximations defined by a corresponding linear problem we start with the following problem. Let Aj (x, t), j = 0, 1, . . . , N be symmetric m × m matrices in IRN × (0, T ) with some T > 0, f (x, t) and v 0 (x), m-dimensional vector functions defined in IRN × (0, T ) and IRN , respectively. Let us consider the problem N A0 (x, t)∂t v + j=1 Aj (x, t)∂j v + B(x, t)v = f (x, t), x ∈ IRN , t ∈ (0, T ), (2.2.47) v(x, 0) = v 0 (x), x ∈ IRN . (2.2.48) Theorem 2.16 Let m×m , j = 0, 1, . . . , N, Aj ∈ C([0, T ], H s (IRN )) ∩ C 1 ((0, T ), H s−1 (IRN ))

A0 (x, t) invertible, inf A0 (x, t)IRm ×IRm > 0, B ∈ C((0, T ), H s−1 (IRN ))m×m , x,t

s

N

f ∈ C((0, T ), H (IR ))m , v 0 ∈ H s (IRN )m , where s > N2 + 1 is an integer. Then there exists a unique solution to (2.2.47), (2.2.48), i.e. a function m v ∈ C([0, T ), H s (IRN )) ∩ C 1 ((0, T ), H s−1 (IRN ))

satisfying (2.2.47) and (2.2.48) pointwise (i.e. in the classical sense).

Before we start with the proof of Theorem 2.16 we need the following Proposition 2.17 Let L = L(t), t ∈ [0, T ] (T > 0) be a family of linear bounded operators in a Banach space B and let L ∈ C([0, T ], L(B)). Then for any v0 ∈ B and f ∈ C([0, T ], B) the problem v ′ (t) = L(t)v(t) + f (t), v(0) = v0

t ∈ (0, T ),

(2.2.49) (2.2.50)

has a unique solution v ∈ C([0, T ), B)∩C 1 (0, T, B). This means in particular that the derivative v ′ (t) exists with respect to the norm of B in (0, T ) and equation (2.2.49) is satisfied pointwise. Proof Define for v ∈ C([0, T ], B)

t (Gv)(t) = v0 + 0 L(s)v(s) + f (s) ds,

t ∈ [0, T ].

(2.2.51)

Then G maps F := C([0, T ], B) into itself and given k > 0, the norm vF := sup {e−kt v(t)B }, t∈[0,T ]

v∈F

(2.2.52)

90


is an equivalent norm to the natural supremal norm on F. Moreover, for v1 , v2 ∈ F we have t

G(v1 ) − G(v2 )F = supt∈[0,T ] 0 e−k(t−s) L(s)e−ks v1 (s) − v2 (s) ds

t ≤ supt∈[0,T ] L(t)L(B) supt∈[0,T ] 0 e−k(t−s) ds v1 − v2 F ≤

1 k

supt∈[0,T ] L(t)L(B) v1 − v2 F .

So if k > supt∈[0,T ] L(t)L(B) , then G defined by (2.2.51) is contractive in F under the norm defined by (2.2.52). By the Banach contraction principle (see Section 1.4.11.2), G has in F a unique fixed point v = G(v). It is a routine matter to verify that v solves problem (2.2.49), (2.2.50) in the sense of Proposition 2.17. 2 Proof of Theorem 2.16. The proof will be divided into several steps. 2.2.10

Approximations

Define approximations v ε = v ε (x, t) of the sought solution v of (2.2.47), (2.2.48) as a solution of the problem N (2.2.53) A0 (x, t)∂t v ε + j=1 Jε Aj (x, t)∂j (Jε v ε ) + B(x, t)v ε = f (x, t), v ε (x, 0) = v 0 (x),

x ∈ IRN t ∈ (0, T ).

(2.2.54)

Here Jε is the usual mollifier (see Section 1.3.4.4), i.e. for w = w(x),

(Jε w)(x) = IRN ωε (x − y)w(y) dy,

(2.2.55)

where

ωε (x) :=

1 εN

ω( xε ),

∞

ω(x) = ω(−x), x ∈ IRN N

(2.2.56)

with a positive C -function ω such that supp ω ⊂ (−1, 1) and IRN ωε (x) dx = 1. (Notice that under these conditions the operator Jε is symmetric in L2 (IRN ).) 2.2.11

Existence of approximations

It can be easily derived from Lemma 1.34 that for each ε > 0 the operator Dα Jε is bounded in Lp (IRN ) for any fixed multiindex α and exponent p ∈ [1, ∞). So system (2.2.53), (2.2.54) can be written in the form (2.2.49), (2.2.50), where

N L(t)w (x) = −A0 (x, t)−1 B(x, t)w(x, t) + j=1 Jε Aj (x, t)∂j (Jε w)(x, t) . (2.2.57) By assumptions L(·) ∈ C((0, T ), L(H s (IRN )))m×m , v 0 ∈ H s (IRN )m , f ∈ C((0, T ), H s (IRN ))m and so by Proposition 2.17 for each ε > 0 there is a unique solution v ε ∈ C([0, T ], H s (IRN ))m ∩ C 1 ((0, T ), H s−1 (IRN ))m of problem (2.2.49), (2.2.50) with L given by (2.2.57). Multiplying equation (2.2.49) by A0 (x, t) we find that v ε satisfies (2.2.53) and, of course, (2.2.54).


2.2.12

91

Energy estimate

Define now Eε (t) :=

1 2

IRN

A0 (x, t)v ε (x, t) · v ε (x, t) dx,

t ∈ [0, T ].

(2.2.58)

Then by the symmetry of A0 and (2.2.53) we get

ε

1 dEε ε · v + A0 ∂t v ε · v ε dx dt (t) = IRN 2 ∂t A0 v

N ε εv · v ε dx. = IRN 12 ∂t A0 v ε · v ε + f · v ε − Bv ε · v ε − j=1 Jε Aj ∂J∂x j (2.2.59) Further, since Jε and Aj are symmetric, by integration by parts we find

ε ε Jε Aj ∂J∂xε vj · v ε dx = IRN Aj ∂J∂xε vj Jε v ε dx IRN

∂A

ε = − IRN ∂xjj · Jε v ε · Jε v ε dx − IRN Aj Jε v ε · ∂J∂xε vj dx. (2.2.60) But symmetry of Jε , Aj and (2.2.60) imply

IRN

Jε Aj ∂J∂xε vj

for j = 1, . . . , N. Denote

ε

· v ε dx = − 21

IRN

∂Aj ∂xj

Jε v ε · Jε v ε dx

(2.2.61)

C0 := 1 + (N + 2) sup Jε L(L2 (IRN )) sup{∂t A0 (·, t)L2 (IRN )m×m , ε∈(0,1]

B(·, t)L2 (IRN )m×m , Aj (·, t)H 1 (IRN )m×m ;

t ∈ [0, T ], j = 1, . . . , N }

(see Lemma 1.34, (iii)). Then by (2.2.59), (2.2.61) we have dEε (t) ≤ C0 v ε (t)2L2 (IRN )m + f (·, t)2L2 (IRN )m , dt

t ∈ [0, T ], ε ∈ (0, 1],

(2.2.62) where C0 is a constant independent of t and ε. By integration of (2.2.62) with the help of the positivity of A0 (x, t) we get

C1 v ε (t)2L2 (IRN )m ≤ 12 A0 (·, 0)v 0 , v 0

t +C0 0 v ε (s)2L2 (IRN )m + f (s)2L2 (IRN )m ds with a positive constant C1 which yields

t v ε (t)2L2 (IRN ) ≤ C2 1 + 0 v ε (s)2L2 (IRN ) ds . By the Gronwall inequality (see Lemma 1.2) we have v ε (t)2L2 (IRN )m ≤ C2 eC2 t ≤ C(T ),

ε ∈ (0, 1].

(2.2.63)

92


2.2.13

Convergence of approximations to a generalized solution

By (2.2.63) and Section 1.4.5.26 there is a sequence εk → 0 such that v k := v εk → v

weakly-∗ in L∞ ((0, T ), L2 (IRN ))m

as k → ∞.

(2.2.64)

Let ϕ ∈ C0∞ (IRN × [0, T )) be arbitrary. Multiply (2.2.53) with ε = εk by ϕ and integrate over IRN × (0, T ). Using again the licence k := εk , the symmetry of Jk and integrating by parts we obtain

T N ∂t (A0 ϕ) · v k + j=1 (Jk ∂j Jk Aj ϕ) − BT ϕ · v k + f · ϕ dx dt 0 IRN

+ IRN v 0 ϕ(x, 0) dx = 0. (2.2.65) By Lemma 1.34, (iii), the Lebesgue theorem and with regard to the smoothness of Aj we have ∂j Jk (Aj ϕ) = Jk ∂j (Aj ϕ) → ∂j Aj ϕ

strongly in L2 ((0, T ), L2 (IRN )).

Since {Jk } is bounded in L(L2 (IRN )), it follows that Jk ∂j (Jk Aj ϕ) → ∂j (Aj ϕ) strongly in L2 (IRN ) for all t ∈ (0, T ). Using (2.2.65), the Lebesgue theorem and Theorem 1.4.5.16, we can pass in (2.2.65) to the limit concluding with

T N ∂t A0 ϕ + A0 ∂t ϕ + j=1 ∂j (Aj ϕ) − BT ϕ · v + f · ϕ dx dt 0 IRN

+ IRN A0 v 0 (x) · ϕ(x, 0) dx = 0 (2.2.66) for all ϕ ∈ C0∞ (IRN × [0, T )).

Identity (2.2.66) can be considered as a generalized version of (2.2.47), (2.2.48). Indeed, if we know a priori that v is sufficiently smooth, then it is an easy excercise in integration by parts and use of a density argument (see Section 1.1.14) to show that v satisfies (2.2.47), (2.2.48) in the classical sense. So the objective of the next section will be to prove more regularity of v. 2.2.14

Regularity of the generalized solution

To prove that v is more regular, we proceed by induction. Let σ ∈ [1, s] be an integer and assume that we have already proved that {v ε } is bounded in L∞ ((0, T ), H σ−1 (IRN )). Let us show then that v ∈ L∞ ((0, T ), H σ (IRN )) ∩ W 1,∞ ((0, T ), H σ−1 (IRN )). To this purpose, let ℓ be a multiindex such that |ℓ| ≤ σ and assume that there is a constant C(σ) such that v ε (t)H σ−1 (IRN ) ≤ C(σ) for t ∈ [0, T ) and say ε ∈ (0, 1].

(2.2.67)


93

Denote wεℓ := Dℓ v ε . Then applying Dℓ to (2.2.53) we get A0 ∂t wεℓ +

N

j=1

Jε Aj ∂j Jε wεℓ + Bwεℓ = g εℓ ,

(2.2.68)

where it may be easily checked that

gℓε (t)L2 (IRN ) ≤ C 1 + v ε (t)H σ (IRN ) .

(2.2.69)

Notice that from (2.2.53) we get

N

f − Dα A0 Dβ ∂t v ε = Dα A0 Dβ A−1 Jε Aj ∂j Jε v ε − Bv ε for |α| + |β| ≤ |σ| − 1. 0 j=1

Define Eεℓ (t) := we can prove

IRN

dEεℓ dt

ε

A0 wℓ · wℓε dx. Then similarly as in Section 2.2.12 for v ε

≤ C0 wℓε (t)2L2 (IRN ) + gℓε (t)2L2 (IRN )

ε 2 ≤C |α|=σ wα (t)L2 (IRN ) + C(σ) ,

(2.2.70)

the left inequality being a consequence of (2.2.62) and (2.2.67). Summing up (2.2.70) over |ℓ| = σ, integrating over (0, t) and using the Gronwall inequality (see Lemma 1.1) we get wεℓ (t)2L2 (IRN ) ≤ const (1 + T ) ≤ const < ∞,

ε ∈ (0, 1], |ℓ| = σ.

(2.2.71)

According to Section 1.4.5.26 we can choose a subsequence {v k } so that v k → v and Dℓ v k → wℓ , |ℓ| ≤ σ weakly-∗ in L∞ ((0, T ), L2 (IRN )). Then wℓ = Dℓ v and v k → v weakly-∗ in L∞ ((0, T ), H σ (IRN )). So by induction we have v ∈ L∞ ((0, T ), H s (IRN ))m .

(2.2.72)

With this smoothness in hand we can carry out integration by parts with respect to x in (2.2.66) to obtain

T

0

IRN

∂t ϕ · A0 v dx dt

T N = 0 IRN −f + Bv + j=1 Aj ∂j v − ∂t A0 v · ϕ dx dt (2.2.73) for ϕ ∈ C0∞ (IRN × (0, T ))m .

So ∂t (A0 v)∈ L2 (IRN × (0, T ))m and N

Aj ∂j v + ∂t A0 v ∈ L∞ ((0, T ), H s−1 (IRN ))m . (2.2.74) As the sequences v k and ∂t v k are respectively bounded in L∞ ((0, T ), H s (IRN ))m and L∞ ((0, T ), H s−1 (IRN ))m , we can choose in Theorem 1.72, X = H s (IRN )m , B = Y = H s−1 (IRN )m and θ ∈ (0, 1) arbitrary to obtain that {v k } is compact ∂t (A0 v) = f − Bv −

j=1

94


in C((0, T ), H s−1 (IRN ))m . Let us note that X ֒→֒→ Y and the interpolation inequality (1.3.127) is here satisfied for any θ ∈ [0, 1], since H s (IRN )m ֒→֒→ H s−1 (IRN )m (cf. Section 1.3.5.8, b). Hence v k can be chosen so that v k → v in C([0, T ], H s−1 (IRN ))m . Further, by the Sobolev interpolation theorem (see Theorem 1.48), if 0 ≤ r′ ≤ r, then there is a constant Cr such that 1− r

′

r′

wH r′ (IRN )m ≤ Cr wL2 (IrRN )m wHr r (IRN )m

for any

w ∈ H r (IRN )m . (2.2.75)

Hence for s′ < s, 1− s

′

s′

v k − vH s′ (IRN )m ≤ Cs v k − vL2 (IsRN )m v k − vHs s (IRN )m 1− s

′

≤ C s v k − vL2 (IsRN )m → 0 in C([0, T ]). ′

So v ∈ C([0, T ], H s (IRN ))m and consequently, by (2.2.74), ′ ∂(A0 v) ∈ C([0, T ], H s −1 (IRN ))m . ∂t ′

An elementary argument using the imbedding H s −1 (IRN ) ֒→ C(IRN ) for s′ suitably close to s yields m v ∈ C((0, T ), C 1 (IRN )) ∩ C 1 ((0, T ), C(IRN )) . Theorem 2.16 is proved. 2.2.15

2

Quasilinear system

Let us now consider a more general system than (2.1.5), namely A0 (u)∂t u +

N

j=1

Aj (u)∂j u + Bu = f , u(x, 0) = u0 (x),

x ∈ IRN , t ∈ (0, T ) (T > 0), (2.2.76) x ∈ IRN . (2.2.77)

Here u = (u1 (x, t), . . . , um (x, t)), Aj (u), j = 1, . . . , N, B(x, t) are given m × m matrices, f = f (x, t), and u0 = u0 (x) are given m-vectors. Assume for simplicity that Aj (v), j = 0, 1, . . . , N are symmetric s times continuously differentiable for (2.2.78) v ∈ D, such that inf |det A0 (v)| > 0, where s ≥ 1 is an integer; v ∈D D is an open domain in IRN ;

{u0 (x); x ∈ IRN } ⊂ D;

B, f and u0 are s times continuously differentiable for x ∈ IRN , t ∈ (0, T0 ). (2.2.79)


2.2.16

95

Local existence for a quasilinear system

We are in a position to prove the following local existence theorem for problem (2.2.76), (2.2.77). Theorem 2.18 Let assumptions (2.2.78), (2.2.79) be satisfied with s > N2 + 1. Then there exists T ∈ (0, T0 ) such that problem (2.2.76), (2.2.77) has a unique solution m u ∈ C 1 (IRN × (0, T )) ∩ C 1 ((0, T ), H s−1 (IRN )) ∩ C((0, T ), H s (IRN )) with values in D.

Proof We use the standard approximation method. 2.2.17

Second grade approximations

Let u0 (x, t) := u0 (x) k+1

and u

(x, t) for k = 0, 1, . . . be defined as a solution of the system

A0 (uk )∂t uk+1 +

N

Aj (uk )∂j uk+1 + Buk+1 = f , x ∈ IRN , t ∈ (0, T0 ) (2.2.80) uk+1 (x, 0) = u0 (x), x ∈ IRN . (2.2.81)

j=1

By induction, Theorem 2.16 and assumptions (2.2.78), (2.2.79) for any k ≥ 0 there is a solution m uk+1 ∈ C 1 (IRN × (0, T0 )) ∩ C 1 ((0, T0 ), H s−1 (IRN )) ∩ C((0, T0 ), H s (IRN )) of (2.2.80), (2.2.81). Denote for simplicity

u := uk , v := uk+1 − u0 , v α := Dα v for |α| ≤ s.

(2.2.82)

Then we have A0 (u)∂t v +

2.2.18

N

j=1

Aj (u)∂j v + Bv = f −

N

j=1

Aj (u)∂j u0 − Bu0 ,

v(·, 0) = 0.

(2.2.83) (2.2.84)

Higher order energy estimates

If we apply to equation (2.2.83) the operator A0 (u)Dα A0 (u)−1 , then we get A0 (u)∂t v α +

N

−1

= A0 D α A0

where

Aj (u)∂j v α + Bv α

N f − j=1 Aj (u)∂j u0 − Bu0 + f α ,

j=1

v α (x, 0) = 0,

(2.2.85) (2.2.86)

96


fα =

N

A0 (u) A−1 0 Aj (u)∂j v α

−Dα A−1 + Bv α − A0 Dα A−1 (u)Bv, |α| ≤ s. 0 Aj (u)∂j v 0

j=1

Define analogously as in (2.2.58) E α (t) :=

1 2

IRN

A0 (u)v α · v α dx.

Then, following (2.2.59)–(2.2.62), we obtain

dE α 2 t ∈ [0, T ] (0 < T ≤ T0 ), dt (t) ≤ C0 1 + v α (t)L2 (IRN ) ,

(2.2.87)

(2.2.88)

where now

2 C0 := supt∈[0,T ] {4 + ∂t A0 (u)(·, t)C(IRN )m×m + A0 Dα A−1 0 f (·, t)L2 (IRN )m

N 2 0 (u) j=1 Aj (u) ∂u + A0 Dα A−1 0 ∂xj (·, t)L2 (IRN )m 2 2 +A0 Dα A−1 0 Bu0 L2 (IRN )m + f α L2 (IRN )m N ∂A + j=1 ∂xjj (u(·, t))C(IRN )m×m + B(·, t)C(IRN )m×m }.

(2.2.89)

For f α we find from (2.2.87) the estimate (we omit the arguments) f α L2 (IRN )m ≤ +

N j=1

β| 0. Then there is a T > 0 such that problem (2.2.108)–(2.2.111) has a unique 4 solution (p, v) ∈ C 1 (IR3 × (0, T )) ∩ C(IR3 × [0, T )) . Proof Define

D := {(p, v) ∈ IRN +1 ;

d0 < p < 2D0 }, 2

where D0 := supx∈IRN p0 (x). Then D is open in IRN +1 and in D, the coefficients of Aj (p, v), j = 0, 1, . . . , N are infinitely differentiable with respect to p, v. Moreover, for (p, v) ∈ D we have det A0 (p) = (γp)−1

p 3/γ a

≥ c0 > 0.

(2.2.116)

Hence all assumptions of Theorem 2.2.26 are satisfied and we conclude that there exists T > 0 such that the symmetrized problem has a unique solution u = (p, v) ∈ C 1 (IRN × (0, T )). By multiplying the symmetrized system by A0 (p)−1 given by (2.2.112) we obtain the original equations (2.2.108), (2.2.109) which completes the proof. 2 2.2.22

Existence of global smooth solutions for nonlinear hyperbolic systems

The natural question which arises after having proved the local existence of smooth solutions to the Euler equations is whether there is some hope for the existence of classical solutions in the large. In what follows we will show that generically the answer is no and only in particular cases (for example for special initial data) do smooth solutions exist without time restriction. First, let us investigate what happens when the initially smooth solution ceases to continue smoothly in time. 2.2.23

2 × 2 system of conservation laws, Riemann invariants

As an example we consider a pair of nonlinear conservation laws in one space dimension. Example 2.20 Let us consider the 2 × 2 system of conservation laws ∂t w + ∂x f (w) = 0, w(x, 0) = ϕ,

x ∈ IRN , t > 0,

(2.2.117)

x ∈ IRN .

(2.2.118)

Here w = w(x, t) = (w1 (x, t), w2 (x, t))T , f (w) = (f1 (w), f2 (w))T , ϕ = (ϕ1 (x), ϕ2 (x))T . Let A(w) be the Jacobian matrix of the mapping f : IR2 → IR2 . (Here Df (w), and (2.2.117) can be we assume f and ϕ smooth enough.) So A(w) = Dw written in the form ∂t w + A(w)∂x w = 0. (2.2.119) Assume that ∂w2 f1 = ∂w1 f2

in IR2 ,


101

and the eigenvalues λ1 (w) < λ2 (w) of A(w) correspond to left eigenvectors ℓ1 (w) and ℓ2 (w), respectively. If ψ = ψ(w) satisfies (2.2.26), then ∇w ψj (w) = µj ℓj (w),

j = 1, 2, w ∈ IR2

(2.2.120)

with suitable multiplicative factors µj = µj (w), j = 1, 2. For two conservation laws there is an effective procedure which allows us to construct so-called Riemann invariants (cf. Definition 2.32). Let us describe it briefly. Deleting the indices in (2.2.120), we have to satisfy ' ( ℓ ∇w ψ(w) = µℓ(w) = µ 1 (w). (2.2.121) ℓ2 Assuming the necessary smoothness, (2.2.121) can be satisfied only if the Cauchy–Riemann conditions are fulfilled, namely ∂w1 (µℓ2 ) = ∂w2 (µℓ1 ).

(2.2.122)

To resolve (2.2.122) with respect to µ we can use the method of characteristics. This yields a solution provided certain natural assumptions hold. The details of the construction of the solution to (2.2.122) can be found for example in (Bart´ ak et al., 1991), II.5.1. Having found µ = µ(w) we have from (2.2.121) ψj (w) = ψj (0, w2 ) + ∂ψj ∂w2 (0, w2 )

w1 0

(µℓj1 )(η, w2 ) dη,

= (µℓj2 )(0, w2 ),

j = 1, 2

which yields ψj (w) =

w2 0

(µℓj2 )(0, η) dη +

w1 0

(µℓj1 )(η, w2 ) dη,

j = 1, 2,

if we choose the reference line {w; w1 = 0} and ψj (0, 0) = 0. Define now v j (x, t) = ψj (w(x, t)).

(2.2.123)

(2.2.124)

Then by (2.2.117) and (2.2.120)

∂t v j + λj (w)∂x v j = ∇w ψj (w) · ∂t w + λj (w)∂x w = ∇w ψj (w) −∂x f (w) + λj (w)∂x w) (2.2.125)

j = µ(w) ℓ (w) · −A(w) + λj (w) · ∂x w = 0, j = 1, 2.

Let us note that if ξj = ξj (s) (s ∈ J ⊂ IR, J interval) is a classical solution of the problem

dξj (2.2.126) s ∈ J ⊂ IR ds (s) = λj w(ξj (s), s) ,

with w = w(x, t), a classical solution of (2.2.117), then

102


d j ds v (ξj (s), s)

= ∂t v j (ξj (s), s) + λj w(ξj (s), s) ∂x v j (ξj (s), s) = 0,

j = 1, 2 (2.2.127)

which means that

v j (ξj (s), s) = const,

s ∈ J.

(2.2.128) 1

2

By (2.2.120) the Jacobian of the transformation w → (v , v ) is regular and so v 1 , v 2 can be used as new dependent variables. Consider λj as functions of v 1 , v 2 . Assume now that ϕ ∈ C01 (IR2 )2 , ∂vk λj > 0 in IR2 for j = 1, 2, j = k

(2.2.129)

and that there exists an x0 ∈ IR2 such that either ∂x ψ1 (ϕ)(x0 ) < 0

or ∂x ψ2 (ϕ)(x0 ) < 0.

(2.2.130)

We will show that the classical solution cannot exist for all t > 0 in this case. To prove the above assertion we follow the argument of (Evans, 1998), p. 597. Let a := ∂x v 1 ,

b := ∂x v 2

(2.2.131)

with v 1 , v 2 given by (2.2.124). Differentiating the first equation (j = 1) in (2.2.125) with respect to x we obtain ∂t a + λ1 ∂x a + ∂v1 λ1 a2 + ∂v2 λ1 ab = 0.

(2.2.132)

The second equation in (2.2.125) yields ∂t v 2 + λ1 ∂x v 2 = (λ1 − λ2 )b. Eliminating b from (2.2.132) with the help of the last relation gives

1 ∂v2 λ1 ∂t v 2 + λ1 ∂x v 2 a = 0. ∂t a + λ1 ∂x a + ∂v1 λ1 a2 + λ1 −λ 2 To integrate (2.2.133), fix ξ0 ∈ IR and set t 1

η(t) := exp 0 λ1 −λ ∂v2 λ1 ∂t v 2 + λ1 ∂x v 2 (ξ(s), s) ds , 2

(2.2.133)

(2.2.134)

where (v := (v 1 , v 2 ))

dξ (s) = λ1 v(ξ(s), s) , ds ξ(0) = ξ0 := x0 .

s > 0,

(2.2.135)

Observe now that by (2.2.128) v 1 (ξ(s), s) = v 1 (ξ0 , 0) =: v0 .

1 ∂v2 λ1 depends only on v 2 if we We use the fact that the expression λ1 −λ 2 confine ourselves to the curve {(ξ(s), s)}.


Put γ(z) :=

z 0

1 2 λ1 −λ2 ∂v λ1

103

(v0 , η) dη.

Then (2.2.134), (2.2.135) imply

t d 2 γ v (ξ(s), s) ds = exp γ v 2 (ξ(t), t) − γ v 2 (ξ0 , 0) . η(t) = exp 0 ds (2.2.136) Put α(t) = a(ξ(t), t), assuming α = 0, insert it into (2.2.133) and multiply the resulting equation by α(t)−2 η(t)−1 . We obtain, with the help of (2.2.134),

∂λ1 −1 d −1 = ∂v1 η dt (αη) and by integration

−1

t = α(0)−1 + 0 α(t)η(t)

∂λ1 2 −1 ∂v 2 (v0 , v (ξ(s), s))η(s)

ds,

or

t α(t) = α(0)η(t)−1 1 + α(0) 0

∂λ1 2 −1 ∂v 1 (v0 , v (ξ(s), s))η(s)

ds

−1

.

(2.2.137)

According to (2.2.128), v is bounded in IR2 × (0, ∞). So by (2.2.136), 0 < η ≤ η(t) ≤ η for all t > 0 with appropriate constants η, η. If α(0) ≡ ∂x v 1 (x0 , 0) < 0 then by the assumption ∂v1 λ1 > 0 and (2.2.137) there exists t∗ > 0 such that the denominator of (2.2.137) equals zero. Taking the minimal such t∗ we have limtրt∗ ∂x v 1 (ξ(t), t) = ∞, which is a contradiction to the assumption of smoothness of the solution in the whole of IR2 × (0, ∞). A similar calculation holds with v 2 replacing v 1 . The details are left to the reader. 2.2.24 Plane wave solutions Consider the system of conservation laws (2.1.4) in Ω = IRN and let us seek its solution in the form

w(x, t) = z σ(x · ω, t) , (2.2.138)

where σ = σ(y, t) is a scalar function and ω ∈ IRN a constant vector. The functions z, σ and the vector ω are to be found so that (2.1.4) holds true. Denote by Aj (w) the Jacobian matrix of f j , i.e. Aj (w) :=

Df j Dw (w),

(2.2.139)

and define also A(w, ω) :=

N

j=1

ωj Aj (w) for ω ∈ IRN .

Then system (2.1.4) takes the form N ∂t w + j=1 Aj (w)∂j w = 0.

(2.2.140)

(2.2.141)

Our next considerations will be formal (i.e. valid at least for C 1 -solutions) since they serve to derive heuristically the solution procedure as a basis for further rigorous treatment.

104


From (2.2.138) we clearly get ∂t w = z ′ · ∂t σ, ∂j w = z ′ · ∂y σ · ωj ,

j = 1, . . . , N,

which leads us to the system ∂t σz ′ + ∂y σA(z(σ), ω)z ′ = 0.

(2.2.142)

Let r k = r k (z, ω)(k ∈ {1, . . . , m}) be an eigenvector of the matrix A(z, ω) defined for z ∈ D (a definition region of f j , j = 1, . . . , N ), and ω ∈ IRN , the corresponding eigenvalue being denoted by λk (z, ω). Given a z 0 ∈ D let us solve the problem (2.2.143) z ′ (σ) = r k (z(σ), ω), z(0) = z 0 , in the neighborhood of σ = 0. It follows from classical existence theorems for ordinary differential systems (see e.g. (Kurzweil, 1986), Theorem 10.1.1), that such a solution exists on some maximal interval σ− < σ < σ+ , where σ− < 0, σ+ > 0 and gives us the curve {z(σ) ∈ D; σ− < σ < σ+ }. Now, having the solution z = z(σ, ω) of (2.2.143) let σ = σ(y, t) be a solution of the problem ∂t σ + λk (z(σ, ω))∂y σ = 0 (σ− < σ(y, t) < σ+ ), σ(y, 0) = σ0 (y), σ− < σ0 (y) < σ+ , y ∈ (−∞, ∞)

(2.2.144) (2.2.145)

with a given function σ0 = σ0 (y). Integrating (2.2.144) along the characteristic we obtain (see Section 2.2.5)

σ(y, t) = σ0 y − λk (z(σ(y, t), ω)) · t (2.2.146)

which represents a functional equation for σ = σ(y, t). After resolving (2.2.146) with respect to σ we find that w given by (2.2.138) is a solution of (2.1.4). Of course, as we have already seen, in general we cannot expect the global existence of σ satisfying (2.2.144), (2.2.145) in the classical sense. On the other hand as long as the inequality

k (2.2.147) σ0′ y − λk (w(σ, ω))t dλ dσ (w(σ, ω)) < 0 holds, σ can be continued smoothly in time (cf. Section 2.2.5). If (2.2.147) does not hold for all t > 0 then the classical solution σ(y, t) may break down in a (arbitrarily small) finite time. In Section 2.3.13 we will see how even in this case the solution can be constructed globally in time; of course it can no longer be a classical solution. 2.2.25

Plane waves for the Euler system in 2D

Let us demonstrate the above procedure on the Euler system in two space dimensions. Setting qj := ρvj , j = 1, 2, two-dimensional Euler system can be written in the form


q12

+ ∂x2 ( q1ρq2 ) + ∂x1 p(ρ) = 0,

q2 ∂t q2 + ∂x1 q1ρq2 + ∂x2 ρ2 + ∂x2 p(ρ) = 0, ∂t ρ + ∂x1 q1 + ∂x2 q2 = 0. ∂t q1 + ∂x1

105

ρ

(2.2.148)

Putting w := (q1 , q2 , ρ), after carrying out the space differentiations, we get from (2.2.148) the system 2 ∂t w + j=1 Aj (w)∂j w = 0, with

 q2 p′ (ρ) − ρ12   A1 (w) =  − qρ1 q22  1 0 0   q2 q1 q1 q2 − ρ2 ρ ρ 2   A2 (w) =  0 2qρ2 p′ (ρ) − ρq22  . 0 1 0 

2q1 ρ 0 q2 q1 ρ ρ

(2.2.149)

Consequently, for ω = (ω1 , ω2 ) ∈ IR2 we have 

 q2 p′ (ρ) − ρ12 ω1 − qρ1 q22 ω2    q22 A(w, ω) =  q2 q1 q2 2q2 q1 ′  ω1 + p (ρ) − ρ2 ω2  . ρ ω1 ρ ω1 + ρ ω2 − ρ2 0 ω1 ω2 (2.2.150) For the sake of simplicity, assume ω1 = 1, ω2 = 0. The general case can be transformed to this one by the transformation x → x′ satisfying the condition x · ω = x′ · (1, 0) with a slight change in the matrices A1 , A2 . For this choice and with the convention a = a(ρ) = p′ (ρ) we have (w := z)   z12 2z1 2 z3 0 a − z32   A(z, (1, 0)) =  zz2 zz1 − z1 z2 2  . (2.2.151) 2q1 ω1 +q2 ω2 ρ

ω2 q1 ρ

3

3

1

0

z3

0

The eigenvalues for matrix (2.2.151) are λ1,2 =

z1 z3

± a(z3 ),

with the corresponding eigenvectors   z1 z3 ± a(z3 ) z2 , r 1,2 (z) =  z3 1

λ3 =

z1 z3

  0 r3 =  1  . 0

According to our procedure we should solve the systems

(2.2.152)

(2.2.153)

106


z(0) = z 0 ,

z σ (σ) = r k (z),

k = 1, 2, 3.

(2.2.154)

Starting with k = 1, 2 we find the systems z1′ =

z1 ± a(z3 ), z3

z2′ =

z2 , z3

z(0) = z 0 .

z3′ = 1,

Successive integration starting from z3 over z2 to z1 , respectively, yields z3 (σ) = σ + z30 ,

z2 (σ) =

z0 σ z1 (σ) = (σ + z30 ) z10 ± 0 3

z20 (σ z30

+ z30 ),

a(τ +z30 ) τ +z30

For k = 3 system (2.2.154) is trivially solved by   z10 z(σ) =  z20 + σ  . z30

dτ .

(2.2.155)

Next step is to solve equations (2.2.144), (2.2.145). For k = 1, 2, 3 we get respectively the problems ∂t σ +

z0 1

z30

±

σ 0

a(τ +z30 ) τ +z30

dτ ± a(σ + z30 ) ∂y σ = 0, σ(y, 0) = σ0 (y),

∂t σ +

z10 ∂ σ z30 y

= 0,

σ(y, 0) = σ0 (y).

(2.2.156) (2.2.157)

For the solution of (2.2.156) the general theory of single conservation laws explained in Section 2.3.13 should be applied. Having resolved respectively (2.2.156) and (2.2.157) we obtain the corresponding special solution in the form (2.2.138) which means in this case    0 σ(x1 ,t) a(τ +ρ0 )  (ρv1 )(x, t)

v1 ± 0 τ +ρ0 dτ   (ρv2 )(x, t)  = σ(x1 , t) + ρ0  v20 ρ(x, t) 1     0 (ρv1 )(x, t) ρ0 v1  (ρv2 )(x, t)  =  ρ0 v20 + σ(x1 − v10 t)  . (2.2.158) ρ0 ρ(x, t) 2.3

Weak solutions

In this section we are concerned with the situation when the system of the type (2.1.4) does not possess a globally defined smooth solution despite the fact that the data may be smooth enough, for example so smooth that Theorem 2.18 holds true. Such a situation is typical for even very simple quasilinear hyperbolic equations as, for example, (2.1.6). This fact brings our attention to a weaker

WEAK SOLUTIONS

107

formulation of systems like (2.1.4) than the classical one. We have already illustrated the nonexistence of global classical solutions to the systems of conservation laws in Section 2.2.22. First we summarize our knowledge about the blow up of solutions. Then we introduce weak solutions and comment on their typical structure composed of shock waves and rarefaction waves. The intermediate boundary constraints known as Rankine–Hugoniot conditions will be derived and the role of the so-called entropy pairs will be exhibited on the Riemann problem for a system of conservation laws which play an important role in their theoretical treatment. Fundamental existence theorems for a single conservation law, two conservation laws and several conservation laws in one dimension will be given. 2.3.1

Blow up of classical solutions

As we already mentioned nonlinear hyperbolic systems exhibit a typical nonlinear phenomenon: break down of the classical solutions. Let us consider the situation as in Theorem 2.18. We have data smooth enough and existence of a classical solution on some interval (0, T ) (T > 0). We do not know how large (or small) T is but due to uniqueness we can consider a unique maximal classical solution on the interval (0, Tmax ) with Tmax = sup{T > 0; the C 1 solution from Theorem 2.18 exists on (0, T )}. (2.3.1) Without additional information we do not know whether Tmax = ∞

(2.3.2)

Tmax < ∞.

(2.3.3)

or If (2.3.2) holds, then no problem arises in this respect. This is the case of Example 2.1 with s = 1. If we have (2.3.3), then there are several possibilities. It is well known from the theory of nonlinear ordinary differential equations that the solution can tend to infinity in a finite time. A similar phenomenon is possible here, i.e. for example for w a maximal solution of (2.1.4) on (0, Tmax ) there may exist x0 ∈ IRN such that lim inf |w(x0 , t)| = ∞. t↑Tmax

(2.3.4)

This is the case of Example 2.1. Note that in that case we also have lim |∂x w(x0 , t)| = ∞ (Tmax = 1)

t↑Tmax

and the same for ∂t w(x0 , t). On the other hand, it is also possible that |w(x, t)| ≤ C < ∞,

for all x and t ∈ [0, Tmax )

with a constant C independent of x and t, but

(2.3.5)

108


lim inf t↑Tmax

m N ∂k wj (x0 , t) + ∂t wj (x0 , t) = ∞ j=1

k=1

(2.3.6)

for some x0 . This situation occurs generically. In fact, it can be shown that for any nontrivial initial data with compact support in IRN there exists a solution to (2.1.4) with the initial data which develops a singularity of the type (2.3.6). The detailed discussion of this issue is given in (Majda, 1984), Chapter 3. The question arises whether relaxing the notion of a classical solution to a generalized one would allow to continue the solution in time in some reasonable sense. This is our next subject. 2.3.2

Generalized formulation for systems of conservation laws

N m Definition 2.21 Let w0 ∈ [L∞ loc (IR )] . A vector function w is called a weak N solution of problem (2.2.10) if w ∈ [L∞ × [0, ∞))]m and the following loc (IR integral identity holds:

∞

0

IRN

w · ∂t ϕ +

N

j=1

f j (w) · ∂j ϕ dx dt + IRN w0 (x) · ϕ(x, 0) dx = 0

for all ϕ ∈ [C0∞ (IRN × [0, ∞))]m .

c

(2.3.7)

Theorem 2.22 Any classical solution of problem (2.2.10) is a weak one. Conversely, if a weak solution w of (2.2.10) belongs to the class C 1 (IRN ×(0, ∞))m ∩ C(IRN × [0, ∞))m , then it is a classical solution. Proof Let w be a classical solution of (2.2.10). Multiplying (2.2.10)1 by ϕ ∈ [C0∞ (IRN × [0, ∞))]m , and using Green’s theorem (see Section 1.1.13) we obtain (2.3.7). Conversely, if (2.3.7) holds and w ∈ C 1 (IRN × (0, ∞))m ∩ C(IRN × [0, ∞))m , then again by Green’s theorem we get

∞

0

IRN

∂t w +

N

j=1

∂j f j (w) · ϕ dx dt = 0 ∀ϕ ∈ [C0∞ (IRN × [0, ∞))]m .

The last relation yields that the first equation in (2.2.10) is satisfied pointwise. Using this fact and Green’s theorem in (2.3.7), we obtain

IRN

w(x, 0) − w0 · ϕ(x, 0) dx = 0 ∀ϕ ∈ [C0∞ (IRN × [0, ∞))]m .

Then, with the help of Lemma 1.47, the second relation in (2.2.10) easily follows. 2 Notice that the weak solution satisfies the first equation in (2.2.10) in the sense of distributions. 2.3.3

Piecewise smooth solutions

An important class of special solutions is the class of piecewise smooth weak solutions which admit discontinuities.

WEAK SOLUTIONS

109

Definition 2.23 We say that a function w is piecewise smooth if there exists a finite number of smooth hypersurfaces Γ in IRN × [0, ∞) outside of which the function w is of class C 1 and on which w and its first derivatives have one-sided limits w± (x, t) ((x, t) ∈ Γ), defined in the following way. Let (x, t) ∈ Γ and let U = Bε (x, t) be a ball with center at (x, t) and so small radius ε > 0 that U \ Γ has exactly two components U ± . Then we set w± (x, t) =

lim

(y,ϑ)→(x,t) (y,ϑ)∈U ±

w(y, ϑ),

(2.3.8)

provided this limit exists. (Similarly for derivatives of w.) (If N = 1, Γ is a curve in IR × [0, ∞).) A weak solution w of (2.2.10) is called a piecewise smooth weak solution, if the function w is piecewise smooth. The following theorem is a well known characterization of piecewise smooth solutions with the help of the so-called Rankine–Hugoniot conditions. Theorem 2.24 A function w : IRN × [0, ∞) → IRm is a piecewise smooth weak solution of system (2.2.10)1 if and only if the following conditions are satisfied: a) w is a classical solution in any Lipschitz domain where w is of class C 1 , b) w satisfies the condition N (2.3.9) (w+ − w− )nt + j=1 (f j (w+ ) − f j (w− ))nj = 0

on smooth hypersurfaces Γ of discontinuity. Here n = (n1 , . . . , nN , nt ) denotes the normal to Γ.

Proof The proof is a classical exercise in the use of Green’s theorem. Details can be found for example in (Feistauer et al., 2003) or (Smoller, 1983). 2 Definition 2.25 Relation (2.3.9) is called the Rankine–Hugoniot condition. or jump condition. In the case (n1 , . . . , nN ) = 0, we can choose a normal n = (ν, −s) to Γ, where s ∈ IR and ν ∈ IRN is a unit vector. Then (2.3.9) can be written in the form N (2.3.10) s[w] = j=1 νj [f j (w)], where

[w] = w+ − w− ,

[f j (w)] = f j (w+ ) − f j (w− ).

(2.3.11)

The vector ν and the number s can be interpreted as the direction and the speed of propagation of the discontinuity Γ, respectively. This is quite clear in the case N = 1. If we express the discontinuity Γ as a curve x = ξ(t), we have s = dξ/dt and n = (1, −s). It is clear that s represents the speed of propagation of the discontinuity as a function of time. Then (under the notation f = f 1 ) the Rankine–Hugoniot condition becomes s[w] = [f (w)].

(2.3.12)

Theorem 2.24 can be used for the construction of weak solutions to some simple hyperbolic problems. Examples of piecewise smooth weak solutions can be

110


found, e.g., in (Feistauer et al., 2003), Chapter 2 or (Chorin and Marsden, 1979), Section 3.2. 2.3.4

Entropy condition

Since weak solutions might not be unique, some additional conditions are sought to select an appropriate weak solution. Motivation is found in the physics of fluids, where the second law of thermodynamics (see Section 1.2.8) imposes further restriction to the sought solution. Roughly speaking, this law implies that along trajectories of individual particles of the fluid the entropy cannot decrease in the course of time. That is why the condition we are going to introduce is called the entropy condition. Let η, Gs : D → IR (s = 1, . . . , N ) be sufficiently smooth functions satisfying the conditions ∇w η(w)T As (w) = ∇w Gs (w),

w ∈ D, s = 1, . . . , N,

(2.3.13)

where ∇w η = (∂η/∂w1 , . . . , ∂η/∂wm )T and As are the matrices defined by As (w) =

Dfs (w) Dw

m = ∂wj fsi i,j=1 .

(2.3.14)

If w is a classical solution of system (2.2.10)1 , then the function η(w) = η ◦w satisfies the conservation law equation ∂t η(w) +

N

s=1

∂xs Gs (w) = 0,

(2.3.15)

as easily follows from (2.2.10)1 and (2.3.13). On the other hand, a weak solution of (2.2.10)1 may not satisfy (2.3.15). If the weak solution is piecewise smooth, then it satisfies the Rankine–Hugoniot condition (2.3.9). It is again an easy exercise in the use of Green’s theorem to prove that if this w satisfies (2.3.15) in the sense of distributions, i.e.

∞ N η(w)∂t ϕ + s=1 Gs (w)∂xs ϕ dx dt = 0, ∀ ϕ ∈ C0∞ (IRN × (0, ∞)), 0 IRN

then the jump condition

nt [η(w)] +

N

s=1

ns [Gs (w)] = 0.

(2.3.16)

holds. Conditions (2.3.9) and (2.3.16) are not, in general, compatible, as can be shown on the example of the Burgers equation (2.1.6) whose classical solution also satisfies the equation p+2 p+1 (2.3.17) p = 1, 2, . . . . ∂t up+1 + ∂x up+2 = 0, However, it is possible to show that a discontinuous piecewise smooth weak solution of (2.1.6) may not be a weak solution of the above equation (2.3.17) because it might not satisfy the corresponding Rankine–Hugoniot condition.

WEAK SOLUTIONS

111

Definition 2.26 Let D be a domain in IRN . A function η : D → IR (η ∈ C 1 (D)) is called the entropy of the system (2.2.10)1 if there exist functions G1 , . . . , GN : D → IR, called entropy fluxes, such that relations (2.3.13) hold. If the domain D is convex and the function η convex (see Definition 1.1.2), then we call η convex entropy. The pair (η, G) is called the (convex) entropy–entropy flux pair. The concept of entropy allows us to specify physically admissible solutions of the hyperbolic system (2.2.10)1 : Definition 2.27 We say that a weak solution w of (2.2.10) is a (convex) entropy solution if for every entropy η of system (2.2.10)1 the condition ∂t η(w) +

N

j=1

∂j Gj (w) ≤ 0

(2.3.18)

is satisfied in the sense of distributions on IRN × (0, ∞). This means that

∞

0

IRN

η(w)∂t ϕ +

N

j=1

Gj (w)∂j ϕ dx dt ≥ 0

∀ ϕ ∈ C0∞ (IRN × (0, ∞)), ϕ ≥ 0.

(2.3.19)

Inequality (2.3.19) is called the (convex) entropy condition. Theorem 2.28 1. Every classical solution of (2.2.10)1 is a weak (convex) entropy solution and satisfies (2.3.15). 2. A piecewise smooth solution is a weak (convex) entropy solution if and only if the condition N (2.3.20) nt [η(w)] + j=1 nj [Gs (w)] ≤ 0

is satisfied on every Lipschitz-continuous hypersurface Γ of discontinuity. The normal n = (n1 , . . . , nN , nt ) to Γ is oriented in the direction from η − to η + .

Proof 1. Obviously, any classical solution satisfies (2.3.15) and consequently (2.3.19). 2. The proof of the second assertion is quite analogous to the proof of Theorem 2.24. It is an elementary exercise in the application of Green’s theorem. 2 In the one-dimensional case (N = 1) condition (2.3.20) takes the form s[η(w)] ≥ [G(w)]

on Γ,

(2.3.21)

where s is the speed of propagation of the discontinuity Γ. Inequality (2.3.21) is called the Lax shock entropy condition.

112


In the case of one scalar equation N ∂t w + j=1 ∂j fj (w) = 0,

(2.3.22)

every (convex) function η ∈ C 1 (D) is a (convex) entropy of this equation. Actually, the entropy fluxes Gj should satisfy G′j = η ′ fj′ ,

j = 1, . . . , N,

(2.3.23)

and thus can be easily obtained by integration. 2.3.5 Physical entropy Let us consider system (2.1.1). If the entropy condition is not to be empty, then the existence of at least one entropy has to be guaranteed. In (1.2.61) we introduced the physical entropy p/p0 S = cv ln (ρ/ρ κ + const 0)

(where p0 , ρ0 are fixed reference values of the pressure and density, respectively) which can be rewritten in the form 2 N S = cv ln e10 e − i=1 (ρv2ρi ) + const. (2.3.24) − κ ln ρρ0

We can set const = 0. Putting ui = ρvi , we can prove that the function N u2 (ρ, u1 , . . . , uN , e) → −ρS = −cv ρ ln e10 e − i=1 2ρi − κ ln ρρ0 (2.3.25) is strictly convex. After a lengthy calculation we find that the functions Gs = −ρvs S, s = 1, . . . , N , are entropy fluxes of system (2.1.1). This means that the physical entropy defines a mathematical entropy for the system of conservation laws of an inviscid gas. Using equation (1.2.64) and the continuity equation, we obtain the identity N (2.3.26) ∂t (ρS) + j=1 ∂j (ρvj S) = 0.

for adiabatic flow. This means that a smooth solution of system (2.1.1) satisfies the entropy equation (2.3.26). For a weak solution (2.3.26) becomes the inequality N (2.3.27) ∂t (ρS) + s=1 ∂s (ρvs S) ≥ 0

(in the sense of distributions) which corresponds to the entropy condition implied by the second law of thermodynamics (1.2.40) (where q = 0 and q = 0 in the case of adiabatic flow). It is now seen that the mathematical concept of entropy is in agreement with the physical one. As follows from the above considerations, our goal is to find physically admissible weak entropy solutions of problem (2.2.10). To this end, we can use the method of artificial viscosity proposed in (Lax, 1954). This method is based on introducing an additional “viscous” (dissipative) term, multiplied by a small parameter, to the right-hand side of (2.2.10)1 . We will work here with a more general approximation than that of P.D. Lax.

WEAK SOLUTIONS

2.3.6

113

General parabolic approximation and the entropy condition

To begin with, let us return once more to the system of conservation laws N (2.3.28) ∂t w + j=1 ∂j f j (w) = 0, x ∈ Ω ⊂ IRN , t > 0 (w(x, t) ∈ IRm ).

We already know that (2.3.28) typically arises in fluid mechanics from more complex dissipative equations containing viscous and heat conductive effects (see the complete Newtonian system given by (1.2.77)–(1.2.80)). In the Euler equations these effects are neglected and the mechanism of the change of mechanical energy into heat and diffusion of heat is substituted by imposing entropy conditions. These conditions in the analogy are motivated by the second law of thermodynamics. In what follows we show a heuristic argument to obtain the entropy condition by a limit process from a parabolic problem to the hyperbolic one. We will imitate the vanishing of dissipative mechanisms by the vanishing of artificial, more virtual than correctly physical, viscosity term input on the right-hand side of (2.3.28). So, let w = (w1 , . . . , wm ), f j = (fj1 (w), . . . , fjm (w)) and η = η(w) be, provisionally arbitrary, twice continuously differentiable function of w ∈ IRm . Assuming obvious regularity of fjk we can write (2.1.4) in the form ∂t wk +

N

j=1

∂j fjk (w) = 0,

k = 1, . . . , m.

(2.3.29)

Assume now (cf. (2.3.13)) that there are Gj (w), j = 1, . . . , m such that m ∂wℓ Gj = k=1 ∂wℓ fjk ∂wk η, j = 1, . . . , N ; ℓ = 1, · · · , m. (2.3.30)

Multiplying the k-th equation in (2.3.29) by ∂wk η, summing up over k and using the chain rule we obtain N (2.3.31) ∂t η(w) + j=1 ∂j Gj (w) = 0.

Thus, under assumption (2.3.30), classical solutions of (2.3.29) satisfy (2.3.31). Since for weak solutions we cannot in general use the chain rule the above assertion might no longer be true. Now consider (2.3.28) with an artificial viscosity term. Let Bjk (w), j, k = 1, . . . , N be m × m matrices with continuously differentiable elements, ε > 0 and consider a smooth solution w of the system

N N (2.3.32) ∂t w + j=1 ∂j f j (w) − ε j,k=1 ∂j Bjk (w)∂k w = 0.

If we make the same manipulation with system (2.3.31) as we did above with (2.3.29), after obvious arrangements we obtain

N N m ∂t η + j=1 ∂j Gj (w) − ε j,k=1 ℓ=1 ∂j ∂wℓ ηBjk (w)∂k w ℓ (2.3.33)

N m = −ε j,k=1 ℓ,n=1 ∂wℓ ∂wn η∂j wn Bjk (w)∂k w ℓ .

114


Assuming that N m

∂wl ∂wn η(z)ξnj Bjk (z)ξ k ℓ ≥ 0, z, ξ r ∈ IRm , r = 1, . . . , N, (2.3.34) we get from (2.3.33)

N m N ∂t η + j=1 ∂j Gj (w) − ε j,k=1 ℓ=1 ∂j ∂wℓ η Bjk (w)∂k w ℓ ≤ 0. (2.3.35) j,k=1

ℓ,n=1

The weak form of (2.3.35) is

T

N η(w)∂t ϕ + j=1 Gj (w)∂j ϕ

N m −ε j,k=1 ℓ=1 ∂wℓ η Bjk (w)∂k w ℓ ∂j ϕ dxdt ≥ 0 0

Ω

(2.3.36)

for any ϕ ∈ C0∞ (Ω × (0, T )), ϕ ≥ 0. If we knew the solution w = wε (x, t) of (2.3.32) for ε > 0 small and were able to show that, e.g., wε is bounded in L∞ (Ω × (0, T )), compact in L1 (Ω × (0, T )) and satisfied εn

T N 0

Ω

j,k=1

m

ℓ=1

∂wℓ η(wεn ) Bjk (wεn )∂k wεn ℓ ∂j ϕ dx dt → 0

for some εn → 0 + for all ϕ ∈ C0∞ (Ω × (0, T )),

then we would have

T 0

Ω

η(w∗ )∂t ϕ +

N

j=1

Gj (w∗ )∂j ϕ dx dt ≥ 0

for ϕ ∈ C0∞ (Ω × (0, T )), ϕ ≥ 0

(2.3.37)

(2.3.38)

and some w∗ = L1 − limn→∞ wεn with some εn → 0+ . If we were analogously able to pass to the limit in the weak form of (2.3.32), then w = w∗ would be a weak solution of (2.3.28) satisfying the weak entropy inequality (2.3.38). So the entropy condition (2.3.38) is necessary for the weak solution of (2.3.28) to arise from the solutions of the viscous laws (2.3.39) by the above limit procedure. Of course, to establish the above limit procedure is a difficult task to discover a sufficiently strong compactness argument. But this requires sufficiently strong a priori estimates which we are not able to derive for more general equations than a scalar law in IRN or 2 × 2 system in IR, unless assuming special structures like space symmetry, etc. Notice that if Bjk (w) = I, where I is the m × m identity matrix, then (2.3.37) assumes the form

T N εn 0 Ω j.k=1 η(wεn )∂j ∂k ϕ dx dt → 0 (εn → 0+ ) for all ϕ ∈ C0∞ (Ω × (0, T ))

and (2.3.34) becomes

N

j,k=1

m

ℓ,n=1

∂wℓ ∂wn η(z)ξnj ξℓk ≥ 0

for all ξ r ∈ IRm , r = 1, . . . , N and z ∈ IRm .

(2.3.39)

WEAK SOLUTIONS

115

Finally, the one-dimensional form of (2.3.39), i.e. for N = 1, reads m m k,ℓ=1 ∂wk ∂wℓ η(z)ξk ξℓ ≥ 0 for all ξr ∈ IR, r = 1, . . . , m, z ∈ IR . (2.3.40)

Condition (2.3.40) expresses the convexity of the function η with respect to w ∈ IRm . 2.3.7

Entropy for a general scalar conservation law

The simplest illustration of the entropy condition and viscosity method is the case of the scalar law N (2.3.41) ∂t w + j=1 ∂j fj (w) = 0.

The viscous form of (2.3.41) is ∂t w +

N

j=1

∂j fj (w) = ε

N

j,k=1

∂j bk (w)∂k w

(2.3.42)

with continuous functions bk (w), k = 1, . . . , N. Given a function η = η(w), assumption (2.3.30) assumes in this case the existence of functions Gj , j = 1, . . . , N such that dGj dfj dη (2.3.43) j = 1, . . . , N. dw = dw dw , All such functions are given by Gj (z) = gj0 +

z

dfj dη (ζ) dw (ζ) dζ, z0j dw

z ∈ IR, j = 1, . . . , N

(2.3.44)

with arbitrarily chosen gj0 , z0j ∈ IR. Condition (2.3.34) now reads d2 η(z) dw2 bjk (z)ξj ξk

≥ 0 for all ξ ∈ IRN , z ∈ IR.

(2.3.45)

Anticipating the resolvability of (2.3.42) we would like to have equation (2.3.42) parabolic. This leads us to the natural ellipticity condition N

j,k=1 bjk (z)ξj ξk

≥ 0,

ξ ∈ IRN , z ∈ IR.

(2.3.46)

But then (2.3.45) implies d2 η dw2 (z)

≥ 0 for z ∈ IR.

(2.3.47)

Any convex function η ∈ C 2 (IR) may serve for the entropy function for the scalar conservation law (2.3.41). Having chosen η, the corresponding fluxes are 2,1 (IR), by integration by given by (2.3.44). Let us note that assuming fj ∈ Wloc parts in (2.3.44) we can remove the differentiation from η and assume just η convex instead of (2.3.47), and η ∈ L∞ loc (R) which slightly broadens the class of entropies taking into account. This is in coincidence with the following definition of admissible weak solutions.

116


Definition 2.29 Let w0 ∈ L∞ (IRN ), T > 0. A function w ∈ L∞ (IRN × (0, T )) is called an admissible weak solution of (2.3.41) with the initial condition w(x, 0) = w0 (x),

x ∈ IRN

(2.3.48)

if

T

0

IRN

η(w)∂t ϕ +

N

j=1

Gj (w)∂j ϕ dx dt + IRN w0 (x)ϕ(x, 0) dx ≥ 0

for all ϕ ∈ C0∞ (IRN × [0, T )), ϕ ≥ 0

0,1 and every convex function η ∈ Cloc (IRN ) with the fluxes

Gj (z) = cj + fj′ (z)η(z) −

z

zj

fj′′ (ζ)η(ζ) dζ,

z ∈ IR

(2.3.49)

(2.3.50)

and some constants cj ∈ IR, zj ∈ [−∞, ∞). If in addition w(·, t) ∈ BVloc (IRN ) for a.e. t > 0, then w is called an admissible BV-solution. (For the definition of the spaces BV see Definition 1.58.) Remark 2.30 If we choose in (2.3.49) η(w) := ±w, Gj (w) := ±fj (w), j = 1, . . . , N, then we get

T

0

IRN

w∂t ϕ +

N

j=1

fj (w)∂j ϕ dx dt + IRN w0 (x)ϕ(x, 0) dx = 0

for all ϕ ∈ C0∞ (IRN × [0, T )).

(2.3.51)

Further, any Lipschitz-continuous function h(z) is the locally uniform limit of piecewise linear functions hj (z) = aj0 +

ℓj

i=1

aji (z − ki )+ ,

j = 1, 2, . . .

with some ℓj ∈ IN and aj0 , aji , ki ∈ IR, aji > 0, i = 1, . . . , ℓj , j = 1, 2, . . . , where for, say r ∈ IR, we denote r+ := max{r, 0}. These facts suggest we restrict ourselves to entropies of the form η(z) := (z − k)+

for z ∈ IR with arbitrary k ∈ IR but fixed.

(2.3.52)

The corresponding fluxes determined by (2.3.50) are then given by Gj (z) =

fj (z) − fj (k), z ≥ k, j = 1, . . . , N. 0, z < k,

(2.3.53)

The fact that in the investigation of scalar conservation laws it suffices to use the entropies given by (2.3.52) only was first observed in (Volpert, 1967) (see also (Bressan, 2000), Chapter 6) and used in the pioneering paper (Kruzhkov, 1970) on the existence of admissible weak solutions obtained by the viscosity method.

WEAK SOLUTIONS

2.3.8

117

Entropy for a 2 × 2 system of conservation laws in 1D

Consider now in (2.3.28) m = 2, N = 1. Then we have the system (f 1 := f := (f1 , f2 )T ) ∂t w1 + ∂x f1 (w1 , w2 ) = 0, (2.3.54) ∂t w2 + ∂x f2 (w1 , w2 ) = 0. According to Section 2.3.6, if a given function η = η(w1 , w2 ) is to be an entropy function, then there has to exist a corresponding flux function G(w1 , w2 ), j = 1, 2 such that (cf. (2.3.30)) ∂wℓ G = ∂w1 η∂wℓ f1 + ∂w2 η∂wℓ f2 =: hℓ ,

ℓ = 1, 2.

(2.3.55)

Assume that ∂w2 h1 = ∂w1 h2

for w ∈ D,

(2.3.56)

where D ⊂ IR2 is an open set with a point z 0 such that any z ∈ D can be connected to z 0 by a piecewise linear curve Γ ⊂ D. Notice that for example convex sets fall into this class. Let z ∈ D be arbitrary and assume that Γ := {ξ(s); s ∈ (0, 1), ξ(0) = z 0 , ξ(1) = z, ξ piecewise linear} ⊂ D. Put h = (h1 , h2 ) and

1 (2.3.57) G(z) = 0 h(ξ(s)) dξ(s),

and prove that under condition (2.3.56) G satisfies (2.3.55). Indeed, assuming for simplicity Γ = {ξ(s) = (1 − s)z 0 + sz; s ∈ (0, 1)} (the generalization to arbitrary number of sections is easy), (2.3.57) becomes

1

h1 ((1 − s)z 0 + sz)(z1 − z10 ) + h2 ((1 − s)z 0 + sz)(z2 − z20 ) ds (2.3.58) and taking into account (2.3.56) and ∂zk ξj = δjk , after an easy calculation we obtain

1

1 ∂z1 G = 0 s ∂ξ1 h1 (z1 − z10 ) + ∂ξ2 h1 (z2 − z20 ) ds + 0 h1 (ξ) ds (2.3.59)

1 d = 0 s ds h1 (ξ) + h1 (ξ) ds = h1 (ξ(1)) = h1 (z). G(z) =

0

Similarly we proceed with ∂z2 G. Condition (2.3.56) gives one linear partial differential equation with variable coefficients

∂w1 ∂w1 η∂w2 f1 + ∂w2 η∂w2 f2 = ∂w2 ∂w1 η∂w1 f1 + ∂w2 η∂w1 f2

(2.3.60)

for one unknown function η = η(w1 , w2 ). To solve (2.3.60) the application of the theory of linear partial differential equations with variable coefficients is relevant. Having resolved (2.3.60) we find the flux for η for example from (2.3.58). More about the resolution of (2.3.60) will be given in Section 2.3.21.

118

2.3.9


Entropy function for a p-system

As already mentioned in Section 2.2.6 the so-called p-system ∂t u + ∂x p(V ) = 0,

∂t V − ∂x u = 0

(2.3.61)

governs the isentropic one-dimensional compressible flow in Lagrangian mass coordinates with velocity u, specific volume V and pressure law p = p(V ). It may be easily checked that for the p-system (2.3.61) and its entropy function η(u, V ), equation (2.3.60) reads ∂V2 η + p′ (V )∂u2 η = 0.

(2.3.62)

The usual assumption for the function p(·) is continuous differentiability and p′ (V ) < 0 for V > 0.

(2.3.63)

This assumption has a natural physical interpretation. The larger the density the larger the pressure. With (2.3.63) relation (2.3.62) becomes the wave equation for η which is hyperbolic as long as (2.3.63) holds true. It is an interesting excercise to solve (2.3.62) by means of the method of characteristics and Riemann invariants. One of the possible solutions is

V 2 (2.3.64) η(u, V ) = u2 − 1 p(s) ds. The corresponding flux G can be extracted from the formula (2.3.58) which, after some rearrangement, yields G(u, V ) = u · p(V ).

(2.3.65)

The entropy inequality for (2.3.61) with entropy function (2.3.64) now reads (cf. (2.3.38))

T u2 V ( 2 − 1 p(s)ds)∂t ϕ + up(V )∂x ϕ dxdt ≥ 0, ϕ ∈ C0∞ (Ω × (0, T )), ϕ ≥ 0. 0 Ω (2.3.66) Let us note that under condition (2.3.63) the entropy function η given by (2.3.64) is convex. 2.3.10

Riemann problem

The Riemann problem for a hyperbolic system ∂t w + ∂x f (w) = 0,

(x, t) ∈ IR × (0, ∞)

(2.3.67)

consists in finding its weak solution which satisfies the initial condition formed by two constant states wL , wR ∈ D: ! wL , x < 0, 0 (2.3.68) w(x, 0) = w (x) = wR , x > 0. We again assume that D is an open subset of IRm and f ∈ C 1 (D)m .

WEAK SOLUTIONS

119

Theorem 2.31 If the Riemann problem (2.3.67), (2.3.68) has a unique piecewise smooth weak solution w, then w can be written for t > 0 in the similarity : IR → IRm . form w(x, t) = w(x/t) where w

Proof We easily find that for any fixed α > 0 the function w(αx, αt) is also a weak solution of (2.3.67), (2.3.68). Due to the uniqueness, we have w(αx, αt) = w(x, t), which means that w is a homogeneous vector function of order zero. Hence, for any fixed x and t, taking α = 1/t, we see that w(x, t) = w(x/t, 1) =: w(x/t). 2

In connection with the Riemann problem (2.3.67), (2.3.68) some concepts play an important role. Assuming hyperbolicity of (2.3.67), we have A(w) =

Df Dw (w)

= T Λ\ T−1 ,

w ∈ D,

(2.3.69)

where Λ\ = Λ\(w) = diag (λ1 (w), . . . , λm (w)) and λj = λj (w) ∈ IR, j = 1, . . . , m, are the eigenvalues of the matrix A = A(w). The columns of the matrix T = T(w), denoted by r s = r s (w), s = 1, . . . , m, are the eigenvectors of A(w) associated with the eigenvalues λs (w) and form a basis in IRm . Definition 2.32 An eigenvector r k = r k (w) of the matrix A = A(w) is called genuinely nonlinear (in the domain D), if ∇λk (w)T · r k (w) = 0

∀ w ∈ D.

(2.3.70)

(We write ∇ = ∇w = (∂/∂w1 , . . . , ∂/∂wm )T .) Further, we say that r k is linearly degenerate if ∇λk (w)T · r k (w) = 0 ∀ w ∈ D. (2.3.71) We call ψk : D → IR a k-Riemann invariant if ψk ∈ C 1 (D) and ∇ψk (w)T · r k (w) = 0

∀ w ∈ D.

(2.3.72)

It is obvious that if an eigenvector r k is linearly degenerate, then the corresponding eigenvalue λk is a k-Riemann invariant. It may be proved (see (Godlewski and Raviart, 1996), Lemma 3.1), at least locally, that there exist (m−1) k-Riemann invariants whose gradients are linearly independent. Compare relation (2.3.71) for the k-Riemann invariant in Definition 2.32 with relation (2.2.26) for the complementary Riemann invariant in Definition 2.13. By (2.3.72), r k is orthogonal in IRN to ∇ψkT . Denoting by ℓj the left eigenvector corresponding to the eigenvalue λj (j = 1, . . . , m), we know that ℓj · r k = δjk . Hence, by (2.3.72), ∇ψkT = j=k µj ℓj with some µj and we can normalize this choice by ∇ψkT := ℓjk for some jk = k. On the other hand, in Definition 2.13, we choose ∇ψkT := ℓk (notice that ℓj are also right eigenvectors of AT ) which is quite complementary, as it is expressed by the term complementary Riemann invariant.

120


Solutions of the Riemann problem can be constructed in similarity form (cf. Theorem 2.31) for some special couples of initial states wL and wR . It is obvious that every similarity solution is constant on any line x/t = const and this fact plays an important role in the process of resolution of the Riemann problem. For general results on the existence and form of solutions to the Riemann problem we refer to (Feistauer et al., 2003) and the standard monographs (Smoller, 1983), (Dafermos, 2000) and (Bressan, 2000) 2.3.11

Riemann problem for 2 × 2 isentropic gas dynamics equations

In Section 2.3.14 below, the existence of weak admissible solutions to the 2 × 2 Euler system with initial data is studied thoroughly. The approximations of the exact solution are constructed with the help of the Lax–Friedrichs difference scheme which uses a sequence of Riemann problems as a tool to their construction. To this purpose a global existence result for the Riemann problem with the initial data which are not necessarily a small perturbation of one constant state on the left by another constant state on the right, is applied. So, consider the system 2

(2.3.73) ∂t ρ + ∂x m = 0, ∂t m + ∂x mρ + p(ρ) = 0, x ∈ IR, t > 0,

with the initial data

(ρ, m)|t=0 =

!

(ρL , mL ), x < 0, (ρR , mR ), x > 0.

(2.3.74)

Here we denote by m := ρv the momentum and assume that ρJ ≥ 0 and mJ ∈ IR are constants, and |mJ | ≤ C0 ρJ < ∞, J = L, R. We also assume that for ρ > 0, p(ρ) > 0, p′ (ρ) > 0 (hyperbolicity), ρp′′ (ρ) + 2p′ (ρ) > 0 (genuine nonlinearity),

(2.3.75)

p(0+ ) = p′ (0+ ) = 0.

(2.3.76)

and We use the convention m ρ

m ρ

= v if ρ = 0,

= 0 if ρ = 0.

The eigenvalues of system (2.3.73) are λi (ρ, m) =

m ρ

+ (−1)i

p′ (ρ),

i = 1, 2,

(2.3.77)

and the corresponding right eigenvectors are

T r i (ρ, m) = αi (ρ) 1, λi (ρ, m) ,

√ 2ρ p′ (ρ) αi (ρ) = (−1)i ρp′′ (ρ)+2p′ (ρ)

so that (∇(ρ,m) λi )T · r i = 1, i = 1, 2. The Riemann invariants are

(2.3.78)

WEAK SOLUTIONS

ψi (ρ, m) =

m ρ

From (2.3.77), (2.3.76) we get

+ (−1)i+1

ρ √p′ (s) 0

s

121

ds,

i = 1, 2.

(2.3.79)

λ2 (ρ, m) − λ1 (ρ, m) = 2 p′ (ρ) → 0, ρ → 0+ .

This implies that system (2.3.73) is strictly hyperbolic in the domain {(ρ, v); ρ > 0, |v| ≤ C0 }. Denote w := (ρ, m)T , wJ = (ρJ , mJ )T , J = L, R. Theorem 2.33 Let p ∈ C 2 ((0, ∞)) satisfy (2.3.75), (2.3.76). Then the Riemann problem (2.3.73), (2.3.74) has a unique, globally defined, piecewise smooth weak entropy solution w(x/t) = (ρ, m)(x/t) for x ∈ IR, t > 0, satisfying ψ1 (w(x/t)) ≤ ψ1 (wR ), ψ2 (w(x/t)) ≥ ψ2 (wL ), ψ1 (w(x/t)) − ψ2 (w(x/t)) ≥ 0.

(2.3.80)

This Riemann solution can be constructed for the case ψi (wR ) ≥ ψi (wL ), i = 1, 2 as follows. If ρL > 0 and ρR = 0, then there exists a unique vc such that  x/t < λ1 (wL ), wL ,    (2.3.81) w(x/t) = V 1 (x/t), λ1 (wL ) ≤ x/t ≤ vc ,    T (0, 0) , x/t > vc ,

where V 1 (ξ) is the solution of the initial–value problem

V ′1 (ξ) = r 1 V 1 (ξ) , ξ > λ1 (wL ), V 1 |ξ=λ1 (wL ) = wL .

(2.3.82)

where V 2 (ξ) is the solution of the initial–value problem

V ′2 (ξ) = r 2 V 2 (ξ) , ξ < λ2 (wR ), V 2 |ξ=λ2 (wR ) = wR .

(2.3.83)

If ρL = 0 and ρR > 0, then there exists vc such that  (0, 0)T , x/t < vc ,    w(x/t) = V 2 (x/t), vc ≤ x/t ≤ λ2 (wR ),    wR , x/t > λ2 (wR ),

If ρL , ρR > 0, then we distinguish two cases: (a) There exist unique vc1 < vc2 , such that the Riemann solution has the form  wL , x/t < λ1 (wL ),       V 1 (x/t), λ1 (wL ) ≤ x/t ≤ vc1 ,    w(x/t) = (0, 0)T , vc1 < x/t < vc2 ,     V 2 (x/t), vc2 < x/t < λ2 (wR ),      wR , x/t > λ2 (wR ),

122


where V 1 (ξ) and V 2 (ξ) are the solutions of the initial–value problems (2.3.82) and (2.3.83), respectively. (b) There exists a unique wc = (ρc , mc ), ρc > 0, such that the Riemann solution has the form  x/t < λ1 (wL ), wL ,       V 1 (x/t), λ1 (wL ) ≤ x/t ≤ λ1 (wc ),    λ1 (wc ) < x/t < λ2 (wc ), w(x/t) = wc ,     V 2 (x/t), λ2 (wc ) ≤ x/t ≤ λ2 (wR ),      wR , x/t > λ2 (wR ),

where V1 (ξ) and V2 (ξ) are again solutions of the initial–value problems (2.3.82), (2.3.83), respectively. In the remaining cases ψi (wR ) < ψi (wL ), ψi (wR ) ≥ ψj (wL ), i, j = 1, 2, the Riemann solutions can be constructed similarly (see (Dafermos, 2000), (Serre, 1996), (Smoller, 1983) for details). Proof Theorem 2.33 is taken over from (Chen and Wang, 2001), Section 6.3. Let, for example, ρL > 0, ρR = 0 and w = w(x/t) be given by (2.3.81). 2 Denote f (w) = (m, mρ + p(ρ))T . First, we have to verify the identity

∞

0

IR

(w · ∂t ϕ + f (w)∂x ϕ) dx dt + ϕ∈

C0∞ (IR

where w0 (x) =

IR

w0 (x)ϕ(x, 0) dx = 0,

× [0, ∞)),

wL = (ρL , mL )T , x < 0 wR = (ρR , mR )T , x > 0.

(2.3.84)

(2.3.85)

Substituting y = x/t in (2.3.84) we get

∞ w(y)∂t ϕ(yt, t) + f (w(y))∂x ϕ(yt, t) t dy dt 0 IR

0

∞ +wL −∞ ϕ(y, 0) dy + wR 0 ϕ(y, 0) dy = 0.

(2.3.86)

Since

d dt ϕ(ty, t)

we have

= y∂x ϕ(ty, t) + ∂t ϕ(ty, t) and

d dy ϕ(ty, t)

= t∂x ϕ(ty, t),

d d t∂t ϕ(ty, t) = t dt ϕ(ty, t) − y dy ϕ(ty, t).

Consequently, the first integral in (2.3.86) assumes the form

d

∞ d d ϕ(ty, t) − y dy ϕ(ty, t) + f (w(y)) dy ϕ(ty, t) dy dt. I := 0 IR w(y) t dt

Split I into the three integrals:

WEAK SOLUTIONS

I=

∞ λ1 (wL ) 0

−∞

+

vc

λ1 (w L )

+

∞ vc

123

dy dt = I1 + I2 + I3 ,

assuming vc > λ1 (wL ) given, for a moment. Clearly, if w is of class C 1 in (a, b) × (0, ∞), then d

∞ b d d w(y) t dt ϕ(ty, t) − y dy ϕ(ty, t) + f (w(y)) dy ϕ(ty, t) dy dt a 0

b ∞

b = − limt→0+ a tw(y)ϕ(ty, t) dy + 0 −yw(y) + f (w(y)) ϕ(ty, t) dt y=a

∞ b Dϕ (w(y)) w′ (y)ϕ(ty, t) dy dt. + 0 a y − Dw Using this formula and (2.3.81) we obtain

0

∞ I1 = − −∞ wL ϕ(x, 0) dx + 0 −λ1 (wL )wL + f (wL ) ϕ(tλ1 (wL ), t) dt,

∞ I2 = − 0 −λ1 (wL )wL + f (wL ) ϕ(tλ1 (wL ), t) dt

∞ + 0 −vc V 1 (vc ) + f (V 1 (vc )) ϕ(tvc , t) dt

∞ v + 0 λ1c(wL ) y − λ1 (y) r 1 V 1 (y) ϕ(ty, t) dy dt,

∞

∞ I3 = − 0 wR ϕ(x, 0) dx − 0 −vc wR + f (wR ) ϕ(tvc , t) dt.

Summing up these equalities we arrive at

∞ I = wR − V 1 (vc ) vc + f (V 1 (vc )) − f (wR ) 0 ϕ(tvc , t) dt

∞ v + 0 λ1c(wL ) y − λ1 (y) r 1 V 1 (y) ϕ(ty, t) dy dt

0

∞ −wL −∞ ϕ(x, 0) dx − wR 0 ϕ(x, 0) dx.

Inserting the last form of I into (2.3.86), we should verify

∞ wR − V 1 (vc ) vc + f (V 1 (vc )) − f (wR ) 0 ϕ(tvc , t) dt

∞ v + 0 λ1c(wL ) y − λ1 (V 1 (y) r 1 (V 1 (y))ϕ(ty, t)dydt = 0, ϕ ∈ C0∞ (IR × [0, ∞))2 . (2.3.87) The condition that the first bracket equals zero represents the Rankine–Hugoniot condition between the states w = V 1 (vc ) and w = wR which should be satisfied both for the continuous as well as for the discontinuous case. Since ρR = 0, this condition reads vc ρ(vc ) = m(vc ) vc m(vc ) =

m(vc )2 ρ(vc )

+ p(ρ(vc ))

which implies p(ρ(vc )) = 0, i.e. ρ(vc ) = 0. We show that V 1 and vc exist. System (2.3.82) in the variables ρ = ρ(ξ), m = m(ξ) reads √ 2ρ p′ (ρ) dρ = α (ρ) = − 1 dξ ρp′′ (ρ)+2p′ (ρ) , (2.3.88) √ 2ρ p′ (ρ) dm m ′ p (ρ) . dξ = α1 (ρ)λ1 (ρ, m) = − ρp′′ (ρ)+2p′ (ρ) ρ −

124


The integration of the first equation yields

ρ(ξ) √p′ (s) p′ (ρ(ξ)) − p′ (ρ(ξ0 )) + ρ(ξ0 ) s ds = −(ξ − ξ0 ).

(2.3.89)

Putting ξ0 = λ1 (wL ), ξ = vc we obtain (ρ(vc ) = 0!)

ρ √p′ (s) vc = λ1 (wL ) + p′ (ρL ) + 0 L s ds.

By the second inequality in (2.3.75) the function on the left-hand side of (2.3.89) is increasing with respect to ρ = ρ(ξ) and so equation (2.3.89) is solvable with respect to ρ(ξ). The proof of (2.3.87) will be complete if we show that y = λ1 (V 1 (y)). But a short computation using (2.3.77) and (2.3.78) yields the identity ∇(ρ,m) λ1 (ρ, m) · r 1 (ρ, m) = 1.

Since (ρ, m) = V 1 and V ′1 (ξ) = r(V 1 (ξ)), we find

d dξ λ1 V 1 (ξ) = ∇(ρ,m) λ1 V 1 (ξ) · r 1 V 1 (ξ) = 1 T

which yields λ1 (V 1 (ξ)) = ξ. Inequalities (2.3.80) are trivial for constant states. For w = V 1 , (2.3.80) follows from d dξ ψj (V 1 (ξ))

≥ 0,

λ1 (wL ) ≤ ξ ≤ vc , j = 1, 2

which is obtained by differentiation with the help of (2.3.88), (2.3.78), (2.3.77). Quite analogously, the remaining definitions of w can be treated. The uniqueness can be shown by the method used in (Chen and Wang, 2001), Section 8.2 to nonisentropic Euler equations (see also (Chen et al., 2002)). Let us note that for p(ρ) = κργ , γ > 1 conditions (2.3.75), (2.3.76) are satisfied. 2 The explicit formulae for the Riemann solution expressed in Theorem 2.33 have the following consequence. Theorem 2.34 The regions R(ψ10 , ψ20 ) = {(ρ, m); ψ1 ≤ ψ10 , ψ2 ≥ ψ20 , ψ1 − ψ2 ≥ 0} are invariant regions of the Riemann problem (2.3.73) and (2.3.74). That is, if the Riemann initial data lie in R(ψ10 , ψ20 ), then the solution of the Riemann problem has its values also in R(ψ10 , ψ20 ). Proof Since

d dξ ψj (V k (ξ))

we have ψ1 (V j (ξ)) ≤ ψ1 (wR ) ≤ ψ10 , The inequality ψ1 ≥ ψ2 is trivial.

≥ 0 for j, k = 1, 2,

ψ2 (V j (ξ)) ≥ ψ2 (wL ) ≥ ψ20 ,

j = 1, 2. 2

WEAK SOLUTIONS

2.3.12

125

Existence and uniqueness of admissible weak solution for a scalar conservation law

The theory of the existence and qualitative properties of scalar conservation laws is now well developed. The following theorem is a typical representive of the existence and uniqueness result available in the literature. Theorem 2.35 For any w0 ∈ L∞ (IRN ) there exists a unique admissible weak solution w of (2.3.41), (2.3.48) on IRN × [0, ∞) and w ∈ C([0, ∞), L1loc (IRN )),

w(·, t)L∞ (IRN ) ≤ w0 L∞ (IRN ) .

(2.3.90)

If in addition, w0 ∈ BVloc (IRN ), then w ∈ L∞ ((0, ∞), BVloc (IRN )) ∩ BVloc (IRN × (0, ∞))

(2.3.91)

and T V{|x| 0

with some s > 0 depending only on ess inf{w(x, t); (x, t) ∈ IRN × (0, ∞)} and ess sup{w(x, t); (x, t) ∈ IRN × (0, ∞)}. Finally, if w− < w0 (x) < w+ for x ∈ IR, then w− < w(x, t) < w+ a.e. in IR × (0, ∞). The proof of the first assertion can be found in (Dafermos, 2000), where different typical methods are used. The second result with w as in (2.3.91) is proved, e.g., in (M´ alek et al., 1996) by the viscosity method (see also (Bressan, 2000) and (Serre, 1996)). 2.3.13

Plane waves admitting discontinuities

We return to the final remark in Section 2.2.24. If we choose σ ∈ (σ− , σ+ ), where σ± are the numbers from (2.2.144), (2.2.145), and put

σ fk (σ) = σ λk (z(η)) dη, then (2.2.144) reads

∂t σ + ∂x fk (σ) = 0 and Theorem 2.35 can be applied to obtain a solution σ ∈ C([0, ∞), L1loc (IR)) ∩ L∞ ((0, ∞), L∞ (IR)) which belongs also to L∞ ((0, ∞), BVloc (IR)) ∩ BVloc (IRN × (0, ∞)). Notice that σ− < σ(y, t) < σ+ , y ∈ IR, t > 0 so that there is no other restriction on the time interval of existence due to the requirement that σ(y, t) lies in the definition region of z. 2.3.14

Existence of solutions to the 2 × 2 Euler system for an isentropic gas

Recent developments in weak compactness techniques have allowed a development of a rigorous proof of the existence of weak solutions for a large class of 2 × 2 conservation laws. Here we describe representative results for the Euler

126


system of an isentropic gas in one space dimension. Consider again the system (cf. (2.1.9)) ∂t ρ + ∂x m = 0, ∂t m + ∂x

m2 ρ

+ p(ρ) = 0,

x ∈ IR, t > 0,

(2.3.92)

where we denote by m = ρv, the momentum of the gas; ρ ≥ 0, the density; and p = p(ρ) ≥ 0, the pressure (a given function of ρ). We will consider the resolution of system (2.3.92) in the physical region D := {(ρ, m); ρ ≥ 0, |m| ≤ Cρ} with a given fixed constant C depending on initial data and state equation for p. 2 In the case ρ = 0 we put by definition mρ = 0 so that there would be no doubt about the interpretation of this function when the denominator is zero. Next, we assign to (2.3.92) the initial data (2.3.93) (ρ, m)t=0 = (ρ0 , m0 ),

measurable functions with values in the region D, and we are interested in resolution of problem (2.3.92), (2.3.93) in the sense of distributions. In what follows we keep sufficiently general assumptions on the function p(·) which include the γ-law p(ρ) = κργ (κ > 0, γ > 1 constants), to admit more general pressure laws, especially those having higher derivatives unbounded near the vacuum. This is of course possible only thanks to results available in the literature, and our main sources here are (Chen and LeFloch, 2000) and (Dafermos, 2000). Assume the following: p(·) ∈ C 4 (0, ∞); (2.3.94) ∃ γ ∈ (1, 3), C > 0, δ > 0 such that (2.3.95)

p(ρ) = κργ 1 + P (ρ) , |P (n) (ρ)| ≤ Cρ1−n for n = 0, 1, 2, 3, 4 and ρ ∈ (0, δ), (define p(0) := p(0+ ) = 0, p′ (0) := p′ (0+ ) = 0, so that p(·) ∈ C 1 ([0, ∞))); p′ (ρ) > 0, 2p′ (ρ) + ρp′′ (ρ) > 0 for ρ > 0.

(2.3.96)

Note that the first assumption in (2.3.96) implies strict hyperbolicity of system (2.3.92) away from the vacuum (while in the vacuum the two characteristic speeds may coincide and the system will be nonstrictly hyperbolic), and the second inequality in (2.3.96) guarantees genuine nonlinearity (also away from the vacuum). In order to formulate the existence result we define the notion of admissible solution to (2.3.92), (2.3.93). First we recall that the entropy–entropy flux pair for (2.3.92) is a couple (η, G), where (cf. (2.3.55)) ∂ρ G = ∂m η p′ (ρ) −

m2 ρ2

, ∂m G = ∂ρ η +

2m ρ ∂m η.

(2.3.97)

WEAK SOLUTIONS

127

The compatibility condition analogous to (2.3.60) (in fact the Cauchy–Riemann condition of interchangeability of partial derivatives) now, after a short computation, yields the following wave equation for η: 2 2m (2.3.98) ∂ρρ η − p′ (ρ) − m ρ2 ∂mm η + ρ ∂ρm η = 0.

The method for determining G by representation (2.3.57) does not seem to be sufficiently general for our purposes and we return to this issue later. The pairs (η, G) satisfying (2.3.97), (2.3.98) are called weak entropy–entropy flux pairs because of their singularities near the vacuum, i.e. the line {ρ = 0}, in the phase space D. The analysis of weak entropy–entropy flux pairs will be performed in Section 2.3.21. Definition 2.36 A couple (ρ, m) ∈ L∞ (IR)2 is called an admissible solution to problem (2.3.92), (2.3.93) if the equations

∞

ρ∂t ϕ + m∂x ϕ dx dt + IR ρ0 (x)ϕ(x, 0) dx = 0, 0 IR (2.3.99) 2

∞ m∂t ϕ + mρ + p(ρ) ∂x ϕ dx dt + IR m0 (x)ϕ(x, 0) dx = 0 0 IR

hold for any ϕ ∈ C0∞ (IR × [0, ∞)) and if for any convex entropy function η ∈ C 2 (D) satisfying (2.3.98) and a corresponding entropy flux G ∈ C 2 (D) satisfying (2.3.97) we have

∞ η(ρ, m)∂t ϕ + G(ρ, m)∂x ϕ dx dt ≥ 0, ϕ ∈ C0∞ (IR × (0, ∞)), ϕ ≥ 0. 0 IR (2.3.100)

Now we are in a position to state the existence theorem.

Theorem 2.37 Let assumptions (2.3.94)–(2.3.96) be satisfied and let (ρ0 , m0 ) 2 ∈ L∞ (IR × (0, ∞)) be such that 0 ≤ ρ0 (x) ≤ C0 , m0 (x) ≤ C0 ρ0 (x) for a.e. x ∈ IR and some C0 > 0. Then there exists an admissible solution (ρ, m) ∈ ∞ 2 L (IR × (0, ∞)) of the Cauchy problem (2.3.92), (2.3.93) satisfying 0 ≤ ρ(x, t) ≤ K(C0 ), m(x, t) ≤ K(C0 )ρ(x, t) for a.e. (x, t) ∈ IR × (0, ∞) (2.3.101) with a constant K(C0 ) > 0 depending only on C0 . Further, if {(ρε , mε )}ε∈(0,1] ⊂ ∞ 2 L (IR × (0, ∞)) is a family of couples satisfying 0 ≤ ρε (x, t) ≤ K, |mε (x, t)| ≤ Kρε (x, t) for a.e. (x, t) ∈ IR × (0, ∞) and all ε ∈ (0, 1]

such that for any entropy pair (η, G) described above the family {∂t η(ρε , mε ) + ∂x G(ρε , mε )}ε∈(0,1]

−1 is relatively compact in Hloc (IR × (0, ∞)),

(2.3.102)

then the family {(ρε , mε )}ε∈(0,1] is compact in Lrloc (IR × (0, ∞)) for any fixed r ∈ [1, ∞), but arbitrary.

128


−1 Let us note that by relative compactness of a set in Hloc (IR×(0, ∞)) we mean that given a fixed (but arbitrary) bounded set Ω ⊂ IR × (0, ∞), the set in question is relatively compact in H −1 (Ω).

Proof The proof is very technical but we will give here the most crucial steps at least for the γ-law. Some details will be left to the reader with regard to the good state of the availability of references. We start with an approximation scheme providing us with a sequence which will finally appropriately converge to an admissible solution to the problem (2.3.92), (2.3.93). 2.3.15

Lax–Friedrichs difference approximations

We introduce the following difference approximations (ρh (x, t), mh (x, t)) to system (2.3.92), (2.3.93), called in the literature theLax–Friedrichs approximations. First, define v = m ρ whenever ρ > 0 and v = 0 otherwise. Second, choose a time step τ > 0 and the space step h > 0 and assume that the ratio hτ is constant. Define the sound speed a(ρ) := p′ (ρ). (2.3.103)

It is important that the ratio τ /h and the approximations fulfil at each step the so-called Courant–Friedrichs–Loewy stability condition τ · h

sup (x,t)∈IR×(0,∞)

|v h (x, t) ± a(ρh (x, t))| < 1.

(2.3.104)

Put xj = jh, tn = nτ, j = 0, ±1, ±2, . . . ; n = 1, 2, . . . , and Jn = {j; j is an integer such that n + j is even} for each fixed n. In the first strip {(x, t); 0 < t < t1 , xj−1 < x < xj+1 , j odd}, we define (ρh (x, t), mh (x, t)) with the help of solutions of the sequence of Riemann problems for (2.3.92) corresponding to the Riemann data ! 0 (ρj−1 , m0j−1 ), x < xj , h h (ρ , m )(x, 0) = (2.3.105) (ρ0j+1 , m0j+1 ), x > xj with

(ρ0j+1 , m0j+1 ) =

1 2h

xj+2 xj

(ρ0 , m0 )(x) dx.

Now, having defined (ρh , mh ) for t < tn , we set

xj+1 h h 1 (ρ , m )(x, tn − ) dx. (ρnj , mnj ) = 2h xj−1

(2.3.106)

In the region {(x, t) : xj < x < xj+2 , tn < t < tn+1 , j ∈ Jn }, we define (ρh (x, t), mh (x, t)) with the help of Riemann problems for (2.3.92) with the data ! n n x < xj+1 , (ρj , mj ), h h (ρ , m )(x, tn ) = (2.3.107) n n (ρj+2 , mj+2 ), x > xj+1 .

WEAK SOLUTIONS

129

2.3.16 Existence of approximations The admissible solutions to the Riemann problems used above for the definition of approximations (ρh , mh ) can be composed of rarefaction waves, entropy shocks and contact discontinuities as shown in Theorem 2.33. Let us note that due to condition (2.3.104) the solutions do not mutually interact in the adjacent cells. We must check only that in the course of continuing the approximations in time, condition (2.3.104) is satisfied without changing the value of τ /h. This can be shown from the form of the solution of the Riemann problem described in Theorem 2.33. 2.3.17 Invariant regions for Riemann invariants Recalling Definition 2.13 we can find Riemann invariants for system (2.3.92). Taking w := (ρ, m) in (2.2.26), + * 0 1 2 , A(ρ, m) = 2m a(ρ)2 − m ρ2 ρ where a(ρ) is given by (2.3.103), we obtain λ(ρ, m) = v ± a(ρ), recalling the relation m = ρv. Integrating the equations

T ∇(ρ,m) ψ ± = − ρm2 ± aρ , ρ1

we obtain

ψ± =

m ρ

±

ρ 0

a(σ) σ

dσ = v ±

ρ 0

a(σ) σ

dσ.

(2.3.108)

Notice that we had to choose appropriate multipliers (integrating factors) of the unit eigenvectors corresponding to v ± a, respectively, to be able to integrate equations for ψ ± . Theorem 2.33 gives us bounds for the solution of the Riemann problem for t > 0. Next, Theorem 2.34 yields that given ψ ± : ψ0+ > ψ0− , the region R(ψ0± ) := {(ρ, m); ψ + ≤ ψ0+ , ψ − ≥ ψ0− , ψ + − ψ − ≥ 0}

(2.3.109)

is invariant. But then it is invariant for Lax–Friedrichs approximations since the data for the Riemann problems are given by (2.3.106) and if initially (ρh , mh )(x) ∈ R(ψ0± ) for all x ∈ IR then by Jensen’s integral inequality (see (1.3.8)), since R(ψ0 ±) is convex, (ρnj , mnj ) ∈ R(ψ0± ), as well. In particular, it follows that ρh ≥ 0 and ρh L∞ (IR) ≤ K(C0 ), mh L∞ (IR) ≤ K(C0 )ρh L∞ (IR) , (2.3.110) where K(C0 ) > 0 is independent of h. Indeed,

ρ 0 ≤ ψ + − ψ − = 2 0 a(σ) σ dσ implies ρ ≥ 0 and

v+

ρ 0

a(σ) σ

ρ − − + dσ ≤ ψ0+ and v − 0 a(σ) σ dσ ≥ ψ0 imply ψ0 ≤ v ≤ ψ0 ,

ρ + − 1 (0 ≤) 0 a(σ) σ dσ ≤ 2 (ψ0 − ψ0 ).

130


2.3.18

Compactness argument

Take τk → 0, hk → 0, τk /hk = const, such that (2.3.104) is satisfied for h := hk and all k = 1, 2, . . . . Uniform estimates (2.3.110) allow us to select {(ρk , mk )} ⊂ {(ρhk , mhk )} so that ρk → 0, mk → m

weak-∗ in L∞ (IR × (0, ∞)).

(2.3.111)

Since for wk → w weak-∗ in L∞ does not in general imply f (wk ) → f (w) in D′ when f is nonlinear, we need additional information to pass to the limit in 2 the term mρ + p(ρ) in the second equation of (2.3.92). This is one of the most complicated steps in the proof of Theorem 2.37. The first observation to be made is the following proposition. For simplicity, we prove it only for p(ρ) = κργ , κ > 0, γ ∈ (1, 2] constants, and refer to (Chen and LeFloch, 2000) for the general case. Proposition 2.38 For any weak entropy–entropy flux pair (η, G) of (2.3.92) the sequence of distributions {∂t η(ρk , mk ) + ∂x G(ρk , mk )}∞ k=1 −1 is a compact subset of Hloc (IR × (0, ∞)).

(2.3.112)

Proof of Proposition 2.38. Setting q h := (ρh , mh ), choose T = Kτ for some integer K > 1, an entropy pair (η, G) and ϕ ∈ C0∞ (IR × (0, T )). Using the structure of q h as compositions of noninteracting solutions of the Riemann problems and the Green formula we can write

T

η(q h )∂t ϕ + G(q h )∂x ϕ dx dt = M h (ϕ) + S h (ϕ) + Lh1 (ϕ) + Lh2 (ϕ), (2.3.113) where (note that for example [η](t) := η(q(x(t), t)) − η(q(x(t)− , t)) normalizing η(q(x(t), t)) := η(q(x(t)+ , t))), 0

IR

η(q h (x, T ))ϕ(x, T ) dx − IR η(q h (x, 0))ϕ(x, 0) dx = 0, ′

T S h (ϕ) := 0 shocks x(t) x (t)[η](t) − [G](t) ϕ(x(t), t) dt,

xj+1 η(q h (x, tk − )) − η(q h (xj , tk )) dx, Lh1 (ϕ) := j,k ϕ(xj , tk ) xj−1

xj+1 η(q h (x, tk − )) − η(q h (xj , tk )) ϕ(x, tk ) − ϕ(xj , tk ) dx. Lh2 (ϕ) := j,k xj−1 (2.3.114) Given a bounded set Ω ⊂ IR × [0, T ] and ϕ ∈ C0 (Ω), by a technical procedure it is possible to get the following estimates: M h (ϕ) :=

IR

|S h (ϕ)| ≤ CϕC0 (Ω)

T 0

′ shocks x(t) {x (t)[η∗ ]

− [G∗ ]} dt ≤ CϕC0 (Ω) , (2.3.115)

WEAK SOLUTIONS

131

xj+1 1

T q h (x, tk − ) − q h (xk , tk ) ∇2 η∗ q h (xj , tk )

+s(q h (x, tk − ) − q h (xj , tk )) q h (x, tk − ) − q h (xj , tk ) (1 − s) ds dx

|Lh1 (ϕ)| ≤ CϕC0 (Ω)

j,k

xj−1

0

≤ CϕC0 (Ω) ,

|Lh2 (ϕ)|

(2.3.116)

2 1/2 ≤ h ϕC0α (Ω) k dx j xj−1 η(q (x, tk )) − η(q (xj , tk )) xj+1 h

1/2 1 − h 2 ≤ hα− 2 ∇ηL∞ ϕC0α j,k xj−1 |q (x, tk ) − q (xj , tk )| dx xj+1

α

h

1

≤ Chα− 2 ϕW 1,p (Ω) , for all p >

2 1−α ,

−

h

ϕ ∈ C0α (Ω) with

1 2

< α < 1, (2.3.117) where the couple (η∗ , G∗ ) is a reference entropy pair which will be specified at the end of this section together with other facts concerning the characterization of all weak entropy pairs for system (2.3.92). The analysis leading to estimate (2.3.115), which we have skipped, relies on this characterization. On the other hand, the remaining estimates (2.3.116), (2.3.117) follow from (2.3.110) and Sobolev imbeddings. Now, from M h (ϕ) = 0 and (2.3.113)–(2.3.117) we deduce (M (Ω) := CB (Ω)∗ , see Section 1.3.8) M h + S h + Lh1 M (Ω) ≤ C. (2.3.118) 0

1,q ′

Since W0 0 is compactly imbedded into C0 (Ω), if q0′ > 2, by duality of imbeddings (see Section 1.4.10.7), we have that M (Ω) is compactly imbedded into W −1,q0 (Ω) with any q0 ∈ (1, 2). So from (2.3.118) we find that the family of measures {M h + S h + Lh1 }h∈(0,1]

is compact in W −1,q0 (Ω), (1 < q0 < 2).

(2.3.119)

On the other hand, estimate (2.3.117) yields 2 < 2. 1+α (2.3.120) Combining (2.3.119), (2.3.120) we get M h +S h +Lh1 +Lh2 is compact in W −1,q0 (Ω), 1 < q0 < 2/(1 + α). Estimating the left-hand side of (2.3.113) with the help of the uniform estimate q h L∞ (Ω)2 ≤ K we get also that M h + S h + Lh1 + Lh2 is bounded in W −1,r (Ω) for any r > 2. By interpolation (see (Ding et al., 1985)) we find that M h + S h + Lh1 + Lh2 is relatively compact in W −1,2 (Ω) which, by (2.3.113), implies (2.3.112). The proof of Proposition 2.38 is finished. 2 1

Lh2 W −1,q0 (Ω) ≤ Chα− 2 → 0, when h → 0+ , for 1 < q0
0 are equivalent to (2.3.92). Looking for weak entropy–entropy flux pairs (η, G) with respect to variables v and ρ we have to fulfil the equations ∂ρ G = v∂ρ η + ρk ′ (ρ)2 ∂v η, ∂v G = ρ∂ρ η + v∂v η.

(2.3.133)

Eliminating G we obtain the equation for η in the form ∂ρ2 η − k ′ (ρ)2 ∂v2 η = 0.

(2.3.134)

This is a second order linear hyperbolic partial differential equation. For resolution of equation (2.3.134) standard methods for linear partial differential equations can be used. The solution is found with the help of the Fourier transform and assumes the form

η(ρ, v) = IR χ(ρ, v, s)θ(s) ds, (2.3.135) where θ ∈ L1loc (IR) is a given but arbitrary function and χ is to be determined so that η is a solution to (2.3.134). It is not surprising that χ is chosen as the fundamental solution for (2.3.134), i.e. a distribution satisfying ∂ρ2 χ − k ′ (ρ)2 ∂v2 χ = 0, χ(0, v, s) = 0, ∂ρ χ(0, v, s) = δv=s , ρ > 0, v ∈ IR, s ∈ IR.

(2.3.136)

By a solution of (2.3.136) we mean a function χ(·, ·, s) ∈ L1loc ((0, ∞) × IR), s ∈ IR such that for any ϕ ∈ C0∞ ([0, ∞) × IR) the following integral equality holds true:

∞

χ(ρ, v, s) ∂ρ2 ϕ(ρ, v) − k ′ (ρ)2 ∂v2 ϕ(ρ, v) dρ dv − ϕ(0, s) = 0 for s ∈ IR. IR 0 (2.3.137) For the entropy flux G, after a short computation, we get from (2.3.133), using equation (2.3.134), the equation ∂ρ2 G − k ′ (ρ)2 ∂v2 G = This suggests we express G in the form

2p′′ (ρ) ∂v η. ρ

(2.3.138)

136


G(ρ, v) =

IR

σ(ρ, v, s)θ(s) ds

(2.3.139)

with the given function θ as above in (2.3.135), and the kernel σ satisfying the equations ∂ρ2 σ − k ′ (ρ)2 ∂v2 σ =

p′′ (ρ) ρ ∂v χ,

σ(0, v, s) = 0, ∂ρ σ(0, v, s) = vδv=s ,

(2.3.140)

ρ > 0, v ∈ IR, s ∈ IR.

The definition of the solution to (2.3.140) is analogous to that for χ. The following existence result for problems (2.3.136), (2.3.140) holds true. Proposition 2.43 ((Chen and LeFloch, 2000), Theorem 2.1) Problem (2.3.136) admits a unique generalized solution χ(ρ, v, s) = χ(ρ, v − s) which is H¨ older continuous in [0, ∞) × IR2 , it is supported in the sets K(s) := {(ρ, v); ρ ≥ 0, |s − v| ≤ k(ρ)} = {ψ; ψ + ≥ s, ψ − ≤ s}, s ∈ IR and positive in the interior of K(s), s ∈ IR. Problem (2.3.140) admits a unique solution σ(ρ, v, s), H¨ older continuous in [0, ∞) × IR2 , and supported with respect to (ρ, v) in the set K(s) for each fixed s ∈ IR with σ − vχ depending only on (ρ, v − s). Proof of Proposition 2.43. The proof for the general case is very technical and can be found in detail in the paper cited in the theorem. It uses the Fourier transform with respect to v, energy estimates and asymptotic expansion separating singular and regular parts of the solutions χ resp. σ. Here we restrict ourselves to the case p = p∗ (ρ) = κργ with constants κ > 0, γ ∈ (1, 3). Then the solution of (2.3.136) can be written in the form λ χ = χ∗ (ρ, v, s) = M αρβ − (v − s)2 + ,

(2.3.141)

where [y]+ := max{0, y} and M, α, β and λ are appropriate constants dependent on κ and γ. Inserting (2.3.141) for αρβ > (v − s)2 into the first equation in (2.3.136), after a standard calculation we obtain for α, β, λ the values α=

4κγ (γ−1)2 ,

β = γ − 1, λ =

3−γ 2(γ−1) .

(2.3.142)

We show that with an appropriate choice of M, function (2.3.141) satisfies (2.3.136) in the sense (2.3.137). First, notice that if in (2.3.137) ϕ is odd with respect to v relative to s, then the function χ(ϕρρ − k ′ (ρ)2 ϕvv ) has the same property and consequently

∞

χ ∂ρ2 ϕ − k ′ (ρ)2 ∂v2 ϕ dv dρ = 0, and ϕ(0, s) = 0. 0 |v−s|2 ≤αρβ

Since for any s we can write ϕ(ρ, v) = 12 ϕ(ρ, v) + ϕ(ρ, 2s − v) + 21 ϕ(ρ, v) − ϕ(ρ, 2s − v)

WEAK SOLUTIONS

137

which is the sum of an even and odd function with respect to the point s, by linearity, we may assume that ϕ is even. Then we have

∞

I := 0 IR χ(ρ, v, s) ∂ρ2 ϕ(ρ, v) − k ′ (ρ)2 ∂v2 ϕ(ρ, v) dv dρ ∞

λ = M |v−s|2 2ε.

This proves I1ε → 0 weak-∗ in M (IR). Further,

v+k v−k

v+k v−k

+ε

ϕj (s1 − sj ) dsj =

s1 −v+k ε s1 −v−k ε

sj ϕεj (s1 − sj ) dsj = s1


ϕj (σ) dσ → 1(v−k,v+k) (s1 ) pointwise in IR,

v+k v−k

ϕεj (s1 − sj ) dsj

σϕj (σ) dσ → s1 1(v−k,v+k) (s1 ) pointwise in IR,

and given w ∈ IR we also have

ϕε (s − w)η(s1 ) ds1 = IR ϕεj (−w − s1 )η(−s1 ) ds1 → η(w), IR j 1

(2.3.158)

so that

ε I (s )η(s1 ) ds1 → (v − k) IR η(v − k)1I (s1 ) − η(v − k)1I (s1 ) ds1 IR 2 1

+(v + k) IR η(v + k)1I (s1 ) − η(v + k)1I (s1 ) ds1 = 0.

142


This proves I2ε → 0 weak-∗ in M (IR). Finally, from (2.3.158) we find I3ε (s1 ) → s1 1I (s1 ) − s1 1I (s1 ) = 0 pointwise in IR, and by the Lebesgue theorem (see Theorem 1.16) also weak-∗ in M (IR). So, by (2.3.156), (2.3.157) we have proved as ε → 0+ weak-∗ in M (IR) µϕε2 ∗ ϕε3 ∗ (s2 − s3 )∂22 χ2 ∂32 χ3 (ρ, v, ·) → 0 uniformly with respect to ρ, v.

(2.3.159)

Let us find the weak-∗ limit of the expression (∂22 χε2 ∂3 χε3 −∂2 χε2 ∂32 χε3 ) in (2.3.156). According to (2.3.152) we have

= 4 k ϕε2 (s1 − v + k) + ϕε2 (s1 − v − k) ∂22 χε2 ∂3 χε3 − ∂2 χε2 ∂32 χε3 s=s1

+


ϕ2 (σ) dσ

v+k v−k

ϕε3 (s1 − s3 )(s3 − v) ds3

−4 k ϕε3 (s1 − v + k) + ϕε3 (s1 − v − k)

s1 −v+k v+k ε + s1 −v−k ϕ3 (σ) dσ v−k ϕε2 (s1 − s2 )(s2 − v) ds2 .

(2.3.160)

ε

By (2.3.158)

v+k v−k

(sj − v)ϕεj (s1 − sj ) dsj = (s1 − v)

v+k v−k

ϕεj (s1 − sj ) dsj + εrjε , where |rjε | ≤ C < ∞.

Clearly, ε∂j2 χεj → 4k 1{v−k} (s1 ) + 1{v+k} (s1 ) , so that

ε r3ε ∂22 χε2 − r2ε ∂32 χε3 → 0 pointwise,

(2.3.161)

(2.3.162)

and so by the Lebesgue theorem weakly-∗ in M (IR) (uniformly with respect to ρ, v). Further, given y, w, z ∈ IR, η ∈ CB (IR), we have ε Iij :=

=

IR

IR

ϕi (s1 − y) ϕi (σ)

s1 −z ε s1 −w ε

y−z ε +σ y−w ε +σ

ϕj (τ ) dτ η(s1 )(s1 − v) ds1

ϕj (τ ) dτ η(y + εσ)(y + εσ − v) dσ.

Putting here respectively y = v + k, w = v + k, z = v − k and y = v − k; w, z the same, we respectively get

WEAK SOLUTIONS

σ ϕi (σ) −∞ ϕj (τ ) dτ dσ η(v + k), and

∞

ε = −k IR ϕi (σ) σ ϕj (τ ) dτ dσ η(v − k) limε→0+ Iij

σ

= −k IR ϕj (σ) −∞ ϕi (τ ) dτ dσ η(v − k).

ε =k limε→0+ Iij

143

IR

(2.3.163)

Putting together relations (2.3.156)–(2.3.163) we finally obtain

∂22 χε2 ∂32 σ3ε − ∂32 χε3 ∂22 σ2ε → 4k(ρ)2 Y (ϕ2 , ϕ3 ) δs1 =v+k(ρ) + δs1 =v−k(ρ) , (2.3.164)

where

Y (ϕ2 , ϕ3 ) :=

s2 IR −∞

ϕ2 (s2 )ϕ3 (s3 ) − ϕ3 (s2 )ϕ2 (s3 ) ds3 ds2 .

(2.3.165)

By integration with respect to the measure ν we get

% $ χ1 P2 χε2 P3 σ3ε − P3 χε3 P2 χε2 → 4χ1 Y (ϕ2 , ϕ3 )k(ρ) δs1 =v−k + δs1 =v+k $

% = 4χ1 Y (ϕ2 , ϕ3 ) k(ρ) δs1 =v−k + δs1 =v+k . So for any η ∈ CB (IR) we have

∞ Y (ϕ2 , ϕ3 ) 0 IR χ1 (v + k(ρ))k(ρ)η(v + k(ρ))

+χ1 (v − k(ρ)k(ρ)η(v − k(ρ) dν(ρ, v) = 0.

(2.3.166)

0

−1 s3 −1 ϕ2 2 , ϕ2 such that ϕ2 (s) > Choose, for example, ϕ3 (s3 ) = 2 −1 ϕ2 (s)ds 0 for s ∈ (−1, 1). Then ϕ3 is a regularization kernel whenever ϕ2 is, and

−1 s2 −1 0 2 (2.3.167) Y (ϕ2 , ϕ3 ) = −1 ϕ2 (s) ds ϕ (s )ϕ2 (s) ds ds2 = 0 IR 2s2 +1 2 2

since the intersection of the range of integration with the cube (−1, 1)2 has a positive two-dimensional measure. Dividing (2.3.166) by Y (ϕ2 , ϕ3 ), using nonnegativity of summands and the definition of the symbol ·, we can write (2.3.166) in the form

∞

∞ v±k(ρ)+k(ρ ) k(ρ) 0 v±k(ρ)−k(ρ11) k(ρ1 )2 0 {ρ>0} (2.3.168)

2 − v1 − (v ± k(ρ)) dν(ρ1 , v1 ) dν(ρ, v) = 0.

By positivity, ν = 0 on the set

2

∞ v±k(ρ)+k(ρ1 ) k(ρ1 )2 − (v1 − (v ± k(ρ)) dν(ρ1 , v1 ) > 0 . ± (ρ, v); 0 v±k(ρ)−k(ρ1 ) (2.3.169) ± ± Denoting ψ := v ±k(ρ ) and, as before, ψ = v±k(ρ), we see that the functions 1 1 1 k(ρ1 )2 − (v1 − ψ ± )2 + are respectively strictly positive when |v1 − ψ ± | < k(ρ1 ), respectively. This yields ν = 0 in the set − + ± ± − + (ρ , (2.3.170) , v ); ψ < ψ , ψ > ψ − + 1 1 1 1 ± ψ ≤ψ 0. In addition, w ∈ C([0, ∞), L1loc (IR)) and the initial condition (2.2.10)2 is satisfied in the sense of the traces in this space, i.e. w(·, 0) = w0 in L1loc (IR).

146


Proof In the original proof of J.Glimm (Glimm, 1965), an approximative family wh is constructed by means of a sequence of solutions of Riemann problems ∂t wh +

N

wh (x, nτ ) =

i=1

!

∂i f i (wh ) = 0,

wnm−1 , x < mh, wnm+1 , x ≥ mh.

,

w0m = w0 (mh), where m, n are integers, h and τ are the space step and the time step, respectively, satisfying the Courant–Friedrichs–Loewy condition (cf. (2.3.104) for the 2 × 2 system)

supk,x,t |λk wh (x, t) | < hτ , (λk (w) are the eigenvalues of

Df Dw (w)).

The condition ensures that the solutions corresponding to adjacent jumps have equal outer constant state and thus a single-valued approximation can be constructed. Then the following apriori estimates are established: wh (·, t) − w0,∞ ≤ C0 w0 − w0,∞ , t ∈ [0, ∞),

T VIR wh (·, t) ≤ C0 T VIR (w0 ), t ∈ [0, ∞),

wh (·, t1 ) − wh (·, t2 )0,1 ≤ C0 |t1 − t2 | + h T VIR (w0 ), t1 , t2 ∈ [0, ∞),

for some constant C0 > 0. After that a compactness argument using the Helly theorem (see Theorem 1.59) is applied to obtain an accumulation point of the family {wh }0 0, sup x∈IR,j=0,1

(j)

(j)

{ρ0 , |v0 |} < ∞,

and the Riemann invariants evaluated at the initial data satisfy γ−1 γ−1 2 2 d d 2 ρ0 2 v0 − γ−1 ≥ 0, dx ≥ 0, x ∈ IR. dx v0 + γ−1 ρ0

(2.4.1)

Then Cauchy problem (2.3.92), (2.3.93) has a unique global solution (ρ, v) ∈ C 1 (IR)2 with ρ > 0 in IR × (0, ∞). The proof of Theorem 2.45 can be found in (Chen and Wang, 2001). 2 For the global existence of smooth solutions of general one-dimensional hyperbolic systems of conservation laws we refer to (Li, 1994) which gives some

148


implications also for the Euler system. The discussion of the global existence of Lipschitz–continuous solutions developing from discontinuous initial data that may not stay away from the vacuum is performed in (Lin, 1987). For the three-dimensional Euler equations (1.2.68)–(1.2.71) Serre (1997) studied the existence of global smooth solutions under appropriate assumptions on the initial data. Similar analysis was performed in (Grassin and Serre, 1997) for the isentropic Euler equations (2.3.92) in nonconservative form. Finally, it was shown in (Godunov and Romenskij, 1998) that the three-dimensional Euler equations for a polytropic gas, given by (1.2.68), (1.2.69), (1.2.64) and (1.2.91), have a global smooth solution under the condition that the initial entropy S0 and the initial density ρ0 are small enough and the initial velocity forces particles to spread out. This condition is of a similar nature to condition (2.4.1). 2.4.3

Blow up and the lifespan of smooth solution

By the lifespan of a classical solution to a general (abstract) differential equation, say of the type dw (2.4.2) t ∈ (0, T0 ), T0 > 0, dt (t) = A(w)(t),

where A is an operator in a Banach space X, we mean

Tmax := sup{T > 0; ∃ w ∈ C 1 ((0, T ), X) ∩ C([0, T ), X) satisfying (2.4.2)}, (cf. definition of Tmax in (2.3.1)). Consider again problem (2.3.92), (2.3.93) with p(ρ) = ργ /γ, γ ∈ (1, 3). Then by (2.3.108) the Riemann invariants evaluated at initial data are given by ψ ± = ψ0± = v0 ±

γ−1 2 ρ0 2 . γ−1

The following theorem, the proof of which can be found in (Chen and Wang, 2001), holds true: Theorem 2.46 The lifespan of any C 1 solution of problem (2.3.92), (2.3.93) satisfying ρ > 0 is finite for any initial data (ρ0 , v0 ), ρ0 > 0 with finite C 1 norm and dψ0− dψ0+ dx (β) < 0 or dx (β) < 0, for some point β ∈ IR. Furthermore, if there exist positive constants δ and ε such that inf x∈IR ψ0+ (x) − supx∈IR ψ0− (x) = δ

and, for some point β ∈ IR,

dψ0+ dx (β)

≤ −ε,

or

dψ0− dx (β)

≤ −ε,

then the lifespan of any C 1 solution of (2.3.92), (2.3.93) does not exceed T∗ =

4 (γ+1)ε

γ−1 4 δ

γ−3 2(γ−1)

supx∈IR |ρ0 (x)|

γ−3 4

.

FINAL COMMENTS

149

Results on the lifespan and blow up for general systems of quasilinear hyperbolic systems in one dimension are collected in (Alinhac, 1995) and references therein. The following representative result is due to John (1974). Theorem 2.47 Consider the Cauchy problem (2.2.10) under the assumption of genuine nonlinearity. Let the nontrivial initial data w0 ∈C 2 (IR)N have compact support. Then there exists a positive constant δ > 0 such that, if 0 < sup |w′′0 (x)| ≤ δ, x∈IR

then the solution of (2.2.10) cannot exist in the class C 2 for all positive t. This result was generalized in (Liu, 1979) to include systems with linearly degenerate characteristic fields such as the Euler equations. The next result concerns the lifespan of the classical solution to the Cauchy problem for the threedimensional Euler equations for a polytropic gas. Consider equations (1.2.68)– (1.2.70), (1.2.64) with p = p(ρ, S) = ργ eS , γ > 1 and the initial data (ρ, v, S)|t=0 = (ρ0 , v 0 , S0 ) ∈ C(IR)5 , ρ0 > 0 in IR3

(2.4.3)

satisfying (ρ0 , v 0 , S0 )(x) = (ρ, 0, S) for |x| ≥ R, where ρ > 0, S, and R ≥ 0 are given constants. It may be deduced from (John, 1981) that the support of (ρ0 (x) − ρ, v 0 , S0 − S) propagates ) with the sound speed corresponding to the state (ρ, 0, S), i.e. with the speed a = ∂ρ p(ρ, S). This fact implies (ρ, v, S)(x, t) = (ρ, 0, S) for |x| ≥ R + at.

Take p := p(ρ, S) and define

P (t) := IR3 p(x, t)1/γ − p1/γ dx

= IR3 ρ(x, t) exp S(x, t)/γ − ρ exp S/γ dx,

F (t) = IR3 x · ρv(x, t) dx.

The following result on the lifespan of the maximal solution to the problem (1.2.69), (1.2.70), (1.2.64), (2.4.3) can be found in (Sideris, 1985).

Theorem 2.48 Suppose that (ρ, v, S) is a C 1 solution of (1.2.69), (1.2.70), (2.4.2) on IR3 × (0, T )(T > 0) with p = ργ eS , γ > 1, and P (0) ≥ 0,

F (0) >

16π 4 3 aR

supx∈IR3 ρ0 (x).

Then the lifespan of the C 1 solution is finite; more precisely, −1 4 . (R + aT )4 < R4 F (0) F (0) − 16π 3 aR supx∈IR3 ρ0 (x)

(2.4.4)

150


For a refinement of the proof of Theorem 2.48 see (Chen and Wang, 2001), Theorem 4.3. It is clear (Chen and Wang, 2001), Remark 4.3, that condition (2.4.4) can be satisfied if we choose, for example, ρ0 = ρ, S0 = S. Then P (0) = 0 and it suffices to assume

4 x · v 0 (x) dx > 16π 3 aR . |x| 4a,

|x| 1;

151

q|r=1 = 0.

(2.4.7)

Notice that system (2.4.6) also has a self-contained interpretation as a model for transonic nozzle flow with variable cross-sectional area A(r). For the stationary problem corresponding to (2.4.6) with p = ργ (γ > 1), explicit solutions can be found in (Chen and Wang, 2001), p. 515. If in (2.4.6), A′ (r) ≡ 0, then the system becomes the one-dimensional isentropic Euler equations studied in Section 2.3.14. For A′ (r) = 0, the existence of global solutions for the transonic nozzle flow problem is surveyed in (Chen and Wang, 2001), p. 515. For system (2.4.6) the following theorem analogous to Theorem 2.37, for isentropic gas dynamics system, has been proved in Chen and Glimm (1996). Theorem 2.49 Let p(ρ) = κργ , κ > 0, γ > 1. There exists a family of approximate solutions (ρε , q ε ) of (2.4.6), (2.4.7) such that for any T ∈ (0, ∞), there is C = C(T ) < ∞ independent of ε so that, when t ∈ [0, T ], (i) 0 ≤ ρε (r, t) ≤ C, |q ε (r, t)| ≤ Cρε (r, t), r > 1; −1 (ii) {∂t η(ρεk , q εk ) + ∂r G(ρεk , q εk )}∞ k=1 is relatively compact in Hloc (Ω) for any + εk → 0 and any weak entropy–entropy flux pair (η, G), where Ω ⊂ IR × (0, ∞) or Ω ⊂ (1, ∞) × (0, ∞). Furthermore, there is a convergent subsequence (ρεℓ , q εℓ )(r, t) of the approximate family of solutions (ρε , q ε )(r, t) such that (ρεℓ , q εℓ )(r, t) → (ρ, q)(r, t),

a.e.

and the limit function (ρ, q)(r, t) is a global admissible weak solution of (2.4.6) with the assigned initial data in L∞ and satisfies 0 ≤ ρ(r, t) ≤ C,

|q(r, t)| ≤ Cρ(r, t).

Moreover, (ρ, q)(x, t), defined in (2.4.5) through the solution (ρ, q)(r, t) (2.4.6)– (2.4.7) is a global admissible (entropy) weak solution of (1.2.68), (1.2.69) and (1.2.102) with spherical symmetry outside the solid core for the initial data in L∞ . Further related results concerning entropy solutions with symmetric structure can be found in (Chen, 1997), (Courant and Friedrichs, 1962), (Makino et al., 1992), (Slemrod, 1996) (Whitham, 1974), (Zhang and Zheng, 1998), (Zhang and Zheng, 1997). 2.4.5

Riemann problem

The Riemann problem is an important theoretical and numerical tool for construction of approximations of exact or approximative solutions. Nevertheless, it is of importance also as a model in physical situations. One example is the so-called shock tube problem which consists in the description of the evolution of the gas of two different densities initially distributed respectively left and right

152


of a membrane which is removed at time t = 0, or the piston problem, if the membrane is substituted by a piston moving with the gas due to the difference of pressures between both its sides (Chorin and Marsden, 1979), Chapter 3, p. 142. In Section 2.3.14 we used a sequence of Riemann problems for the definition of approximations of the exact solution to the 2 × 2 Euler system for an isentropic gas. The theory of the Riemann problem is a huge thema difficult to grasp and we restrict ourselves to pointing out a few monographs and further references. The basic theory of the Riemann problem for a one-dimensional system of conservation laws with applications to isentropic gas dynamics is presented, e.g., in (Bressan, 2000), Chapter 5. For the two-dimensional Riemann problem in gas dynamics we refer to (Jiequan et al., 1998). Here one can find results for the Riemann problem for zero-pressure gas dynamics, i.e. p = 0 in (1.2.69), (1.2.70) in both one and two dimensions, similar results for pressure gradient equations of the Euler system, and analysis of the Riemann problem for equations (1.2.69)–(1.2.71) with p = (γ − 1)ρe, (polytropic gas). A special study of the axisymmetric case is included. Further extensive references to the Riemann problem are given in (Chen and Wang, 2001), Section 10.2. A connection of uniqueness for 2 × 2 laws to hysteresis is shown in (Krejˇc´ı and Straˇskraba, 1997) and (Krejˇc´ı, 1996), Chapter IV. A simple uniqueness criterion is obtained generalizing other known uniqueness criteria. Nonconvex state functions and nonsmooth flow functions are included. 2.4.6 Euler equations with source terms By a system of conservation laws with a source term we mean the system of the form N (2.4.8) ∂t w + j=1 ∂j f j (w) = g(w), x ∈ IRN , t ∈ (0, T ).

Physically, source terms arise when an additional supply of physical quantities (momentum, energy, heat, etc.) is provided into the whole region of the flow (i.e. not only through the boundary of the flow region). There are at least two important cases to be mentioned: (i) The Euler equations with relaxation, where g(w) = −ε−1 S(w), with a short relaxation time ε and S(w) physically suitable; (ii) the Euler equations for chemically reacting fluids, if a chemical reaction like burning or combustion can arise. For a survey of the Euler equations with source terms see (Chen and Wang, 2001), Section 11. 2.4.7 Comments on the 2 × 2 Euler system for an isentropic fluid In Section 2.3.14 an L∞ entropy (admissible) solution to the system for an isentropic fluid (2.3.92), (2.3.93) is constructed via Lax–Friedrichs difference approximations. As pointed out in Section 2.1.3, system (2.3.92), (2.3.93) can be

FINAL COMMENTS

153

written in the form (2.1.13) if transformed to Lagrangian mass coordinates, when excluding vacuum states. This transformation assigns to each regular solution of the Euler system in Euler coordinates a regular solution of the Euler system in Lagrangian mass coordinates (p-system), and vice versa. An analogous relation is not evident for weak solutions but it holds as shown in (Wagner, 1987) and (Mizohata, 1994). Thus the results for weak solutions of the p-system have corresponding consequences for the Euler equations in Lagrangian mass coordinates and vice versa. The system of nonlinear elasticity is usually written in the form (2.1.13) with p(V ) := −σ(V ) under the assumption

(2.4.9) sgn τ σ ′′ (τ ) > 0, if τ = 0. The mere condition (2.4.9) does not guarrantee that the system is genuinely nonlinear. Nevertheless, the existence of a global entropy solution in L∞ has been proved in (DiPerna, 1983a). Analogous results in the Lp (p < ∞) framework are reviewed in (Dafermos, 2000), Theorem 15.7.4 and Section 15.9, and details can be found in (Shearer, 1994), (Lin, 1987). In addition to discrete approximation methods like the Lax–Friedrichs difference scheme, the so-called vanishing viscosity method is often used. For example, using parabolic regularity existence results we can construct approximations of solutions to the p-system (2.1.13) as solutions of the system ∂t Vµ − ∂x uµ = ε(µ)∂xx Vµ ,

∂t uµ + ∂x p(Vµ ) = µ∂xx uµ ,

(2.4.10)

where µ > 0 and ε(µ) ≥ 0 are small. The initial data, if not smooth enough, should also be approximated by smoother functions depending on µ. If ε(µ) = µ, then it is known (Smoller, 1983), Chapter 21, §B, that problem (2.4.10) has bounded invariant regions in a similar sense as in Section 2.3.17. Consequently, the family {(Vµ , uµ )}0 0 and a similar compactness argument as in Section 2.3.14 can be used to obtain an admissible solution to problem (2.1.13). For more details about the use of these approximations we refer to (DiPerna, 1983b), (Lions et al., 1996), (Serre, 1996), Vol. I, Chapter 6 and Vol. II, Chapter 15. It is worth noting that a natural physical approximation corresponds to the choice ε(µ) ≡ 0 in (2.4.10) which leads to a qualitatively similar problem to onedimensional Navier–Stokes equations in Lagrangian mass coordinates. But it is not known whether in that case system (2.4.10) enjoys the invariance property as in the former case. Only a partial result is known (Straˇskraba, 1996), namely 1 = Vµ ≥ V > 0 independently of µ, ρµ and so we are not able to establish the L∞ scheme for construction of a solution with the help of this kind of approximations (Vµ , uµ ), and thus only the Lp (p < ∞) framework is available as mentioned above.

154


The global solutions to the system of isentropic gas dynamics (2.1.13) with large data were first treated by the method of compensated compactness in (DiPerna, 1983a), for the special values γ = 1 + n2 , n = 2ℓ + 1, for the adiabatic exponent. Further history of this problem which ends up with the recent result of Chen and LeFloch (2000) contained in Theorem 2.37, can be found in (Dafermos, 2000). The viscosity method is elaborated in detail in a recent monograph (Lu, 2003), where also other special systems connected with the Euler equations are studied. An abstract semigroup approach for a strictly hyperbolic 2 × 2 system of conservation laws in one space dimension has been used in (Bressan and Colombo, 1995). A nonlinear one-parameter semigroup assigning to each initial function from a subset of the space L1 (IR)2 of functions, having small total variation over IR, the value of the solution at a given time t, is constructed and shown to be Lipschitz–continuous with respect to time and the initial states. 2.4.8

Euler equations for a nonisentropic fluid

Equations (1.2.68)–(1.2.71) in Lagrangian mass coordinates read ∂t V − ∂x u = 0,

∂t u − ∂x p = 0,

u2 + ∂xx (pV ) = 0. ∂t e + 2

It is shown in (Chen and Dafermos, 1995), (see also (Chen and Wang, 2001), Section 9.8 for short version), that under an appropriate choice of constitutive relations the model can be regarded as a “first-order correction” to the general constitutive laws, and existence and compactness for weak entropy formulation of the Cauchy problem and the decay of periodic solutions can be established. For more general constitutive relations like those for polytropic gases with p = (γ − 1)ρe the problems of existence, compactness, and decay of entropy solutions with arbitrarily large data are still open. Another method is the relaxation approximation scheme which is formulated in (Dafermos, 2000), Section 15.5 for scalar conservation laws and reviewed therein in Section 15.9 The role of dissipative mechanisms in quasilinear hyperbolic systems is addressed in (Hsiao, 1997).

3 SOME MATHEMATICAL TOOLS FOR COMPRESSIBLE FLOWS This chapter deals with several auxiliary results of functional analysis which will be broadly used especially in Chapters 4 and 7. In Section 3.1 we prove Friedrich’s lemma about commutators and investigate the renormalized solutions to the steady continuity equation. Section 3.2 is devoted to spaces of vector fields with summable divergence. In Section 3.3 we investigate the equation div v = f in various domains with compact or noncompact boundaries. In Section 3.4 we recall some useful results from the theory of monotone operators and from convex analysis. Let us note that since in what follows, involved integral expressions are abundant, we will omit differentials under integral signs whenever there will be no doubt about the interpretation of the integration in question. Thus we write e.g.

simply Ω f instead of Ω f (x)dx, etc.

3.1

Renormalized solutions of the steady continuity equation

In this section we shall list several results concerning the continuity equation which will be needed to build up the mathematical theory of the steady compressible barotropic Navier–Stokes equations. These results mostly follow from DiPerna–Lions transport theory (DiPerna and Lions, 1989). Their proofs, as presented here, are adaptations of the proofs published in (DiPerna and Lions, 1989), and we perform them only for the sake of completness. They rely on DiPerna’s and Lions’ generalization of Friedrichs’ commutator lemma (see Lemma 3.1). Lemma 3.2 about the prolongation of the continuity equation to the whole space is the steady version of the same result proved for the nonstationary case, e.g., by Feireisl, Novotn´ y and Petzeltov´ a in (Feireisl et al., 2001). 3.1.1

Friedrichs’ lemma about commutators

We recall the following generalization of the so-called Friedrichs’ lemma about commutators.. Lemma 3.1 Suppose that N ≥ 2. Let 1 ≤ q, β ≤ ∞, (q, β) = (1, ∞), and let 1,q (IRN ))N . ρ ∈ Lβloc (IRN ), u ∈ (Wloc

1 q

+

1 β

≤1

Then Sǫ (u·∇ρ)−u·∇Sǫ (ρ) → 0 strongly in Lrloc (IRN ), 155

r ∈ [1, q) if β = ∞, q ∈ (1, ∞] 1 1 1 β + q ≤ r ≤ 1 otherwise,

156

SOME MATHEMATICAL TOOLS

where Sǫ is the usual mollifier (see Section 1.3.4.4) and u · ∇ρ := div (ρu) − ρdiv u. Proof (i) Using the definition of the distribution u · ∇ρ and the definition of convolution of a distribution with a smooth function with compact support (cf. Section 1.3.4.3), we easily find that Sǫ (u · ∇ρ), ϕ =

[ρu](y) · IRN ∇ωǫ (x − y)ϕ(x) dx dy

− IRN [ρdiv u](y) IRN ωǫ (x − y)ϕ(x) dx dy, ϕ ∈ D(IRN ).

IRN

In virtue of Tonelli’s theorem (cf. Theorem 1.9), we have F, G ∈ L1 (IRN × IRN ), where F (x, y) = [ρu](y)·∇ωǫ (x−y)ϕ(x) and G(x, y) = [ρdiv u](y)ωǫ (x−y)ϕ(x). Therefore, Fubini’s theorem can be applied to both integrals on the right-hand side; one thus obtains

Sǫ (u · ∇ρ), ϕ = IRN ϕ(x) IRN [ρu](y) · ∇ωǫ (x − y) dy dx

− IRN ϕ(x) IRN ωǫ (x − y)[ρdiv u](y) dy dx. On the other hand, since

u · ∇Sǫ (ρ), ϕ = we can write

IRN

ϕ(x) IRN u(x) · ∇ωǫ (x − y)ρ(y) dy dx,

Sǫ (u · ∇ρ) − u · ∇Sǫ (ρ), ϕ = where

IRN

Iǫ (x)ϕ(x) dx −

IRN

Jǫ (x)ϕ(x) dx, (3.1.1)

ρ(y)[u(y) − u(x)] · ∇ωǫ (x − y) dy,

Jǫ (x) = IRN ωǫ (x − y)[ρdiv u](y) dy.

Iǫ (x) =

IRN

(3.1.2)

(ii) In this part of the proof we show that

˜ Iǫ 0,r0 ,BR ≤ c(R, β)ρ ˜ R+1 ∇u0,q,BR+2 , 0,β,B

(3.1.3)

where   ∈ [1, ∞) r0 : ∈ (1, q)  1 = 1+ r0 β

 if β = q = ∞  r0 if β = q = ∞ r0 q if β = ∞, q ∈ (1, ∞) , β˜ = q−r0 if β = ∞, q ∈ (1, ∞)  1 β if 1 ≤ β < ∞. q otherwise (3.1.4) Taking into account the position of the support of ω and the definition of ωǫ , we can write

RENORMALIZED SOLUTIONS OF THE CONTINUITY EQUATION 0 = Iǫ r0,r 0 ,BR

=

ρ(y)[u(y) − u(x)] · BR |x−y|≤ǫ

u(x) ρ(x − ǫz) u(x−ǫz)− ǫ BR |z|≤1

157

r 0 ) dy ∇ω( x−y dx ǫ r0 · ∇ω(z) dz dx 1

ǫN +1

r0′ r0′

r0 r0 u(x−ǫz)−u(x) r0 − ǫz) dz dz dx ∇ω(z) ρ(x ǫ |z|≤1 BR |z|≤1

r0 u(ξ) r0 dz dξ =: cA . ≤ c BR+1 |z|≤1 ρ(ξ) u(ξ+ǫz)− ǫ ǫ

≤

Here, to pass from the first line to the second one, we used the change of variables older’s z = x−y ǫ , to pass from the second line to the third one, we applied H¨ inequality and, finally, to pass from the third line to the last one, we employed the change of variables ξ = x − ǫz and assumed that ǫ ∈ (0, 1). The integral Aǫ can be majorized as follows:  r0

rq0 r 0

β˜ u(ξ+ǫz)−u(ξ) q   dξ B dz ρ(ξ) 1  ǫ BR+1 |z|≤1    if 1 ≤ β, q < ∞ or if 1 < q < ∞, β = ∞, Aǫ ≤ r0  u(ξ+ǫz)−u(ξ) r0

  | ess sup |B dξ ρ(ξ) 1  (ξ,z)∈BR+1 ×B1 ǫ BR+1   if q = ∞, 1 ≤ β ≤ ∞  r0

r0

rq0

β˜ u(ξ+ǫz)−u(ξ) q  β˜ β˜  |ρ| [ dz] dξ B 1  ǫ BR+1 BR+1 |z|≤1    if 1 ≤ β, q < ∞ or if 1 < q < ∞, β = ∞, ≤ r0  u(ξ+ǫz)−u(ξ) r0

  |B | ess sup dξ ρ(ξ) 1  (ξ,z)∈B ×B 1 ǫ R+1 BR+1   if q = ∞, 1 ≤ β ≤ ∞.

To get the first bound, we used H¨ older’s inequality in the inner integral; for the passage to the next bound, we employed once more Hölder’s inequality, but now in the outer integral. Now we apply Lagrange’s formula u(ξ + ǫz) − u(ξ) =

1 ǫ 0 z · ∇u(ξ + tǫz)dt (cf. formula (1.3.50)) to get (3.1.3) in all considered cases. (iii) By elementary properties of mollifiers, we have

Jǫ → ρdiv u strongly in Lr0 (IRN )

(3.1.5)

(cf. Lemma 1.34). By virtue of (3.1.2), it is therefore enough to prove 0 Iǫ → ρdiv u strongly in Lrloc (IRN ).

(3.1.6)

By virtue of (3.1.3) it is enough to show that (3.1.6) is valid with any ρ ∈ C0∞ (IRN ). ˜

Indeed, with ρn ∈ C0∞ (IRN ), ρn → ρ strongly in Lβ (BR+1 ), there holds

(3.1.7)

158


Iǫ − ρdiv u0,r0 ,BR+1 ≤

(ρ(y) − ρn (y))(u(y)

−u(·)) · ∇ωǫ (· − y) dy0,r0 ,BR+1 + IRN ρn (y)(u(y) IRN

−u(·)) · ∇ωǫ (· − y) dy − ρn div u0,r0 ,BR+1 + (ρn − ρ)div u0,r0 ,BR+1 . The first norm at the right-hand side is bounded by ˜ n − ρ ˜ c(R, β)ρ 0,β,BR+2 ∇u0,q,BR+3 (cf. (3.1.3)), where we replace ρ by ρn − ρ), and so it tends to 0 as n → ∞. The second norm tends to 0 as ǫ → 0+ (cf. (3.1.7)). The last norm is bounded by ˜ n − ρ ˜ c(R, β)ρ 0,β,BR+1 div u0,q,BR+1

and tends to 0 as n → ∞.

(iv) Thus, it remains to prove (3.1.7). Effecting the change of variables z = we can write

u(x) · ∇ω(z) dz. Iǫ (x) = |z|≤1 ρ(x − ǫz) u(x−ǫz)− ǫ

x−y ǫ ,

1,q Due to the hypothesis u ∈ (Wloc (IRN ))N , we have u(x−ǫz)−u(x) = −z · 1 ∇u(x − ǫtz) dt ǫ 0

→ z · ∇u(x) for a.a. (x, z) ∈ IRN × B1

(see viii) in Section 1.1.14.1). By virtue of the regularity of ρ, ρ(x − ǫz) → ρ(x), (x, z) ∈ BR+1 × B1 . Therefore, by Vitali’s convergence theorem, we obtain

I ϕ → − z ∂ ω(z) dz ρ(x)ϕ(x)∂i uj (x) dx = IRN ρdiv uϕ ǫ i j N IR B1 IRN

and

Iǫ 0,r0 ,BR+1 → ρdiv u0,r0 ,BR+1 .

This completes the proof.

2

3.1.2 Continuity equation and its prolongation Lemma 3.2 Let N ≥ 2 and let K be a compact subset of a bounded Lipschitz ′ domain Ω ⊂ IRN . Let ρ ∈ L2 (Ω) ∩ L2 (Ω \ K) (see (1.3.64)), u ∈ (W01,2 (Ω))N and f ∈ L1 (Ω). Suppose that div (ρu) = f in D′ (Ω).

(3.1.8)

Then, prolonging ρ, u and f by zero outside Ω and denoting the new functions again by ρ, u and f , we have div (ρu) = f in D′ (IRN ).

RENORMALIZED SOLUTIONS OF THE CONTINUITY EQUATION

Proof We have to show

− IRN ρu · ∇η = IRN f η, η ∈ D(IRN )

159

(3.1.9)

provided ρ, u and f are prolonged by 0 outside Ω. (Due to the assumptions, ρu ∈ L1 (IRN )N and the integral on the left-hand side of (3.1.9) makes sense.) To this end consider a sequence of functions Φm ∈ D(Ω), m ∈ IN, 0 ≤ Φm ≤ 1, Φm (x) = 1, x ∈ {y ∈ Ω; dist(y, ∂Ω) ≥

1 m },

(3.1.10)

|∇Φm (x)| ≤ 2m, x ∈ Ω. Clearly, Φm → 1 pointwise in Ω, supp∇Φm ⊂ Ω \ K for m ≥ m0 (K) ∈ IN , |supp∇Φm | → 0.

(3.1.11) (3.1.12)

Due to (3.1.8), we have

f Φm η + IRN Φm ρu · ∇η + IRN ηρu · ∇Φm = 0. (3.1.13) IRN

The first two integrals in (3.1.13) tend to Ω f η + Ω ρu · ∇η = IRN f η + IRN ρu · ∇η by virtue of (3.1.11) and the Lebesgue dominated convergence theorem (see Section 1.1.18.4). The third integral is majorized by 2 sup |η(x)| ρ0,2,supp∇Φm x∈IRN

u 0,2,Ω . dist(·, ∂Ω)

(3.1.14)

In accordance with (3.1.12) and since ρ ∈ L2 (Ω \ K), ρ0,2,supp∇Φm → 0 (cf. u ∈ (L2 (Ω))N by Hardy’s Section 1.1.17). Since u ∈ (W01,2 (Ω))N , we have, dist(·,∂Ω) inequality (see Section 1.3.5.9). Therefore

ηρu · ∇Φm → 0. (3.1.15) IRN Hence (3.1.13) yields (3.1.9) as m → ∞, completing the proof of Lemma 3.2.

3.1.3 3.1.3.1 ctions

2

Renormalized solutions of the continuity equation Renormalizing functions with continuous derivatives Consider the fun-

b ∈ C 0 ([0, ∞)) ∩ C 1 ((0, ∞)), |b′ (t)| ≤ ct−λ0 , t ∈ (0, 1], λ0 < 1

(3.1.16)

with growth conditions at infinity |b′ (t)| ≤ ctλ1 , t ≥ 1, where c > 0, −1 < λ1 < ∞.

(3.1.17)

160


Lemma 3.3 Let N ≥ 2, 2 ≤ β < ∞, and λ1 ≤

β − 1. 2

(3.1.18)

1,2 Further, let ρ ∈ Lβloc (IRN ), ρ ≥ 0 a.e. in IRN , u ∈ (Wloc (IRN ))N and f ∈ ′ Lzloc (IRN ), where z = λβ1 if λ1 > 0 and z = 1 if λ1 ≤ 0. Suppose that

div (ρu) = f in D′ (IRN ).

(3.1.19)

(i) Then for any function b ∈ C 1 ([0, ∞)) satisfying (3.1.17), (3.1.18), div (b(ρ)u) + {ρb′ (ρ) − b(ρ)}div u = f b′ (ρ) in D′ (IRN ).

(3.1.20)

(ii) If f = 0, then (3.1.20) holds with any b satisfying (3.1.16), (3.1.17), (3.1.18). Proof Due to the possible singularity of b′ at 0, the proof of part (ii) of the lemma is more difficult than that of part (i). We therefore concentrate on this part and we leave the proof of part (i) to the interested reader as an exercise. Regularizing (3.1.19) over the space variables using the standard mollifier Sǫ , ǫ > 0 (cf. Section 1.3.4.4), we get div (Sǫ (ρ)u) = rǫ a.e. in IRN ,

(3.1.21)

rǫ = div (Sǫ (ρ)u) − div (Sǫ (ρu)).

(3.1.22)

where Due to Lemma 3.1, rǫ → 0 in Lrloc (IRN ), 1r = β1 + 21 . We multiply equation (3.1.21) by b′(h) (Sǫ (ρ)), where b(h) (·) = b(· + h), h ∈ (0, 1), to get div (b(h) (Sǫ (ρ))u) + {Sǫ (ρ)b′(h) (Sǫ (ρ)) − b(h) (Sǫ (ρ))}div u = rǫ b′(h) (Sǫ (ρ)) a.e. in IRN .

(3.1.23)

Now, we pass to the limit ǫ → 0+ . Clearly, Sǫ (ρ) → ρ in Lβloc (IRN ) and therefore a.e. in IRN . By Vitali’s convergence theorem (see Section 1.1.18.4), by virtue of the growth conditions (3.1.17), (3.1.18), we have b(h) (Sǫ (ρ)) → b(h) (ρ) and

{Sǫ (ρ)b′(h) (Sǫ (ρ))

− b(h) (Sǫ (ρ))} → {ρb′(h) (ρ) − b(h) (ρ)}

(3.1.24)

in Lploc (IRN ), 1 ≤ p < 2. Since these sequences are bounded in L2 (Ω′ ) for any bounded subdomain Ω′ ⊂ IRN , we conclude that the limits (3.1.24) are L2weak (Ω′ )-limits as well. This gives the convergence to the corresponding terms on the left-hand side of (3.1.23). The L1 -norm of the right-hand side over any

RENORMALIZED SOLUTIONS OF THE CONTINUITY EQUATION

161

bounded measurable Ω′ ⊂ IRN is bounded by b′(h) (Sǫ (ρ))0,r′ ,Ω′ rǫ 0,r,Ω′ and therefore tends to zero. Hence, equation (3.1.23) yields div (b(h) (ρ)u) + {ρb′(h) (ρ) − b(h) (ρ)}div u = 0 in D′ (IRN ).

(3.1.25)

Now we let h → 0+ . One realizes that b(h) (t) → b(t), tb′(h) (t) − b(h) (t) → tb (t) − b(t) for all t ∈ [0, +∞). Further, for any k > 1, ′

max |b(h) (t)| ≤ max |b(t)| [0,k]

[0,2k]

and also, due to (3.1.16), we have max |tb′(h) (t) − b(h) (t)| ≤ max |tb′ (t) − b(t)| + max |b′ (t)| [0,k]

[0,2k]

[0,2k]

for all h ∈ (0, 1). Finally, for any k > 1 and for any Ω′ a bounded subdomain of IRN , by H¨ older’s inequality, there holds |{ρ ≥ k} ∩ Ω′ | ≤ k −β ρβ0,β,{ρ≥k}∩Ω′ .

(3.1.26)

Consequently, we have {ρb′(h) (ρ) − b(h) (ρ)}1{ρ≥k} p0,p,Ω′ ≤ ck (λ1 +1)p−β ρβ0,β,{ρ≥k}∩Ω′ , and also

b(h) (ρ)1{ρ≥k} p0,p,Ω′ ≤ ck (λ1 +1)p−β ρβ0,β,{ρ≥k}∩Ω′

with any 1 ≤ p < ∞, β −

(λ1 + 1)p > 0. Thus, the first Nterm at the left-hand ′ side of (3.1.25), namely IRN b(h) (ρ)u · ∇ψ (ψ ∈ C0∞ (I

R ), suppψ ⊂ Ω ) can be written as the sum of IRN b(h) (ρ)1

{ρ≤k} u · ∇ψ and IRN b(h) (ρ)1{ρ>k} u · ∇ψ, k > 1. The first integral tends to IRN b(ρ)1{ρ≤k} u · ∇ψ due to the Lebesgue β

dominated convergence theorem. The second integral is majorized by ck λ1 +1− 2′ β ′

2 ρ0,β,{ρ>k}∩Ω ′ (1 + u0,2,Ω′ ) supx∈Ω′ |∇ψ(x)| (cf. Section 1.3.5.8); it tends to

zero as k → ∞. Proof of convergence IRN b(ρ)1{ρ≤k} u · ∇ψ → IRN b(ρ)u · ∇ψ follows the same argument. The analysis of the convergence of the other term is similar. 2

The reader can modify the statements of Lemma 3.2 and Lemma 3.3 to the case of inequalities: Exercise 3.4 (i) Let K, Ω, ρ, u and f satisfy the assumptions of Lemma 3.2. Suppose that div (ρu) ≥ f in D′ (Ω).

Then, prolonging ρ, u and f by zero outside Ω and denoting the new functions again by ρ, u, f , we have div (ρu) ≥ f in D′ (IRN ).

(3.1.27)

162


(ii) Let ρ, u, f and b satisfy the assumptions of Lemma 3.3. Moreover let b be a nondecreasing function. Suppose that (3.1.27) holds. Then div (b(ρ)u) + {ρb′ (ρ) − b(ρ)}div u ≥ f b′ (ρ) in D′ (IRN ). 3.1.3.2

(3.1.28)

Renormalizing functions with jump in derivatives For k > 0, we set bk (t) =

b(t) if t ∈ [0, k), where b ∈ C 1 ([0, ∞)). b(k) if t ∈ [k, ∞)

(3.1.29)

Then b′k ∈ C 0 ([0, k) ∪ (k, ∞)) and

lim b′k (t) = b′ (k),

t→k−

lim b′k (t) = 0.

t→k+

(3.1.30)

We see that the right derivative (bk )′+ (t) =

b′ (t) if t ∈ [0, k), 0 if t ∈ [k, ∞)

(3.1.31)

exists on [0, ∞) and [(bk )′+ ◦ ρ](x) =

b′k (ρ(x)) if x ∈ {ρ = k}, 0 if x ∈ {ρ = k}.

(3.1.32)

The renormalized continuity equation holds also if we replace b by bk and b′ by (bk )′+ . More precisely, we have: Lemma 3.5 Let N ≥ 2. Let β, ρ, u satisfy the assumptions of Lemma 3.3 and let f ∈ L1loc (IRN ). Then (i) div (bk (ρ)u) + {ρ(bk )′+ (ρ) − bk (ρ)}div u = f b′k (ρ) in D′ (IRN )

(3.1.33)

for any function bk , k > 0 satisfying (3.1.29). (ii) If f = 0, then the condition on b in (3.1.29) can be relaxed to (3.1.16). Proof The proof of this lemma is based on the following statement: Auxiliary lemma 3.6 Let β, ρ, u, f satisfy the assumptions of Lemma 3.3 and let k > 0. Then kdiv u = f a.e. in {ρ = k}. Proof of Auxiliary lemma 3.6. Take b ∈ D(IR) such that supp b ⊂ (0, ∞), − ǫ ǫ b(t) = t in ( 34 k, 54 k) and then set b+ k,ǫ = S 2 (bk+ǫ ), and bk,ǫ = S 2 (bk−ǫ ), where Sǫ is the standard one-dimensional mollifier. We have as ǫ → 0+ + ′ + ′ ′ b+ k,ǫ (t) → bk (t), t ∈ IR+ , [bk,ǫ ] (t) → bk (t), t = k, [bk,ǫ ] (k) → 1,

− ′ − ′ ′ b− k,ǫ (t) → bk (t), t ∈ IR+ , [bk,ǫ ] (t) → bk (t), t = k, [bk,ǫ ] (k) → 0.

Lemma 3.3 applied to equation (3.1.19) yields

VECTOR FIELDS WITH SUMMABLE DIVERGENCE

163

± ′ ± ± ′ div (b± k,ǫ (ρ)u) + {ρ[bk,ǫ ] (ρ) − bk,ǫ (ρ)}div u = f [bk,ǫ ] (ρ).

Therefore, letting ǫ → 0+ , we have div (bk (ρ)u) + {ρb′k (ρ)1{ρ=k} + k1{ρ=k} − bk (ρ)}div u = {b′k (ρ)1{ρ=k} + 1{ρ=k} }f

and div (bk (ρ)u) + {ρb′k (ρ)1{ρ=k} − bk (ρ)}div u = b′k (ρ)1{ρ=k} f. Subtracting the last two equations, we obtain the result. The proof of Auxiliary lemma 3.6 is thus complete. 2 With Auxiliary lemma 3.6 at hand the reasoning of the proof of Lemma 3.5 copies that of Lemma 3.3. We shall outline the proof of part (i). Lemma 3.3 applied to Sǫ [bk ] (Sǫ denotes the standard one dimensional regularizing operator, and bk is prolonged by b(0) outside IR+ ) gives div ({Sǫ [bk ]}(ρ)u) + {ρ{Sǫ [bk ]}′ (ρ) − {Sǫ [bk ]}(ρ)}div u = f {Sǫ [bk ]}′ (ρ) in D′ (IRN ).

(3.1.34)

Letting ǫ → 0+ , we have {Sǫ [bk ]}(t) → bk (t), t ∈ [0, ∞), {Sǫ [bk ]}′ (t) → b′k (t), t ∈ [0, k) ∪ (k, ∞). In accordance with (3.1.32), there holds {Sǫ [bk ]}(ρ) → bk (ρ) a.e. in IRN , {Sǫ [bk ]}′ (ρ) → b′k (ρ) a.e. in {ρ = k}. Due to Auxiliary lemma 3.6, ρ{Sǫ [bk ]}′ (ρ)div u = {Sǫ [bk ]}′ (ρ)f a.e. in {ρ = k}. Moreover, |Sǫ [bk ](t)| and |{Sǫ [bk ]}′ (t)| are uniformly bounded with respect to ǫ > 0. We pass to the limit ǫ → 0+ in (3.1.34) by using the Lebesgue dominated convergence theorem and we obtain (3.1.33). The proof of Lemma 3.5 is thus complete. 2 The proof of the next observation is left to the reader as an exercise. Exercise 3.7 Suppose that β > 1, and that ρ, u, f , b satisfy the assumptions of Lemma 3.3. If equation (3.1.33) holds with any k > k0 > 0, then equation (3.1.20) holds as well. Hint: Use the Lebesgue dominated convergence theorem. Remark 3.8 Assumption β ≥ 2 in Lemmas 3.3 and 3.5 is essential. We refer to Remark 6.14 for more details. 3.2

Vector fields with summable divergence

In this section we shall deal with the spaces of vector fields with summable divergence introduced in (Temam, 1983). We recall the definition and several properties related to these spaces which will be needed in the sequel as technical tools (as, e.g., density properties, existence of normal traces and the generalized Stokes formula).

164


Throughout this section, Ω is a bounded domain in IRN , N ≥ 2 and 1 < q, p < ∞. We set E q,p (Ω) = {g ∈ (Lq (Ω))N ; div g ∈ Lp (Ω)} ֒→ Lq (Ω), gE q,p = g0,q + div g0,p .

(3.2.1)

It is easily seen that E q,p (Ω) is a Banach space endowed with the norm · E q,p . The following density result is well known: Lemma 3.9 Let Ω be a bounded Lipschitz domain and 1 < p ≤ q < ∞. Then (D(IRN ))N is dense in E q,p (Ω). Proof See (Temam, 1983), Proposition 1.3. p,p

2 p

In the sequel E (Ω) is denoted simply by E (Ω) and similarly for the corresponding norm. If Ω is a bounded Lipschitz domain and functions ψ, φ belong to D(IRN ), then

ψdiv φ = ∂Ω ψφ · n − Ω φ · ∇ψ, Ω and

∂Ω

ψφ · n ≤ φE p ψ1,p′ ≤ c(p, Ω)φE p ψ1− 1′ ,p′ ,∂Ω p

(see Section 1.3.5.6). Hence, the operator

γn : φ → γ0 (φ) · n

(3.2.2)

(where γ0 is the usual operator of traces (see again Section 1.3.5.6)) is a linear 1− 1 ,p′ bounded densely defined (on (D(IRN ))N ) operator from E p (Ω) to [W p′ 1− 1 ,p′ (∂Ω)]∗ . Its closure is a continuous linear operator of E p (Ω) to [W p′ (∂Ω)]∗ . Its value at φ is usually called the normal trace of φ and denoted φ · n|∂Ω or Ω γn (φ) or simply γn (φ) if the domain in question is clear from the context. By using the density argument, we can resume the above discussion in the following lemma. Lemma 3.10 Let Ω be a bounded Lipschitz domain in IRN . Then there exists a unique continuous linear operator γn : E p (Ω) → [W

1− p1′ ,p′

(∂Ω)]∗ , 1 < p < ∞

such that γn (φ) = γ0 (φ) · n for every φ ∈ (D(IRN ))N . The Stokes formula

ψdiv φ = γn (φ), γ0 (ψ) Ω

{[W

1− 1′ ,p′ 1− 1′ ,p′ p p ]∗ ,W } ′

holds with any φ ∈ E p (Ω) and ψ ∈ W 1,p (Ω).

−

Ω

φ · ∇ψ

(3.2.3)

THE EQUATION DIV V = F

165

Proof For a more detailed proof see (Temam, 1983). 1,p

N

1,p

2 N

Let φn ∈ (W (Ω)) , φn → φ strongly in (W (Ω)) . Then, on one hand γ0 (φn ) → γ0 (φ) strongly in (Lp (∂Ω))N and γ0 (φn ) · n → γ0 (φ) · n strongly in Lp (∂Ω), and on the other hand, thanks to Lemma 3.10, we have γ0 (φn ) · n → 1− 1 ,p′ γn (φ) strongly in [W p′ (∂G)]∗ . We have thus obtained the following corollary of Lemma 3.10: Corollary 3.11 Let Ω be a bounded Lipschitz domain and let 1 < p < ∞. If φ ∈ (W 1,p (Ω))N , then γn (φ) ∈ Lp (∂Ω) and γn (φ) = γ0 (φ) · n a.e. in ∂Ω. We define

E q,p

E0q,p (Ω) = D(Ω)

Ep

and E0p = D(Ω)

.

(3.2.4)

The following statement holds: Lemma 3.12 Let 1 < p < ∞, let Ω be a bounded Lipschitz domain and let φ ∈ E p (Ω). Then φ ∈ E0p (Ω) if and only if γn (φ) = 0. Proof See (Temam, 1983).

2

Lemma 3.2 and Lemma 3.12 imply the following result whose proof is left to the reader as an exercise. Exercise 3.13 Let Ω ⊂ IRN be a bounded Lipschitz domain and let 2 ≤ p < ∞. Suppose that ρ ∈ Lp (Ω), u ∈ (W01,2 (Ω))N and div (ρu) ∈ Lt (Ω) for some 1 < t ≤ ∞. Then 2p , t} ρu ∈ E0r (Ω), r = min{ 2+p (cf. Section 1.3.5.8). 3.3

The equation div v = f

In this section we shall discuss the solvability of the equation div v = f in Ω and v = 0 in ∂Ω in various bounded and unbounded domains Ω in IRN , N ≥ 2. In what follows, we need a convenient solution of this auxiliary problem in order to get improved estimates of density (see, e.g., Sections 4.2.1, 4.8.2, 4.10.6, 4.13.3, 4.15, 4.16, 4.17, 7.3.3, 7.8.4, 7.9.5, 7.11.4, 7.12). For our purpose, the convenient solution in bounded domains is given by the so-called Bogovskii operator introduced in (Bogovskii, 1980). Its main properties are formulated in Lemma 3.17 and proved in Sections 3.3.1.1–3.3.1.2. Then, in Section 3.3.1.3 we investigate the nonhomogenous boundary conditions, and in Section 3.3.1.4, the x . These auxiliary results will properties of solutions under the homotopy x → R play an important role when dealing with unbounded domains. In Section 3.3.2, we investigate solutions in the whole space, in an exterior domain Ω and in invading domains ΩR = Ω ∩ BR , R > 0. These results are needed in Sections 4.13 and 7.11.

166


In Section 3.3.3, we introduce certain domains with noncompact boundaries. We shall study solutions in conical and in superconical exits introduced in (Solonnikov, 1983). Then we construct solutions in domains with several conical and superconical exits, and in corresponding invading domains. The main properties of these solutions are formulated in Lemmas 3.26–3.30. These results are needed in Sections 4.16 and 4.17 . 3.3.1

Bounded domains

In this section, we consider the problem div v = f in Ω, v|∂Ω = 0,

(3.3.1)

where Ω is a bounded domain. 3.3.1.1 Preliminaries We say that Ω is a star-shaped domain with respect to a point x0 ∈ Ω if there exists a positive function h : ∂B1 → IR such that x − x 0 Ω = x ∈ IRN ; |x − x0 | < h . |x − x0 | We say that Ω is star-shaped with respect to a ball B if it is star-shaped with respect to any point of this ball. We recall an important auxiliary lemma concerning the decomposition of Lipschitz domains to star-shaped domains. Auxiliary lemma 3.14 a) If Ω is a bounded domain with Lipschitz boundary, then there exists a family of open sets G := {G1 , . . . , Gr , Gr+1 , . . . , Gr+m }, r, m ∈ IN with the following properties: (i) Ω⊂ (ii)

r+m

∂Ω ⊂ (iii) there exists a family of balls

i=1

r

i=1

(3.3.2)

Gi ;

(3.3.3)

Gi ;

(3.3.4)

{B (1) , . . . , B (r) , B (r+1) , . . . , B (r+m) } such that any set Ωi := Ω ∩ Gi , i = 1, . . . , r + m

is star-shaped with respect to the ball B (i) . b) If f ∈ C0∞ (Ω) and

Ω

f = 0, then there exists a family of functions

F := {f1 , . . . , fr , fr+1 , . . . , fr+m }, r, m ∈ IN

(3.3.5)


such that (i) fi ∈ C0∞ (Ωi ), (ii)

Ωi

167

fi = 0 (no summation over i);

f= (iii)

r+m i=1

(3.3.6) (3.3.7)

fi ;

fi s,q,Ωi ≤ c(m, q, Ω1 , . . . , Ωr+m , Ω)f s,q,Ω , s = 0, 1, . . . , 1 < q < ∞.

(3.3.8)

Proof Recalling the local description of the boundary (cf. Section 1.1.11), statements (i), (ii), (iii) of part a) can be proved by using elementary geometrical considerations which we leave to the reader as an exercise. The detailed proof can be found, e.g., in (Galdi, 1994a), Lemma II.3.2. To prove part b), we proceed in the spirit of (Galdi, 1994a), Lemmas III.3.2 and III.3.4. We set Ωk =

r+m

i=k+1

Ωi , Fk = Ωk ∩ Ωk , k = 1, . . . , r + m.

As Ω is a bounded connected set, we immediately see that Fk = ∅. Further we introduce {ψ1 , . . . , ψm+r } a partition of unity corresponding to the covering G, i.e. ψi ∈ C0∞ (Gi ), 0 ≤ ψi ≤ 1, (3.3.9) r+m i=1 ψi (x) = 1, x ∈ Ω. We continue by induction. We set

f1 = ψ1 f − κ1 Ω ψ1 f, g1 = Ψ2 f − κ1 Ω Ψ2 f,

where

κ1 ∈ C0∞ (F1 ),

F1

κ1 = 1, Ψ2 =

r+m i=2

ψi .

Since ψ1 ∈ C0∞ (G1 ) and f ∈ C0∞ (Ω), we have ψ1 f ∈ C0∞ (Ω1 ), and therefore f1 ∈ r+m C0∞ (Ω1 ). Since Ψ2 ∈ C0∞ (∪i=2 Gi ) and f ∈ C0∞ (Ω), we have Ψ2 f ∈ C0∞ (Ω1 ), ∞ 1 and therefore g1 ∈ C0 (Ω ). One verifies by direct calculation that

and that

and

Ω1

f1 = 0,

f1 (x) + g1 (x) = f (x), x ∈ Ω Ω1

g1 = 0. Finally, using H¨ older’s inequality, we get

q

q q q−1 q′ |κ | |f |q ≤ c(q, |Ω1 |, |F1 |) Ω1 |f |q 1 + |Ω |f | ≤ 2 | 1 1 1 F1 Ω1 Ω1

q

|g1 |q ≤ 2q−1 1 + |Ω1 | q′ F1 |κ1 |q Ω1 |f |q ≤ c(q, |Ω1 |, |F1 |) Ω1 |f |q .

Ω1

168


Next we decompose g1 on Ω1 in the same way as we have decomposed f on Ω. More precisely, we shall write g1 (x) = f2 (x) + g2 (x), x ∈ Ω1 , where f2 = ψ2 g1 − κ2 and

Ω1

κ2 ∈ C0∞ (F2 ),

ψ2 g1 , g2 = Ψ3 g1 − κ2

F2

κ2 = 1, Ψ3 =

Ω1

r+m i=3

Ψ3 g1

ψi .

One easily verifies that f2 ∈ C0∞ (Ω2 ), g2 ∈ C0∞ (Ω2 ) and finally that

g dx = 0. Then by direct calculation Ω2 2 and

Ω2

f2 = 0,

q

q q−1 q q′ |f | ≤ 2 | |κ | |f |q ≤ c(q, |Ω2 |, |F2 |) Ω2 |g1 |q 1 + |Ω 2 2 2 Ω2 F2 Ω2

Ω2

q

|g1 |q ≤ 2q−1 1 + |Ω2 | q′ F2 |κ2 |q Ω2 |f |q ≤ c(q, |Ω2 |, |F2 |) Ω2 |f |q .

Repeating the whole procedure (m+r−1)-times, we arrive at the required result, thus completing the proof of Auxiliary lemma 3.14 2 3.3.1.2

The Bogovskii solutions

Auxiliary lemma 3.15 Let Ω be a Lipschitz domain. Then there exists a linear operator

(3.3.10) BΩ : C0∞ (Ω) := {f ∈ C0∞ (Ω); Ω f = 0} → (C0∞ (Ω))N such that

div BΩ (f ) = f, f ∈ C0∞ (Ω);

(3.3.11)

∇BΩ (f )0,q ≤ c(q, Ω)f 0,q , 1 < q < ∞.

(3.3.12)

BΩ (f )0,q ≤ c(q, Ω)g0,q 1 < q < ∞.

(3.3.13)

Moreover, if f = div g, where g ∈ (C0∞ (Ω))N , then

p of BΩ in Lp (Ω) = Remark 3.16 In the sequel, we shall consider the closure BΩ

p p {f ∈ L (Ω); Ω f = 0}, where 1 < p < ∞. Clearly, thanks to (3.3.12), BΩ is a 1,p continuous linear operator from Lp (Ω) to (W0 (Ω))N . Moreover, p1 p2 (f ) = BΩ (f ), for f ∈ Lp1 (Ω) ∩ Lp2 (Ω). BΩ

Due to this fact, in what follows we shall write simply BΩ (or even only B) p . instead of BΩ


169

Using the license of the last remark and the density argument (cf. vii) in Section 1.1.14.1), we can rewrite Auxiliary lemma 3.15 in the form needed in the sequel: Lemma 3.17 Let Ω be a bounded Lipschitz domain. Then there exists a linear 1 N , . . . , BΩ ) with the following properties: operator BΩ = (BΩ (i) (3.3.14) BΩ : Lp (Ω) → (W01,p (Ω))N , 1 < p < ∞; (ii) div BΩ (f ) = f a.e. in Ω, f ∈ Lp (Ω);

(3.3.15)

∇BΩ (f )0,p ≤ c(p, Ω)f 0,p , 1 < p < ∞.

(3.3.16)

(iii) (iv) If f = div g, where g ∈ E0q,p (Ω) with some 1 < q < ∞, then BΩ (f )0,q ≤ c(q, Ω)g0,q .

(v) If f ∈ D(Ω) (and, of course, Ω f = 0), then BΩ (f ) ∈ (D(Ω))N .

(3.3.17)

In the sense of Remark 3.16, the operator BΩ does not depend on q, p but it depends of the domain Ω. Nevertheless, we shall write simply B instead of BΩ , if the domain in question is clear from the context. In accordance with what was told before, Lemma 3.17 is nothing but Auxiliary lemma 3.15 reformulated by using the density argument. Our main task is therefore to prove Auxiliary lemma 3.15. In accordance with Auxiliary lemma 3.14, it is sufficient to do it only for domains which are star-shaped with respect to a ball. In this case it is possible to give a quite convenient estimate for the constant c(q, Ω) in the estimates (3.3.12) and (3.3.13). More precisely, we have: Auxiliary lemma 3.18 Let Ω ⊂ IRN , be a star-shaped domain with respect to a ball BR (x0 ), where BR (x0 ) ⊂ Ω. Then statements (3.3.10)–(3.3.13) of Auxiliary lemma 3.15 hold. Moreover the constant c in estimates (3.3.12) and (3.3.13) has the form c0 (N, q)

diam (Ω) N R

1+

diam (Ω) , R

diam (Ω) = supx,y∈Ω |x − y|.

(3.3.18)

Proof Since the operator div is invariant with respect to translations, it is enough to prove the lemma for domains Ω which are star-shaped with respect to the ball BR (0) centered at 0. One candidate for a solution is the so-called Bogovskii operator introduced in (Bogovskii, 1980) (see also Section III.3 in (Galdi, 1994a)). It reads

N −1 x−y ∞

x−y v(x) = BΩ (f )(x) = Ω f (y) |x−y| s ω y + s |x−y| ds dy, (3.3.19) N |x−y| R where

170


ωR (x) =

1 RN

x ), ω ∈ C0∞ (IRN ), supp ω ∈ B1 (0), ω( R

B1 (0)

ω = 1.

(3.3.20)

Thus ωR possesses the following properties:

supp ωR ⊂ BR (0), BR (0) ωR = 1,

and

|ωR |C 0 ≤

1 RN

|ω|C 0 , |∇ωR |C 0 ≤

1 |∇ω|C 0 . RN +1

It will be useful to have the following equivalent forms of formula (3.3.19):

∞

v(x) = Ω f (y)(x − y) 1 ωR y + r(x − y) rN −1 dr dy (3.3.21)

(to get it, one uses the change of variables r = v(x) =

s |x−y|

in (3.3.19)),

N −1

∞ x−y x−y dr dy |x − y| + r ωR x + r |x−y| f (y) |x−y| N Ω 0

(3.3.22)

(to obtain the last identity, one uses the change of variables s = |x − y| + r in (3.3.19)). As f ∈ C0∞ (Ω), in all formulas (3.3.19)–(3.3.22), the integration over Ω can be replaced by integration over IRN . Therefore, the change of variables z = x − y in (3.3.21) yields

∞

v(x) = IRN f (x − z)z 1 ωR x − z + rz rN −1 dr dz. (3.3.23)

In the sequel we wish to show that (3.3.19) or any of its equivalent forms satisfies all statements of Auxiliary lemmas 3.15 and 3.18. From the theorem about the differentiation of integrals dependent on a parameter (see Section 1.1.18.2) applied to formula (3.3.23), one easily sees that BΩ (f ) ∈ (C ∞ (Ω))N . Next, we realize that supp BΩ (f ) ⊂ A,

(3.3.24)

where A = {z ∈ Ω; z = λz1 + (1 − λ)z2 , z1 ∈ supp f , z2 ∈ BR (0), λ ∈ [0, 1]}. Indeed, if x ∈ Ω \ A, then y + r(x − y) ∈ / BR (0) for all r ≥ 1 and y ∈ supp f . This means that BΩ (f )(x) = 0 and confirms (3.3.24). Since A is compact, we have shown that BΩ (f ) ∈ (C0∞ (Ω))N , thus completing the proof of (3.3.10). Differentiating formula (3.3.23) (cf. Section 1.1.18.2 ), we obtain

∞ ∂i vj (x) = IRN ∂i f (x − z)zj 1 ωR x − z + rz rN −1 dr dz (3.3.25)

∞ + IRN f (x − z)zj 1 ∂i ωR x − z + rz rN −1 dr dz.

Splitting IRN into Bǫ (0) and B ǫ (0), ǫ > 0 and integrating by parts in the first integral over B ǫ (0), we arrive at


∂i vj (x) =

171

"

∞ ∂i f (x − z)zj 1 ωR x − z + rz rN −1 dr

#

∞ + f (x − z)zj 1 ∂i ωR x − z + rz rN −1 dr dz " ∞

+ B ǫ (0) f (x − z) δij 1 ωR x − z + rz rN −1 dr

#

∞ +zj 1 ∂i ωR x − z + rz rN dr dz

∞ zi ωR x − z + rz rN −1 dr dSz + ∂Bǫ (0) f (x − z)zj |z| 1

Bǫ (0)

(3.3.26)

= (Iǫ1 )ij (x) + (Iǫ2 )ij (x) + (Iǫ3 )ij (x). Clearly

lim Iǫ1 (x) = 0.

(3.3.27)

ǫ→0+

Coming back to the original variables, we rewrite the second integral as follows:

N −1

∞ δij x−y (Iǫ2 )ij (x) = B ǫ (x) f (y) |x−y| |x − y| + r ωR x + r |x−y| dr dy N 0

N ∞ xj −yj x−y |x − y| + r dr dy. ∂i ωR x + r |x−y| + B ǫ (x) f (y) |x−y| N +1 0 (3.3.28)

N in the second integral by using Further we develop the expression |x − y| + r the binomic formula and we obtain

(Iǫ2 )ij (x) = B ǫ (x) Kij (x, x − y)f (y) dy + B ǫ (x) Gij (x, y)f (y) dy, (3.3.29) where

Kij (x, z) = z θij (x, |z| ) = δij

and

∞ 0

z θij (x, |z| ) , N |z|

z )rN −1 dr + ωR (x + r |z|

zj |z|

c(N ) |ωR |C 0 (diam (Ω))N −1 |x−y|N −1

∞ 0

z )rN dr ∂i ωR (x + r |z|

+ |∇ωR |C 0 (diam (Ω))N (diam (Ω))N −1 c(N ) 1 + diamR (Ω) , x, y ∈ Ω. ≤ |x−y| N −1 RN

|Gij (x, y)| ≤

(3.3.30)

(3.3.31)

The third integral in (3.3.26) can be rewritten as follows:

∞ zi (Iǫ3 )ij (x) = ∂Bǫ (0) (f (x − z) − f (x))zj |z| ωR x − z + rz rN −1 dr dSz 1

z ∞

i ωR x − z + rz rN −1 dr dSz . +f (x) ∂Bǫ (0) zj |z| 1

The first integral in Iǫ3 tends to 0 as ǫ → 0+ ; it is an easy exercise. The second integral is equal to

∞ ǫN f (x) ∂B1 (0) zi zj 1 ωR (x − ǫz + rǫz)rN −1 dr dSz

∞ = ǫN −1 f (x) ∂B1 (0) zi zj 0 ωR (x + tz)(1 + ǫt )N −1 dt dSz

172


(to get the last identity, the change of variables ǫ(r − 1) = t has been used). Developing (1 + ǫt )N −1 by using the binomic formula, we easily check that the last integral is equal to

zi zj + f (x) IRN |z| 2 ωR (x + z) dz + o(1) as ǫ → 0 . Consequently,

limǫ→0+ (Iǫ3 )ij = f (x)Hij (x), Hij (x) =

zi zj ω (x IRN |z|2 R

+ z) dz.

(3.3.32)

Summarizing (3.3.27), (3.3.29) and (3.3.32), we obtain the following formula:

∂i vj (x) = limǫ→0+ B ǫ (x) Kij (x, x − y) (3.3.33)

+ IRN Gij (x, y)f (y)dy + f (x)Hij (x), x ∈ Ω. Since

d dr

x−y ωR (x + r |x−y| )(|x − y| + r)N

=

xk −yk |x−y| ∂k ωR (x

x−y + r |x−y| )

x−y )(|x − y| + r)N −1 , ×(|x − y| + r)N + N ωR (x + r |x−y|

we have

B ǫ (x)

(Kii (x, x − y) dy + Gii (x, y))f (y) dy = ωR (x)

cf. (3.3.28). Moreover, from (3.3.32), it is evident that

Hii (x) = Ω ωR (y)dy = 1.

Ω

f (y) dy = 0, ǫ > 0,

Formula (3.3.33) therefore yields

div v(x) = f (x), x ∈ Ω which completes the proof of (3.3.11). Due to (3.3.31) the kernel Gij is a weakly singular kernel (cf. Section 1.3.4.5). By virtue of (3.3.31),

IRN

N 1+ Gij (·, y)f (y) dy 0,q,Ω ≤ c(q, N ) diamR (Ω)

diam (Ω) f 0,q,Ω R

(we simply put G(x, y) = 0 if (x, y) ∈ / Ω×Ω and use estimate (1.3.35) of Theorem 1.35 together with H¨ older’s inequality (if q < N ) and estimates (1.3.38), (1.3.39) of Theorem 1.37 (if q ≥ N )). Since ∞

∞

θ(x, z) = |z|=1 δij 0 ωR (x + rz)rN −1 dr + zj 0 ∂i ωR (x + rz)rn dr dS |z|=1

= IRN δij ωR (x + y) + yj ∂i ωR (x + y) dy = 0


173

and since from (3.3.30), one has z supx,z∈IRN |θ(x, |z| )| ≤ c(N )|ω|C 1

diam (Ω) N 1+ R

diam (Ω) R

z (we simply put θ(x, |z| ) = 0 if x ∈ / Ω), all kernels Kij , i, j = 1, . . . , N are singular kernels of Calder´ on–Zygmund type, cf. Section 1.3.4.6. Therefore, according to Theorem 1.38, we have

B ǫ (.)

Finally,

N 1+ Kij (·, · − y)f (y)dy 0,q,Ω ≤ c(q, N, ω) diam(Ω) R supx∈Ω |Hij (x)| ≤

IRN

diam(Ω) f 0,q,Ω . R

ωR (y) dy = 1.

Coming back to formula (3.3.33), with these inequalities at hand, we obtain immediately estimate (3.3.12) of Auxiliary lemma 3.15. It remains to prove (3.3.13). To this end, let us suppose that f = div g, g ∈ (C0∞ (Ω))N and let us come back to formula (3.3.23). Again, splitting the integration into the integration over Bǫ (0) and B ǫ (0), and integrating by parts in the integral over B ǫ (0), we obtain

∞ div g(x − z)zj 1 ωR x − z + rz rN −1 dr

∞

+ B ǫ (0) div g(x − z)zj 1 ωR x − z + rz rN −1 dr dz

∞ = Bǫ (0) div g(x − z)zj 1 ωR x − z + rz rN −1 dr dz "

∞

+ B ǫ (0) gi (x − z) δij 1 ωR x − z + rz rN −1 dr

#

∞ +zj 1 ∂i ωR x − z + rz (r − 1)rN −1 dr dz

∞ zi + ∂Bǫ (0) gi (x − z)zj |z| ωR x − z + rz rN −1 dr dSz 1

vj (x) =

Bǫ (0)

= (Iǫ1 )ij (x) + (Iǫ2 )ij (x) + (Iǫ3 )ij (x), ǫ > 0.

By similar argument to that of (3.3.27)–(3.3.32), we arrive at the following formula:

vj (x) = limǫ→0+ B ǫ (x) Kij (x, x − y)gi (y) dy (3.3.34)

˜ ij (x, y)gi (y) dy + gj (x)Hij (x), x ∈ Ω, + IRN G

˜ ij is a weakly singular kernel satisfying where K is given by (3.3.30) and G (3.3.31). Estimate (3.3.13) follows directly from this formula by the same reasoning as before. 2

174


Remark 3.19 Combining (3.3.16), (3.3.17), the Poincaré inequality (see Section 1.3.5.7) and the imbedding theorems (see Section 1.3.5.8), we find that B(div g + f )0,q ≤ c(q, q, Ω)(g0,q + f 0,q ), (3.3.35)

∇B(div g + f )0,p ≤ c(p, Ω)(div g0,p + f 0,p ), E0q,p (Ω)

B(f )0,p ≤ c(p, p, Ω)f 0,p ,

and f ∈ Lq (Ω) ∩ Lp (Ω), 1 < p, q < ∞. In formula (3.3.35), provided g ∈ q p we have denoted q = NN+q if q > NN−1 , 1 < q < ∞ if q ≤ NN−1 and p = NN−p , if p < N , 1 ≤ p < ∞ if p = N , p = ∞ if p > N . 3.3.1.3 Nonhomogenous boundary conditions We shall start with the following nonhomogenous version of Lemma 3.17 Corollary 3.20 Let Ω be a bounded Lipschitz domain. Then there exists a linear operator B˜ = (B˜1 , . . . , BÑ ) with the following properties: (i)

(ii)

˜ a)) = a, ˜ a) = f a.e. in Ω, γ0 (B(f, div B(f,

1 f ∈ Lp (Ω), a ∈ (W 1− p ,p (∂Ω))N , 1 < p < ∞, Ω f = ∂Ω a · n; ˜ )1,p ≤ c(p, Ω)[f 0,p + a1− 1 ,p,∂Ω ], 1 < p < ∞. B(f p

(3.3.36)

(3.3.37)

(iii) Let a = 0 on S ⊂ ∂Ω, where µ ˜(S) = 0 (˜ µ is the measure on the boundary, see Section 1.1.12). Then ˜ )p ≤ c(p, Ω)[f p + I p−1 B(f 0,p 1,p ,p,∂Ω (a)], 1 < p < ∞, 0, p

1

(3.3.38)

1

where [I0, p−1 ,p,∂Ω ] p is the seminorm on (W 1− p ,p (∂Ω))N defined in Section p 1.3.5.11. Proof Let A be a continuous extension of a to (W 1,p (Ω))N (cf. Lemma 1.47), i.e. γ0 (A) = a, A1,p,∂Ω ≤ c(p, Ω)a1− p1 ,p,∂Ω . We set Clearly and

˜ a) = B(f − div A) + A. v = B(f, div v = f − div A + div A = f, γ0 (v) = a v1,p,Ω ≤ c(p, Ω)[A1,p,Ω + f − div A0,p,Ω ] ≤ c(p, Ω)[f 0,p,Ω + a1− p1 ,p,∂Ω ].

This completes the proof of (3.3.36) and (3.3.37). If a vanishes on the set of 1 positive measure on the boundary, then a1− p1 ,p,∂Ω and [I0, p−1 ,p,∂Ω (a)] p are p


175

equivalent norms (cf. Section 1.3.5.11). Therefore (3.3.38) follows directly from (3.3.37). 2 3.3.1.4 Nonhomogenous boundary conditions and invading domains quel, we shall consider the equation G

div v = f in G(R) , γ0 (R) (v) = a,

In the se(3.3.39)

where

x ∈ G}, (3.3.40) R and where G is a bounded domain with Lipschitz boundary. We shall be interested essentially on the dependence of the coefficient of estimates for the Bogovskii solutions on the parameter R. G(R) = {x ∈ IRN ,

Auxiliary lemma 3.21 Let G be a bounded Lipschitz domain, and let S(R) ⊂ ˜(S(R) ) = 0, where {G(R) }R>0 is a family of domains defined in (3.3.40) ∂G(R) , µ and µ ˜ is the measure on the boundary. Then there exists a family of linear oper1 N , . . . , B˜R ), R > 0 with the following properties: ators B˜R = (B˜R (i) div B˜R (f, a) = f a.e. in G(R) , γ0 (B˜R (f )) = a, f ∈ Lp (G(R) ),

1− 1 ,p a ∈ (WS(R)p (∂G(R) ))N , 1 < p < ∞, G(R) f = ∂G(R) a · n,

(3.3.41)

where

1− 1 ,p

1

(WS(R)p (∂G(R) ))N = {u ∈ (W 1− p ,p (∂G(R) ))N ; u = 0 µ ˜-a.e. in S(R) }. (ii) There holds ∇B˜R (f, a)p0,p,G(R) ≤ c(p, S)[f p0,p,G(R) + I0, p−1 ,p,∂G p

(R)

(a)],

(3.3.42)

where S = S(1) = {˜ x ∈ ∂G(1) ; R˜ x ∈ S(R) }.

(3.3.43)

a)]. ∇˜ v p0,p,G ≤ c(p, S)[f˜p0,p,G + I0, p−1 ,p,∂G (˜

(3.3.44)

˜ ), where f˜(˜ ˜ = B˜G (f˜, a x) = Rf (˜ xR), x ˜ ∈ G (recall that G(1) = Proof We set v ˜ (˜ G) and a x) = a(˜ xR), x ˜ ∈ ∂G. According to (3.3.38), there holds p

x ˜( R ) defined a.e. in GR We verify by direct calculation that the quantity v(x) = v satisfies (3.3.41). Writing ∇˜ v (˜ x) = R∇v(˜ xR) and using the change of variables x=x ˜R in (3.3.44) we arrive at the formula (3.3.42) with B˜R (f, a) replaced by v. Now, it suffices to put B˜R (f, a) = v to complete the proof of Auxiliary lemma 3.21. 2

We shall finish this section by the following consequence of the trace theorem:

176


Auxiliary lemma 3.22 Let A ∈ (W 1,p (G(R) ))N , where G(R) is given by (3.3.40) with G a bounded Lipschitz domain and set a = γ0 (A). Suppose that S(R) ⊂ ∂G(R) is a subset of nonzero measure on the boundary and a is an element 1− 1 ,p

of the space (WS(R)p (∂G(R) ))N . Then

I0, p−1 ,p,∂G(R) (a) ≤ c(p, S)∇Ap0,p,G(R) , p

(3.3.45)

where S is defined in (3.3.43). x) = A(˜ ˜ (˜ Proof We put A(˜ xR), x ˜ ∈ G, a x) = a(˜ xR), x ˜ ∈ ∂G. Then clearly G ˜ ˜-a.e. on S. By the Friedrichs–Poincaré inequality, the norms γ0 (A) = 0 µ ˜ 1,p,G and ∇A ˜ 0,p,G are equivalent (cf. Section 1.3.5.7). From the trace theA orem (cf. Section 1.3.5.11), we therefore have ˜ p a) ≤ c(p, S)∇A I0, p−1 ,p,∂G (˜ 0,p,∂G . p

x) = R∇A(˜ Writing ∇A(˜ xR) and using the change of variables x = x ˜R at the right-hand side and at the left-hand side of the above inequality, we arrive at (3.3.45) which completes the proof. 2 3.3.2

Exterior domains

By an exterior domain Ω we mean an open set such that Ωc := IRN \ Ω is a bounded domain.

(3.3.46)

We say that an exterior domain is of class C k,θ , k ∈ IN ∪ {0}, θ ∈ [0, 1] if IRN \ Ω ∈ C k,θ .

(3.3.47)

Before investigating the case of an exterior domain, we shall discuss the case of the whole space. 3.3.2.1 Whole space One possibility to prove existence of a convenient solution to problem (3.3.1) in the whole space Ω = IRN relies on the fundamental solution to the Laplace operator. This result is formulated in the following exercise: Exercise 3.23 Set BIRN (f ) = ∇E ∗ f, where E is the fundamental solution of the Laplace operator (see Exercise 5.12 in Section 5.5). Then (i) BIRN : Lp (IRN ) → (D01,p (IRN ))N , 1 < p < ∞,

where D01,p (IRN ) is the homogenous Sobolev space (see Section 1.3.6.1); (ii) div BIRN (f ) = f, f ∈ Lp (Ω);


177

(iii) ∇BIRN (f )0,p ≤ c(p, N )f 0,p , 1 0.

3.3.2.2 Exterior domains With the result of Exercise 3.23 at hand, we shall prove the existence of solutions in exterior domains. Lemma 3.24 Let Ω be an exterior domain with Lipschitz boundary. Then there 1 N , . . . , BΩ ) with the following properties: exists a linear operator BΩ = (BΩ (i) BΩ : Lp (Ω) → (D01,p (Ω))N , 1 R0 , where R0 is such that Ωc ⊂ BR0 .

c(p,N,R) |x|N −1

for all

Proof We repeat briefly the proof following (Galdi, 1994a), Theorem 3.4. Without loss of generality, we suppose that f ∈ C0∞ (Ω) and we extend it by zero outside Ω so that f ∈ C0∞ (IRN ). (If f ∈ Lp (Ω), we use the density of C0∞ (Ω) in Lp (Ω) in the spirit of Remark 3.16.) We set v = u + w, where u=

∇E

∗ f, 1 < p < N, ∇E ∗ f − ΩR ∇E ∗ f, N ≤ p < ∞ 0

and

w=

BΩR0 (0, a) in ΩR0 , 0 in B R0

/

, a=

/

∈ (D01,p (Ω))N

−u in ∂Ω, 0 in ∂BR0

/

.

We have div u = f in IR N and ∇u 0,p,IRN ≤ c(p,

N )f 0,p,Ω (cf. Exercise 3.23). Since ∂ΩR a · n = ∂Ωc u · n = Ωc div u = Ωc f = 0, w does exist and 0 satisfies the estimate ∇w0,p,Ω = ∇w0,p,ΩR0 ≤ c(p, N, ΩR0 )γ0 (u)1− p1 ,p,∂Ω ≤ cu1,p,ΩR0 ≤ c∇u0,p,ΩR0 (cf. Corollary 3.20, the trace theorem in Section 1.3.5.11 and the Poincaré inequality in Section 1.3.5.7). The two last-mentioned chains of inequalities yield ∇v0,p,Ω ≤ c(p, N, Ω) f 0,p,Ω . Now, we set BΩ (f ) = v thus completing the proof of Lemma 3.24. 2

178


3.3.2.3 Invading domains on exterior domains An important consequence of x Lemma 3.17 essentially due to the scaling x → R is the following result. We shall need it for treating the Navier–Stokes equations in exterior domains by the method of invading domains in Sections 4.12 and 4.13. Lemma 3.25 Let Ω be an exterior domain with Lipschitz boundary such that IR3 \ Ω ⊂ BR0 , where R0 > 0, or let Ω = IR3 . Then there exists a family of linear operators {BR }R>R0 = {(BR )1 , . . . , (BR )N }R>R0 , BR : Lp (ΩR ) → (W01,p (ΩR ))N , 1 0 in estimate (3.3.51) is, in particular, independent of R. Proof (i) If Ω = IR3 , then ΩR = Ω(R) = BR (cf. (3.3.40)), and the scaling x x → R together with Lemma 3.17 ensures the validity of the statement (the reader can use the same argumentation as that of Lemma 3.21). (ii) In the case of general ΩR , we extend f by zero outside ΩR and we set v = u + w,

where u = BBR (f )

(see part (i) of this proof) and w(x) =

[B˜ΩR0 (0, a)](x) if x ∈ ΩR0 R 0 in BR 0

/

, a=

−u in ∂Ω, 0 in ∂BR0

/

.

We have div u = f in BR and ∇u 0,p,BR ≤ c(p, N )f 0,p,Ω

R (cf. part (i) of this proof). Since ∂ΩR a · n = ∂Ωc u · n = Ωc div u = Ωc f = 0, w exists. 0 By virtue of Lemma 3.20, it satisfies the estimate ∇w0,p,ΩR = ∇w0,p,ΩR0 ≤ c(p, N )u1− p1 ,p,∂ΩR0 which is bounded by c∇u0,p,ΩR0 thanks to the trace theorem and the Poincaré inequality. Now, we put BR (f ) = v thus completing the proof. 2 3.3.3

Domains with noncompact boundaries

3.3.3.1

Definitions

We shall deal with domains of the type

E = {x ∈ IRN ; xN > F (|x′ |), x′ = (x1 , . . . , xN −1 ) ∈ IRN −1 }.

(3.3.52)

To simplify, we suppose, without loss of generality, that F (0) = 0. (i) If F is a (globally) Lipschitz function on [0, ∞), i.e. if there exists M > 0 such that |F (t) − F (s)| ≤ M |t − s|, t, s ∈ [0, ∞), (3.3.53)


the infinite cone ′

+ C0,ϑ = {x ∈ IRN , xN > M |x′ |}

179

(3.3.54) π 2 −arctanM

of vertex 0, of axis {(0 , xN ), xN ∈ (0, ∞)} and of half-aperture ϑ = belongs to E. Therefore, in the sequel, domains satisfying (3.3.52), (3.3.53) are called supercones (of axis {(0′ , xN ), xN > 0} and of vertex 0). If F (s) = ks, k ∈ IR, then E is a rotational cone of axis {(0′ , xN ), xN ∈ (0, ∞)} of vertex 0 and of half-aperture ϑC = π2 − arctan k. Sometimes, we shall use the notation − + = IRN \ C0,π−ϑ , ϑ ∈ (0, π). C0,ϑ

It is easily seen that

− + + − C0,ϑ = −C0,ϑ , C0,ϑ = −C0,ϑ .

(ii) If F is locally Lipschitz on [0, ∞) and if

" (s) # = ∞, inf R>0 supt,s>R, t=s F (t)−F t−s

(3.3.55) (3.3.56)

then there does not exist any cone which is contained in E. On the contrary, there exists a cone of axis {(0′ , xN ), xN ∈ (0, ∞)} and vertex x0 which contains the whole exit. Therefore, domains (3.3.52) satisfying (3.3.55) and (3.3.56) are called subcones (of axis {(0′ , xN ), xN > 0} and of vertex 0). In this sense, a limiting case of a subcone is a cylindrical exit of radius r > 0 E = {x ∈ IRN ; xN > 0, |x′ | < r}.

(3.3.57)

˜ a supercone (resp. subcone) of axis a and of vertex x0 if (iii) We shall call E there exists a rotation R such that ˜ − x0 = E, R(E) where E is the supercone (resp. subcone) (3.3.52) of axis {(0′ , x3 ), x3 > 0} and of vertex 0. (iv) Ω is called a Lipschitz domain with M , M ∈ IN ∪ {0} superconical exits, with S ∈ IN ∪ {0} conical exits and with L ∈ IN ∪ {0} subconical exits if L + S + M ≥ 1 and provided there exists R0 such that: ˜R is a bounded Lipschitz domain, ˜ R := Ω ∩ B Ω 0 0 R ˜ 0 := Ω ∩ B ˜ R0 = M +S+L [E ˜i ]R0 , Ω i=1 ˜ R0 ∩ E ˜ R0 = ∅, i = j, E j i

(3.3.58) (3.3.59) (3.3.60)

˜i , i = 1, . . . , M are supercones, E ˜i , i = M + 1, . . . , M + S are cones, E ˜i , where E N i = M +S +1, . . . , M +S +L are subcones (of axis ai and of vertex si ∈ IR ) and ˜R , [E ˜i ]R0 = E ˜i \BR (si ) = E ˜i ∩B R0 (si ). ˜ R0 = IRN \ B ˜R = ∪L+S+M BR (si ), B B 0 0 0 0 i=1

180


3.3.3.2 Solonnikov’s solutions for cones and supercones Lemma 3.26 Let E be a supercone (3.3.52)–(3.3.53). Then there exists a linear 1 N operator BE = (BE , . . . , BE ) with the following properties: (i) (3.3.61) BE : Lp (E) → (D01,p (E))N , 1 < p < ∞,

where D01,p (E) is the homogenous Sobolev space defined in Section 1.3.6.1; (ii) (3.3.62) div BE (f ) = f a.e. in E, f ∈ Lp (E); (iii)

∇BE (f )0,p ≤ c(p, E)f 0,p , 1 0.

∞

N

then B(f ) ∈ (C (Ω))

and |[B(f )](x)| ≤

(3.3.63) c(p,N,R) |x|N −1

for all

− Proof Due to the definition of E, there exists an infinite cone C0 = C0,ϑ ⊂ − − IRN \ E such that the cone Cx = Cx,ϑ := {y ∈ IRN ; y − x ∈ C0,ϑ } belongs to IRN \ E for all x ∈ IRN \ E. We suppose, without loss of generality, that f ∈ C0∞ (E). (Then one can complete the proof by the density argument, in the spirit of Remark 3.16.) As usual, we extend f by zero outside E so that it is defined in the whole IRN . We set x−y

x−y v(x) = [BE (f )](x) = Cx |x−y| N ω |x−y| f (y) dy (3.3.64) z

f (x − z) dz, = C + |z|zN ω |z| 0,ϑ

where

+ ), ω ∈ C01 (∂B1 (0) ∩ C0,ϑ

∂B1 (0)

ω dS = 1.

(3.3.65)

We want to prove that v satisfies all properties enumerated in Lemma 3.26. We find by using similar argumentation as we have already used in Section 3.3.1.2, that "

z z f (x − z) dz ∂i vj (x) = limǫ→0+ C + \Bǫ (0) ∂i |z|jN ω |z| 0,ϑ (3.3.66)

#

z zj z ω |z| + C + ∩∂Bǫ (0) |z|iN +1 f (x − z) dSz . 0,ϑ

It is easy to show that

limǫ→0+ C +

where

z zi zj N +1 ω |z| f (x |z| ∩∂B (0) ǫ 0,ϑ Hij =

Since

zi zj z ω |z| dS ∂B1 (0) |z|2

∂i

and since

− z) dSz = Hij f (x),

zi |z|N

ω

z |z|

∈ IR.

=0

Hii = 1,

we conclude from (3.3.66) that div v = f , thus completing the proof of statement (ii).


181

To prove statements (i) and

by virtue of formula (3.3.66), it suffices to z(iii), z is a Calderón–Zygmund singular kernel (cf. verify that Kij (z) = ∂i |z|jN ω |z| Section 1.3.4.6). This cumbersome but direct calculation is left to the reader as an exercise (see (Novo et al., 2003) for more details). Evidently, the statement (iv) of the lemma holds true. It is an easy application of the properties of integrals dependent on a parameter (see Section 1.1.18.2). 2 Remark 3.27 Lemma 3.26 and Solonnikov’s proof apply without changes also to nonsymmetric supercones E = {x ∈ IRN ; xN > F (x′ )}, where |F (x′ ) − F (y ′ )| ≤ M |x′ − y ′ |, x′ , y ′ ∈ IRN −1 , see Solonnikov (Solonnikov, 1983). To our knowledge, the validity of Lemma 3.26 for subcones is an open problem. 3.3.3.3 Domains with finite number of superconical and conical exits With the result of Lemma 3.26 at hand, we shall prove the existence of solutions to the problem div v = f , v|∂Ω = 0 in domains with several conical and superconical exits. We shall prove the following lemma: Lemma 3.28 Let Ω be a domain with several superconical (and/or conical) exits Ei , i = 1, . . . , M (see (iv) in Section 3.3.3). Then there exists a linear operator N 1 ) with the following properties: BΩ = (BΩ , . . . , BΩ (i) BΩ : Lp (Ω) → (D01,p (Ω))N , 1 < p < ∞; (3.3.67) (ii) div BΩ (f ) = f a.e. in Ω, f ∈ Lp (Ω);

(3.3.68)

∇BΩ (f )0,p ≤ c(p, Ω)f 0,p , 1 0.

c(p,N,R) |x|N −1

for all

Proof As usual, without loss of generality, we suppose that f ∈ C0∞ (Ω) (if f ∈ Lp (Ω) we use the density of C0∞ (Ω) in Lp (Ω) in the spirit of Remark 3.16) and we extend f by zero outside Ω so that f ∈ C0∞ (IRN ). We set F= and F (x) =

˜2R \ (∪M Ei )] ∪ B ˜ 2R0 f (x) if x ∈ [B 0 i=1 1 M ˜ 2 f (x) if x ∈ B2R0 ∩ (∪i=1 Ei ) φ(x) ˜ Ω 2R0

We take v 0 (x) =

φ

!

˜ 2R Ω 0

˜ 2R \ Ω ˜ R ). F, φ ∈ C0∞ (Ω 0 0

˜ 2R BΩ˜ 2R (F − F ) if x ∈ Ω 0 0 0 otherwise

182


Since Ω˜ 2R (F −F ) = 0, due to Lemma 3.17, v 0 exists and satisfies ∇v 0 0,p,Ω˜ 2R 0 0 ≤ ∇v 0 0,p,Ω ≤ c(p, R0 , Ω)F − F 0,p,Ω˜ R . Further we take 0

v i (x) =

BEi (F + F ) if x ∈ Ei 0 otherwise

(cf. Lemma 3.26). Due to the same lemma ∇v i 0,p,Ω ≤ c(p, N )F + F 0,p,Ei . With these facts at hand, recalling the definitions of F and F , we easily complete the proof by setting M BΩ (f ) = i=0 v i . 2

Remark 3.29 Lemma 3.28 holds without changes also for domains with several superconical nonsymmetric exits evoked in Remark 3.27. It is not known to us whether Lemma 3.28 holds if one or more exits are subconical. 3.3.3.4 Invading domains on conical or superconical domains Lemma 3.30 Let Ω be a domain with one conical or one superconical exit E. Suppose further that the function F characterizing E (cf. (3.3.52)) is monotone. N 1 , . . . , BR }, Then there exists R0 > 0 and a family of linear operators {BR } := {BR BR : Lp (ΩR ) → (W01,p (ΩR ))N , 1 R0 satisfying div BR (f ) = f

a.e. in ΩR , f ∈ Lp (ΩR )

and ∇B(f )0,p,ΩR ≤ c(p, Ω)f 0,p,ΩR , f ∈ Lp (ΩR ), 1 < p < ∞.

(3.3.70)

The constant in estimate (3.3.70) is, in particular, independent of R. Proof For the sake of simplicity, we perform the proof only with nondecreasing F . If F is nonincreasing, the proof is slightly more technical and we let it to the reader as an exercise. We observe that there exists R0 > 0 such that if R > R0 the algebraic equation |F (r)|2 + r2 = R2 possesses just one positive solution r0 . We set + ∩ B1 ; xN > cos ϑ}, ϑ = arccos HR = G(R) , where G(1) = {C0,ϑ

F (r0 ) R

(for the definition of G(R) , see (3.3.40)). Clearly, there exists ϑ0 such that 0 < ϑ0 ≤ ϑ ≤ π2 . As usual, we extend f by zero outside ΩR and we set v = u + w,

where

u(x) =

[BΩ (f )](x) if x ∈ Ω 0 otherwise

SOME RESULTS FOR MONOTONE AND CONVEX OPERATORS

and w = B˜HR (0, a), a =

183

0 in ∂HR \ ∂BR , −u in E ∩ ∂BR .

≤ c(p, N )f 0,p,ΩR (cf. Lemma 3.28). We have

div u = f in Ω and ∇u0,p,Ω Since ∂HR a · n = − ∂ΩR u · n = − ΩR div u = ΩR f = 0, w exists. By virtue of Lemmas 3.21 and 3.18, it satisfies the estimate ∇wp0,p,HR ≤ c(p, N, ϑ0 ) I0, p−1 ,p,∂HR (u). Let us take p

z=

u in HR 0 in BR \ HR

Due to Auxiliary Lemma 3.22 I0, p−1 ,p,∂HR (u) = I0, p−1 ,p,∂(BR \C − π ) (z) ≤ c∇z p0,p,B p

p

0,

2

− R \C0, π 2

.

Of course, the last bound is less than c∇up0,p,BR = c∇u p0,p,Ω . This completes the proof. 2 Remark 3.31 If Ω is a domain with one subconical exit, the above proof fails. In this case, under certain conditions on F , Lemma 3.30 remains true, with the exception that the coefficient c in estimate (3.3.70) depends on R (and explodes as R → ∞). The precise formula can be deduced from (3.3.18). It is not known to us, whether Lemma 3.30 holds for domains with more than one conical or superconical exit. 3.4

Some results for monotone and convex operators

In the first part of this section we formulate and prove several consequences of well-known general theorems from convex analysis and from the theory of lower (upper) semicontinuous functionals. (Most of these general theorems and basic notions related to them have been recalled in Sections 1.4.2.7, 1.4.5.10, 1.4.5.11.) In the second part of this section we prove several properties of monotone operators in the form needed in the sequel. We shall do it by using the so-called Minty trick. These results will be needed as auxiliary (but important) tools in Sections 4.9, 4.10, 4.11, 4.14, 4.15, 4.16, 4.17, 7.9, 7.10, 7.11 and 7.12, whenever we want to prove the strong convergence of the density and whenever we want to pass to the limit in the energy inequalities. Throughout this section, if not stated explicitly otherwise, G is a domain in IRN , where N ≥ 1. 3.4.1

Some results from convex analysis

Very often it is not easy to verify whether a functional defined on a Banach space is weakly lower semicontinuous. It is much easier to show strong lower semicontinuity. For convex linear functionals, both properties are equivalent.

184


The following lemma deals with this problem. It is nothing but a particular case of one of the main classical results in the calculus of variation (see Sections 1.4.2.7, 1.4.5.11). It reads: Lemma 3.32 Let 1 ≤ p < ∞ and let G be a domain of IRN . (i) Suppose that F : Lp (G) → IR ∪ {∞} is a convex lower semicontinuous functional on Lp (G). Then F is weakly lower semicontinuous. In particular, F(u) ≤ lim inf F(un ) whenever un → u weakly in Lp (G). n→∞

(ii) Suppose that F : Lp (G) → IR ∪ {−∞} is a concave upper semicontinuous functional on Lp (G). Then F is weakly upper semicontinuous. In particular, F(u) ≥ lim sup F(un ) whenever un → u weakly in Lp (G). n→∞

This lemma has several direct consequences needed in the sequel: Corollary (i) Let G be a domain in IRN and let 1 ≤ p < ∞. We set

3.33 p F(u) = K |u| , u ∈ Lp (K), where K is a measurable subset of G. Then

K

|u|p ≤ lim inf n→∞

K

|un |p whenever un → u weakly in Lp (G).

(ii) Let G be a bounded domain in IRN , I an interval in IR and 1 ≤ p < ∞. Let f : I → IR be a convex lower semicontinuous (resp. a concave upper semicontinuous) function on I which satisfies |f (t)| ≤ c1 + c2 tp , t ∈ I with some c1 , c2 > 0.

(3.4.1)

We set p

F± : L (G) →

IR ∪ {∞} , F± (u) = IR ∪ {−∞}

!

G

f (u) if |{u ∈ / I}| = 0

±∞

otherwise

(3.4.2)

Then F+ and F− are covex weakly lower semicontinuous and concave weakly upper semicontinuous functionals, respectively. (iii) Let G be a domain in IRN , I an interval in IR and f a convex lower semicontinuous (resp. concave upper semicontinuous) function on I. Let 1 ≤ p < ∞. Suppose that un is a sequence of nonnegative functions from Lp (G) with values in I such that un → u weakly in Lp (G) and f (un ) → f (u) weakly in L1 (G). Then f (u) ≤ f (u) (resp. f (u) ≥ f (u)) a.e. in G.


185

Proof Part (i) is an immediate consequence of Lemma 3.32. We prove statement (ii) in the case of F+ . The proof in the case F− is left to the reader as an exercise. Suppose that un → u strongly in Lp (G).

(3.4.3)

/ I}| = 0, then If there exists m ∈ IN such that for all n > m, n ∈ IN , |{un ∈ lim inf n→∞ F+ (un ) = ∞ and the statement is obvious. Otherwise, there exists a subsequence denoted again by un such that |{un ∈ / I}| = 0 and such that un → u a.e. in G.A convex and lower semicontinuous function on I is continuous on I. In virtue of this continuity there also holds f ◦ un → f ◦ u a.e. in G. According to Egoroff’s theorem about uniform convergence (see again Section 1.1.18.4), for any k ∈ IN there exists a measurable subset Ak of G such that |G \ Ak | < k1 , f ◦ un → f ◦ u uniformly on Ak . (3.4.4)

Due to (3.4.1), (3.4.3), lim inf n→∞ G\Ak |f ◦ un | ≤ ck1 + c2 G\Ak |u|p , G\Ak |f ◦

u| ≤ ck1 +c2 G\Ak |u|p and also, according to (3.4.4), limn→∞ Ak (f ◦un −f ◦u) = 0. The identity

F+ (u) = G f (un ) + Ak (f ◦ u − f ◦ un ) − G\Ak f ◦ un + G\Ak f ◦ u thus yields, as n → ∞, the following inequality

F+ (u) = lim inf n→∞ F+ (un ) + ξ(k), |ξ(k)| ≤ 2 ck1 + 2c2

G\Ak

|u|p .

Of course, ξ(k) → 0 as k → ∞. This gives the result about the strong lower semicontinuity of F+ . It is a consequence of the definition to show that F+ is convex provided f is convex. In this situation, according to Lemma 3.32, F+ is also weakly lower semicontinuous. Now we prove statement (iii) in the case of convex f , letting again the ”concave case” to the reader. Let J be a compact interval in int(I), let B be a ball in G and Z a bounded measurable set in G. Any convex function on I is Lipschitz continuous on J. Therefore the functional F+ defined in (3.4.2), where we replace G by Z and I by J is strongly lower semicontinuous in Lp (Z). Due to Lemma 3.32 it is weakly lower semicontinuous on Lp (Z) as well. In particular, for the sequence un and its weak limit u, there holds

f (u)1{u∈J} ≤ B∩{u∈J} f (u) ≤ B

lim inf n→∞ B∩{u∈J} f (un ) = B f (u)1{u∈J} .

This implies that f (u) ≤ f (u) almost everywhere in {u ∈ J} (see Section 1.1.15) as well as almost everywhere in {u ∈ int(I)}, and completes the proof if I is open. If I is not open, there exists α ∈ I \ int(I). We have either un ≥ α or un ≤ α a.e. in G and the weak convergence of un implies strong convergence in 2 L1 ({u = α} ∩ Z). This observation completes the proof.

186


If a strictly convex function “commutes with weak convergence”, this weak convergence may be, in fact, strong. An overall discussion of this phenomenon can be found, e.g., in (Visintin, 1984). More precisely, there holds: Lemma 3.34 Let G be a bounded domain in IRN , I an interval in IR, 1 < p < ∞ and f : I → IR a strictly convex function. Let un be a sequence of functions in Lp (G) with values in I. If un → u weakly in Lp (G) and f (un ) → f (u) weakly in L1 (G), then un → u strongly in L1 (G). Proof A convex function on I admits at any point of int(I) a right and left ′ ′ and f− . If it is strictly convex, these derivatives are increasing derivatives f+ functions on int(I) and there holds f (t) − f (a) − fS′ (a)(t − a) > 0, t ∈ I, a ∈ int(I), t = a f ′ +f ′

where fS′ = + 2 − . Therefore, there exists a continuous increasing function ψa on [0, max{sup I − a, a − inf I}) vanishing at 0 and such that f (t) − f (a) − fS′ (a)(t − a) ≥ ψa (|t − a|), t ∈ I, a ∈ int(I). (In fact, one can take ψa (s) = min{f (a + s) − f (a) − fS′ (a)s, f (a − s) − f (a) + fS′ (a)s}, s ∈ [0, min{sup I − a, a − inf I}) and extend it continuously onto [0, max{sup I − a, a − inf I}) by the difference of f (a + s) (or of f (a − s)) with a convenient affine function. The details are left to the reader.) For the sequence un and its weak limit u, we thus obtain

(f (un ) − f (u) − fS′ (u)(un − u)) ≥ {u∈J} ψu (|un − u|), {u∈J} where J is a closed interval of int(I). Letting n → ∞ in the last formula, we observe that the left hand side tends to 0. Consequently, {u∈J} ψu (|un − u|) → 0 as well. Therefore, there exists a set M in G of measure zero such that ψu(x) (|un (x) − u(x)|) → 0 for all x ∈ {u ∈ int(I)} \ M . Due to the strict monotonicity of ψu(x) , we conclude that un → u almost everywhere in {u ∈ int(I)} and consequently un → u strongly in L1 ({u ∈ int(I)}) (see Theorem 1.18). This completes the proof if I is open. If I is not open, we get un → u strongly in L1 ({u = α}), where α ∈ I \ int(I)}) in the same manner as in the previous proof. 2 3.4.2

Some results from monotone operators

We shall start with the following lemma. Lemma 3.35 Let G be a domain in IRN , I an interval in IR and let P : I → IR be a nondecreasing function (defined on I). Let un be a sequence of functions from L1 (G) with values in I such that


187

un → u weakly in L1 (G), P (un ) → P (u) weakly in L1 (G), P (un )u → P (u) u weakly in L1 (G),

(3.4.5)

P (un )un → P (u)u weakly in L1 (G), P (u)(un − u) → 0 weakly in L1 (G).

Then P (u)u ≥ P (u) u a.e. in G.

(3.4.6)

Remark 3.36 The assumptions of Lemma 3.35 can be slightly weakened: If we suppose that P (un )un , P (un )u, P (un ), P (u)(un − u) ∈ L1 (G), then in all assumptions of the lemma, “weakly in L1 (G)” can be replaced by “weakly-∗ in [C 0 (G)]∗ with the weak-∗ limit belonging to L1 (G)”. Proof Let ǫ > 0 and B ⊂ B(ǫ) ⊂ G be two balls of the same center and radii rB and rB + ǫ. Let κǫ ∈ D(IRN ) be such that 0 ≤ κǫ ≤ 1, κǫ = 1 in B and κǫ = 0 outside B(ǫ) so that κǫ → 1B pointwise. Since P is nondecreasing, we have

(P (un ) − P (u))(un − u)κǫ ≥ 0 G

and, consequently, letting n → ∞, we get

Letting ǫ → 0+ , this yields

G

(P (u)u − P (u) u)κǫ ≥ 0.

(P (u)u − P (u) u) ≥ 0.

B

Since B is an arbitrary ball in G, the last formula implies the statement.

2

We shall also need the following modification of Lemma 3.35. Exercise 3.37 Let G be a domain in IRN , I an interval in IR and let P, Q : I → IR be two nondecreasing functions on I. Let un be a sequence of functions defined a.e. in G with values in I. Suppose that P (un ) → P (u), Q(un ) → Q(u), P (un )Q(u) → P (u) Q(u), P (un )Q(un ) → P (u)Q(u), P (u)(Q(un ) − Q(u)) → 0 weakly in L1 (G). Then P (u)Q(u) ≥ P (u) Q(u). In the sequel, we shall use the above results in the following context.

188


Exercise 3.38 Let G be a domain of IRN . Suppose that 1 ≤ s < ∞ and 0 < θ < ∞ and that ρn is a nonnegative sequence in Ls+θ (G) satisfying ρθn → ρθ weakly in L ρsn → ρs weakly in L

s+θ θ

(G),

s+θ s

(G),

ρs+θ → ρs+θ weakly in L1 (G). n Then ρs+θ ≥ ρs ρθ a.e. in G.

(3.4.7)

We finish this section with the following lemma. Lemma 3.39 Let G, P and un satisfy the assumptions of Lemma 3.35. Suppose moreover that P is continuous, P (u + η) ∈ L1 (G), P (u + η)u ∈ L1 (G), η ∈ D(G), P (u + η)un → P (u + η)u weakly in L1 (G),

and P (u)u = P (u) u a.e. in G.

(3.4.8)

Then we have P (u) = P (u) a.e. in G. Proof We shall perform the proof via the Minty trick. Since P is nondecreasing, we have

(P (un ) − P (u ± αη))(un − (u ± αη)) ≥ 0 G for all α > 0 and η ∈ D(G). Letting n → ∞, we get

[P (u)u − P (u) u ∓ αηP (u) ± αηP (u ± αη)] ≥ 0. G

By virtue of (3.4.8), the last formula yields

±α G [P (u ± αη) − P (u)]η ≥ 0.

Dividing the last inequality by α and then letting α → 0+ , one obtains

(P (u) − P (u)) η ≥ 0, η ∈ D(G). G This implies the statement (cf. Lemma 1.33 in Section 1.3.4).

2

This lemma will be used in the sequel in the following context: Exercise 3.40 Under the assumptions of Exercise 3.38 and provided ρs+θ = ρs ρθ , we have s ρs = [ρθ ] θ a.e. in G.

4 WEAK SOLUTIONS FOR STEADY NAVIER–STOKES EQUATIONS OF COMPRESSIBLE BAROTROPIC FLOW This chapter is devoted to the weak solvability of steady compressible Navier– Stokes equations in the barotropic regime in different geometrical situations. After the formulation of the problem and the main statements for bounded and exterior domains in the first section, we explain, in the second section, the main ideas of the proof, and in the third section, the way the original system is approximated. In Section 4.4 we investigate weak compactness of the effective viscous flux which is the main tool of the proofs. Subsequent Sections 4.5, 4.7 recall some classical results for the Neumann problem with Laplacian for the relaxed continuity equation with dissipation and for the Lamé system, needed throughout the proofs. In Sections 4.8–4.10 the systems on different levels of approximations are solved and estimates independent of parameters of approximations are derived. Finally, in Section 4.11, the complete system of a viscous barotropic gas is solved in a bounded domain. Sections 4.12–4.14 are devoted to weak solutions of the same problem in exterior domains. In Section 4.15 we generalize all results to domains with only Lipschitz boundaries. Finally, in Section 4.16, the well-posedness of compressible flows in domains with noncompact boundaries is investigated. In this Chapter, if not stated explicitly otherwise, c, ci , L denote generic positive constants which may depend on various parameters of the problem. They may take different values in different formulas. Sometimes, if we judge it useful for better understanding or if it is not clear from the context, we explicit the dependence on some of the parameters in the argument of the corresponding constants. 4.1

Formulation of problems in bounded and exterior domains and main results

In this section we formulate the problems which we are going to solve in the sequel. We define weak solutions, renormalized weak solutions and bounded energy weak solutions in bounded and unbounded domains and explain how these definitions are motivated. We shall also formulate two existence theorems. The first one is concerned with the homogeneous Dirichlet problem in bounded domains and the second one deals with the homogeneous Dirichlet problem in exterior domains. They will be proved in Sections 4.3–4.14. We shall discuss possible generalizations to other unbounded domains, to more complicated pressure laws and to other boundary conditions in Sections 4.16 and 4.17. 189

190

4.1.1

WEAK SOLUTIONS FOR STEADY NAVIER–STOKES EQUATIONS

Definition of weak solutions

4.1.1.1 Homogeneous Dirichlet problem We shall study the following system of equations: ∂j (ρuuj ) − µ∆u − (µ + λ)∇div u + ∇p(ρ) = ρf + g in Ω, div (ρu) = 0 in Ω,

(4.1.1) (4.1.2)

where Ω is a domain of IR3 , ρ(x), u(x) = (u1 (x), u2 (x), u3 (x)), x ∈ Ω are unknown functions and f (x) = (f 1 (x), f 2 (x), f 3 (x)), g(x) = (g 1 (x), g 2 (x), g 3 (x)) are given data. The map s → p(s) is a prescribed function, the so-called constitutive law for pressure. In (4.1.1), (4.1.2), we shall suppose5 2 µ > 0, λ + µ ≥ 0, 3

(4.1.3)

and p(ρ) = ργ , γ>

3 2

and curl f = 0 if γ ≤ 53 .

(4.1.4)

System (4.1.1) and (4.1.2) will be complemented with the Dirichlet boundary condition u(x) = 0, x ∈ ∂Ω.

(4.1.5)

For physical background to these equations and conditions see Sections 1.2.3, 1.2.4, 1.2.6, 1.2.8, 1.2.13 and 1.2.21. If Ω is unbounded, one has to prescribe the behavior of (ρ, u) at infinity. One reasonable option is to require u(x) → 0, ρ(x) → ρ∞ as |x| → ∞, x ∈ Ω,

(4.1.6)

where ρ∞ is a given nonnegative constant. Other reasonable options for conditions at infinity are discussed in Section 4.16. 5 These conditions on µ, λ are consequence of the Clausius-Duhem inequality, see Section 1.2.13. Conditions imposed on µ, λ from the point of view of existence theory may be different. For example, they may be weaker in the case of Dirichlet boundary conditions (4.1.5), see (4.7.2) and they are stronger in the case of slip boundary conditions (4.17.14), see (4.17.12).

FORMULATION OF PROBLEMS AND MAIN RESULTS

191

4.1.1.2 Renormalized continuity equation and energy inequality Now, we perform some formal calculation. To this end, suppose that the couple (ρ, u) is a sufficiently smooth solution of problem (4.1.1)–(4.1.6) and that ρ is positive throughout Ω. Multiplying equation (4.1.2) by b′ (ρ), where b ∈ C 1 ((0, ∞)), we obtain div (b(ρ)u) + {ρb′ (ρ) − b(ρ)}div u = 0.

(4.1.7)

This is the so-called renormalized continuity equation. If Ω is a bounded Lipschitz domain, the steady form of the energy equation (1.2.89) integrated over Ω reads µ

Ω

Ω

|∇u|2 + (µ + λ)

Ω

Ω

|div u|2 =

Ω

Ω

(ρf + g) · u.

(4.1.8)

As we have explained in Section 1.2.18, it can also be obtained by scalar multiplying equation (4.1.1) by u and integrating over Ω. Indeed, after a straightforward but long calculation which uses essentially integration by parts and identity (4.1.2), we obtain (4.1.8) as well. Due to the lower weak semicontinuity of norms at the left-hand side of (4.1.8), for weak solutions, we can expect inequality rather than equality, namely µ

|∇u|2 + (µ + λ)

|div u|2 ≤

(ρf + g) · u.

(4.1.9)

This is the so-called energy inequality. If Ω is an unbounded locally Lipschitz domain, the steady version of the energy equation (1.2.89) integrated over ΩR , R > 0 reads:

|∇u|2 + (µ + λ) ΩR |div u|2

= ΩR (ρf + g) · u − ∂BR ∩Ω [E(ρ, u) + p(ρ)]u · ndS

+ ∂BR ∩Ω [λu · ndiv u + µ(u · ∇u) · n + µ(n · ∇u) · u]dS. µ

ΩR

(4.1.10)

At the right hand side, we are left with four integrals over ∂BR ∩ Ω: γ − γ−1

∂BR ∩Ω

∂BR ∩Ω

1 (ργ−1 − ργ−1 ∞ )ρu · ndS, − 2

∂BR ∩Ω

ρ|u|2 u · ndS,

[λu · ndiv u + µ(u · ∇u) · n + µ(n · ∇u) · u]dS,

γ −ργ−1 ∞ γ−1 ∂BR ∩Ω ρu · ndS.

From physical reasons, due to the expected decay of ρ − ρ∞ and u, it is natural to suppose that the first three integrals tend to 0 as R → ∞ (the heuristic arguments can be taken from Sections 5.2.2 and 5.5). By virtue of (4.1.2), the fourth integral is equal to 0. Therefore, for weak solutions, we can again expect that the energy inequality (4.1.9) holds.

192


4.1.1.3 Weak solutions In this section, we define weak solutions, renormalized weak solutions and bounded energy weak solutions. Definition 4.1 (i) A couple (ρ, u) is called a weak solution of problem (4.1.1)– (4.1.6) if: ρ ∈ Lsloc (Ω) for some γ ≤ s ≤ ∞, ρ ≥ 0 a.e. in Ω, u ∈ (D01,2 (Ω))3 ,

(4.1.11)

ρ − ρ∞ ∈ Lr (Ω) for some 1 ≤ r < ∞ if Ω is unbounded. Equation (4.1.1) holds in (D′ (Ω))3 .

(4.1.12)

Equation (4.1.2) holds in D′ (Ω).

(4.1.13)

Equation (4.1.2) holds in D′ (IR3 ) provided (ρ, u) is prolonged by zero outside Ω.

(4.1.14)

(ii) A couple (ρ, u) is called a renormalized weak solution of problem (4.1.1)– (4.1.6) if in addition to (4.1.11), (4.1.12), (4.1.13):

Equation (4.1.7) holds in D′ (IR3 ) provided (ρ, u) is prolonged by zero outside Ω, for any function b satisfying (3.1.16), (3.1.17) and (3.1.18) with β = s.

(4.1.15)

(iii) A couple (ρ, u) is called a bounded energy weak solution of problem (4.1.1)– (4.1.6) if in addition to (4.1.11), (4.1.12), (4.1.13), it satisfies the energy inequality (4.1.9). Exercise 4.2 (i) If Ω is a bounded domain then u ∈ (D01,2 (Ω))3 is equivalent to saying that u ∈ (W01,2 (Ω))3 (a consequence of the Poincaré inequality). (ii) If Ω is a bounded Lipschitz domain, then u ∈ (W01,2 (Ω))3 is equivalent to saying that u ∈ (W 1,2 (IR3 ))3 , u = 0 a.e. in IR3 \ Ω (a consequence of the Stokes formula). 4.1.2 Existence of weak solutions In this section, we shall state existence theorems for system (4.1.1)–(4.1.6) in a bounded and in an exterior domain. 4.1.2.1 Bounded domains Theorem 4.3 Assume that Ω is a bounded domain of class C 2 ,

(4.1.16)

f ∈ (L∞ (Ω))3 , g ∈ (L∞ (Ω))3 , m > 0.

(4.1.17)

Then there exists a renormalized bounded energy weak solution (ρ, u) to problem (4.1.1)–(4.1.5) such that ρ ∈ Ls(γ) (Ω), s(γ) =

3(γ − 1) if 3/2 < γ < 3 2γ if γ ≥ 3,

ρ = m, u ∈ (W01,2 (Ω))3 . Ω

(4.1.18)


193

Proof The underlying ideas of the proof of Theorem 4.3 are explained in Section 4.2. The detailed proof is given in Sections 4.3–4.11. 2 Remark 4.4 By virtue of Lemma 3.2 and Lemma 3.3, any weak solution of problem (4.1.1)–(4.1.5) is automatically a renormalized weak one provided s(γ) ≥ 2 (i.e. provided γ ≥ 35 ).

Remark 4.5 The condition γ > 23 marks the limit of the present theory in the sense that for γ ≤ 32 , one cannot guarantee ρu ⊗ u ∈ (L1loc (Ω))3×3 . In this situation, the interpretation of the convective term div (ρu ⊗ u) in the weak formulation of momentum equation (4.1.1) is questionable. Remark 4.6 Conditions (4.1.17) imposed on f and g are not optimal. Their optimization is left to the reader as an exercise. See also Section 4.17.2.1. Remark 4.7 In Section 4.15, we shall explain how to generalize Theorem 4.3 to the case of Ω a Lipschitz domain. The generalization of Theorem 4.3 to the case of more general monotone pressure laws, and to the case of slip boundary conditions described in Section 1.2.21, will be discussed in Section 4.17.

4.1.3 Exterior domains We recall that exterior domains have been defined in (3.3.46) and (3.3.47) in Section 3.3.2. We shall prove the following result. Theorem 4.8 Assume that Ω is an exterior domain belonging to class C 2 , 1

and

∞

3

1

∞

3

f ∈ (L (Ω) ∩ L (Ω)) , g ∈ (L (Ω) ∩ L (Ω)) , ρ∞ > 0, γ > 3.

(4.1.19) (4.1.20) (4.1.21)

Then there exists a renormalized bounded energy weak solution (ρ, u) to problem (4.1.1)–(4.1.6) which is such that ρ − ρ∞ ∈ L3 (Ω) ∩ L2γ (Ω).

(4.1.22)

Proof The main ideas of the proof are presented again in Section 4.2. The detailed proof of Theorem 4.8 is performed in Sections 4.12–4.14. 2 Remark 4.9 Conditions (4.1.20) imposed on f and g are not optimal. Their optimization is left to the reader as an exercise. Remark 4.10 In Sections 4.15 and 4.17 we explain how to generalize Theorem 4.8 to the case of only Lipschitz domains, to the case of more general monotone pressure laws, and to the case of slip boundary conditions. We shall also give examples of nonexistence if ρ∞ = 0. In Section 4.16, we shall consider domains with several exits at infinity. In this situation, we introduce the notion of renormalized bounded energy weak solutions including pressure drops and fluxes through the exits. We shall discuss existence of these solutions in the case of domains with several conical and superconical exits and nonexistence in the case of domains with several cylindrical and subconical exits.

194


4.2 Heuristic approach This section is intended to be a brief guide to the proofs of existence of weak solutions to the steady, compressible barotropic Navier–Stokes equations. In Sections 4.3–4.14 we shall prove Theorems 4.3 and 4.8 via several levels of approximations. On any level, we use the same underlying ideas. They are often hidden behind different technicalities. We shall explain these ideas in the present section. To this end, we shall investigate the weak compactness of the set of bounded energy renormalized weak solutions to system (4.1.1)–(4.1.5). This property is formulated in Theorem 4.11. All known variants of proofs of this theorem (and consequently those of Theorems 4.3 and 4.8) rely on P.-L. Lions’ ideas published in (Lions, 1998) (see also (Lions, 1993b), (Lions, 1993a)). Here, we shall describe Lions’ approach of (Lions, 1998), Chapter 6 (which applies to values γ ≥ 35 ), modified to treat the case y, 2002), by adapting Feireisl’s nonsteady approach γ > 32 in (Novo and Novotn´ (Feireisl, 2001) to the case of steady equations. The theorem reads: Theorem 4.11 Let Ω be a bounded domain with smooth boundary and let m > 0. Let (ρn , un ) ∈(4.1.18) be a sequence of bounded energy renormalized weak solutions to problem (4.1.1)–(4.1.5) with the right-hand side of (4.1.1) of the form ρn f n + g n , where g n → g, f n → f strongly in L∞ (Ω). Then there is a subsequence such that ρn → ρ weakly in Ls(γ) (Ω),

un → u weakly in (W01,2 (Ω))3 ,

where (ρ, u) is a bounded energy renormalized weak solution of the same problem which satisfies (4.1.18). In the sequel, we explain the main lines of the proof of the above theorem. 4.2.1 Estimates due to the energy inequality and improved estimates of density The energy inequality (4.1.9), by itself, does not give any reasonable bound for (ρn , un ) only by the external data. If we test the momentum equation (4.1.1) written for (ρn , un ) by the test function

φ = B(ρθn − Ω ρθn ), (4.2.1)

where B is the Bogovskii operator on Ω (see Lemma 3.17) and θ > 0, after using (4.1.9), we obtain un L2 (I,W 1,2 (Ω))3 ≤ L < ∞, 0

ργ+θ ≤ L, Ω n

θ = 2γ − 3, 32 < γ < 3 θ = γ, γ ≥ 3.

(4.2.2)

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195

To get the above estimate, one does not even need that (ρn , un ) satisfies the renormalized continuity equation. However, the growth conditions in assumption (4.1.4) are crucial. For all details of this proof, see Section 4.10.6. In various different contexts, these estimates were well known in the mathematical literature. In the context of weak solutions, they were obtained by P.-L. Lions in (Lions, 1998) by another means rather than by the Bogovskii operator. Here, we have described the proof taken over from (Novotn´ y, 1996). Estimate (4.2.2) is the first step to the existence of weak solutions. Among others, it ensures that the limiting pressure cannot be a mere measure. We shall emphasize two points: 1) The stationary case seems to be worse than the nonsteady case in the sense that the energy inequality by itself does not give any information about the sequence of weak solutions (compare with the nonstationary case, Sections 7.3.2 and 7.3.3). 2) Due to the use of the Bogovskii operator, the constant L in (4.2.2) depends on Ω. This is to be compared with the nonstationary case, Section 7.3.2, where the constant L does not depend on Ω. Some consequences of this fact will be discussed in Sections 4.15 and 7.12. 4.2.2

Limit passage

Estimates (4.2.2) yield, at least for an appropriately chosen subsequence denoted again by (ρn , un ): ρn → ρ weakly in Lγ+θ (Ω),

un → u weakly in (W01,2 (Ω))3 , 2γ

ρn un → ρu weakly in (L γ+1 (Ω))3 , ρn uin ujn

i j

(4.2.3)

′

→ ρu u in D (Ω),

ργn → ργ weakly in L

γ+θ γ

(Ω).

Consequently, by virtue of (4.1.1), (4.1.2), (ρ, u, ργ ) prolonged by (0, 0, 0) outside Ω, satisfies (4.2.4) div (ρu) = 0 in D′ (IR3 ), and div (ρu ⊗ u) − µ∆u − (µ + λ)∇div u + ∇ργ = ρf + g in (D′ (Ω))3 .

(4.2.5)

Moreover, due to (4.1.15), we have div [b(ρ)u] + [ρb′ (ρ) − b(ρ)]div u = 0 in D′ (IR3 ),

(4.2.6)

where b satisfies (3.1.16), (3.1.17) with λ1 + 1 < γ+θ 2 , and where the overlined quantities denote corresponding weak limits. The results presented in this section rely on standard analysis and do not give anything new. The reader can consult Section 4.11.1 for all details.

196

4.2.3


Effective viscous flux

Now, the main difficulty in the proof is to show that ργ = ργ . This requires some sort of compactness of the sequence ργn which cannot be deduced from estimates (4.2.2). This missing information will be replaced by the observation that the quantity6 P = ργ − (2µ + λ)div u,

usually called the effective viscous flux or effective pressure, possesses such property. The following result is absolutely crucial in all known proofs of existence of weak solutions to system (4.1.1)–(4.1.2). Lemma 4.12 If γ > 32 , then

limn→∞ Ω η[ργn − (2µ + λ)div un ]b(ρn ) = Ω η[ργ b(ρ) − (2µ + λ)b(ρ) div u], (4.2.7) where η is any function belonging to D(Ω) and b satisfies (3.1.16), (3.1.17) with λ1 + 1 ≤ min{ γ+θ 2 , θ}.

This type of identities was discovered by P.-L. Lions in (Lions, 1993a) and proved in (Lions, 1998). For a detailed proof see Section 4.4.4 (and also 4.9.2, 4.10.2, 4.11.2). We shall see in Section 7.3.5 that a similar statement holds also in the nonstationary case. It plays the same crucial role in the problems of evolution of compressible barotropic flows. In fact, in the steady case, a stronger property of the sequence Pn = ργn − (2µ + λ)div un than property (4.2.7) is known. Indeed, effecting div of (4.1.1), we arrive at the identity ∆P = −div (ρu · ∇u) + div (ρf + g).

(4.2.8)

Using a version of div–curl lemma due to (Coifman et al., 1993) to show that 3γ−3 ρu · ∇u ∈ h 2γ−1 (Ω′ ), and then employing elliptic regularity theory along with bounds (4.2.2), one obtains ∇Pn

3γ−3

h 2γ−1 (Ω′ )

≤ L(Ω′ ) < ∞, γ >

3 2

(4.2.9)

for any subdomain Ω′ such that Ω′ ⊂ Ω. Here hs (Ω′ ) is a so-called local Hardy space. The above bound implies Lemma 4.12 by using compact imbeddings and interpolations. The interested reader can consult (Novotn´ y, 1996), (Lions, 1998) for the missing definition of Hardy spaces and for missing proofs related to (4.2.9). In this book, we give another proof which does not rely on (4.2.9). This proof is based on a version of the commutator lemma due to E. Feireisl (cf. Section 4.4.3), a result in the spirit of (Coifman et al., 1993). One of its advantages 6 The same quantity plays an essential role in the proof of existence of strong solutions, see Section 5.2, especially formula (5.2.10).

HEURISTIC APPROACH

197

is the fact that it follows the same lines both for steady and nonsteady cases. Its steady version as presented in this book is taken over from (Novo and Novotn´ y, 2002). We finish this section by remarking that no result like (4.2.9) is available in the nonstationary case. 4.2.4

Strong convergence of density – Lions’ approach

In this section, we explain schematically Lions’ approach after Lemma 4.12. On one hand, equation (4.2.6) with b(s) = sϑ , 0 < ϑ < 1, reads div (ρϑ u) = (1 − ϑ)ρϑ div u.

(4.2.10)

On the other hand, we can apply to equation (4.2.10) the DiPerna–Lions 1 transport theory (DiPerna and Lions, 1989). Thus, Lemma 3.3 with b(s) = s ϑ and identity (4.2.7) will give, after a long calculation, div

ϑ1 ρϑ u =

1−ϑ ϑ(2µ+λ)

1 −1 ργ+ϑ − ργ ρϑ ρϑ ϑ .

(4.2.11)

However, due to (4.2.2) and (3.1.18), the application of Lemma 3.3 to equation (4.2.10) is rigorous only provided s(γ) ≥ 2, i.e., provided γ ≥ 35 . This is the first point, where this restriction on γ appears. Performing the difference of (4.2.4) and (4.2.11), we obtain div

ρϑ

ϑ1

−ρ u =

1−ϑ ϑ(2µ+λ)

1 −1 ργ+ϑ − ργ ρϑ ρϑ ϑ

which, when integrated over Ω, implies 1 −1

ργ+ϑ − ργ ρϑ ρϑ ϑ = 0. Ω

(4.2.12)

(4.2.13)

Via some arguments of the theory of monotone operators which are explained in Section 3.4, the last identity yields ϑγ ργ = ρϑ a.e. in Ω.

At the end, this together with the second bound in (4.2.2), yields the strong convergence of ρn in Lp (Ω), 1 ≤ p < γ + θ. Throughout this section, we have used some formal arguments. All details can be found in Sections 4.9.3, 4.9.4, 4.10.3, 4.10.4. Starting from a sequence of bounded energy renormalized weak solutions, we obtained a weak solution provided γ ≥ 53 . This solution is also a renormalized one provided we can apply to it the DiPerna–Lions transport theory (see Section 3.1). This is possible provided ρ ∈ L2 (Ω), i.e., provided, again, γ ≥ 53 (see (4.2.2)). The assumption γ ≥ 53 can be relaxed up to (4.1.4) by applying a different approach based on the adaptation by Novo and Novotn´ y (see (Novo and Novotn´ y, 2002)) of Feireisl’s method (Feireisl, 2001), developed originally for nonsteady equations. This approach is explained in the following section.

198

4.2.5


Strong convergence of density – Feireisl’s approach

In the steady case, the crucial observation of (Feireisl, 2001) is rephrased as follows: Lemma 4.13 Let k > 0 and let Tk : [0, ∞) → [0, ∞) be such that Tk (s) = s if s ∈ [0, k), Tk (s) = k if s ∈ [k, ∞). Then we have: If γ > 32 , then supk>0 lim supn→∞ Tk (ρn ) − Tk (ρ)γ+1 0,γ+1,Ω

γ γ ≤ limn→∞ Ω (ρn Tk (ρn ) − ρ Tk (ρ)) < ∞.

This formula is an immediate consequence of Lemma 4.12. For its detailed proof see Section 4.11.3. Its meaning can be translated as follows: Although the sequence ρn is bounded only in Lγ+θ (Ω), its oscillations, measured by the left-hand side of the above formula, are always bounded in “better” space than L2 (Ω). This is the property which will replace the missing condition ρ ∈ L2 (Ω) in the DiPerna–Lions transport theory (compare with Sections 7.3.7.1 and 7.3.7.2). Indeed, with Lemma 4.13 at hand, one can prove that the renormalized continuity equation holds: Lemma 4.14 If γ > 23 , then the equation div (b(ρ)u) + (ρb′ (ρ) − b(ρ))div u = 0 holds in D′ (IR3 ) with any b satisfying (3.1.16), (3.1.17), (3.1.18), where we have taken β = γ + θ. For a detailed proof of this statement, see Section 4.11.4. Of course, due to (4.2.4), Lemma 4.14 still holds if we replace b(s) by cs+b(s), c ∈ IR. Now, it is enough to write the renormalized continuity equation with function Lk , where Lk (s) =

s ln s, s ∈ [0, k), s ln k + s − k, s ∈ [k, ∞),

and subtract it from equation (4.2.6). One gets div [(Lk (ρ) − Lk (ρ))u] + [Tk (ρ)div u − Tk (ρ)div u] = 0. This identity, when manipulated conveniently (integrating over Ω, letting the first term disappear, using Lemmas 4.12 and 4.13 in the second term), implies lim lim sup Tk (ρn ) − Tk (ρ)Lγ+1 (Ω) ≤ 0.

k→∞ n→∞

This and the second bound in (4.2.2) imply the strong convergence of density in Lp (Ω), 1 ≤ p < γ + θ. We have presented a rather formal sketch of the last part of the proof. For all details see Section 4.11.5. Both methods can be applied to more general or to slightly different situations. The reader can consult Sections 4.12–4.17 to learn more about it.

HEURISTIC APPROACH

4.2.6

199

Remarks to approximations

4.2.6.1 Proof by using Lions’ existence theorem One point of view would be to apply Theorem 6.7 and the results of Section 6.10 of Lions’ book (Lions, 1998) to system (4.1.1)–(4.1.3), (4.1.5) with p(ρ) = ργ + δρβ , γ >

5 3 , β > , δ > 0. 2 3

Passage to the limit δ → 0+ , using Lions’ and Feireisl’s compactness arguments explained in Sections 4.2.1–4.2.3 and 4.2.5 would then yield the statement of Theorem 4.3. This is, however, not the path we want to follow. Our goal is to present a consistent theory starting with basic approximations and ending up with the weak solutions of the original system. 4.2.6.2 Sequence of approximations used in the steady case Since the continuity equation (4.1.2) is of degenerate hyperbolic type, it is natural to relax its degeneracy by adding to the left-hand side the term αρ (α → 0+ ). At the same time, we have to add to the right-hand side the term αh with h a convenient

function. If Ω is bounded, this term has to be chosen in such a way that Ω h = m (conservation of mass). If Ω, an unbounded domain, is approximated by invading domains (and the boundary condition for u on the boundaries of these domains is zero), then h has to generate the required fluxes through given cross-sections (as, e.g., in the case of an exterior domain described in Sections 4.12–4.14 or, better, in the case of a domain with several exits described in Section 4.16). After such modification of the continuity equation, we have to change the momentum equation by adding to its left-hand side the term 12 αhu + 32 αρu in order to keep the information coming from the energy inequality (see (4.3.20) with δ = 0 and Remark 4.21). Equations (4.1.1) and (4.1.2) are now approximated by equations (4.3.13) and (4.3.14). We refer to Section 4.3.2 for more details. After these arrangements, it is natural to use in equation (4.3.14) (it is still hyperbolic) elliptic regularization, by adding to its left-hand side the term −ǫ∆ρ (ǫ → 0+ ). Due to the conservation of mass, we have to complete this modified equation with the Neumann boundary condition ∂n ρ = 0 at ∂Ω. We obtain the elliptic problem (4.3.26), (4.3.27) which can be treated by standard methods and which yields nonnegative density provided h ≥ 0 (see Section 4.6 for all details). Then, we have to modify the momentum equation in order to maintain the energy inequality. This is done by considering the convective term ∂j (ρuuj ) in the form 21 ∂j (ρuuj ) + 21 ρu∇uj . For the whole new system see (4.3.25) with δ = 0 and (4.3.26) in Section 4.3.3. The new energy inequality is given in (4.3.31) and calculated in Section 4.8.1.2. System (4.3.25) with δ = 0, (4.3.26), (4.3.27), (4.1.5) can be solved by the Leray–Schauder fixed point theorem (as explained in Section 4.8). During this process, we need a technical assumption γ > 3. This assumption is not optimal, but convenient for the proof. This is the reason why we modify the pressure

200


p(ρ) = ργ by considering pδ (ρ) = ργ + δ(ρ2 + ρβ ). (The term δρ2 is not needed for the Leray–Schauder theorem but simplifies some parts of the proof.) The result about the solvability of system (4.3.25), (4.3.26), (4.3.27), (4.1.5) is stated in Proposition 4.22. The limit process ǫ → 0+ already requires the Lions compactness argument explained in Sections 4.2.1–4.2.4. The detailed proof of this issue is presented in Section 4.9. The existence result obtained for system (4.3.13), (4.3.14), (4.1.5) is formulated in Proposition 4.18. The limit process α → 0+ again essentially uses Lions’ compactness arguments of Sections 4.2.1–4.2.4. The reader can consult Section 4.10 for all details and Proposition 4.15 for the precise formulation of the existence result for the system (4.3.1), (4.3.2), (4.1.5). Finally, passing from artificial pressure to physical pressure by letting δ → 0+ , already requires the compactness argument described in Section 4.2.10. Indeed, if γ < 53 , one loses the uniform estimate for the sequence ρδ in L2 (Ω) and therefore one cannot apply the DiPerna–Lions transport theory. The reader can see Section 4.2.10 for more explanation concerning this phenomenon, Section 4.11 for all details of the proof, and Theorem 4.3 for the precise formulation of the existence result for the original system (4.1.1)–(4.1.5). To treat the problems on unbounded domains, we shall use the method of invading domains. It consists in solving system (4.3.13), (4.3.14) and (4.1.5) on ΩR = Ω ∩ BR , and then in letting (α, R) → (0+ , ∞). This process is described in all details in Sections 4.12–4.14 in the case of an exterior domain, and in Section 4.16 in the case of domains with noncompact boundaries. 4.3

Approximations in bounded domains

In this section we introduce the chain of approximations which is used to solve problem (4.1.1)–(4.1.5). In Section 4.2.6 we have described the motivation for such a choice. We shall also explain the process leading from approximations to weak solutions of the original system. On any level of approximations, we formulate corresponding statements about the existence of weak solutions and their properties which are needed to carry out rigorously the whole limit procedure. 4.3.1

First level approximation – artificial pressure

We consider the following system of equations ∂j (ρuuj ) − µ∆u − (µ + λ)∇divu + ∇pδ (ρ) = ρf + g in Ω, div(ρu) = 0 in Ω

(4.3.1) (4.3.2)

with boundary conditions (4.1.5) and with pδ (ρ) = ργ + δ(ρ2 + ρβ ), δ > 0, β > max{γ, 3}.

(4.3.3)

The quantity pδ is called the artificial pressure. A weak solution of the original problem (4.1.1), (4.1.2), (4.1.5) will be obtained as a weak limit (ρ, u) as δ → 0+

APPROXIMATIONS IN BOUNDED DOMAINS

201

of a subsequence (ρδ , uδ ) of weak solutions to system (4.3.1), (4.3.2), (4.1.5). This limit process will be carried out in Section 4.11; its main ingredient will be the following proposition which concerns the existence of solutions and estimates independent of δ for system (4.3.1), (4.3.2), (4.1.5). It will be proved in Section 4.10. Proposition 4.15 Let f , g, m satisfy (4.1.17), let µ, λ satisfy (4.1.3), let γ be as in (4.1.4) and let δ, β satisfy (4.3.3). Suppose that Ω belongs to the class (4.1.16). Then there exists a couple (ρδ , uδ ) with the following properties: (i)

ρδ ∈ L2β (IR3 ), Ω ρδ = m, ρδ ≥ 0 a.e. in Ω, ρδ = 0 in IR3 \ Ω, (4.3.4) 6β uδ ∈ (W 1,2 (IR3 ))3 , uδ = 0 in IR3 \ Ω, ρδ uδ ∈ (L β+3 (IR3 ))3 . (ii) ∂j (ρδ uδ ujδ ) − µ∆uδ − (µ + λ)∇div uδ

+∇ργδ + δ∇(ρ2δ + ρβδ ) = ρf + g in (D′ (Ω))3 .

(4.3.5)

(iii) div(ρδ uδ ) = 0 in D′ (IR3 ).

(4.3.6)

(iv) For any b satisfying (3.1.16), (3.1.17), (3.1.18) with β replaced by 2β, we have div (b(ρδ )uδ ) + {ρδ b′ (ρδ } − b(ρδ ))div uδ = 0 in D′ (IR3 ).

(4.3.7)

For any bk , k > 0 defined by (3.1.29) with b satisfying (3.1.16), there holds div (bk (ρδ )uδ ) + (ρδ (bk )′+ (ρδ ) − bk (ρδ ))div uδ = 0 in D′ (IR3 ). (v) µ

Ω

|∇uδ |2 + (µ + λ)

Ω

(vi) The following estimates hold:

|div uδ |2 ≤

Ω

(ρδ f + g) · uδ .

uδ 1,2,IR3 ≤ L(Ω, f , g, m), ρδ 0,s(γ),IR3 ≤ L(Ω, f , g, m), s(γ) = 1

3(γ − 1) if γ < 3 2γ if γ ≥ 3,

δ β ρδ 0, s(γ) β,IR3 ≤ L(Ω, f , g). γ

(4.3.8) (4.3.9) (4.3.10) (4.3.11) (4.3.12)

Here L is a positive constant which is, in particular, independent of δ. Remark 4.16 One easy verifies by a density argument and by (4.3.4) that equation (4.3.5) holds with any test function φ ∈ (W01,2 (Ω))3 , and equation (4.3.6) holds with any test function η ∈ W 1,2 (Ω). Remark 4.17 A similar statement would hold with β = max {2, γ} and with ∇ρβ instead of ∇(ρ2 + ρβ ). This choice is not essential and influences only the technicalities of proofs.

202

4.3.2


Second level approximation – relaxation in the continuity equation

Let α be a positive constant and h a nonnegative function. A weak solution (ρδ , uδ ) of Proposition 4.15 will be obtained as a weak limit of a subsequence (ρα , uα ) of solutions to the following family of partial differential equations with m : h = |Ω| 1 3 2 αhu + 2 αρu + ∂j (ρuuj ) − µ∆u (4.3.13) −(µ + λ)∇div u + ∇pδ (ρ) = ρf + g in Ω, αρ + div (ρu) = αh in Ω

(4.3.14)

with boundary conditions (4.1.5). This limit process is carried out in Section 4.10. Existence of the sequence (ρα , uα ) with suitable properties is guaranteed by the following proposition. It will be proved in Section 4.9. Proposition 4.18 Let f , g satisfy (4.1.17), let µ, λ satisfy (4.1.3), let γ be as in (4.1.4), let δ, β satisfy (4.3.3) and let α ∈ (0, 1), h ≥ 0, h ∈ L∞ (Ω).

(4.3.15)

Suppose that Ω belongs to class (4.1.16). Then there exists a couple (ρα , uα ) with the following properties: (i) We have

ρα ∈ L2β (IR3 ), Ω ρα = Ω h, ρα ≥ 0 a.e. in Ω, ρα = 0 in IR3 \ Ω,

uα ∈ (W 1,2 (IR3 ))3 , uα = 0 in IR3 \ Ω,

(4.3.16)

6β

ρα uα ∈ (L β+3 (IR3 ))3 , div (ρα uα ) ∈ L2β (IR3 ). (ii) We have 1 2 αhuα

+ 23 αρα uα + ∂j (ρα uα ujα ) − µ∆uα − (µ + λ)∇div uα

+∇ργα + δ∇(ρ2α + ρβα ) = ρα f + g in (D′ (Ω))3 . (4.3.17) (iii) Provided h is prolonged by 0 outside Ω, αρα + div (ρα uα ) = αh in D′ (IR3 ).

(4.3.18)

(iv) Provided h is prolonged by 0 outside Ω, for any b from C 1 ([0, ∞)) satisfying (3.1.17), (3.1.18) with β replaced by 2β, then div (b(ρα )uα ) + {ρα b′ (ρα ) − b(ρα )}div uα = α(h − ρα )b′ (ρα ) in D′ (IR3 ).

(v)

(4.3.19)


α

Ω

γ ρα |uα |2 + α γ−1 (ρ − h)(ργ−1 − hγ−1 ) α Ω α

β +αδ β−1 (ρ − h)(ρβ−1 − hβ−1 ) + 2αδ Ω (ρα − h)2 α Ω α

+µ Ω |∇uα |2 + (µ + λ) Ω |div uα |2

γ (h − ρα )hγ−1 ≤ Ω (ρα f + g) · uα + α γ−1 Ω

β +αδ β−1 (h − ρα )hβ−1 + 2αδ Ω (h − ρα )h. Ω

h|uα |2 + α

203

Ω

(4.3.20)

(vi) The following estimates hold:

uα 1,2,IR3 ≤ L(Ω, f , g, h, δ),

(4.3.21)

ρα 0,2β,IR3 ≤ L(Ω, f , g, h, δ),

(4.3.22)

ρα uα 0,

6β 3 β+3 ,IR

≤ L(Ω, f , g, h, δ).

(4.3.23)

Here L is a positive constant which is, in particular, independent of α. Remark 4.19 If h = const, then (4.3.20) reads simply

γ (ρ − h)(ργ−1 − hγ−1 ) α Ω h|uα |2 + α Ω ρα |uα |2 + α γ−1 α Ω α

β (ρ − h)(ρβ−1 − hβ−1 ) + 2αδ Ω (ρα − h)2 +αδ β−1 α Ω α

+µ Ω |∇uα |2 + (µ + λ) Ω |div uα |2

≤ Ω (ρα f + g) · uα .

(4.3.24)

Remark 4.20 It is an easy exercise to show that equation (4.3.17) holds with any test function φ ∈ (W01,2 (Ω))3 , and equation (4.3.18) with any test function η ∈ W 1,2 (Ω). Remark 4.21 Proposition 4.18 is valid with δ = 0 provided γ > 3. In this case, we replace β by γ in all its statements.

4.3.3 Third level approximation – relaxed continuity equation with dissipation The sequence (ρα , uα ) satisfying Proposition 4.18 will be obtained as a weak limit as ǫ → 0+ of the sequence (ρǫ , uǫ ) of weak solutions to the following system of partial differential equations αhu + αρu + 21 ∂j (ρuuj ) + 21 ρu · ∇u − µ∆u −(µ + λ)∇div u + ∇pδ (ρ) = ρf + g,

(4.3.25)

αρ + div (ρu) − ǫ∆ρ = αh in Ω

(4.3.26)

∂n ρ = 0 in ∂Ω

(4.3.27)

with boundary conditions (4.1.5) for u and

for ρ. This limit process is carried out in Section 4.9. Existence of the sequence (ρǫ , uǫ ) with suitable properties is guaranteed by the following proposition. Its proof is performed in Section 4.8.

204


Proposition 4.22 Let f , g satisfy (4.1.17), let µ, λ satisfy (4.1.3), let γ be as in (4.1.4), let δ, β satisfy (4.3.3), and let α, h satisfy (4.3.15). Suppose that Ω belongs to class (4.1.16) and ǫ > 0. Then there exists a couple (ρǫ , uǫ ) with the following properties: (i) β

ρǫ ∈ W 2,2 (Ω), ρǫ2 ∈ W 1,2 (Ω), Ω ρǫ = Ω h, (4.3.28) ρǫ ≥ 0 in Ω, uǫ ∈ (W01,2 (Ω))3 . (ii) αhuǫ + αρǫ uǫ + 12 ∂j (ρǫ uǫ ujǫ ) + 12 ρǫ uǫ · ∇uǫ − µ∆uǫ

−(µ + λ)∇div uǫ + ∇ργǫ + δ∇(ρ2ǫ + ρβǫ ) = ρǫ f + g in (D′ (Ω))3 . (4.3.29)

(iii) α (iv)

Ω

(ρǫ − h)η −

Ω

ρǫ uǫ · ∇η + ǫ

Ω

∇ρǫ · ∇η = 0, η ∈ C ∞ (IR3 ).

(4.3.30)

γ h|uǫ |2 + α Ω ρǫ |uǫ |2 + α γ−1 (ρ − h)(ργ−1 − hγ−1 ) ǫ Ω ǫ

β (ρ − h)(ρβ−1 − hβ−1 ) + 2αδ Ω (ρǫ − h)2 +αδ β−1 ǫ Ω ǫ

+2ǫδ Ω |∇ρǫ |2 + βǫδ Ω ρβ−2 |∇ρǫ |2 + µ Ω |∇uǫ |2 + (µ + λ) Ω |div uǫ |2 ǫ

β ≤ Ω (ρǫ f + g) · uǫ + αδ β−1 (h − ρǫ )hβ−1 Ω

γ (h − ρǫ )hγ−1 + 2αδ Ω (h − ρǫ )h. +α γ−1 Ω (4.3.31) (v) The following estimates hold: α

Ω

uǫ 1,2 ≤ L(Ω, f , g, h, δ),

(4.3.32)

ρǫ 0,2β ≤ L(Ω, f , g, h, δ),

(4.3.33)

ǫ∇ρǫ 20,2

≤ L(Ω, f , g, h, δ).

(4.3.34)

Here L is a positive constant which is, in particular, independent of ǫ and α. Remark 4.23 One easily verifies by using a density argument and by (4.3.28) that equation (4.3.29) holds with any test function φ ∈ W01,2 (Ω)3 and equation (4.3.30) holds with any test function η ∈ W 1,2 (Ω). 4.4

Effective viscous flux

The main goal of this section is to prove the weak compactness of the effective viscous flux. We have already emphasized that this is the key property for the existence proofs. For system (4.1.1), (4.1.2), this property was formulated in

EFFECTIVE VISCOUS FLUX

205

Section 4.2.3. Here we deal with a more general formulation which applies to all levels of approximating equations described in Section 4.3. The precise statement is given in Proposition 4.26 in Section 4.4.4. To prove Proposition 4.26, one needs some nontrivial preliminary results. They are discussed in Section 4.4.1 devoted to the Riesz operator and in Section 4.4.2 devoted to the div–curl lemma. The div–curl lemma implies the so-called commutator lemma as proved in Section 4.4.3, and the commutator lemma plays an essential role in the proof of the weak compactness of the effective viscous flux, as shown in Section 4.4.4. This scheme of proof is taken over from (Feireisl, 2001). The original proof in (Lions, 1998) in the steady case is based on estimate (4.2.9). For simplicity, and since the applications we have in mind concern only threedimensional domains, all statements of this section are formulated only in IR3 . However, they do hold and they can be easily reformulated in N dimensions (N ≥ 2). This task is left to the reader as an exercise. 4.4.1

Riesz operators

We introduce the operators Aj : S(IR3 ) → S ′ (IR3 ), j = 1, 2, 3, Aj (g) = −F −1

iξ

j F(g) |ξ|2

,

(4.4.1)

where F is the Fourier transform and F −1 its inverse (see Section 1.3.7.1). The Marcinkiewicz theorem about multipliers (see Theorem 1.56) yields ∇A(g)0,r,IR3 ≤ c(r)g0,r,IR3 , 1 < r < ∞.

(4.4.2)

Due to Sobolev’s imbedding theorem (see Section 1.3.6.4), one has A(g)0,r∗ ,IR3 ≤ c(r)g0,r,IR3 , 1 < r < 3.

(4.4.3)

The closure of the operator A (denoted again by A) is therefore a continuous linear operator from Lr (IR3 ) into the homogeneous Sobolev space D01,r (IR3 ) (which satisfies estimates (4.4.2), (4.4.3)). By using duality, we can extend the operator Ai to a continuous linear operator on D−1,r (IR3 ), 1 < r < ∞, by setting ′

A(v), φ = v, A(φ), φ ∈ D01,r (IR3 ), v ∈ D−1,r (IR3 ).

(4.4.4)

(For the definition of homogenous Sobolev spaces and their duals see Sections 1.3.6.1, 1.3.6.4 and 1.3.6.5.) We observe that for a real function g, Ai (g) is real. Due to this fact and due to the Parseval–Plancherel formula (see Theorem 1.55), we have

Ai (f )g = − IR3 Ai (g)f, f, g ∈ S(IR3 ). (4.4.5) IR3 Next, we denote

Rij = ∂i Aj .

(4.4.6)

206


This the is the so-called Riesz operator. We recall some of its evident properties needed in the sequel, namely (4.4.7) Rij = Rji , and 4.4.2

Rii (g) = g,

IR3

Rij (f )g =

Div–curl lemma

(4.4.8)

′

IR3

Rij (g)f, f ∈ Lr (IR3 ), g ∈ Lr (IR3 ).

(4.4.9)

The div–curl lemma is a classical statement of the compensated compactness due to (Murat, 1978) and (Tartar, 1975). One of its most general formulation is the following assertion. Lemma 4.24 Let 1 < p1 , p2 , , q1 , q2 < ∞, domain in IR3 . Suppose that

1 p1

+

1 p2

=

1 r

< 1 and let Ω be a

f n → f weakly in (Lp1 (Ω))3 ,

(4.4.10)

div f n → div f strongly in W −1,q1 (Ω),

(4.4.11)

g n → g weakly in (Lp2 (Ω))3 ,

and

Then

curl g n → curl g strongly in (W −1,q2 (Ω))3 . f n · g n → f · g weakly in Lr (Ω).

(4.4.12)

Proof All we need in the sequel is a trivial version of this assertion, where (4.4.11) is replaced by div f n = 0, curl g n = 0. We shall therefore prove only this simple statement referring to (Yi, 1992) for a more complicated full proof. Since it is enough to prove (4.4.12) only in the sense of distributions, we may suppose, without loss of generality, that Ω is bounded and smooth. Using the properties of Helmholtz decomposition (see Section 5.3.2 or Chapter III, Lemma 1.2 in (Galdi, 1994b)), we observe that g n = ∇Gn , where Gn is a unique solution of the weak Neumann problem

− Ω ∇Gn · ∇η = Ω g n · ∇η, η ∈ C ∞ (IR3 )

in W 1,p2 (Ω). By virtue of (4.4.10), the sequence Gn is bounded in the same space, cf. Lemma 4.27. Since the imbedding W 1,p2 (Ω) ֒→ Lp2 (Ω) is compact, Gn → G strongly in Lp2 (Ω) and g = ∇G. Integrating by parts and using these facts we get

f · g n η = Ω Gn f n · ∇η → Ω Gf · ∇η = Ω f · gη, η ∈ D(Ω). Ω n

This yields (4.4.12) by a density argument.

2


207

4.4.3 Commutator lemma The following lemma plays a crucial role in the proof of the weak compactness of the effective viscous flux presented in the next section. Its proof is taken over from (Feireisl, 2001). Lemma 4.25 Suppose that 1 < p, q < ∞, fn → f

1 p

+

1 q

=

1 r

< 1 and

weakly in Lp (IR3 ),

gn → g weakly in Lq (IR3 ).

(4.4.13)

Then fn Rij (gn ) − gn Rij (fn ) → f Rij (g) − gRij (f ) weakly in Lr (IR3 ).

(4.4.14)

Proof The reader easily verifies that the conclusion of Lemma 4.25 is a particular case of the more general statement: 3 i j j i i,j=1 {vn Rij (wn ) − wn Rij (vn )} → (4.4.15) 3 i j j i ′ 3 i,j=1 {v Rij (w ) − w Rij (v )} weakly in D (IR ) provided

v n → v weakly in (Lp (IR3 ))3 , wn → w weakly in (Lq (IR3 ))3 . We therefore show (4.4.15). To do so, we use symmetry (4.4.7) to deduce 3 i j j i i,j=1 {vn Rij (wn ) − wn Rij (vn )} 3 3 3 = i=1 [{ j=1 Rij (wnj )}{vni − k=1 Rik (vnk )}] 3 3 3 − j=1 [{ i=1 Rji (vni )}{wnj − k=1 Rjk (wnk )}] = Un · Vn − Xn · Yn .

In accordance with (4.4.6) and (4.4.8), we have curl Un = curl Xn = 0, div Vn = div Yn = 0. We can thus use Lemma 4.24 to conclude Un · Vn → U · V, Xn · Yn → X · Y weakly in Lr (BR ), R > 0,

where

1 r

=

1 p

+

1 q

and Ui =

3

Rij (wj ), V i = v i −

3

Rij (v j ), Y i = wi −

j=1

Xi =

j=1

This completes the proof.

3

Rik (v k ),

3

Rik (wk ).

k=1

k=1

2

208


4.4.4 Effective viscous flux The crucial result of Section 4.4 reads: Proposition 4.26 Let Ω be a domain of IR3 . Let 1 < r < ∞, max{2, r′ } ≤ q ≤ ∞, 1s + q1∗ < 1 and 1t + 1q < 56 . Assume that for any bounded subdomain Ω′ , such that Ω′ ⊂ Ω, there holds q n → q weakly-∗ in (Lt (Ω′ ))3 ,

(4.4.16)

un → u weakly in (L2 (Ω′ ))3 ,

(4.4.17)

div q n → div q strongly in W −1,t (Ω′ ), ∇un → ∇u weakly in (L2 (Ω′ ))3×3 , pn → p weakly in Lr (Ω′ ), s

′

q

′

(4.4.18)

3

F n → F weakly-∗ in (L (Ω )) ,

(4.4.19)

gn → g weakly-∗ in L (Ω ),

and suppose that

(4.4.20)

∂j (qni ujn ) − µ∆uin − (µ + λ)∂i div un + ∂i pn = Fni in D′ (Ω) (i = 1, 2, 3). (4.4.21) Then: (i)

limn→∞ Ω η[pn gn − (2µ + λ)gn div un ]

= Ω η[pg − (2µ + λ)gdiv u], η ∈ D(Ω).

(4.4.22)

(ii) If q > max{r′ , 2}, then there exists a subsequence such that gn div un → 2q rq gdiv u weakly in L 2+q (Ω′ ) and pn gn → pg weakly in L r+q (Ω′ ). Moreover, pg − (2µ + λ)gdiv u = pg − (2µ + λ)gdiv u a.e. in Ω.

(4.4.23)

Proof By virtue of (4.4.16)–(4.4.20), system (4.4.21) holds with any test func1,max{2,

6t

}

′

′

5t−6 (Ω))3 , div φ ∈ Lr (Ω) and φ ∈ Ls (Ω)3 which has comtion φ ∈ (W0 pact support in Ω. We use in it the test function φ = ηA(˜ η gn ), where η ∈ D(Ω), η˜ ∈ D(Ω), and where A is defined by (4.4.1). Under the assumptions on r, q, s and t, this is evidently an admissible test function. After a long but straightforward calculation, when using conveniently (4.4.6), (4.4.7), (4.4.8), (4.4.9), we arrive at

η η˜(pn − (2µ + λ)div un )gn Ω

η gn ) η gn ) + (µ + λ) Ω ∂i ηdiv un Ai (˜ = − Ω ∂i ηpn Ai (˜

i i +µ Ω ∂j η∂j un Ai (˜ η gn ) − µ Ω ∂j ηun ∂i Aj (˜ η gn ) (4.4.24)

+µ Ω ∂i ηuin η˜gn − Ω ηFni Ai (˜ η gn ) − Ω ∂j ηqni ujn Ai (˜ η gn )

j i j i i − Ω ηRij (˜ η gn un )qn + Ω un [˜ η gn Rij (qn η) − ηqn Rij (˜ η gn )].

Before mechanical continuation of the proof, one comment is appropriate at this place: It is a consequence of the basic functional analysis that all terms at


209

the right-hand side, except the last two terms, are compact, i.e., up to selection of a subsequence, they converge to the corresponding expected limits. The compactness of the last two terms follows on one side from the div–curl lemma and on the other side, from Feireisl’s commutator lemma. This is just to emphasize, how important it is to write (4.4.24) in the form which lets the commutator appear. This is the crucial observation which makes the proof possible.7 Now let us continue with the details. We shall pass to the limit n → ∞ in identity (4.4.24). Firstly, we recall some consequences of assumptions (4.4.16)– (4.4.20). Due to the compact imbedding W 1,2 (Ω′ ) ֒→ Lp (Ω′ ) (on condition 1 ≤ p < 6), (4.4.25) un → u strongly in (Lp (Ω′ ))3 , 1 ≤ p < 6, hence

gn un → gu weakly in (Lc˜(Ω′ ))3 , c˜ =

6q 6+q ,

(4.4.26)

6t

q n ujn → quj weakly in (L 6+t (Ω′ ))3 .

By virtue of (4.4.3) and due to the compact imbedding W 1,q (Ω′ ) ֒→ Lp (Ω′ ), for arbitrary 1 ≤ p < q ∗ , A(˜ η gn ) → A(˜ η g) strongly in (Lp (Ω′ ))3 .

(4.4.27)

By virtue of (4.4.2) η g) weakly in (Lq (Ω′ ))3×3 if q < ∞, R(˜ η gn ) → R(˜ weakly in (Lp (Ω′ ))3×3 , 1 < p < ∞ if q = ∞ ,

R(qni η) → R(q i η) weakly in (Lt (Ω′ ))3×3 if t < ∞, p

′

3×3

weakly in (L (Ω ))

(4.4.28)

, 1 < p < ∞ if t = ∞ , 6q

η gui ) weakly in (L 6+q (Ω′ ))3×3 . R(˜ η gn uin ) → R(˜ We put in Lemma 4.24 f n = q n and g n = ∇Aj (gn ujn ) and get under the condition c˜′ < t, i.e. 1q + 1t < 65 , Rij (˜ η guj )q i weakly in Lp (Ω′ ), 1 < p ≤ η gn ujn )qni → Rij (˜

c˜t c˜+t .

(4.4.29)

Lemma 4.25 with fn = ηqni and gn = gn yields η gn ) → η˜gRij (ηq i ) − ηq i Rij (˜ η g) η˜gn Rij (ηqni ) − ηqni Rij (˜

(4.4.30)

tq if t, q < ∞ (on condition t > q ′ ) and weakly in Lp (Ω′ ) for all 1 < p ≤ t+q 1 < p < min{t, q} if q = ∞ or t = ∞. If p can be chosen greater than 6/5, i.e. 7 Unlike the nonsteady case, in the steady case there exists another possibility to prove Proposition 4.26, via the Laplace equation or via the Stokes problem, satisfied by P , see Sections 4.2.3 and 4.17.1.

210

WEAK SOLUTIONS FOR STEADY NAVIER–STOKES EQUATIONS ′

6q if t > 5q−6 , the imbedding W 1,2 (Ω′ ) ֒→ Lp (Ω′ ) is compact, and the last limit implies by virtue of (4.4.25)

j u [˜ η gn Rij (ηqni ) − ηqni Rij (˜ η gn )] → Ω uj [˜ η gRij (ηq i ) − ηq i Rij (˜ η g)]. Ω n (4.4.31) We are now in position to pass to the limit in identity (4.4.24). We obtain:

limn→∞ Ω η η˜[pn gn − (2µ + λ)gn div un ]

= − Ω ∂i ηpAi (˜ η g) + (µ + λ) Ω ∂i ηdiv uAi (˜ η g) + µ Ω ∂j η∂j ui Ai (˜ η g)

i i i −µ Ω ∂j ηu ∂i Aj (˜ η g) + µ Ω ∂i ηu η˜g − Ω ηF Ai (˜ η g)

− Ω ∂j ηq i uj Ai (˜ η guj )q i + Ω uj [˜ η gRij (ηq i ) − ηq i Rij (˜ η g)]. η g) − Ω ηRij (˜ (4.4.32) To get the first term at the right-hand side, one uses (4.4.18) and (4.4.27), for the second and third terms we employ (4.4.17) and (4.4.27), for the fourth term (4.4.28) and (4.4.25) are applied, the fifth one follows from (4.4.20) and (4.4.25) and the sixth one from (4.4.19) and (4.4.27); the seventh limiting term is obtained with the help of (4.4.26) and (4.4.27), the eighth one comes from (4.4.29) and the last one from (4.4.31). Using (4.4.16)–(4.4.31), the reader easily convinces himself that all these limits are rigorous provided r, q, s, t satisfy the assumptions of Proposition 4.26. Passing to the limit in (4.4.21), we obtain

∂j (q i uj ) − µ∆ui − (µ + λ)∂i div u + ∂i p = F i in D′ (Ω).

(4.4.33)

We use in (4.4.33) the test function φ = ηA(˜ η g) with η, η˜ and A as before. One easily verifies using (4.4.2), (4.4.3) that it is an admissible test function. After a long but straightforward calculation, when using conveniently (4.4.6), (4.4.7), (4.4.8) and (4.4.9), we arrive at

η η˜[p − (2µ + λ)div u]g Ω

= Ω ∂i ηpAi (˜ η g) + (µ + λ) Ω ∂i ηdiv uAi (˜ η g) + µ Ω ∂j η∂j ui Ai (˜ η g)

i i i −µ Ω ∂j ηu ∂i Aj (˜ η g) + µ Ω ∂i ηu η˜g − Ω ηFn Ai (˜ η g)

− Ω ∂j ηq i uj Ai (˜ η g) − Ω ηRij (˜ η guj )q i + Ω uj [˜ η gRij (ηq i ) − ηq i Rij (˜ η g)]. (4.4.34) Subtracting (4.4.32) and (4.4.34) we obtain part (i) of Proposition 4.26. By (4.4.17), (4.4.18) and (4.4.20), pn gn → pg weakly in La˜ (Ω′ ), a ˜= and

˜

rq r+q

gn div un → gdiv u weakly in Lb (Ω′ ), ˜b =

(4.4.35) 2q 2+q .

(4.4.36)

Using these formulas at the left-hand side of formula (4.4.22), we obtain (4.4.23). This completes the proof. 2

NEUMANN PROBLEM FOR THE LAPLACIAN

4.5

211

Neumann problem for the Laplacian

In this section we shall deal with the Neumann problem for the Laplacian. In the first part, we recall some results about the existence, uniqueness and regularity of weak solutions. In the second part we shall be concerned with the spectral properties of the corresponding operator.8 Both results are classical and well known. First one will be used in Section 4.6 as an auxiliary tool when we study the stationary continuity equation with dissipation. We need the spectral properties when we construct the solutions to the nonsteady continuity equation with dissipation in Section 7.6. 4.5.1

Existence, uniqueness and regularity

We consider the weak version of the boundary value problem −ǫ∆ρ = div b in Ω, ∂n ρ = b · n in ∂Ω

(4.5.37)

with ǫ a positive constant, b a given vector field and ρ an unknown function. We shall need the following result which is well known. Lemma 4.27 Let 1 < p < ∞ and let Ω be a bounded domain. (i) If Ω ∈ C 2 , b ∈ (Lp (Ω))3 , then there exists ρ ∈ W 1,p (Ω) satisfying

ǫ Ω ∇ρ · ∇η = − Ω b · ∇η, η ∈ C ∞ (IR3 )

(4.5.38)

(4.5.39)

and the estimate

c(p, Ω) b0,p . ǫ Any other solution in the same class is of the form ρ + d, d ∈ IR. (ii) If k = 0, 1, . . . and ∇ρ0,p ≤

Ω ∈ C k+2 , b ∈ E0p (Ω), div b ∈ W k,p (Ω),

(4.5.40)

(4.5.41)

then moreover ∇ρ ∈ E0p (Ω) ∩ (W k+1,p (Ω))3 and ∇ρk+1,p ≤

c(p, Ω) [b0,p + div bk,p ]. ǫ

(4.5.42)

Proof The proof of part (i) can be found in (Simader and Sohr, 1992a), Theorem 1.4. Part (ii) is a consequence of general results on elliptic regularity due to Agmon, Douglis and Nirenberg (Agmon et al., 1959) or see, e.g., (Tanabe, 1961), Chapter 4 and Sections 5.1 and 5.2. 2 8 Assumptions about the regularity of the boundary in both results are not optimal. The reader can consult (Neˇcas, 1967) or (Mitrea and Mitrea, 2001) for more precise answers.

212

4.5.2


Eigenvalue problem

We consider −∆ as an operator from D(Ω) ⊂ L2 (Ω) → L2 (Ω) with the domain of definition given by D(−∆) := W 2,2 (Ω) ∩ {ρ; ∇ρ ∈ E02 (Ω)}. Due to the Stokes formula (3.2.3), −∆ is a symmetric operator, and by virtue of Lemma 4.27, its range is equal to L2 (Ω). By the Stokes formula − Ω ∆ρρ =

∂ ρ∂i ρ ≥ 0, and by the Poincaré inequality on W 1,2 (Ω), we conclude that Ω i there exists c > 0 such that ∆ρ0,2 ≥ c ρ1,2 , ρ ∈ D(Ω). This means that the inverse operator (−∆)−1 exists and that it is positive and compact. Since −∆ is symmetric and since D((−∆)−1 ) = L2 (Ω), (−∆)−1 is selfadjoint. From the well known results about the eigenvalue problems of compact, selfadjoint and positive operators (see Section 1.4.8) we can conclude: Lemma 4.28 Let Ω ∈ C 2 be a bounded domain. Then the operator −∆ : L2 (Ω) → L2 (Ω), D(−∆) := W 2,2 (Ω) ∩ {ρ; ∇ρ ∈ E02 (Ω)} is a selfadjoint operator in L2 (Ω), and there exists countable sets {λi }i∈IN and {Φi }i∈IN , 0 < λ1 ≤ λ2 ≤ . . . < ∞, Φi ∈ D(−∆) such that −∆Φi = λi Φi (no summation over i).

Moreover, {Φi }i∈IN is at the same time an orthonormal basis in L2 (Ω) with respect to the norm uv and an orthogonal basis in W 1,2 (Ω) with respect to the Ω

scalar product Ω ∂i u∂i v. 4.6

Relaxed continuity equation with dissipation

The relaxed continuity equation with dissipation forms one part of the system at the last (third) level of approximations (see (4.3.25), (4.3.26), (4.3.27), (4.1.5)). In order to solve that system (see Section 4.8), we shall need some elementary properties of this equation. These properties are formulated in Proposition 4.29 in the next section. We shall prove them in Sections 4.6.2–4.6.4 by using the Leray– Schauder fixed point theory. It is quite a standard analysis and we perform it here only for the sake of completeness. 4.6.1

Statement of the problem and results

We consider the problem

ǫ Ω ∇ρ · ∇η + α Ω ρη − Ω ρv · ∇η = α Ω hη, η ∈ C ∞ (IR3 )

(4.6.43)

for unknown function ρ with ǫ, α given positive constants, v a given vector field and h a given function. We shall prove the following result.

RELAXED CONTINUITY EQUATION WITH DISSIPATION

213

Proposition 4.29 Let α, ǫ > 0, Ω be a bounded domain of class C 2 and h ∈ L∞ (Ω). Then there exists a mapping

where

S : (W01,∞ (Ω))3 → W 2,p (Ω), 1 0, then

Ω

h.

ρ2,p ≤ c(ǫ, p, Ω)(1 + K)h0,p , 1 < p < ∞.

(4.6.46)

(4.6.47)

(iii) If h ≥ 0 a.e. in Ω, then ρ ≥ 0 a.e. in Ω. Proof We shall prove the existence of S through several assertions by using the Leray–Schauder fixed point theorem recalled in Sections 1.4.11.7, 1.4.11.8. The proof of (i) and (ii) is given in Sections 4.6.2, 4.6.3. The proof of part (iii) is given in Section 4.6.4. For t ∈ [0, 1], we denote by ρt = Tt (ξ)

(4.6.48)

the solution of the problem −ǫ∆ρ = −tdiv (ξv) + αt(h − ξ) in Ω, ∂n ρ = 0 on ∂Ω.

(4.6.49)

Throughout the proof, in order to simplify the notation, we omit the index t at 2 ρ, whenever it is not confusing. We also denote T1 simply by T. 4.6.2

Estimates for the Leray–Schauder fixed points

Auxiliary lemma 4.30 Let 1 0, v ∈ (W01,∞ (Ω))3 , v1,∞ ≤ K, t ∈ [0, 1]. Assume that Ω, h, ǫ, α satisfy the assumptions

of Proposition 4.29. Suppose that ρ ∈ W 1,p (Ω) satisfies ρ = Tt (ρ) and Ω ρ = Ω h. Then there exists a positive constant C(p, K, ǫ, Ω, h) independent of t such that ρ1,p < C.

(4.6.50)

0∈ / (I − Tt )(∂BC ),

(4.6.51)

In other words, where

"

# BC = ρ ∈ W 1,p (Ω); ρ1,p < C, Ω ρ = Ω h

and I is the identity operator.

(4.6.52)

214


Proof We use in equation (4.6.49) with ξ = ρ subsequently test functions η + = (ρ+ + l)β and η − = −(ρ− + l)β , β ∈ (0, 1). Well known properties of the positive and negative parts of functions belonging to W 1,p (Ω) (see Section 1.3.5.5) and an obvious density argument imply that η ± are admissible test functions. (If p < 2, we have still to bootstrap using (4.6.49) with ξ = ρ and ′ (4.5.42) to verify that ρ and consequently also ρ± and η ± belong to W 1,p (Ω).) We calculate using (i), (ii) of Section 1.3.5.5, integration by parts, and the Schwartz and the Young inequalities:

∇ρ · ∇(ρ± + l)β Ω

= β Ω (ρ± + l)β−1 ∇ρ · ∇ρ± (4.6.53)

= ±β Ω (ρ± + l)β−1 |∇ρ± |2

= ±β Ω (ρ± + l)β−1 |∇(ρ± + l)|2 ,

| Ω ρv · ∇(ρ± + l)β |

= β| Ω ρv · ∇ρ± (ρ± + l)β−1 |

= | ± β Ω (ρ± + l)v · ∇(ρ± + l)(ρ± + l)β−1 ∓ βl Ω v · ∇(ρ± + l)(ρ± + l)β−1 | )

β+1 1 2 ≤ βKρ± + l0,β+1 (ρ± + l)β−1 |∇(ρ± + l)|2 + lK|Ω| β+1 ρ± + lβ0,β+1 Ω

≤ ǫ β2 Ω (ρ± + l)β−1 |∇(ρ± + l)|2 + 1ǫ β2 K 2 ρ± + lβ+1 0,β+1 1

+lK|Ω| β+1 ρ± + lβ0,β+1

(4.6.54)

and

| Ω h(ρ± + l)β | ≤ h0,β+1 ρ± + lβ0,β+1 ≤

β+1 1 β+1 h0,β+1

+

β ± β+1 ρ

Taking into account these calculations, we arrive at

αβ )ρ± + lβ+1 ±α Ω ρ(ρ± + l)β ≤ ( 1ǫ β2 K 2 + β+1 0,β+1 1

β α β+1 ρ± + l + β+1 hβ+1 0,β+1 + lK|Ω| 0,β+1 .

Letting l → 0+ implies α

β+1

−

βK 2 2ǫ

ρ± β+1 0,β+1 ≤

β+1 α β+1 h0,β+1 .

+ lβ+1 0,β+1 . (4.6.55)

(4.6.56)

(4.6.57)

α The choice β < min{1, ǫK 2 } gives the estimate

ρ0,β+1 ≤ c(β)h0,β+1 uniformly with respect to t.

(4.6.58)

Now, we consider (4.6.49) as a weak Neumann problem (4.5.39) with the right-hand side

RELAXED CONTINUITY EQUATION WITH DISSIPATION

215

b = αtB(h − ρ) − tρv,

where the operator B is defined in Lemma 3.17. By that lemma we, in particular, have b0,s ≤ α(ρ0,s + h0,s ) + v1,∞ ρ0,s , s = β + 1

(4.6.59)

uniformly with respect to t. Estimate (4.5.40) of Lemma 4.27 therefore implies ∇ρ0,s ≤ c(α, ǫ, s, Ω)(1 + K)h0,s .

(4.6.60)

If p ≤ β + 1, this combined with the Poincaré inequality for ρ − Ω h finishes the proof. If p > β + 1, using again the Poincaré inequality for ρ − Ω h to estimate the W 1,s -norm of ρ, and then by the Sobolev imbedding theorems, we obtain ρ0,s ≤ c(α, ǫ, s, Ω)(1 + K)h0,s .

(4.6.61)

If necessary, we can go with this better summability back to (4.6.59) replacing s by s > s and improve the summability of ∇ρ through (4.6.60). Repeating such a bootstrap procedure a finite number times, we arrive at ρ1,p ≤ c(α, ǫ, p, Ω)(1 + K)h0,p .

(4.6.62)

Now, it is enough to take C = 1 + c(1 + K)h0,p to obtain (4.6.50) and (4.6.51). This completes the proof of Auxiliary lemma 4.30. 2 4.6.3

Homotopy of compact transformations

Auxiliary lemma 4.31 Let p, t, K, Ω, v, h, ǫ, α be as in Auxiliary lemma 1,p 4.30 and

let B be a ball of center 0 in W (Ω) restricted on functions such that ξ = h. Then Ω Ω (i) for any t ∈ [0, 1], Tt B is a precompact set of W 1,p (Ω); (ii) for any t, s ∈ [0, 1] Tt (ξ) − Ts (ξ)1,p ≤ c|t − s|(ξ1,p + h0,p ). Proof In accordance with Lemma 3.12 and Exercise 3.13, the right-hand side of (4.6.49) is of the form div b with b = αtB(h − ξ) − tξv ∈ E0p (Ω), where B is defined in Lemma 3.17. Due to the last mentioned lemma, bE0p (Ω) ≤ ct[α(ξ0,p + h0,p ) + ξ1,p v1,∞ ]. (4.6.63)

The existence of ρ = Tt (ξ) ∈ W 2,p (Ω) satisfying Ω Tt (ξ) = Ω h follows from Lemma 4.27. Using this lemma for estimating the first and

second gradients of Tt (ξ), and applying the Poincaré inequality for Tt (ξ) − Ω h to estimate the Lp -norm of Tt (ξ), we obtain Tt (ξ)2,p ≤ ct[α(ξ0,p + h0,p ) + ξ1,p v1,∞ ]. This finishes the proof of the first part of Auxiliary lemma 4.31.

(4.6.64)

216


The quantity Tt (ξ) − Ts (ξ) satisfies equation (4.5.37) with b = α(t − s)B(h − ξ) − (t − s)ξv ∈ E0p (Ω) and has zero mean value over Ω. Lemma 4.27 and the Poincaré inequality yield Tt (ξ) − Ts (ξ)1,p ≤ c|t − s|[α(ξ0,p + h0,p ) + ξ1,p v1,∞ ]. This completes the proof of Auxiliary lemma 4.31.

(4.6.65) 2

Due to the existence and uniqueness of solutions to the Neumann problem guaranteed by Lemma 4.27, the problem T0 (ρ0 ) = ρ0 has a unique solution in the

1,p class of functions ρ ∈ W (Ω), ρ = h. As one easily sees, this solution is Ω Ω

/ (I − Tt )(∂BC ). By virtue equal to Ω h. By virtue of Auxiliary lemma 4.30, 0 ∈ of Auxiliary lemma 4.31, I − T· (·) : BC × [0, 1] → W 1,p (Ω) is a homotopy of compact transformations on BC . Thus by the Leray–Schauder fixed point theorem (cf. Section 1.4.11.8), there exists at least one ρt ∈ BC satisfying Tt (ρt ) = ρt , t ∈ [0, 1]. We now put ρ = ρ1 = T1 (ρ). Estimate (4.6.63) in turn with (4.6.62) yields bE0p (Ω) ≤ c(α, ǫ, p, Ω)(1 + K)h0,p . Estimate (4.6.47) thus follows from (4.6.49) with t = 1 and ξ = ρ by Lemma 4.27. 4.6.4 Nonnegativity of the density We employ in (4.6.43) the test function η = −(ρ− +l)β , where l > 0 and β ∈ (0, 1) will be fixed later. Using (4.6.53) and (4.6.54) and instead of (4.6.55) the fact that h ≥ 0, we arrive at

1 β β+1 ρ− + l (4.6.66) −α Ω ρ(ρ− + l)β ≤ 1ǫ β2 K 2 ρ− + lβ+1 0,β+1 . β+1 + lK|Ω| Letting l → 0+ , we get

α β+1

−

βK 2 2ǫ

ρ− β+1 β+1 ≤ 0.

(4.6.67)

αǫ − = 0. This completes the proof of Proposition The choice β < min{1, K 2 } gives ρ 4.29.

4.7 The Lam´ e system The Lamé operator appears on any level of approximations when one linearizes the momentum equation. In the first part of this section we recall well known results about the existence, uniqueness and regularity of solutions to the system induced by this operator complemented with the homogenous Dirichlet boundary conditions. We shall need these results as an auxiliary tool in Section 4.8 for proving the existence of solutions to the system at the third level of approximations (see (4.3.25), (4.3.26), (4.3.27), (4.1.5)). In the second part of this section we recall well known facts about the spectral properties of the Lamé operator. These results are needed in Section 7.7 in order to introduce the Galerkin method in the context of the nonsteady equations.

´ SYSTEM THE LAME

4.7.1

217

Existence, uniqueness and regularity

We recall some basic results which concern the existence and regularity of the system −µ∆u − (µ + λ)∇div u = F in Ω (4.7.1) with boundary conditions (4.1.5). Lemma 4.32 Let 1 0, 4µ + 3λ > 0.

(4.7.2)

(i) If Ω ∈ C 2 , F ∈ (W −1,p (Ω))3 , (W01,p (Ω))3

(4.7.3) ′

3

then there exists a unique u ∈ satisfying (4.7.1) in (D (Ω)) and the estimate (4.7.4) u1,p ≤ c(p, Ω)F −1,p . (ii) If Ω ∈ C k+2 , F ∈ (W k,p (Ω))3 , k = 0, 1, . . . , then moreover u ∈ (W

k+2,p

(4.7.5)

3

(Ω)) and

uk+2,p ≤ c(k, p, Ω)F k,p .

(4.7.6)

Proof It is an exercise to show that µξij ξij + (µ + λ)ξii ξkk ≥ min{µ, 4µ + 3λ}ξij ξij , ξ ∈ IR3 × IR3 .

(4.7.7)

Due to (4.7.7), the operator A := −µ∆ − (µ + λ)∇div

(4.7.8)

is strongly elliptic. If F belongs, e.g., to (D(Ω))3 , existence of a unique weak solution in the space 1,2 (W0 (Ω))3 follows from the Lax–Milgram lemma. The fact that this solution satisfies estimates (4.7.4) and (4.7.5) is a consequence of the classical regularity theory for elliptic systems. We refer the reader, e.g., to (Agmon et al., 1959). Then one uses a density argument to conclude for F satisfying assumptions (4.7.3). 2 4.7.2

Eigenvalue problem

Now, we consider A as an operator from (L2 (Ω))3 to (L2 (Ω))3 with the domain of definition (4.7.9) D(A) := (W01,2 (Ω))3 ∩ (W 2,2 (Ω))3 . It is evident that A is symmetric. Due to Lemma 4.32, the range of A is equal to L2 (Ω). From the chain of inequalities

218


Au0,2 u0,2 ≥ Ω Au · u ≥ min{µ, 4µ + 3λ}∇u0,2 ,

one deduces, with help of the Poincaré inequality,

Au0,2 ≥ cu1,2 , u ∈ D(A) with some c > 0 which means that A−1 exists and that it is compact. Since A is symmetric, A−1 is symmetric too; it is even selfadjoint since D(A−1 ) = (L2 (Ω))3 , and positive, since (Au, u) > 0, u = 0, u ∈ D(A) ((u, v) denotes the scalar product Ω ui v i ). From well known results about the eigenvalue problem for compact selfadjoint operators (see Section 1.4.8), we can conclude the following statement. Lemma 4.33 Let µ, λ satisfy the assumptions of Lemma 4.32. Let Ω ∈ C 2 be a bounded domain. Then: (i) The operator A defined by (4.7.8) and (4.7.9) is a selfadjoint operator in (L2 (Ω))3 . (ii) There exist countable sets 1,2 3 2,2 (Ω))3 {λi }∞ i=1 ⊂ (0, ∞), 0 < λ1 ≤ λ2 ≤ . . . and {φi }i∈IN ⊂ (W0 (Ω)) ∩ (W

such that Aφi = λi φi (no summation over i) and {φi }i∈IN is an orthonormal basis in (L2 (Ω))3 with respect to the scalar product Ω ui v i as well as an orthogonal basis in (W01,2 (Ω))3 with respect to the scalar product Ω (µ∂i uj ∂i v j + (µ + λ)div u div v). 4.8

Complete system with dissipation in the relaxed continuity equation and with artificial pressure

In this section, we shall prove Proposition 4.22. In the first part, we shall prove the existence of bounded energy weak solutions for system (4.3.25), (4.3.26), (4.3.27) and (4.1.5) (i.e., we shall show statements (i)–(iv) of Proposition 4.22). The existence will be shown by using the Leray–Schauder fixed point theory recalled in Section 1.4.11.8, in the way which is described in Section 4.8.1. Verifying the hypothesis of the Leray–Schauder fixed point theorem, one derives rigorously the energy inequality, as well. In the second part, we shall deal with the estimates independent of the parameter ǫ (i.e., we shall show statement (v) of Proposition 4.22). These estimates come from the energy inequality and from testing the momentum equation by a convenient test function involving the Bogovskii operator (cf. Lemma 3.17). The details are described in Section 4.8.2. All results presented in this section rely on classical analysis. We present their proofs only for the reader’s convenience. 4.8.1 4.8.1.1

Existence of solutions Operators Tt

We define the family of operators Tt : (W01,∞ (Ω))3 → (W01,∞ (Ω))3

(4.8.1)

COMPLETE SYSTEM

219

(for the definition of W01,∞ (Ω) see (4.6.45)), where t ∈ [0, 1], as follows: u = Tt v is a solution of the problem −µ∆u − (µ + λ)∇div u = Ft (ρ, v) in Ω, u = 0 in ∂Ω, where

Ft (ρ, v) = −t[α(h + ρ)v + 12 ∂j (ρv j v) + 12 ρv · ∇v

(4.8.3)

ρ = S(v)

(4.8.4)

+∇ργ + δ∇(ρ2 + ρβ ) − ρf − g]

and

(4.8.2)

is given by Proposition 4.29. By virtue of (4.6.47), Ft ∈ Lp (Ω), 1 < p < ∞. Therefore the elliptic regularity from Lemma 4.32 applied to system (4.8.2) yields u ∈ (W 2,p (Ω))3 , 1 ≤ p < ∞. This means that the operators Tt are well defined. 4.8.1.2 Estimates for vector fields u=Tt (u) Auxiliary lemma 4.34 Assume that t ∈ [0, 1] and that Ω, f , g, h, ǫ, α, δ, β, γ satisfy the assumptions of Proposition 4.22. Suppose that u ∈ (W01,∞ (Ω))3 satisfies u = Tt (u). Then there exists a positive constant C(α, β, γ, δ, ǫ, Ω, h, f , g) independent of t such that u1,∞ < C, t ∈ [0, 1].

(4.8.5)

0∈ / (I − Tt )(∂BC ),

(4.8.6)

BC = {u ∈ (W01,∞ (Ω))3 ; u1,∞ < C}.

(4.8.7)

In other words, where

Proof With ρ = S(u), the equation

αtρu + αthu + 2t ∂j (ρuuj ) + 2t ρu · ∇u − µ∆u

−(µ + λ)∇div u + t∇ργ + tδ∇(ρ2 + ρβ ) = tρf + tg

(4.8.8)

holds with any test function φ ∈ (W01,p (Ω))3 , 1 < p < ∞, as one easily sees by a density argument. Taking in particular φ = u, one obtains

γ t Ω (ρ − h)(ργ−1 − hγ−1 ) αt Ω h|u|2 + αt Ω ρ|u|2 + α γ−1

β t Ω (ρ − h)(ρβ−1 − hβ−1 ) + 2αδt Ω (ρ − h)2 +αδ β−1

+βδǫt Ω ρβ−2 |∇ρ|2 + 2δǫt Ω |∇ρ|2 + µ Ω |∇u|2 + (µ + λ) Ω |div u|2

γ ≤ t Ω ρf · u + t Ω g · u + α γ−1 t Ω (h − ρ)hγ−1

β +αδ β−1 t Ω (h − ρ)hβ−1 + 2αt Ω (h − ρ)h. (4.8.9)

220


When establishing this estimate, we have used elementary integration by parts and the formulae

∂ (ρui uj )ui = − Ω ρui uj ∂j ui = − 12 Ω ρu · ∇|u|2 , (4.8.10) Ω j

s [ Ω div (ρu)(ρs−1 − hs−1 ) + Ω div (ρu)hs−1 ] ∇ρs · u = − s−1 Ω

s = α s−1 [ Ω (ρ − h)(ρs−1 − hs−1 ) + Ω (ρ − h)hs−1 ] (4.8.11)

s−1 ǫs − s−1 Ω ρ ∆ρ, 1 < s < ∞,

completed by

ρs−1 ∆ρ ≥ 0, 1 < s < 2,

− Ω ρs−1 ∆ρ ≥ (s − 1) Ω ρs−2 |∇ρ|2 , 2 ≤ s < ∞.

−

Ω

(4.8.12)

The second formula in (4.8.12) is obtained by a simple integration by parts. To get the first formula, we realize that

− Ω (ρ + ϑ)s−1 ∆(ρ + ϑ) = (s − 1) Ω (ρ + ϑ)s−2 |∇ρ|2 ≥ 0, ϑ > 0, 1 < s < ∞ and that, by the Lebesgue dominated convergence theorem,

limϑ→0+ Ω (ρ + ϑ)s−1 ∆(ρ + ϑ) = Ω ρs−1 ∆ρ.

Thanks to (4.6.46) and H¨ older’s inequality, the right-hand side of (4.8.9) can be bounded, e.g., by t[ρ0,6/5 u0,6 f 0,∞ + u0,6 g0,6/5 +c(Ω)(h20,∞ + hγ0,∞ + hβ0,∞ )].

(4.8.13)

After using in the last expression the continuous imbeddings W 1,2 (Ω) ֒→ L6 (Ω), L3β (Ω) ֒→ L6/5 (Ω), after applying to it the Young inequality, and after employ

β ing the identity Ω ρβ−2 |∇ρ|2 = β42 Ω |∇(ρ 2 )|2 , from (4.8.9), we obtain β

(4.8.14) ∇u20,2 + ∇(ρ 2 )20,2 ≤ c(α, δ, ǫ, Ω, f , g, h)(1 + ρ20,3β ).

Now, using the Poincaré inequality (recall that Ω (ρ − h) = 0), by virtue of the imbedding W 1,2 (Ω) ֒→ L6 (Ω), we get ρ0,3β ≤ c(α, δ, ǫ, Ω, f , g, h) uniformly in t.

(4.8.15)

After this, again from (4.8.14), we obtain u1,2 ≤ c(α, δ, ǫ, Ω, f , g, h) uniformly in t.

(4.8.16)

Next, we apply estimate (4.5.40) to equation (4.5.37) with b = −ρu + αB(h − ρ). Using (4.8.15), (4.8.16) and Lemma 3.17, one gets b0,p ≤ c(α, δ, ǫ, Ω, f , g, h), p p 3β + 6 = 1 and consequently ρ1,p ≤ c(α, δ, ǫ, Ω, f , g, h) with the same p.

COMPLETE SYSTEM

221

After this, we find div b0,2 ≤ c(α, δ, ǫ, Ω, f , g, h). Estimate (4.5.42) therefore gives ρ2,2 ≤ c(α, δ, ǫ, Ω, f , g, h). Now, from (4.8.3) we get Ft (ρ, u)0, 23 ≤ c(α, δ, ǫ, Ω, f , g,h)t, and therefore, by Lemma 4.32, u2, 32 ≤ c(α, δ, ǫ, Ω, f , g, h). Next, we obtain Ft (ρ, u)0,2 ≤ c(α, δ, ǫ, Ω, f , g,h)t and therefore the estimate u2,2 ≤ c(α, δ, ǫ, Ω, f , g, h) holds true. Finally Ft (ρ, u)0,6 ≤ c(α, δ, ǫ, Ω, f , g, h)t and u2,6 ≤ tc(α, δ, ǫ, Ω, f , g, h). By Sobolev’s imbeddings u1,∞ ≤ c(α, δ, ǫ, Ω, f , g, h). It is enough to put C = c+1 completing the proof of Auxiliary lemma 4.34. 2 4.8.1.3 Tt u is a homotopy of compact transformations Auxiliary lemma 4.35 Let Ω, α, β, γ, δ, f , g, h satisfy the assumptions of Proposition 4.22. Let B be a ball of center 0 in (W01,∞ (Ω))3 . Then: (i) For any t ∈ [0, 1], Tt B is a precompact set in (W01,∞ (Ω))3 . (ii) For any t, s ∈ [0, 1], we have Tt (v) − Ts (v)1,∞ ≤ c(Ω, B, δ, f , g, h)|t − s|, v ∈ B.

Proof The Lp -norm (1 < p < ∞) of the right-hand side (4.8.3) of equation (4.8.2) can be estimated for all 0 < α, δ < 1 as follows: Ft (ρ, v)0,p ≤ c(Ω, p)t{v1,∞ (h0,∞ + ρ1,p )(1 + v1,∞ )

β−1 +(ρ0,∞ + ργ−1 0,∞ + ρ0,∞ )∇ρ0,p + ρ0,p f 0,∞ + g0,p }. (4.8.17) After using (4.6.47), we obtain

Ft (ρ, v)0,p ≤ c(Ω, B, ǫ, f , g, h)t, v ∈ B.

(4.8.18)

Similarly, we have Ft (ρ, v) − Fs (ρ, v)0,p ≤ c(Ω, B, ǫ, f , g, h)|t − s|, v ∈ B.

(4.8.19)

Lemma 4.32 thus yields Tt (v)2,p ≤ c(Ω, B, ǫ, f , g, h)t, v ∈ B, 1 < p < ∞

(4.8.20)

(this implies statement (i)), and Tt (v) − Ts (v)2,p ≤ c(Ω, B, ǫ, f , g, h)|t − s|, v ∈ B, 1 < p < ∞ which yields statement (ii). Auxiliary lemma 4.35 is thus proved.

(4.8.21) 2

4.8.1.4 Existence of (ρǫ , uǫ ) According to Auxiliary lemma 4.35, H(·, t) = Tt (·) is a homotopy of compact transformations on B C ⊂ (W01,∞ (Ω))3 . According to Auxiliary lemma 4.34, 0 ∈ / (I − Tt )(∂BC ), t ∈ [0, 1]. Moreover, the equation T0 u = u possesses a unique solution in BC , namely u = 0 (cf. Lemma 4.32). The Leray–Schauder theorem (cf. Section 1.4.11.8) thus yields the existence of uǫ ∈ BC satisfying T1 (uǫ ) = uǫ . This uǫ satisfies (4.3.29) with ρǫ = S(uǫ ), where S is defined in Proposition 4.29. In other words, the couple (ρǫ , uǫ ) satisfies equations (4.3.29), (4.3.30). It also belongs to class (4.3.28), and due to (4.8.9), it satisfies the energy inequality (4.3.31) as well.

222

4.8.2


Estimates independent of dissipation

4.8.2.1 Some consequences of the energy inequality Due to the bound (4.8.13) for the right-hand side of the energy inequality (4.3.31), and since W 1,2 (Ω) ֒→ L6 (Ω), L2β (Ω) ֒→ L6/5 (Ω), from (4.3.31), one gets uǫ 1,2 ≤ c(Ω, f , g, h)(1 + ρǫ 0,2β ), and also α

Ω

(4.8.22)

(h + ρǫ )|uǫ |2 ≤ c(Ω, f , g, h)(1 + ρǫ 0,2β )2 ,

ǫ∇ρǫ 20,2 ≤ c(Ω, δ, f , g, h)(1 + ρǫ 0,2β )2 .

4.8.2.2

Estimates of ρǫ

We take in (4.3.29) the test function

φ = B(ρβǫ − ρβǫ ),

(4.8.23) (4.8.24)

(4.8.25)

where the operator B is taken from Lemma 3.17. By virtue of (3.3.16), we have φ0,p ≤ c(p, Ω)ρǫ β0,pβ , 1 < p < ∞, ∇φ0,p ≤ c(p, Ω)ρǫ β0,pβ , 1 < p < ∞

(4.8.26)

(for the definition of p see (1.3.64) in Section 1.3.5.8). Due to Remark 4.23, φ is an admissible test function. From this testing, we obtain the following identity:

2+β

γ+β ρ + δ Ω ρ2β ǫ + δ Ω Ω ρǫ Ω ǫ

= Ω ργǫ Ω ρβǫ + δ Ω ρβǫ Ω ρβǫ + δ Ω ρ2ǫ Ω ρβǫ + α Ω (ρǫ + h)uǫ · φ (4.8.27)

+µ Ω ∂j uǫ · ∂j φ + (µ + λ) Ω div uǫ div φ − 12 Ω ρǫ ui uj ∂j φi

10 + 12 Ω ρǫ uǫ · ∇uǫ · φ − Ω ρǫ f · φ − Ω g · φ = i=1 Ii . If β ≥ max{γ, 3}, the integrals I1 –I10 are estimated as follows:

2β−2

1

2β−1 2β−1 ρǫ 0,2β , by the identity (i) By the interpolation inequality ρǫ 0,β ≤ ρǫ 0,1

2β β+γ 2β h, and by the imbeddings L (Ω) ֒→ L (Ω), L (Ω) ֒→ Lβ+2 (Ω), ρ = Ω Ω ǫ we have 2β−2

β

2β−1 (ρǫ γ0,2β + ρǫ β0,2β + ρǫ 20,2β ). |I1 | + |I2 | + |I3 | ≤ c(Ω, h)ρǫ 0,2β

(4.8.28)

6β

(ii) By H¨ older’s inequality, by the imbedding L2 (Ω) ֒→ L 5β−3 (Ω), by the Poincaré inequality and by (4.8.22), (4.8.26), we obtain |I4 | ≤ α(ρǫ + h)0,2β uǫ 0,6 φ0,

6β 5β−3

≤ αc(Ω, f , g, h)ρǫ β0,2β (1 + ρǫ 0,2β )2 .

(4.8.29)

(iii) By the Schwartz inequality and by (4.8.22), (4.8.26), we majorize I5 and I6 as follows:

SYSTEM WITH RELAXED CONTINUITY EQUATION

223

|I5 | + |I6 | ≤ c∇uǫ 0,2 ∇φ0,2 ≤ c(Ω, f , g, h)ρǫ β0,2β (1 + ρǫ 0,2β ). (4.8.30) 6β

(iv) Employing H¨ older’s inequality, (4.8.22), the imbedding L2 (Ω) ֒→ L 4β−3 (Ω) and (4.8.26) for estimating I7 , and using H¨ older’s inequality (4.8.22), the Sobolev 6β 6β 6β imbedding W 1, 4β−3 (Ω) ֒→ L 2β−3 (Ω), the imbedding L2 (Ω) ֒→ L 4β−3 (Ω) and (4.8.26) for estimating I8 , we get |I7 | ≤ ρǫ 0,2β uǫ 20,6 ∇φ0,

6β 4β−3

2 c(Ω, f , g, h)ρǫ β+1 0,2β (1 + ρǫ 0,2β ) ,

|I8 | ≤ ρǫ 0,2β uǫ 0,6 ∇uǫ 0,2 φ0,

(4.8.31)

6β 2β−3

2 ≤ c(Ω, f , g, h)ρǫ β+1 0,2β (1 + ρǫ 0,2β ) . 2β

(vi) By H¨ older’s inequality, by the imbeddings L2 (Ω) ֒→ L 2β−1 (Ω), L2 (Ω) ֒→ 1 L (Ω) and by (4.8.26), we obtain |I9 | + |I10 | ≤ ρǫ 0,2β f 0,∞ φ0,

2β 2β−1

+ g0,∞ φ0,1

≤ c(Ω, f , g, h)ρǫ β0,2β (1 + ρǫ 0,2β ).

(4.8.32)

Estimates (4.8.28)–(4.8.32) together with (4.8.27) yield the bound (4.3.33) provided β > 3. After this, (4.8.22) yields estimate (4.3.32), and estimate (4.8.24) implies straightforwardly (4.3.34). The proof of Propositon 4.22 is thus complete. 4.9

Complete system with relaxed continuity equation and with artificial pressure

In this section we shall prove Proposition 4.18 letting ǫ → 0+ in Proposition 4.22. In the first step, in Sections 4.9.1–4.9.4, we shall prove the existence of renormalized weak solutions to system (4.3.13), (4.3.14), (4.1.5) (i.e., statements (i)–(iv) of Proposition 4.18) and in the second step, in Section 4.9.5, we shall prove the energy inequality and estimates independent of α (i.e., statements (v), (vi) of Proposition 4.18). After Section 4.9.1 which relies on classical analysis, we are for the first time confronted with the lack of compactness in the sequence of densities. This lack will be compensated by the weak compactness of the effective viscous flux evoked in Section 4.2.3 and studied, in the general context, in Section 4.4. Then the whole problem is solved by using Lions’ approach whose main points are described in Section 4.2.4. It involves the transport theory evoked in Section 3.1 and some elements of convex analysis and of the theory of monotone operators recalled in Section 3.4. All details are available in Sections 4.9.2, 4.9.3, 4.9.4. The proofs of the energy inequality and of the estimates independent of α performed in Section 4.9.5 are again standard.

224


4.9.1

Vanishing dissipation limit

From estimates (4.3.32), (4.3.33), (4.3.34), one obtains at least for an appropriately chosen subsequence, when ρǫ , uǫ are prolonged by 0 outside Ω: uǫ → u weakly in (W 1,2 (IR3 ))3 , uǫ → u strongly in (Lq (Ω))3 , 1 ≤ q < 6, u = 0 a.e. in IR3 \ Ω,

(4.9.1)

ρǫ → ρ weakly in L2β (IR3 ), ρ ≥ 0 a.e. in Ω, ρ = 0 in IR3 \ Ω,

(4.9.2)

2β

ργǫ → ργ weakly in L γ (IR3 ), γ ρ ≥ 0 a.e. in Ω, ργ = 0 in IR3 \ Ω, ρβǫ → ρβ weakly in L2 (IR3 ), ρβ ≥ 0 a.e. in Ω, ρβ = 0 in IR3 \ Ω, ρ2

(4.9.3)

(4.9.4)

ρ2ǫ → ρ2 weakly in Lβ (IR3 ), ≥ 0 a.e. in Ω, ρ2 = 0 in IR3 \ Ω,

(4.9.5)

ǫ∇ρǫ → 0 strongly in (L2 (Ω))3 .

(4.9.6)

These limits have the following properties. Auxiliary lemma 4.36 Let the assumptions of Proposition 4.22 be satisfied and let u, ρ, ργ , ρ2 , ρβ be given by (4.9.1)–(4.9.5). Then we have (i)

ρ = Ω h. (4.9.7) Ω

(ii)

div (ρu) = α(h − ρ) in D′ (IR3 ).

(4.9.8)

(iii) For any b ∈ C 1 ([0, ∞)) satisfying (3.1.17) and (3.1.18) with β replaced by 2β αρb′ (ρ) + div (b(ρ)u) + {ρb′ (ρ) − b(ρ)}div u = αhb′ (ρ) in D′ (IR3 ).

(4.9.9)

(iv) αρu + αhu + 12 ∂j (ρuuj ) + 12 ρu · ∇u − µ∆u −(µ + λ)∇div u + ∇ργ + δ∇(ρ2 + ρβ ) = ρf + g in (D′ (Ω))3 .

(4.9.10)

Remark 4.37 In fact, one easily verifies by a density argument that (4.9.10) holds with any test function φ ∈ (W01,2 (Ω))3 .


225

Proof Statement (i) follows directly from the third formula in (4.3.28) by (4.9.2). By virtue of strong convergence (4.9.1) and weak convergence (4.9.2), there holds r ρǫ uǫ → ρu weakly in (Lr (Ω))3 , 2β + 6r ≤ 1 (4.9.11) and ρǫ uiǫ ujǫ → ρui uj weakly in Ls (Ω),

s 2β

+

s 3

≤ 1.

(4.9.12)

Employing (4.9.2), (4.9.6) and (4.9.11), we can pass to the limit in (4.3.30) and get (4.9.8) in D′ (Ω). Statement (ii) now follows from Lemma 3.2. Finally, statement (iii) follows from (4.9.8) by Lemma 3.3. It remains to prove statement (iv). Using integration by parts and identity (4.3.30), we verify that

ρ u · ∇uǫ · φ Ω ǫ ǫ

= − Ω div (ρǫ uǫ )uǫ · φ − Ω ρǫ uiǫ ujǫ ∂j φi

= −ǫ Ω ∆ρǫ u · φ + α Ω (ρǫ − h)uǫ · φ − Ω ρǫ uiǫ ujǫ ∂j φi

= ǫ Ω ∇ρǫ · ∇uǫ · φ + ǫ Ω ∂j ρǫ uiǫ ∂j φi + α Ω (ρǫ − h)u · φ − Ω ρǫ uiǫ ujǫ ∂j φi . The first two terms of the last and (4.9.6). The

line tend to 0 due i toj (4.9.1) i second two terms tend to α (ρ − h)u · φ − ρu u ∂ φ . By (4.9.8), the last j Ω Ω

limit equals Ω ρu · ∇u · φ . Hence ρǫ uǫ · ∇uǫ → ρu · ∇u weakly in (Lz (Ω))3 ,

z 2β

+

2z 3

≤ 1.

(4.9.13)

Formulae (4.9.1)–(4.9.6), (4.9.12), (4.9.13) make it possible to pass to the limit in (4.3.29) and to obtain (4.9.10). The proof of Auxiliary lemma 4.36 is thus complete. 2 4.9.2


We take in Proposition 4.26 t = r = 2, q = 2β, s =

6β , 4β + 3

(4.9.14)

q n = 12 ρǫ uǫ , un = uǫ , pn = pδ (ρǫ ) := ργǫ + δρβǫ + δρ2ǫ , Fn = ρǫ f + g − α(ρǫ + h)uǫ − 21 ρǫ uǫ · ∇uǫ , gn = b(ρǫ ), ǫ =

1 n,

(4.9.15)

where b ∈ C 1 ([0, ∞)) satisfies (3.1.17) with λ1 = 0 and

p = pδ (ρ) ≡ ργ + δ(ρβ + ρ2 ), q = 21 ρu, F = ρf + g − α(ρ + h)u − 12 ρu · ∇u.

(4.9.16)

(4.9.17)

226


We verify that hypotheses (4.4.16)–(4.4.20) are satisfied. By virtue of (4.9.3)– (4.9.5) (4.9.18) pδ (ρn ) → pδ (ρ) weakly in L2 (IR3 ).

By (4.9.1), (4.9.2), (4.9.11), (4.9.13), Fn is certainly a weakly convergent se6β quence in (L 4β+3 (Ω))3 and converges to F. The sequence q n converges weakly in (L2 (Ω))3 to q = 12 ρu by virtue of (4.9.11). Identity (4.3.30) yields div (ρǫ uǫ ) = ǫ∆ρǫ + α(h − ρǫ ). By (4.9.2) and by (4.9.6), since the imbedding L2 (Ω) ֒→ W −1,2 (Ω) is compact, the right-hand side of the last equation tends to α(ρ − h) strongly in W −1,2 (Ω), i.e. to div (ρu) = 2div q (cf. (4.9.8)). This means that div q n → div q strongly in W −1,2 (Ω). Finally, at least for an appropriately chosen subsequence, (4.9.19) b(ρǫ ) → b(ρ) weakly in L2β (Ω), 2β

b(ρǫ )div uǫ → b(ρ)div u weakly in L β+1 (Ω),

(4.9.20)

pδ (ρǫ )b(ρǫ ) → pδ (ρ)b(ρ) weakly in L

(4.9.21)

2β β+1

(Ω).

Therefore, in the above described situation, Proposition 4.26 can be rephrased as follows: Lemma 4.38 Let the assumptions of Proposition 4.22 be satisfied and let b satisfy (4.9.16). Then the weak limits (4.9.1) and (4.9.17)–(4.9.21) satisfy pδ (ρ)b(ρ) − (2µ + λ)b(ρ)div u = pδ (ρ) b(ρ) − (2µ + λ)b(ρ)div u a.e. in Ω. 4.9.3 Let

Renormalized continuity equation with powers 0 0. By the Lebesgue dominated convergence theorem, the right-hand side of the last inequality tends to 0 as l → 0+ . Therefore (ρ + l)θ → ρθ strongly in L

2β θ

(Ω).

(4.9.36)

Consequently,

(ρ + l)θ p

ϑ

ϑ → ρθ , ϑ ∈ (0, 2β θ ),

strongly in L (Ω), 1 ≤ p
0.

When k → ∞, the Lebesgue dominated convergence theorem together with (4.11.21) gives the convergence in D′ (IR3 ) of the left-hand side of (4.11.27) to div (bM (ρ)u) + {ρ[bM ]′+ (ρ) − bM (ρ)}div u. The L1 (Ω)-norm of the right-hand side of (4.11.27) can be estimated by

maxs∈[0,M ] |b′ (s)| Ωk,M |{ρ[Tk ]′+ (ρ) − Tk (ρ)}div u|, where Ωk,M = {x ∈ Ω, Tk (ρ)(x) ≤ M }. Due to the lower weak semicontinuity of norms, {ρ[Tk ]′+ (ρ) − Tk (ρ)}div u0,1,Ωk,M

≤ lim inf δ→0+ {ρδ [Tk ]′+ (ρδ ) − Tk (ρδ )}div uδ 0,1,Ωk,M . By the Schwartz inequality and the second inequality in (4.11.18), the right-hand side of the last inequality is bounded by lim supδ→0+ Tk (ρδ )1{ρδ ≥k} 0,2,Ωk,M div uδ 0,2,Ω ≤ γ+1 γ−1 2γ 2γ T (ρ )1 c(Ω, f , g, m) lim supδ→0+ Tk (ρδ )1{ρδ ≥k} 0,1,Ω k δ {ρ ≥k} δ 0,γ+1,Ωk,M . k,M

Here we have used the interpolation of L2 (Ω) between L1 (Ω) and Lγ+1 (Ω) and the uniform bound (4.3.10) of div uδ 0,2,Ω . Since Tk (ρδ ) ≤ ρδ , it follows from 9 Since div (ρu) = 0, the renormalized equation (4.11.26) with any function s → cs + b(s), s(γ) c ∈ IR, where b satisfies (3.1.16), (3.1.17) with λ1 + 1 ≤ 2 , holds as well.

244


(4.11.20) with p = 1 that lim supδ→0+ Tk (ρδ )1{ρδ ≥k} 0,1,Ω → 0 as k → ∞. In accordance with Lemma 4.49, we have Tk (ρδ )1{ρδ ≥k} 0,γ+1,Ωk,M ≤ Tk (ρδ ) − Tk (ρ)0,γ+1,Ωk,M + Tk (ρ)0,γ+1,Ωk,M 1

≤ L(Ω, f , g) + M |Ω| γ+1 . Due to these facts {ρTk′ (ρ) − Tk (ρ)}div u → 0 strongly in L1 (Ωk,M ) as k → ∞. Equation (4.11.27) thus yields div (bM (ρ)u) + {ρ[bM ]′+ (ρ) − bM (ρ)}div u = 0 in D′ (IR3 ), M > 0. (4.11.28) At this stage, we let M → ∞ by employing Exercise 3.7, and we obtain the statement of Lemma 4.50. The proof is thus complete. 2 Now, we summarize the situation in the proof of Theorem 4.3. Due to (4.11.1), (4.11.2) and (4.11.10), formulas (4.1.11) and (4.1.18) hold. Formula (4.11.11) completes the proof of (4.1.14) and equation (4.11.26) means that (4.1.15) holds. Letting δ → 0+ in (4.3.9) and using the lower weak semicontinuity of norms together with (4.11.1), (4.11.2), we obtain the energy inequality (4.1.9). It remains to prove (4.1.12) which means to show ργ = ργ in (4.11.12). This is equivalent to the strong convergence of the density in L1 (Ω), a subject treated in the next section. 4.11.5

Strong convergence of the density

For any k > 1, let Lk (t) =

t ln t if t ∈ [0, k), t ln k + t − k if t ∈ [k, ∞).

(4.11.29)

(ln k + 1)t + l(t),

(4.11.30)

Clearly Lk (t) can be written as

where l(t) = t(ln t − ln k)1{t≤k} − t1{t≤k} − k1{t>k} .

(4.11.31)

We observe that l satisfies conditions (3.1.16), (3.1.17) with any λ1 > −1. We check that tL′k (t) − Lk (t) = Tk (t). The continuity equation (4.11.11) and equation (4.11.13) with b = l yield

(4.11.32) T (ρ)div u = 0. Ω k

On the other hand, the continuity equation (4.11.11) and the renormalized continuity equation (4.11.26) with b = l, imply div (Lk (ρ)u) + Tk (ρ)div u = 0 in D′ (IR3 ). Consequently, we have

(4.11.33)

APPROXIMATIONS IN AN EXTERIOR DOMAIN

Ω

Tk (ρ)div u = 0.

245

(4.11.34)

By (4.11.24), (4.11.32) and (4.11.34), we have

lim supδ→0+ Ω |Tk (ρδ ) − Tk (ρ)|γ+1 ≤ (2µ + λ) Ω div u(Tk (ρ) − Tk (ρ)).

The right-hand side of the last inequality is bounded by γ−1

γ+1

2γ 2γ Tk (ρ) − Tk (ρ)0,γ+1 . c(Ω, f , g, m)Tk (ρ) − Tk (ρ)0,2 ≤ cTk (ρ) − Tk (ρ)0,1

By virtue of (4.11.21), (4.11.22) and Lemma 4.49, it tends to zero. We thus have lim lim sup Tk (ρδ ) − Tk (ρ)0,γ+1 = 0.

k→∞ δ→0+

(4.11.35)

Now, we write ρδ − ρ0,1 ≤ ρδ − Tk (ρδ )0,1 + Tk (ρδ ) − Tk (ρ)0,1 + Tk (ρ) − ρ0,1 , and we obtain by using (4.11.21), (4.11.22), (4.11.35) the strong convergence of ρδ in L1 (Ω), at least for a chosen subsequence. Taking into account the bound (4.3.11), we can conclude by the following lemma: Lemma 4.51 Let ρδ be the sequence and ρ its weak limit from (4.11.2). Then, at least for a chosen subsequence, ρδ → ρ strongly in Lp (Ω), 1 ≤ p < s(γ). By virtue of this result, in Assertion 4.47, ργ = ργ . This completes the proof of Theorem 4.3. 4.12

Approximations in an exterior domain

In this section we describe the chain of approximations which we use to solve problem (4.1.1), (4.1.2), (4.1.5) and (4.1.6) in an exterior domain. 4.12.1

Relaxation on invading domains

Let Ω be an exterior domain, cf. (3.3.46). We may suppose without loss of generality that (4.12.1) Ωc := IR3 \ Ω ⊂ B1 . We shall start by solving system (4.3.13), (4.3.14) in ΩR = Ω ∩ BR , where γ > 3, h = ρ∞ and δ = 0, with boundary conditions (4.1.5) on ∂ΩR . Due to Remark 4.21, Proposition 4.18 rephrased to this situation and completed with estimates independent of R and α, reads: Proposition 4.52 Let f , g, ρ∞ satisfy (4.1.20), α ∈ (0, 1), µ, λ satisfy (4.1.3), let γ satisfy (4.1.21) and let Ω belong to the class (4.1.19). Then there exists a

246


couple (ρα,R , uα,R ) with the following properties: (i)

3 ρα,R ∈ L2γ ρ = ρ∞ |ΩR |, loc (IR ), ΩR α,R

ρα,R ≥ 0 a.e. in ΩR , ρα,R = 0 in IR3 \ ΩR ,

uα,R ∈ (W 1,2 (IR3 ))3 , uα,R = 0 in IR3 \ ΩR ,

(4.12.2)

6γ

γ+3 3 ρα,R uα,R ∈ (Lloc (IR3 ))3 ; div (ρα,R uα,R ) ∈ L2γ loc (IR ).

(ii) 1 2 αρ∞ uα,R

+ 23 αρα,R uα,R + ∂j (ρα,R uα,R ujα,R ) − µ∆uα,R

−(µ + λ)∇div uα,R + ∇ργα,R = ρα,R f + g in (D′ (ΩR ))3 .

(4.12.3)

(iii) αρα,R + div (ρα,R uα,R ) = αρ∞ 1ΩR

in D′ (IR3 ).

(4.12.4)

(iv) For any b from C 1 ([0, ∞)) satisfying (3.1.17), (3.1.18) with β replaced by 2γ, div (b(ρα,R )uα,R ) + (ρα,R b′ (ρα,R ) − b(ρα,R ))div uα,R (4.12.5) = α(ρ∞ − ρα,R )b′ (ρα,R )1ΩR in D′ (IR3 ). (v)

ρα,R |uα,R |2 + µ ΩR |∇uα,R |2

γ γ−1 (ρ − ρ∞ )(ργ−1 +(µ + λ) ΩR |div uα,R |2 + α γ−1 α,R − ρ∞ ) ΩR α,R

≤ Ω (ρα,R f + g) · uα,R . αρ∞

Ω

|uα,R |2 + α

ΩR

(4.12.6)

(vi) If we choose R(α) = α−1 and denote ρα = ρα,R(α) , uα = uα,R(α) , the following estimates hold: ∇uα 0,2,IR3 ≤ L(f , g, Ω, ρ∞ ),

(4.12.7)

(ρα − ρ∞ )1ΩR(α) 0,3,IR3 ≤ L(f , g, Ω, ρ∞ ),

(4.12.8)

(ρα − ρ∞ )1ΩR(α) 0,2γ,IR3 ≤ L(f , g, Ω, ρ∞ ).

(4.12.9)

Here L is a positive constant which is, in particular, independent of α. Proposition 4.52 will be proved in Section 4.13. Remark 4.53 One easily verifies by using a density argument and (4.12.2) that equation (4.12.3) holds with any test function φ ∈ (W01,2 (ΩR ))3 and that equation (4.12.4) holds with any test function η ∈ W 1,2 (ΩR ).


247

4.13

Complete system with relaxed continuity equation on an exterior domain Statements (i)–(v) of Proposition 4.52 are, in fact, the corresponding statements of Proposition 4.18 and of Remark 4.21 rephrased to the present situation. It remains to prove estimates (vi). 4.13.1 Some equivalence inequalities It is easily seen that |ts − ρs∞ | |ts − ρs∞ | ρ1−s ∞ lim = 1, lim = 1, s > 0, ρ∞ > 0. t→∞ |t − ρ∞ |s s t→ρ∞ |t − ρ∞ |

(4.13.10)

Therefore, there exist constants c1 (ρ∞ , s), c2 (ρ∞ , s) such that c1 1{|ρ−ρ∞ |≤1} |ρ − ρ∞ | ≤ 1{|ρ−ρ∞ |≤1} |ρs − ρs∞ | ≤ c2 1{|ρ−ρ∞ |≤1} |ρ − ρ∞ |, s c1 1{|ρ−ρ∞ |≥1} |ρ − ρ∞ | ≤ 1{|ρ−ρ∞ |≥1} |ρs − ρs∞ |

≤ c2 1{|ρ−ρ∞ |≥1} |ρ − ρ∞ |s , s > 0. (4.13.11)

4.13.2 Bounds due to the energy inequality The right-hand side of energy inequality (4.12.6) can be majorized by

(1 + ρ∞ ) ΩR (|f · uα,R | + |g · uα,R |)

+ ΩR |ρα,R − ρ∞ |1{|ρα,R −ρ∞ |≥1} |f · uα,R | (4.13.12) ≤ cuα,R 0,6,Ω f 0, 56 ,Ω + g0, 56 ,Ω +(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR f 0, 6γ ,Ω . 5γ−3

Therefore,

∇uα,R 0,2,Ω ≤ c(f , g, ρ∞ ) 1+(ρα,R −ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR . (4.13.13)

Consequently, we also have

α(ρα,R |uα,R |2 0,1,ΩR + uα,R 20,2,ΩR ) 2 ≤ c(f , g, Ω, ρ∞ ) 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR ,

(4.13.14)

and due to (4.13.11),

αρα,R − ρ∞ 20,2,ΩR

2 ≤ c(f , g, Ω, ρ∞ ) 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR ,

(4.13.15)

2 ≤ c(f , g, Ω, ρ∞ ) 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR .

(4.13.16)

αρα,R − ρ∞ γ0,γ,ΩR

248


4.13.3

Estimates independent of invading domains and relaxation

4.13.3.1 First estimate of the density In accordance with Remark 4.53, we use in equation (4.12.3) the test function φ = B sign (ρα,R − ρ∞ )|ρα,R − ρ∞ |γ 1{|ρα,R −ρ∞ |≥1} (4.13.17)

− ΩR sign (ρα,R − ρ∞ )|ρα,R − ρ∞ |γ 1{|ρα,R −ρ∞ |≥1} ,

where B = BΩR is the Bogovskii operator on ΩR defined in Lemma 3.25. Due to (3.3.51), we have ∇φ0,p,ΩR ≤ c(p, Ω)(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} γ0,pγ,ΩR

(4.13.18)

2 pγ

≤ c (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR , 1 < p ≤ 2 with the constant c independent of R. From the testing of (4.12.3) by φ, we obtain

|ργα,R − ργ∞ | |ρα,R − ρ∞ |γ 1{|ρα,R −ρ∞ |≥1} ΩR

γ ≤ ΩR |ργα,R − ργ∞ | |ρ − ρ | 1 α,R ∞ {|ρ −ρ |≥1} ∞ α,R ΩR

3 1 (4.13.19) +α ΩR ( 2 ρα,R + 2 ρ∞ )uα,R · φ

+µ ΩR ∂j uiα,R ∂j φi + (µ + λ) ΩR div uα,R div φ

6 − ΩR ρα,R uiα,R ujα,R ∂j φi − ΩR (ρα,R f + g) · φ = i=1 Ii .

The integrals I1 –I6 are estimated as follows: (i) By virtue of (4.13.16), we have

|ρα,R − ρ∞ |γ 1{|ρα,R −ρ∞ |≥1} ΩR 2 c(f ,g ,Ω,ρ∞ ) 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR ≤ αR3 and due to (4.13.11), (4.13.15), (4.13.16),

|ργα,R − ργ∞ | ≤ c{ ΩR |ρα,R − ρ∞ |1{|ρα,R −ρ∞ |≤1} ΩR

+ ΩR |ρα,R − ρ∞ |γ 1{|ρα,R −ρ∞ |≥1} } 3

≤ c{R 2 ρα,R − ρ∞ 0,2,ΩR + (ρR − ρ∞ )1{|ρα,R −ρ∞ |≥1} γ0,γ,ΩR } 3 √ 2 1 + (ρα,R − ρ∞ )1{|ρ ≤ c(f , g, Ω, ρ∞ ) R 0,2γ,Ω −ρ |≥1} R ∞ α,R α 2 + α1 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR .

We therefore conclude that the first integral is bounded by


249

|I1 | ≤ c(f , g, Ω, ρ∞ ) 1 3 × 3 3 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR (4.13.20) α2 R2 4 1 + 2 3 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR . α R (ii) By using H¨ older’s inequality, Sobolev’s imbedding theorems, (4.13.14) and (4.13.18), we obtain

|I2 | ≤ α(1 + 2ρ∞ ) ΩR |uα,R · φ|(1 + |ρα,R − ρ∞ |1{|ρα,R −ρ∞ |≥1} ) ≤ c(ρ∞ )αuα,R 0,2,Ω (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR ×φ0, 2γ ,ΩR + φ0,2,ΩR γ−1 √ ≤ c(f , g, Ω, ρ∞ ) α ∇φ0, 65 ,ΩR (4.13.21) 6γ +(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR ∇φ 5γ−3 ,ΩR × 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR 5γ

3 ≤ c(f , g, Ω, ρ∞ )(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,Ω R × 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR , α ∈ (0, 1).

(iii) The third and the fourth integrals are bounded by

|I3 | + |I4 | ≤ c(µ, λ)∇uα,R 0,2,ΩR ∇φ0,2,ΩR ≤ c(f , g) 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR (4.13.22) ×(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} γ0,2γ,ΩR .

Here we have used the Schwartz inequality and estimates (4.13.13), (4.13.18). (iv) The next integral can be estimated again by H¨ older’s and Sobolev’s inequalities and by (4.13.13), (4.13.18) as follows:

|I5 | ≤ (1 + ρ∞ ) ΩR |uiα,R ujα,R ∂j φi |

+ ΩR |ρα,R − ρ∞ |1{|ρα,R −ρ∞ |≥1} |uiα,R ujα,R ∂j φi | ≤ c(ρ∞ )uα,R 20,6,Ω ∇φ0, 32 ,ΩR (4.13.23) +(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR ∇φ0, 6γ ,ΩR 4γ−3 2 ≤ c(f , g, Ω, ρ∞ ) 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR 4γ

3 ×(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,Ω . R

250


(v) For the last integral, we have

|I6 | ≤ (1 + ρ∞ ) ΩR (|f · φ| + |g · φ|)

+ ΩR |ρα,R − ρ∞ |1{|ρα,R −ρ∞ |≥1} |f · φ| ≤ c(ρ∞ )φ0,6,ΩR f 0, 65 ,Ω + g0, 65 ,Ω

+(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR f 0, 6γ ,Ω 5γ−3 ≤ c(f , g, Ω, ρ∞ ) 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR

(4.13.24)

×(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR .

Summarizing the results of (4.13.19)–(4.13.24), we get (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 2γ 0,2γ 3 ≤ c(f , g, Ω, ρ∞ ) 31 3 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR

α2 R2

+ α21R3 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR + 1 + (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR 5γ 3 × (ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,Ω R

4

+(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} γ0,2γ,ΩR

+(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,ΩR 2 + 1 + (ρα,R − ρ∞ )1{|ρα, R−ρ∞ |≥1} 0,2γ,ΩR 4γ 3 ×(ρα,R − ρ∞ )1{|ρα,R −ρ∞ |≥1} 0,2γ,Ω , α ∈ (0, 1). R

(4.13.25)

This especially yields (ρα,R(α) − ρ∞ )1{|ρα,R(α) −ρ∞ |≥1} 0,2γ,ΩR(α) ≤ L(f , g, Ω, ρ∞ ) provided R(α) = α−1 , α ∈ (0, 1), γ > 3.

(4.13.26)

The last inequality together with estimates (4.13.13)–(4.13.16) yields the following bounds ∇uα,R 0,2,ΩR ≤ L(f , g, Ω, ρ∞ ), R(α) = α−1 , α ∈ (0, 1),

(4.13.27)

α(1 + ρα,R )|uα,R |2 0,1,ΩR ≤ L(f , g, Ω, ρ∞ ), R(α) = α−1 , α ∈ (0, 1), (4.13.28)


251

αρα,R − ρ∞ 20,2,ΩR ≤ L(f , g, Ω, ρ∞ ), R(α) = α−1 , α ∈ (0, 1),

(4.13.29)

αρα,R − ρ∞ γ0,γ,ΩR ≤ L(f , g, Ω, ρ∞ ), R(α) = α−1 , α ∈ (0, 1).

(4.13.30)

and

4.13.3.2 Second estimate of the density uα,R(α) and ΩR(α) , where

In the sequel we denote ρα,R(α) ,

R = R(α) = α−1 , α ∈ (0, 1),

(4.13.31)

simply by ρα , uα and ΩR . In agreement with Remark 4.53, we can use in equation (4.12.3) the test function φ = B sign (ρα − ρ∞ )|ρα − ρ∞ |ϑ 1{|ρα,R −ρ∞ |≤1} (4.13.32)

− ΩR sign (ρα − ρ∞ )|ρα − ρ∞ |ϑ 1{|ρα −ρ∞ |≤1} , where B = BΩR is defined in Lemma 3.25 and 0 < ϑ ≤ γ. Due to (3.3.51), we have ∇φ0,p,ΩR ≤ c(Ω, p)(ρα − ρ∞ )1{|ρα −ρ∞ |≤1} ϑ0,pϑ,ΩR , 1 < p ≤

2γ . (4.13.33) ϑ

Taking into account the evident inequality 2 1 f 1{|f |≤1} p0,p , 1 ≤ p1 ≤ p2 < ∞, ≥ f 1{|f |≤1} p0,p 2 1

(4.13.34)

we also obtain the bound ϑ+1 p ∇φ0,p,ΩR ≤ c(Ω, p)(ρα − ρ∞ )1{|ρR −ρ∞ |≤1} 0,ϑ+1,Ω , ϑ + 1 ≤ pϑ. (4.13.35) R

Similarly, as in (4.13.19), we find

|ργα − ργ∞ ||ρα − ρ∞ |ϑ 1{|ρα −ρ∞ |≤1} ΩR

≤ ΩR |ργα − ργ∞ | |ρα − ρ∞ |ϑ 1{|ρα −ρ∞ |≤1} ΩR

+α ΩR ( 21 ρα + 23 ρ∞ )uα · φ

+µ ΩR ∂j uiα ∂j φi + (µ + λ) ΩR div uα div φ

− ΩR ρα uiα ujα ∂j φi − ΩR (ρα f + g) · φ 6 = i=1 Ii .

(4.13.36)

Integrals I1 –I6 are estimated as follows: (i) By interpolation, on condition 2 ≤ ϑ ≤ γ, after using (4.13.29), (4.13.30), and supposing at least γ > 3, one obtains

252


ΩR

|ρα − ρ∞ |ϑ 1{|ρα −ρ∞ |≤1} ≤ ≤

c(Ω) R3 (ρα

˜ ϑ(1−λ)

˜

ϑλ − ρ∞ )1{|ρα −ρ∞ |≤1} 0,2 (ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,γ

c(f ,g ,Ω,ρ∞ ) , R3 α

where

˜ λ 2

+

˜ 1−λ γ

=

1 ϑ.

By virtue of (4.13.11), (4.13.29), (4.13.30), we have

γ γ |ρα − ρ∞ |1{|ρα −ρ∞ |≤1} |ρ − ρ | ≤ c(γ, ρ ) ∞ α ∞ ΩR ΩR

+ ΩR |ρα − ρ∞ |γ 1{|ρα −ρ∞ |≥1} 3 ≤ c(γ, ρ∞ , Ω) R 2 ρα − ρ∞ 0,2,ΩR +ρα − ρ∞ γ0,γ,ΩR 3

√ 2 + 1 ). ≤ c(f , g, ρ∞ , Ω)( R α α

Therefore, the first integral at the right-hand side of (4.13.36) is bounded by |I1 | ≤ c(f , g, Ω, ρ∞ ) 31 3 + R31α2 R2 α2 (4.13.37) := cK(α, R), 2 ≤ ϑ ≤ γ, γ > 3. Of course, under conditions (4.13.31), I1 remains bounded. (ii) The second integral is estimated as follows:

|I2 | ≤ α(1 + 2ρ∞ ) ΩR |uα · φ|(1 + |ρα − ρ∞ |1{|ρα −ρ∞ |≥1} ) " ≤ c(ρ∞ )α uα 0,2 φ0,2 + uα 0,6,Ω # ×(ρα − ρ∞ )1{|ρR −ρ∞ |≥1} 0,2,ΩR φ0,3,ΩR 1 ≤ c(f , g, Ω, ρ∞ )α 2 ∇φ0, 65 + ∇φ0, 23 1 ≤ c(f , g, Ω, ρ∞ )α 2 (ρα − ρ∞ )1{|ρα −ρ∞ |≤1} ϑ0, 6 ϑ,ΩR 5 2 (1+ϑ) 3 +(ρα − ρ∞ )1{|ρR −ρ∞ |≤1} 0,1+ϑ,Ω R 1 ¯ ≤ c(f , g, Ω, ρ∞ )α 2 ρα − ρ∞ λϑ 0,2,ΩR ¯ (1−λ)ϑ

×(ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,1+ϑ,ΩR 2

(1+ϑ)

3 +(ρα − ρ∞ )1{|ρR −ρ∞ |≤1} 0,1+ϑ,Ω R

(4.13.38)


253

¯ (1−λ)ϑ ≤ c(f , g, Ω, ρ∞ ) (ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,1+ϑ,ΩR 2 3 (1+ϑ) +(ρα − ρ∞ )1{|ρR −ρ∞ |≤1} 0,1+ϑ,Ω , R γ > 3, 2 ≤ ϑ ≤ min{γ, 5},

¯ λ 2

+

¯ 1−λ 1+ϑ

=

5 6ϑ .

The first inequality is evident. In the next two inequalities, we have used H¨ older’s inequality, Sobolev’s imbedings and bounds (4.13.27)–(4.13.29). Finally, for estimating ∇φ0, 23 ,ΩR , we have employed estimate (4.13.35), and for estimating 6

∇φ0, 56 ,ΩR , we have used (4.13.33) together with the interpolation of L 5 ϑ between L2 and L1+ϑ and together with (4.13.29). (iii) The next two integrals are bounded by |I3 | + |I4 | ≤ c(Ω)∇uα 0,2,Ω ∇φ0,2,Ω ≤ c(f , g, Ω)

(4.13.39) ϑ+1 2

×(ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,ϑ+1,ΩR , γ > 3, 1 ≤ ϑ ≤ γ. (iv) Using H¨ older’s inequality and the Sobolev imbedding theorem together with bounds (4.13.26), (4.13.27), and then employing bound (4.13.35), we obtain

|I5 | ≤ (1 + ρ∞ ) ΩR |uiα ujα ∂j φi |

+ ΩR |ρα − ρ∞ |1{|ρα −ρ∞ |≥1} |uiα ujα ∂j φi | ≤ c(ρ∞ )uα 20,6,Ω {∇φ0, 23

+(ρα − ρ∞ )1{|ρα −ρ∞ |≥1} 0,2γ,ΩR ∇φ0, 6γ 4γ−3 2 3 (ϑ+1) ≤ c(f , g, Ω, ρ∞ ) (ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,ϑ+1,Ω R 4γ−3 6γ (ϑ+1) +(ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,ϑ+1,Ω , γ > 3, 2 ≤ ϑ ≤ γ. R

(v) For the last integral we have,

|I6 | ≤ (1 + ρ∞ ) ΩR (|f · φ| + |g · φ|)

+ ΩR |ρα − ρ∞ |1{|ρα −ρ∞ |≥1} |f · φ| ≤ c(ρ∞ )φ0,6,ΩR f 0, 65 ,Ω + g0, 65 ,Ω

+(ρα − ρ∞ )1{|ρα −ρ∞ |≥1} 0,2γ,ΩR f 0, ϑ+1

6γ 5γ−3 ,Ω

(4.13.40)

≤ c(f , g, Ω)

2 , γ > 3, 1 ≤ ϑ ≤ γ. ×(ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,ϑ+1,Ω R

(4.13.41)

254


Summarizing (4.13.36)–(4.13.41), one gets −1 ) (ρα − ρ∞ )1{|ρα −ρ∞ |≤1} ϑ+1 0,ϑ+1,ΩR ≤ c(f , g, Ω, ρ∞ ) K(α, α ¯ (1−λ)ϑ

+(ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,ϑ+1,ΩR 2(ϑ+1)

3 +(ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,ϑ+1,Ω R 4γ−3

(4.13.42)

(ϑ+1)

6γ +(ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,ϑ+1,Ω R

ϑ+1 2 +(ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,ϑ+1,Ω ] . R

Resuming the conditions imposed on ϑ in (4.13.37)–(4.13.41), and taking into ¯ (see (4.13.38)), we conclude that it is optimal account the admissible values of λ to take ϑ = 2. In this case, from (4.13.42), we obtain (ρα − ρ∞ )1{|ρα −ρ∞ |≤1} 0,3,ΩR ≤ L(f , g, Ω, ρ∞ ), provided R = α−1 , α ∈ (0, 1), γ > 3.

(4.13.43)

Now, we are in a position to summarize the results of the whole Section 4.13.3. Bound (4.13.27) is nothing but estimate (4.12.7) of Proposition 4.52. Bounds (4.13.26) and (4.13.43) imply the estimates (4.12.8) and (4.12.9). Proposition 4.52 is thus proved. 2 4.14

Existence of weak solutions in exterior domains

In this section, we shall complete the proof of Theorem 4.8. 4.14.1

Vanishing relaxation limit

Estimates (4.12.7)–(4.12.9) yield, at least for an appropriately chosen subsequence of the sequence (ρα , uα ), the following properties: uα → u weakly in (L6 (IR3 ))3 ,

∇uα → ∇u weakly in (L2 (IR3 ))3×3 ,

uα → u strongly in (Lq (Bn ))3 , 1 ≤ q < 6, n ∈ IN ,

(4.14.1)

u = 0 a.e. in IR3 \ Ω,

ρα → ρ weakly in L2γ (Bn ),

(ρα − ρ∞ )1ΩR(α) → (ρ − ρ∞ )1Ω weakly in L3 (IR3 ) and L2γ (IR3 ),

(4.14.2)

ρ ≥ 0 a.e. in Ω, ρ = 0 in IR3 \ Ω, ρtα → ρt weakly in L

2γ t

(Bn ), 0 < t < 2γ

ρt ≥ 0 in Ω, ρt = 0 in IR3 \ Ω.

(4.14.3)

EXISTENCE OF WEAK SOLUTIONS IN EXTERIOR DOMAINS

255

By virtue of strong convergence (4.14.1) and weak convergence (4.14.2), r 2γ

ρα uα → ρu weakly in (Lr (Bn ))3 , n ∈ IN ,

+

(ρα − ρ∞ )uα → (ρ − ρ∞ )u weakly in (Lr (IR3 ))3 ,

r 6

≤ 1,

r 2γ

+

+

s 3

r 6

=1

(4.14.4)

and ρα uiα ujα → ρui uj weakly in Ls (Bn ),

s 2γ

≤1

(4.14.5)

Passing to the limit α → 0+ in (4.12.4) and in (4.12.3), we obtain div (ρu) = 0 in D′ (IR3 ) which is the continuity equation (4.1.14) and ∂j (ρuuj ) − µ∆u − (µ + λ)∇div u + ∇ργ = ρf + g in (D′ (Ω))3 .

(4.14.6)

Lemma 3.3 implies the validity of the renormalized equation (4.1.15). Formulae (4.14.1), (4.14.2) and (4.14.4) applied to (4.12.6) combined with the lower weak semicontinuity of norms imply also the energy inequality (4.1.9).10 It remains to prove that ργ = ργ . 4.14.2

Effective viscous flux and renormalized continuity equation

We repeat step by step the reasoning of Section 4.10.2 replacing pδ (ρ) = ργ + δ(ρ2 + ρβ ) by p(ρ) = ργ , pδ (ρ) = ργ + δ(ρ2 + ρβ ) by p(ρ) = ργ , and the weak 2γ limits (4.10.18)–(4.10.20) in Ls (Ω), s = 2γ and s = γ+1 by weak limits in Ls (Ωn ), 2γ n ∈ IN with s = 2γ and s = γ+1 , respectively. We arrive at the usual identity for the effective viscous flux p(ρ)b(ρ) − (2µ + λ)b(ρ)div u = p(ρ) b(ρ) − (2µ + λ)b(ρ)div u a.e. in Ω (4.14.7) with any b satisfying (4.9.16). Next we derive, as in Section 4.10.3, the identity div ((ρθ )1/θ u) =

1−θ θ θ(2µ+λ) (p(ρ)ρ

1

− p(ρ) ρθ )(ρθ ) θ −1 in D′ (IR3 ).

(4.14.8)

Here, θ ∈ (0, 1) and ρθ , p(ρ)ρθ are weak limits of the sequences ρθα and p(ρα )ρθα in 2γ 2γ L θ (Ωn ) and L γ+θ (Ωn ), respectively. Repeating the reasoning of Section 4.10.4, we finally obtain div (rθ u) = where

θ1 − ρ ≤ 0. rθ := ρθ

(4.14.9)

(4.14.10)

get the term Ω (ρf + g) · u, we write Ω (ρα f + g) · uα = Ω [(ρα − ρ∞ )f + g] · uα + Ω f · uα and use (4.14.1), (4.14.2).

10 To

ρ∞

1 1−θ (p(ρ)ρθ − p(ρ) ρθ )(ρθ ) θ −1 in D′ (IR3 ), θ(2µ + λ)

256


4.14.2.1 Global integrability of oscillations By virtue of (4.12.8), (4.12.9) and (4.13.11), we have θ1 2γ ρθ − ρθ∞ ∈ L3 (Ω) ∩ L θ (Ω), ρθ − ρ∞ ∈ L3 (Ω) ∩ L2γ (Ω). This implies

rθ ∈ L3 (Ω) ∩ L2γ (Ω).

(4.14.11)

4.14.2.2 Cut-off functions In the sequel, we shall use the following cut-off functions (cf. Section 1.3.6.3): Φn (x) = Φ nx , n ∈ IN, Φ ∈ D(IR3 ), 0 ≤ Φ(x) ≤ 1, Φ(x) =

(4.14.12)

1 if x ∈ B1 0 if x ∈ B 2 .

It is easy to see that supp Φn ⊂ B2n \ Bn ,

IR3

|∇Φn |p ≤ c(Φ)n3−p , p > 0.

(4.14.13)

If we use the functions Φn as test functions in (4.14.9), we get θ1 −1

1−θ θ θ θ Φn = − IR3 rθ u · ∇Φn . θ(2µ+λ) IR3 (p(ρ)ρ − p(ρ) ρ ) ρ )

(4.14.14)

By using (4.14.13) and H¨ older’s inequality, we observe that

rθ u · ∇Φn ≤ rθ 0,3 u0,6 ∇Φn 0,2 ≤ c(Φ)n 12 . (4.14.15) Ω

Our goal is to prove that | Ω rθ u · ∇Φn | disappears as n → ∞. In order to show this property, we need to improve estimate (4.14.15). To do so, we shall study in more detail the asymptotic behavior of rθ . 4.14.2.3

Asymptotic behavior of oscillations Let us prove the following lemma.

Auxiliary lemma 4.54 Let 3 ≤ p ≤ 2s < ∞, q ∈ IR, ρ∞ > 0, and let ρn be a sequence such that ρn → ρ weakly in L2s (Bn ),

ρn − ρ∞ → ρ − ρ∞ weakly in Lp (IR3 ) and L2s (IR3 ). Suppose that 1

1

Bm 1

1

1

(ρs+ 2 − ρs ρ 2 )ρ 2 ≤ cmq ,

(4.14.16) s+ 21

where c > 0 and ρs+ 2 , ρs , ρ 2 are weak limits of sequences ρn tively. Then

p−3 3 p−2 |r| ≤ cmmax{q,3 p , 2 p } , Bm

1 2

, ρsn , ρn , respec(4.14.17)

where r = r 12 is defined in (4.14.10) and c is a positive constant independent of m.

EXISTENCE OF WEAK SOLUTIONS IN EXTERIOR DOMAINS

257

Proof Due to H¨ older’s inequality, we have K

ρα ≤

K

ρsα ≤

and

1

s+ 12 2s

K

ρα

K

ρα

1

K

1

s+ 12 1− 2s

ρα2

1

1− 2s 1

K

ρα2

where K is any measurable subset of Ω. Therefore,

1

2s

,

1 1 1 1 2s 1− 2s 1− 2s 2s 1 1 1 1 ρ2 ρ2 , ρs ≤ ρs+ 2 ρ ≤ ρs+ 2

and 1

1

ρ ρs ≤ ρs+ 2 ρ 2 . Consequently, 1

1

1

ρs |r| ≤ (ρs+ 2 − ρs ρ 2 )ρ 2 a.e. in IR3 .

(4.14.18)

Further we observe that r ∈ Lp (IR3 ) (cf. (4.13.11) and the reasoning leading to formula (4.14.11)). Now, we write ρs∞

Bm

|r| =

Bm

(ρs∞ − ρs )|r| +

Bm

ρs |r|.

Employing H¨ older’s inequality, formula (4.13.11) and the lower weak semicontinuity of norms, we can majorize the first term by 1

1

3 p−3 p

2 2 2 c lim inf n→∞ (ρn − ρ∞ )1{|ρn −ρ∞ |≤1} 0,p,IR3 r0,p,I R3 r0,1,Bm m

3 p−2 p

+c lim inf n→∞ (ρn − ρ∞ )1{|ρn −ρ∞ |>1} s0,2s,IR3 r0,p,IR3 m 2 12 3 p−3

3 p−2 2 p 2 p m r ≤c + m . Bm

Taking into account the last inequality, and using (4.14.16), (4.14.18) to estimate the second term, we arrive at the following inequality

Bm

1 3 p−3 3 p−2 |r| ≤ c Bm r 2 m 2 p + m 2 p + mq .

Now, we apply to the first term at the right-hand side of the last estimate Young’s inequality thus completing the proof. 2 Auxiliary lemma 4.55 Let 56 ≤ p < ∞ and let q ≥ 0. Suppose that u ∈ (L6 (IR3 ))3 , r ∈ Lp (IR3 ) and B2m |r| ≤ cmq . Then

IR3

ru · ∇Φm ≤ cmq−1

(4.14.19)

258


Proof The following chain of inequalities holds true |

IR3

B2m

c m

ru · ∇Φm | = | ≤ ≤

c(Φ)

m

ru · ∇Φm | B2m

r|u|

r|u|1{|u|≤1} + B2m

r|u|1{|u|>1} B2m

c ′ r0,p ≤m |r| + |u|1 0,p {| u |>1} B2m

≤ c(mq−1 + m−1 ).

To get the last inequality, we have used the evident bound u1{|u|>1} 0,p′ ,IR3 6 ′

p ′ ≤ u0,6,I R3 , provided p ≤ 6. The proof of Lemma 4.55 is thus complete.

Using (4.14.15) and Lemmas 4.54, 4.55, we deduce that

1 then that | IR3 ru · ∇Φn | ≤ cn− 2 . Consequently, we obtain

Bn

Since by Lemma 3.35,

2 1

B2n

|r| ≤ cn 2 and

1 1 1 p(ρ)ρ 2 − p(ρ)ρ 2 ρ 2 ≤ 0, n ∈ IN.

(4.14.20)

1 1 1 p(ρ)ρ 2 − p(ρ)ρ 2 ρ 2 ≥ 0 a.e. in IR3 ,

(4.14.21)

we conclude that 1

1

1

(p(ρ)ρ 2 − p(ρ)ρ 2 )ρ 2 = 0 a.e. in IR3 .

(4.14.22)

This is the same formula as the formula (4.9.47) in Section 4.9.4. Repeating word for word the argumentation of that section starting from (4.9.47), we arrive first 1

1

at the strong convergence of an appropriately chosen subsequence ρα2 to ρ 2 in L2γ (Bn ), n ∈ IN . Finally, taking into account the bound ρα 0,2γ,Bn ≤ L(n, f , g, ρ∞ , Ω) (cf. (4.12.9)), we easily prove the following lemma Lemma 4.56 Let ρα and ρ be a sequence and its weak limit from (4.14.2). Then, at least for a chosen subsequence, ρα → ρ strongly in Lp (Bn ), n ∈ IN , 1 ≤ p < 2γ. This implies ργ = ργ . By virtue of (4.14.6), the proof of equation (4.1.12) and consequently that of Theorem 4.8, is thus complete.

EXISTENCE IN BOUNDED AND EXTERIOR LIPSCHITZ DOMAINS

4.15

259

Existence of weak solutions in bounded and in exterior Lipschitz domains

Theorems 4.3 and 4.8 hold provided Ω is a bounded or an exterior domain with only Lipschitz boundary. More precisely, we have the following theorem Theorem 4.57 Assume that Ω is a bounded domain belonging to the class C 0,1 , f ∈ (L∞ (Ω))3 , g ∈ (L∞ (Ω))3 , m > 0.

(4.15.1) (4.15.2)

Then there exists a renormalized bounded energy weak solution (ρ, u) to problem (4.1.1)–(4.1.5) such that 0 ρ ∈ Ls(γ) (Ω), (4.15.3) ρ = m, u ∈ (W01,2 (Ω))3 . Ω

Theorem 4.58 Assume that Ω is an exterior domain belonging to class C 0,1 , f ∈ (L1 (Ω) ∩ L∞ (Ω))3 , g ∈ (L1 (Ω) ∩ L∞ (Ω))3 , ρ∞ > 0,

(4.15.4) (4.15.5)

and γ > 3.

(4.15.6)

Then there exists a renormalized weak solution (ρ, u) to problem (4.1.1)–(4.1.6) which is such that (4.15.7) ρ − ρ∞ ∈ L3 (Ω) ∩ L2γ (Ω). The proof of these two theorems can be found in (Novo and Novotn´ y, 2003b). We explain briefly its main points. We shall start with Theorem 4.57. First, we recall one classical result about the approximation of a bounded domain Ω by smooth domains: Let ωn := {x ∈ IR3 ; dist (x, Ω) < n1 }, and φn ∈ D(ωn ), 0 ≤ φn ≤ 1, φn = 1 on ωn+1 . Due to the Morse–Sard theorem (see Section 1.1.5), the intersection of sets Ωn = {x ∈ IR3 ; φn > t} and {x ∈ IR3 ; ∇φn = 0} is empty for almost all t ∈ (0, 1). Due to this fact and due to the implicit function theorem (see again Section 1.1.5), there exists a sequence tn ∈ (0, 1) such that Ωn := {x ∈ IR3 ; φn > tn } ∈ C ∞ , Ω ⊂ Ωn+1 ⊂ Ωn .

(4.15.8)

If moreover Ω is a Lipschitz domain, then |Ωn \ Ω| → 0.

(4.15.9)

Using these results, we shall first show that Proposition 4.18 still holds for Ω only a Lipschitz domain. More precisely, we shall have:

260


Proposition 4.59 Let f , g satisfy (4.15.2), let µ, λ satisfy (4.1.3), let γ be as in (4.1.4), let δ, β satisfy (4.3.3) and let α ∈ (0, 1), h ≥ 0, h ∈ L∞ (Ω).

(4.15.10)

Suppose that Ω is a bounded domain belonging to the class C 0,1 . Then there exists a couple (ρα , uα ) satisfying statements (i)–(vi) of Proposition 4.18. Proof We start by taking a sequence (ρn , un ) ∈ L2β (Ωn ) × (W01,2 (Ωn ))3 of renormalized bounded energy weak solutions of the problem (4.3.13), (4.3.14) and (4.1.5) on Ωn with f , g, h replaced by fn =

f in Ω , gn = 0 in Ωn \ Ω

g in Ω , hn = 0 in Ωn \ Ω

h in Ω 0 in Ωn \ Ω.

(4.15.11)

The existence of these solutions is guaranteed by Proposition 4.18. The energy inequality (4.3.20) on Ωn with these new functions, yields, similarly as in (4.8.22), un 1,2,Ω ≤ un 1,2,Ωn ≤ c(Ω, f , g, h)(1 + ρn 0,2β,Ω ).

(4.15.12)

By virtue of (4.15.11) the constant c in (4.15.12) really does depend only on Ω (and not on Ωn ) and there appears the L2β -norm of ρ over the set Ω only (and

not over Ωn ). Then we test equation (4.3.13) written on Ωn by φ = BΩ (ρβn − Ω ρβn ) (see Lemma 3.17) which we extend by 0 outside Ω. Repeating the argumentation of Section 4.8.2.2, we obtain ρn 0,2β,Ω ≤ L(Ω, f , g, h, δ).

(4.15.13)

Consequently, by virtue of (4.15.12), one gets un 1,2,Ωn ≤ L(Ω, f , g, h, δ).

(4.15.14)

Due to the last two bounds, there exists a subsequence (ρn , un ) such that ρn → ρ weakly in L2β (Ω),

un → u weakly in (W 1,2 (Ω))3 .

Letting n → ∞ in the Stokes formula Ωn ∇un · φ = − Ωn ∇φ · un , φ ∈

(C ∞ (IR3 ))3 , by using (4.15.9) and (4.15.14), we conclude that Ω ∇u · φ = − Ω ∇φ · u, φ ∈ (C ∞ (IR3 ))3 which means that γ0Ω (u) = 0, i.e. u ∈ W01,2 (Ω).

Now, we use Lions’ approach described in Sections 4.2.3, 4.2.4 adapted to system (4.3.13), (4.3.14) and (4.1.5). Copying word for word the argumentation

DOMAINS WITH NONCOMPACT BOUNDARIES

261

of Section 4.9, we show that (ρ, u) satisfies Proposition 4.59 on Ω with one exception: Since from the sole information (4.15.13) we cannot conclude

it is not a priori clear whether

Ωn \Ω

Ω

ρn → 0,

ρ=

Ω

h.

(4.15.15)

(4.15.16)

Fortunately, the last formula follows from equation (4.3.18).

2

Once Proposition 4.59 is shown, Theorem 4.57 will be proved by letting α → 0+ and then δ → 0+ in (4.3.13), (4.3.14), repeating step by step the argumentation of Sections 4.10–4.11. In order to prove Theorem 4.58, we repeat word for word the reasoning of Sections 4.12–4.14. The only matter which changes is the fact that invading domains ΩR (cf. Section 4.12.1) are no longer regular but only Lipschitz. Therefore, existence of solutions to problem (4.3.13), (4.3.14) with γ > 3, h = ρ∞ and δ = 0 is guaranteed by Proposition 4.59 in place of Proposition 4.18. Remark 4.60 Since, in general, the implication (4.15.13)⇒(4.15.15) fails, the limit process Ωn → Ω on the level of the original system (4.1.1), (4.1.2) or on the level of approximations (4.3.1), (4.3.2) has not been successful. Remark 4.61 To obtain estimate (4.15.13), we have used, among others, the Bogovskii operator. Due to this fact, this estimate does depend of Ω and it is available only provided Ω is a Lipschitz domain (compare with the “nonsteady” estimates (7.3.2) which do not depend on Ω and do hold for any domain). This is the main obstacle for Theorems 4.3 and 4.8 to be true for Ω less smooth than Lipschitz domains, and also for the steady version of Theorem 7.72 to be valid. 4.16

Existence of weak solutions in domains with noncompact boundaries

In this section we shall discuss the compressible barotropic flows in domains with several cylindrical, conical, superconical or subconical exits at infinity. In Section 4.16.1, we define fluxes through exits in the context of weak solutions and we explain what we mean under renormalized finite energy weak solutions in the situation of prescribed nonzero pressure drops in different exits. In Section 4.16.2, we formulate the main results. Theorem 4.63 deals with the existence of finite energy renormalized weak solutions in domains with several conical or superconical exits. Its proof is performed in Section 4.16.3. A nonexistence result in the case when the exits are subconical is formulated in Theorem 4.64 and proved in Section 4.16.4. The main ideas of this section are taken over from (Novo and Novotn´ y, 2003a) and (Novo et al., 2003).

262


4.16.1

Formulation of the problem, fluxes

Throughout this section, Ω ⊂ IR3 is a Lipschitz domain with M ∈ IN cylindrical, subconical, conical or superconical exits Ei at infinity as defined in Section 3.3.3 (see the formulae (3.3.52)–(3.3.57) and (3.3.58)–(3.3.60)). Without loss of generality, and to simplify, we set R0 = 1 in these formulae, and we omit the tilde ˜i ). We shall however conserve the notation over Ei (i.e., in the sequel Ei means E ˜R (see (3.3.58)–(3.3.60)), i.e., B ˜R = ∪M BR (si ) and Ω ˜R = Ω ∩ B ˜R . ˜ R and B for Ω i=1 In this domain, we consider problem (4.1.1), (4.1.2), (4.1.5) with the conditions at infinity lim

|x|→∞, x∈Ei

ρ(x) = ρi ,

lim

|x|→∞, x∈Ω

u(x) = 0,

(4.16.1)

where we suppose that 0 < ρM ≤ . . . ≤ ρ1 < ∞.

(4.16.2)

We introduce the notion of the flux through the i-th exit. To this end, suppose 6p ) that (ρ, u) ∈ Lploc (IR3 ) × (D01,2 (Ω))3 , p > 65 and f ∈ Lqloc (IR3 ) (1 < q ≤ 6+p satisfy div (ρu) = f in D′ (IR3 ), and introduce the sets 3

˜S ); ξ = 1 in [Ei ]23 , ξ = 0 in [Ej ] 2 , j = i}, ASi := {ξ ∈ D(B 2

˜ 2 ); ξ = 1 in [Ei ]23 , ξ = 0 in [Ej ]23 , j = i}, Bi := {ξ ∈ C ∞ (Ω 2

2 [Ei ]R R1

2

R1

= [Ei ] ∩ BR2 (si ), where [Ei ]R1 = Ei \ where S > 2. We recall that q BR1 (si ), 1 < R1 < R2 . Notice that ρu ∈ Eloc (Ω). Therefore, in accordance with Lemma 3.10 there exist continuous linear operators S : E q ([Ei ]S2 ) → [W γn,i

1− q1′ ,q ′

˜ 2 ) → [W γñ : E q (Ω

1− q1′ ,q ′

(∂[Ei ]S2 )]∗ ,

˜ 2 )]∗ (∂ Ω

such that S γn,i (v) = v · n|∂[Ei ]S2 , v ∈ C ∞ ([Ei ]S2 ),

˜ 2) γñ (v) = v · n|∂ Ω˜ 2 , v ∈ C ∞ (Ω and the corresponding Stokes formulae (see (3.2.3)) hold. By these formulae, one easily verifies that ˜ γn (ρu), ξ1 |∂ Ω˜ 2 = ˜ γn (ρu), ξ2 |∂ Ω˜ 2 , ξ1 , ξ2 ∈ Bi . Therefore, it is natural to define the outgoing flux through the i-th exit by Φi = ˜ γn (ρu), ξ|∂ Ω˜ 2 , ξ ∈ Bi .

(4.16.3)


Again by the Stokes formula, this flux is equal to

Φi = Ω˜ 2 ρu · ∇ξ + Ω˜ 2 f ξ, ξ ∈ Bi

263

(4.16.4)

and also to

S Φi = −γn,i (ρu), ξ|∂[Ei ]S2 , S > 2, ξ ∈ ASi

(4.16.5)

(no summation over repeated i) or to

Φi = − [Ei ]S ρu · ∇ξ − [Ei ]S f ξ, S > 2, ξ ∈ ASi .

(4.16.6)

2

2

(The last two formulas can be easily deduced from identities − Ω˜ S ρu · ∇ξ =

S γn (ρu), ξ|∂ Ω˜ 2 + γn,i (ρu), ξ|∂[Ei ]S2 − Ω˜ S ρu · ∇ξ, ξ ∈ ASi .) ˜ S f ξ and Ω ˜ S f ξ = ˜ Ω M Moreover, since ˜ γn (ρu), [( i=1 ξi ) − 1]|∂ Ω˜ 2 = 0, ξi ∈ Bi , i = 1, . . . , M , we have M

i=1

Φi =

˜2 Ω

f.

(4.16.7)

The reasonable definition of the renormalized finite energy weak solution in this situation reads: Definition 4.62 A couple (ρ, u) is called a renormalized finite energy weak solution of problem (4.1.1)–(4.1.5) and (4.16.1), (4.16.2) if: (i) ρ ∈ Lsloc (Ω) with some γ ≤ s < ∞, ρ ≥ 0 a.e. in Ω,

u ∈ (D01,2 (Ω))3 , ρ − ρi ∈ Lr ([Ei ]1 ) for some 1 ≤ r < ∞.

(4.16.8)

(ii) Equation (4.1.1) holds in (D′ (Ω))3 .

(4.16.9)

Equation (4.1.2) holds in D′ (IR3 ) provided (ρ, u) is prolonged by zero outside Ω.

(4.16.10)

(iii)

(iv) Equation (4.1.7) holds in D′ (IR3 ) provided (ρ, u) is prolonged by zero outside Ω, for any function b satisfying (3.1.16), (3.1.17) and (3.1.18) with β = s. (v) The energy inequality

µ Ω |∇u|2 + (µ + λ) Ω |div u|2

γ−1 γ M ≤ Ω (ρf + g) · u + γ−1 − ργ−1 )Φi i i=2 (ρ1 with Φi given by (4.16.3), holds.

(4.16.11)

(4.16.12)

264


We shall finish with a comment which concerns the energy inequality. To this end, we suppose that (ρ, u) is sufficiently smooth. The steady version of ˜ R , R > 1, is again equation (4.1.10), energy equation (1.2.89) integrated over Ω ˜ R and the where the integrals over ΩR are replaced by the same integrals over Ω ˜R ∩Ei ]. Thus, [∂ B integrals over ∂BR are replaced by the same integrals over ∪M i=1 at the right-hand side of (4.1.10), we are left with the following surface integrals: M γ

γ−1 − ργ−1 )ρu · ndS, − i=1 γ−1 ˜ R ∩Ei (ρ i ∂B

− 21 ∪M [∂ B˜R ∩Ei ] ρ|u|2 u · ndS, i=1

[λu · ndiv u + µ(u · ∇u) · n + µ(n · ∇u) · u]dS, ˜ ∪M i=1 [∂ BR ∩Ei ]

M γ − i=1 ργ−1 ˜ R ∩Ei ρu · ndS. i γ−1 ∂ B

From physical considerations it is reasonable to suppose that the first three expressions tend to 0 as R → ∞ (the heuristic arguments can be taken again from Sections 5.2.2 and 5.5). By virtue of (4.1.2), the fourth integral is equal to γ−1 γ M − ργ−1 )Φi which justifies (4.16.12). i i=2 (ρ1 γ−1 The reader should compare Definition 4.62 with Definition 4.1 and see Section 1.2.18 for the physical background. For an overall discussion of arguments leading to energy inequality (4.16.12) see (Padula, 1997). 4.16.2

Main results

If all exits are conical or superconical, we have the following existence theorem: Theorem 4.63 Assume that Ω is a Lipschitz domain with M ∈ IN conical or superconical exits Ei (see (3.3.52), (3.3.53), (3.3.58)–(3.3.60) and that the functions Fi caracterizing these exits are monotone. Suppose further that f ∈ (L1 (Ω) ∩ L∞ (Ω))3 , g ∈ (L1 (Ω) ∩ L∞ (Ω))3 ,

(4.16.13)

and γ > 3.

(4.16.14)

Then there exists a renormalized finite energy weak solution (ρ, u) ∈ L2γ loc (Ω) × (D01,2 (Ω))3 to problem (4.1.1)–(4.1.5), (4.16.1), (4.16.2) which is such that ρ − ρi ∈ L3 ([Ei ]1 ) ∩ L2γ ([Ei ]1 ). Proof Theorem 4.63 is proved in Section 4.16.3.

(4.16.15) 2

If one of the exits is subconical or cylindrical, the above existence theorem is not, in general, true. This fact is illustrated by the following result: Theorem 4.64 Let Ω be a Lipschitz domain with M ∈ IN , M > 1 subconical or cylindrical exits Ei , i = 1, . . . , M . Suppose that for all exits Ei which are not cylindrical, we have


Fi (t) = ctα , t > t0 > 0, where α ≥ 4, c > 0

265

(4.16.16)

(cf. (3.3.52), (3.3.53)). Then the problem in the sense of Definition 4.62 is illposed. More precisely, setting f = g = 0 and taking ρi , i = 1, . . . , M not all the same, there does not exist any finite energy renormalized weak solution of problem (4.1.1)–(4.1.5), (4.16.1), (4.16.2) which is such that ρ − ρi ∈ L3 ([Ei ]1 ), i = 1, . . . , M . Proof Theorem 4.64 is proved in Section 4.16.4. 4.16.3

2

Domains with conical or superconical exits

In this section, we shall prove Theorem 4.63. The strategy of the proof of this theorem is similar to the strategy of the proof of Theorem 4.8 presented in Sections 4.12–4.14. The original system (4.1.1), (4.1.2), (4.1.5), (4.16.1) is approximated by systems (4.3.13), (4.3.14), (4.1.5) with δ = 0 and with h ∈ C ∞ (IR3 ), ρM ≤ h ≤ ρ1 , h =

˜ 1, ρM in Ω ρi in [Ei ]2 ,

(4.16.17)

˜R = Ω ∩ B ˜R , R > 2 (which are bounded and on the family of invading domains Ω Lipschitz). A renormalized finite energy weak solution (ρα,R , uα,R ) of this system exists due to Proposition 4.59. It is evident that this solution satisfies statements (i)–

˜ R in place of ΩR , where the formula (v) of Proposition 4.52 with Ω ρ = ΩR α,R

ρ∞ |ΩR | in (4.12.2) is replaced by Ω˜ R ρα,R = ΩR h and where energy inequality (4.12.6) is replaced by

α Ω˜ R h|uα,R |2 + α Ω˜ R ρα,R |uα,R |2 + µ Ω˜ R |∇uα,R |2

γ−1 γ γ−1 +(µ + λ) Ω˜ R |div uα,R |2 + α γ−1 ) (4.16.18) ˜ R (ρα,R − h)(ρα,R − h Ω

αγ γ−1 ≤ Ω˜ R (ρα,R f + g) · uα,R + γ−1 ˜ R (h − ρα,R )h Ω

(cf. (4.3.20) and Remark 4.19). Statement (vi) of Proposition 4.52, where the dependence on ρ∞ in L is replaced by the dependence on ρ1 , . . . , ρM , holds, as well. Its proof is however less evident, and we shall perform it in what follows. The first term at the right-hand side of the energy inequality (4.16.18) is ˜ R in place of ΩR and with h in place of ρ∞ . If we estimated in (4.13.12) with Ω use the continuity equation (4.12.4) we find that the second term is bounded as follows:

αγ γ−1 | = γ| Ω˜ R ρα,R uα,R · ∇hhγ−2 | | γ−1 ˜ R (h − ρα,R )h Ω ≤ c(h)∇uα,R 0,2,Ω˜ R (1 + (ρ − h)1{|ρ−h|≥1} 0,2γ,Ω˜ R ).

Therefore, as in the case of an exterior domain, we obtain bounds (4.13.13)– ˜ R and ρ∞ by h). (4.13.16) (where, of course, we replace ΩR by Ω

266


˜ R ) subsequently by the test Then we test equation (4.12.3) (written on Ω functions    B ˜ i sign (ρα,R − h)|ρα,R − h|γ 1{|ρ −h|≥1}    α,R Ω  ˜i , in Ω

R R φi =  − ˜ i sign (ρα,R − h)|ρα,R − h|γ 1{|ρ −h|≥1}  α,R  ΩR   0 otherwise,

R ˜i = Ω ˜ R \ (∪M i = 1, . . . , M , where Ω ˜ i is the Bogovskii R k=1,k=i [Ek ]1 ) and where BΩ R i ˜ operator on Ω defined in Lemma 3.30. As in Section 4.13, one finds the bounds R

(ρα,R − h)1{|ρα,R −h|≥1} 2γ ˜i 0,2γ,Ω ≤ c(f , g, Ω, ρ1 , . . . , ρM )

R

3 α2

1

3 R2

1 + (ρα,R − h)1{|ρα,R −h|≥1} 0,2γ,Ω˜ R

3

4 + α21R3 1 + (ρα,R − h)1{|ρα,R −h|≥1} 0,2γ,Ω˜ R + 1 + (ρα,R − h)1{|ρα,R −h|≥1} 0,2γ,Ω˜ R 5γ γ 3 × (ρα,R − h)1{|ρα,R −h|≥1} 0,2γ, ˜R ˜ R + (ρα,R − h)1{|ρα,R −h|≥1} 0,2γ,Ω Ω +(ρα,R − h)1{|ρα,R −h|≥1} 0,2γ,Ω˜ R 2 + 1 + (ρα,R − h)1{|ρα, R−h|≥1} 0,2γ,Ω˜ R 4γ 3 ×(ρα,R − h)1{|ρα,R −h|≥1} 0,2γ, , i = 1, . . . , M, α ∈ (0, 1), R = α−1 . ˜ Ω R

Summing these estimates up, we finally get

(ρα,R(α) − h)1{|ρα,R(α) −h|≥1} 0,2γ,Ω˜ R(α) ≤ L(f , g, Ω, ρ1 , . . . , ρM ) provided R(α) = α−1 , α ∈ (0, 1), γ > 3.

(4.16.19)

Consequently, as in Section 4.13.3.1, we find that estimates (4.13.27)–(4.13.30) ˜ R ) hold true. (with ρ∞ replaced by h and ΩR replaced by Ω ˜ R ) subsequently by the test Finally, we test equation (4.12.3) (written on Ω functions    B ˜ i sign (ρα,R − h)|ρα,R − h|2 1{|ρ −h|≤1}    α,R ΩR  ˜i in Ω

R φi =  − ˜ i sign (ρα,R − h)|ρα,R − h|2 1{|ρ −h|≤1}  α,R  ΩR   0 otherwise, i = 1, . . . , M , and as in the previous situation, we obtain M estimates (see Section 4.13.3.2 for all details) which, after being summed up, yield


(ρα,R(α) − h)1{|ρα,R(α) −h|≤1} 0,3,Ω˜ R(α) ≤ L(f , g, Ω, ρ1 , . . . , ρM ) provided R(α) = α−1 , α ∈ (0, 1), γ > 3.

267

(4.16.20)

Resuming, we have ∇uα,R(α) 0,2,Ω˜ R(α) ≤ L(f , g, ρ1 , . . . , ρM , Ω), R(α) = α−1 , α ∈ (0, 1), ρα,R(α) − h0,2γ,Ω˜ R(α) ≤ L(f , g, ρ1 , . . . , ρM , Ω), R(α) = α−1 , α ∈ (0, 1), ρα,R(α) − h0,3,Ω˜ R(α) ≤ L(f , g, ρ1 , . . . , ρM , Ω), R(α) = α−1 , α ∈ (0, 1). (4.16.21) This completes the proof of statement (vi) in Proposition 4.52 in the present situation. After this, as in Section 4.14, there exists a subsequence α → 0+ such that uα,R(α) → u weakly in (D01,2 (Ω))3 , R(α) = α−1

ρα,R(α) → ρ weakly in L2γ (K) for any compact K ⊂ Ω, ρα,R(α) − h → ρ − h weakly in L3 (Ω) and in L2γ (Ω), ργα → ργ weakly in L2 (Ω).

Repeating word for word the reasoning of Section 4.14 starting by (4.14.4) and finishing by Lemma 4.56, we show that ργ = ργ and that (ρ, u) is a renormalized weak solution of problem (4.3.1), (4.3.2), (4.1.5), (4.16.1). It remains to prove the energy inequality. γ M (ργ−1 − ργ−1 )Φi in the energy inequality (4.16.12) comes The term γ−1 1 i=2

i γ from the term α γ−1 ΩR (h − ρα,R )hγ−1 in the energy inequality (4.16.18). Indeed, due to the Stokes formula and (4.16.3)–(4.16.7), and since uα,R(α) = 0 in R(α) ∪M , there holds i=1 [Ei ]

M α Ω˜ R(α) (h − ρα,R(α) )hγ−1 = i=1 ργ−1 2R(α) div (ρα,R(α) uα,R(α) )ξi i [Ei ]2 2

M = i=2 (ργ−1 − ργ−1 )Φi,α + αργ−1 ˜ 2 (ρα,R(α) − h), 1 1 i Ω 2R(α)

ξi ∈ C ∞ (IR3 ) ∩ Ai

R(α)

∩ Bi , ξi = 1 in [Ei ]2

,

˜ R(α) = (Ω \ B ˜2 ) ∩ B ˜R(α) and where Ω 2 Φi,α =< γñ (ρα,R(α) uα,R(α) ), ξ|∂ Ω˜ 2 >, ξ ∈ Bi . Therefore,

α Ω˜ R(α) (h − ρα,R(α) )hγ−1

M = i=2 (ργ−1 − ργ−1 )Φi,α + α Ω˜ 2 (ρα,R(α) − h)(ργ−1 − hγ−1 ). 1 1 i M − ργ−1 )Φi and the second one As α → 0+ , the first integral tends to i=2 (ργ−1 1 i disappears. This completes the proof of Theorem 4.63.

268


4.16.4

Domains with cylindrical or subconical exits

In this section, we prove Theorem 4.64. In view of energy inequality (4.16.12), and in view of (4.16.7) (where f = 0), it suffices to show that Φi = 0, i = 2, . . . , M . By virtue of formula (4.16.6),

Φi = − [Ei ]S (ρ − ρi )u · ∇ξ − ρi [Ei ]S u · ∇ξ, S > 2, ξ ∈ ASi . 2

2

If we take ξ such that ξ = 1 in [Ei ]S−1 and ξ = 0 in [Ei ]S we obtain the bound

|Φi | ≤ c(ξ) [Ei ]S |ρ − ρi ||u| + ρi [Ei ]S |u| S−1 S−1 1 1 S S ≤ c ρ − ρi 0,3,[Ei ]SS−1 u0,2,[Ei ]SS−1 |ES−1 | 6 + u0,2,[Ei ]SS−1 |ES−1 |2 .

(4.16.22) 2 First, we take into account the fact that |[Ei ]SS−1 | is equivalent to S α as S → 1 ∞. Second, we use the Poincaré inequality u0,2,[Ei ]SS−1 ≤ cS α ∇u0,2,[Ei ]SS−1 . Third, due to ρ − ρi ∈ L3 ([Ei ]1 ) and ∇u ∈ L2 ([Ei ]1 ), there exists a sequence S → ∞ such that

|ρ − ρi |3 = o( S1 ), [Ei ]S |∇u|2 = o( S1 ). [Ei ]S S−1

S−1

Using all these facts in (4.16.22) we arrive to the conclusion of Theorem 4.64.2

Remark 4.65 The same argumentation applies to domains with M ∈ IN , M > 1 exits at infinity provided M − 1 of these exits are cylindrical or subconical satisfying (4.16.16). In this case, the nonexistence result in the sense of Theorem 4.64 holds, whatever the type of the remaining exit. 4.17 4.17.1

Further results, comments and bibliographic remarks Weak compactness

The results about weak compactness similar to Theorem 4.11 and the crucial observation about the compactness of the effective viscous flux similar to Lemma 4.12 were announced in the note (Lions, 1993a), which concerns the more complicated nonstationary case. Their complete proofs were published in the monograph (Lions, 1998). Roughly speaking, as far as the steady case is concerned, Lions’ original proof of results similar to Lemma 4.12 relies on the local regularity properties of elliptic equation (4.2.8) and on various observations in compensated compactness from (Coifman et al., 1993). In the context of small initial data, importance of effective viscous flux for the existence theory was observed in (Novotn´ y and Padula, 1994). Knowing (Lions, 1993a), in the spirit of (Novotn´ y and Padula, 1994), Novotn´ y obtained boundedness of the effective viscous flux in Triebel–Lizorkin spaces and proved also a version of Theorem 4.11, see (Novotn´ y, 1996). In that paper it is observed that the effective viscous flux P satisfies the Stokes problem −µ∆v + ∇P = −ρu · ∇u + ρf + g, div v = 0,

FURTHER RESULTS, COMMENTS AND BIBLIOGRAPHIC REMARKS

269

where v is the divergence-free part of the Helmholtz decomposition of the velocity field u (see Section 5.2, where a similar method is used for the investigation of strong solutions). Then, the compactness of P is obtained by the local regularity properties of the Stokes system in Hardy spaces in the same manner as that one briefly described before estimate (4.2.9). In this book, we present another proof of Lemma 4.12 (see also Proposition 4.26), an adaptation to the steady case in (Novo and Novotn´ y, 2002) of Feireisl’s “nonsteady approach” (Feireisl, 2001) which is based directly on the div–curl lemma. The observation about the importance of the renormalized continuity equation within the theory of compressible flow is due to P.-L. Lions (Lions, 1998). The theory of renormalized solutions to the transport (and in particular to the continuity) equations is due to (DiPerna and Lions, 1989). In the proof of Theorem 4.11, the condition γ ≥ 35 marks the limit of the applicability of the DiPerna–Lions transport theory to the steady continuity equation. The approach explained in Section 4.2.5 which makes it possible to overcome this difficulty, is due to (Feireisl, 2001). 4.17.2

Bounded domains

4.17.2.1 Dirichlet problem, periodic boundary conditions The first breakthrough into the existence of weak solutions to steady compressible Navier–Stokes equations in the isentropic regime is made in the book (Lions, 1998). It introduces all the principal ideas needed for further development of the theory. The first result about the existence of bounded energy renormalized weak solutions (in the terminology of the present book) is Theorem 6.7 in (Lions, 1998). This theorem is nothing but Theorem 4.3 with the hypothesis that Ω is a bounded smooth three-dimensional domain and γ > 35 . In fact, in Theorem 6.7, Lions considered also N -dimensional bounded domains and proved existence under the assumptions γ > 1 if N = 2 and γ > N2 if N > 3. The generalization of the Lions theorem to the case γ > 32 (if N = 3), potential volume and arbitrary nonvolume forces (if γ ≤ 53 ), as formulated in Theorem 4.3, is due to (Novo and Novotn´ y, 2002). Further generalization to bounded domains with only Lipschitz boundaries, as formulated in Theorem 4.57, is again due to Novo and Novotn´ y (Novo and Novotn´ y, 2003b). The existence theorem will continue to hold if we take for the pressure law p(ρ) any monotone and differentiable function on [0, ∞) such that p(0) = 0 and lims→∞ p(s) sγ = c > 0, see Section 6.10 in (Lions, 1998). An explicit detailed proof of this clear fact, however, does not exist in the mathematical literature. Existence of weak solutions for nonmonotone pressure in the steady case is a difficult open problem. Compare with Section 7.12.3, where nonmonotone pressure laws are considered in the nonstationary regime. If γ = 1, we are concerned with so-called isothermal flows. In this situation, the pressure term in the momentum equation does not cause any trouble. All difficulties are related to the convective term div (ρuui ). Even in two dimensions, this problem is a very comlicated open problem. For more details see the paper (Padula, 1986) which concerns the nonsteady situation. Unfortunately, there

270


is a gap which has not yet been overcome ((Padula, 1988a), (Padula, 1988b)). Nevertheless, it reflects well the nature of the difficulties. For the notion of measure valued solutions to the steady compressible Navier–Stokes equations in the isothermal regime see (Plotnikov and Sokolowski, 2002). If we replace the homogenous Dirichlet boundary conditions u|∂Ω = 0 with the nonhomogenous boundary conditions u|∂Ω = u∞ , where u∞ · n = 0 on ∂Ω

(4.17.1)

and ρ = ρ∞ on the inflow portion of the boundary, i.e. ρ = ρ∞ in ΓI := {x ∈ ∂Ω; u∞ (x) · n(x) < 0}

(4.17.2)

(see Section 1.2.21), the question about the correct formulation and about the existence of weak solutions is a very difficult open problem. Compare with (Kweon and Kellogg, 2003) for some partial results about the existence of strong solutions with small data and with Section 7.12.5, where this problem is discussed and partially solved in the nonstationary situation. If Ω is a periodic cell and if we change the Dirichlet boundary conditions u|∂Ω = 0 to the periodic boundary conditions, weak solutions, in general, do not exist. Indeed, if we integrate the momentum equation (4.1.1) over the periodic cell Ω, we obtain

(ρfi + g i ) = 0, i = 1, 2, 3 (4.17.3) Ω

which is contradictory provided, e.g., fi ≥ 0, fi = 0, gi ≥ 0, i = 1, 2, 3. For more details see Example 6.3 in (Lions, 1998). An inspiring discussion of ill- and well- posed problems for steady barotropic viscous flows can be found in Padula (Padula, 1997). In the spirit of Section 6.7 in (Lions, 1998), in the case 23 < γ ≤ 53 Theorem 4.3 (resp.Theorem 4.57) continues to be true for small perturbations of arbitrarily large potential forces. More precisely, the following result holds: Let 23 < γ ≤ 35 , p(ρ) = ργ and f = ∇Φ + b, Φ ∈ W 1,∞ (Ω), b ∈ (L∞ (Ω))3 . Suppose that λ, µ, g and m satisfy the hypothesis of Theorem 4.3. Then there exists k(µ, λ, m) > 0 such that if b0,∞ ≤ k, then problem (4.1.1), (4.1.2), (4.1.5) admits a bounded energy renormalized weak solution which satisfies (4.1.18). The proof of this statement is left to the reader as an exercise. Conditions (4.1.16) imposed on f and g in Theorem 4.3 (resp. in Theorem 4.57) are not optimal. Both theorems continue to hold provided 6

g ∈ (L 5 (Ω))3 , f ∈ (Lp (Ω))3 , p > p0 (γ) > 1.

(4.17.4)

The precise value of p0 (γ) is left to the reader as an exercise. For example, if γ = 2, then p0 (γ) = 3. 4.17.2.2 Nonuniqueness of weak solutions, equilibria In general, the weak solutions constructed in Theorem 4.3 are not unique. A simple example of nonuniqueness can be constructed in the following way. Let Φ ∈ W 1,∞ (Ω) be a function


271

with the following properties: (1) There exist s, t ∈ IR, s < t, such that for any k ∈ (s, t) the level sets (k) (k) (k) {x ∈ Ω; Φ(x) > k} = O1 ∪ O2 , where Oi , i = 1, 2, are nonempty disjoint domains. (k ) (k ) (t) (2) There holds Oi = ∅ and if k1 , k2 ∈ (s, t), k1 < k2 , then Oi 2 ⊂ Oi 1 , i = 1, 2. In this case, the couple ρki (x) =

γ−1 γ

(Φ(x) − k)+

1 γ−1

1O(k) , u = 0, k ∈ (s, t)

(4.17.5)

i

is a bounded energy renormalized weak solution to problem (4.1.1)–(4.1.5) with

(k) (k) g = 0 and f = ∇Φ satisfying (4.1.18) with the mass mi = Ω ρi . Functions (k)

(t)

k → mi , i = 1, 2, are continuous decreasing functions on (s, t) and mi = 0. (s) Therefore, for any m ∈ (0, mc ), where mc = mini=1,2 mi there exists just one (k ) ki ∈ (s, t) such that mi i = m. By this procedure, for any m ∈ (0, mc ) we have constructed two different bounded energy renormalized weak solutions to problem (4.1.1)–(4.1.5) with g = 0 and f = ∇Φ satisfying (4.1.18) with the mass m. These solutions are (k1 )

(ρ = ρ1

(k2 )

, u = 0) and (ρ = ρ2

, u = 0).

Solutions of type (4.17.5) are called rest states or equilibria. The reader can consult Sections 8.10 and 8.18 to learn more about them. The question about the uniqueness of bounded energy renormalized weak solutions, provided m is given and the norms g0,p , f 0,p are “small” is, to our knowledge, an interesting open problem. Compare with Chapter 5, where the uniqueness for small data is proved in the class of strong solutions. 4.17.2.3 Regularity of weak solutions It can be proved by a bootstrap argument that for γ ≥ 3 (N = 3), the solutions constructed in Theorem 4.3 (resp. 4.57) possess the following further summability ρ ∈ Lploc (Ω), curl u ∈ (Lploc (Ω))3 ,

1,p (Ω), 1 ≤ p < ∞. P = ργ − (2µ + λ)div u ∈ Wloc

(4.17.6)

The same results hold if N = 2 with γ > 1. For all details see (Novotn´ y, 1996), (Lions, 1998). The same statement holds true for solutions constructed in Theorem 4.8 (resp. 4.58) in exterior domains and in Theorem 4.63 in domains with noncompact boundaries. Regardless the value of coefficient γ ≥ 1 and whatever is the regularity of Ω, f and g, one cannot expect much better regularity of (ρ, u) than (4.17.6). In particular, density may exhibit discontinuities over two-dimensional surfaces in Ω (so called shocks). This fact is illustrated by the following example which is taken over with slight modifications from Example 6.4 in (Lions, 1998).

272


We take Ω = B1 , ρ = 1B1/2 , u =

v in B1/2 w in B1 \ B1/2

/

,

(4.17.7)

where v ∈ D(B1/2 ), div v = 0

(4.17.8)

and w solves system ∆2 w = 0 in B1 \ B1/2 , w = 0 in ∂B1/2 ∪ ∂B1 ,

(4.17.9)

x x x µ |x| · ∇w + (µ + λ)div w |x| = − |x| in ∂B1/2 ∪ ∂B1 .

Problem (4.17.9) is an elliptic system of Agmon–Douglis–Nirenberg type and due to general theory, it admits a unique solution w ∈ (C ∞ (B1 \ B1/2 ))3 , see (Agmon et al., 1959) or (Neˇcas, 1967). Due to (4.17.7)–(4.17.9), 1,1 2,1 (ρ, u) ∈ L∞ (Ω) × (W 1,∞ (Ω))3 but (ρ, u) ∈ / Wloc (Ω) × Wloc (Ω).

(4.17.10)

We set f ∈ D(B1 ), f =

ρv · ∇v − µ∆v − (µ + λ)∇div v in B1/2 , 0 in B1 \ B1/2

g ∈ C ∞ (B1 ), g =

/

,

0 in B1/2 , −µ∆w − (µ + λ)∇div w in B1 \ B1/2 .

By construction, (ρ, u) satisfies the continuity equation (4.1.14). Consequently, it satisfies also (4.1.15). We verify that ρu · ∇u − µ∇u − (µ + λ)∇u = ρf + g

(4.17.11)

in B1 \ ∂B1/2 . Taking into account boundary values of v and w on ∂B1/2 from (4.17.8), (4.17.9) we easily verify that the momentum equation (4.1.1) is satisfied as well. Scalar multiplication of (4.17.11) by u and integration over B1 , yields the energy inequality (4.1.9). Thus the couple (ρ, u) is really a renormalized bounded energy weak solution to our system. To our knowledge, question whether a bounded renormalized weak solution (ρ, u) belonging to (4.17.6) and satisfying infessx∈Ω ρ(x) > α > 0 must be more regular than (4.17.6) is an open problem. Some partial results in this direction can be found in Section 6.9 of (Lions, 1998).


4.17.2.4

273

Slip boundary conditions If µ > 0, 2µ + 3λ > 0,

(4.17.12)

Ω is not axially symmetric,

(4.17.13)

and if then Theorem 4.3 holds with slip boundary conditions u · n|∂Ω = 0, [τi dij (u)nj ]|∂Ω = 0,

(4.17.14)

where dij (u) = 12 (∂j ui + ∂i uj ), see Section 1.2.21. In this case, we have to change the weak formulation of the momentum equation (4.1.12) in Definition 4.1 to

− Ω ρui uj ∂j φi + λ Ω div u div φ + 2µ Ω dij (u)dij (φ)

− Ω p(ρ)div φ = Ω (ρf + g) · φ, (4.17.15) φ ∈ (C ∞ (Ω))3 , φ · n|∂Ω = 0

and the energy inequality (4.1.9) to

λ Ω |div u|2 + 2µ Ω dij (u)dij (u) ≤ Ω (ρf + g) · u.

(4.17.16)

In (4.1.11), the assumptions for ρ are maintained while the condition u ∈ (D01,2 (Ω))3 is replaced by u ∈ (W 1,2 (Ω))3 ∩ E02 (Ω). Other requirements (4.1.13)– (4.1.15) are kept unchanged. We have the following theorem: Theorem 4.66 Let Ω be a Lipschitz bounded domain satisfying (4.17.13), let µ, λ satisfy (4.17.12) and let m, f , g, γ satisfy assumptions of Theorem 4.3. Then there exists a renormalized bounded energy weak solution to problem (4.1.1)– (4.1.2), (4.1.4), (4.17.14) satisfying ρ ∈ Ls(γ) , s(γ) =

Ω

3(γ − 1) if 32 < γ < 3 2γ if γ ≥ 3,

ρ = m, u ∈ (W

1,2

3

(Ω)) ∩

(4.17.17)

E02 (Ω).

We have not found any rigorous detailed proof of this result in the mathematical literature except some hints in (Lions, 1998). In the sequel, we explain the most important point. First, we realize that the Korn inequality

c1 u21,2 ≤ Ω dij (u)dij (u) + c0 u20,2 , u ∈ (W 1,2 (Ω))3 (4.17.18) holds with some c0 , c1 > 0, see (Nitsche, 1981). Now, via a standard compactness argument we deduce the Poincaré–Morrey inequality

c2 u20,2 ≤ Ω dij (u)dij (u) + ∂Ω |u · n|2 dS, u ∈ (W 1,2 (Ω))3 |R(Ω) , (4.17.19)

where c2 > 0 and where (W 1,2 (Ω))3 |R(Ω) is the factor space of (W 1,2 (Ω))3 with respect to rigid rotations R(Ω) = {u ∈ (W 1,2 (Ω))3 ∩ E02 (Ω); d(u) = 0} = {u =

274


a × (x − x0 ), u · n|∂Ω = 0}, a, x0 ∈ IR3 . Clearly, R(Ω) = {0}, if Ω is axially symmetric. Otherwise R(Ω) = {0}. We verify by an easy algebraic calculations, that λ|div u|2 + 2µdij (u)dij (u) ≥ min{µ, 2µ + 3λ}dij (u)dij (u).

(4.17.20)

Therefore, energy inequality (4.17.16) together with formulae (4.17.12), (4.17.18) and (4.17.19), yields

c3 u21,2 ≤ Ω (ρf + g) · u

with some c3 > 0. After this observation, the proof follows the same lines as the proof of Theorem 4.3. The details are left to the reader. 4.17.3

Exterior domains

4.17.3.1 Exterior domains with nonzero density at infinity Theorem 4.8 with Ω a smooth exterior domain is due to (Lions, 1998). Its generalization to exterior domains with only Lipschitz boundaries, as formulated in Theorem 4.58, is due to (Novo and Novotn´ y, 2003b). The double limit passage (α, R) → (0+ , ∞) is taken over from (Novo and Novotn´ y, 2003a) (the same method is used for treating fluxes in domains with noncompact boundaries in Section 4.16). Other parts of the proof are taken over from Section 6.8 in (Lions, 1998). The question whether problem (4.1.1), (4.1.2), (4.1.5), (4.1.6) is well posed in a two-dimensional exterior domain is a very difficult open problem. See (Dutto and Novotn´ y, 2001), where the two-dimensional problem is considered in the case of small data. Another very difficult open problem is to prove the existence of weak solutions of system (4.1.1), (4.1.2), (4.1.5) in an exterior domain with the following conditions at infinity: lim u(x) = u∞ ∈ IR3 (= 0),

|x|→∞

lim ρ(x) = ρ∞ > 0.

|x|→∞

(4.17.21)

Compare with Section 7.12.6 in Section 7.12, where similar problems in the nonstationary case are dealt with. See also Sections 5.1, 5.5 and 5.6, where the same problems are considered in the case of small data. 4.17.3.2 Exterior domains with zero density at infinity In general, problem (4.1.1)–(4.1.5), (4.1.6), where Ω is an exterior domain or Ω = IR3 and ρ∞ = 0, does not admit any renormalized energy weak solution with positive finite mass. The following example is taken over from (Lions, 1998). Assume that ρ∞ = 0, g = 0, f = ∇Φ, Φ ∈ C ∞ (Ω), Φ < 0, lim|x|→∞, x∈Ω Φ = 0; for any c > 0 the set {x ∈ Ω; Φ > −c} is connected and unbounded. (A concrete example of such couple Ω, Φ is, e.g., Ω = IR3 or Ω = B 1 (0) and 2 1 .) Suppose that Φ(x) = −e−|x| or Ω = B 1 (0) and Φ(x) = − |x|


(ρ, u), ρ = 0

275

(4.17.22)

is a renormalized energy weak solution of problem (4.1.1)–(4.1.5), (4.1.6) with the above data. Due to energy inequality (4.1.9), ∇u = 0, and since u|∂Ω = 0, we have u = 0. Now, by virtue of (4.1.1), ρ satisfies the ordinary differential equation ∇ργ = ρ∇Φ in D′ (Ω). The list of all solutions to this equation is 1 γ − 1 γ−1 ρ = 0, ρ = ρc = (Φ + c)+ , c > 0, γ

i.e. ρ = 0, ρc =

!

1

γ−1 on {Φ > −c}, c > 0 [ γ−1 γ (Φ + c)] 0 otherwise.

Since ρc ∈ C ∞ ({Φ > −c}) and since {Φ > −c} is an unbounded domain, we have 1 γ−1 > 0 which contradicts the condition ρ ∈ Lr (Ω) lim|x|→∞, x∈{Φ>−c} = [ γ−1 γ c] (see (4.1.11)). 4.17.4

Domains with noncompact boundaries

4.17.4.1 Domains with conical or superconical exits Theorem 4.63 is due to Novo, Novotn´ y, Pokorn´ y (Novo et al., 2003). It was originally proved in (Novo and Novotn´ y, 2003a) for domains with only conical exists. It generalizes Theorem 6.12 in (Lions, 1998) (see Theorem 4.8) which deals with exterior domains to the case of domains with several conical exits and to the case of nonzero fluxes and density drops. It also generalizes the same results known for incompressible Navier–Stokes equations to the case of compressible isentropic Navier–Stokes equations. In the case of incompressible Navier–Stokes equations, the main idea coming from (Heywood, 1976) was developed and generalized in (Solonnikov, 1981), (Ladyzhenskaya and Solonnikov, 1976), (Ladyzhenskaya and Solonnikov, 1977), (Pileckas and Solonnikov, 1977) and (Pileckas, 1980) to the family of domains with finite number of “wide exits”. Domains with superconical exits belong to this family. See (Galdi, 1994a) and (Pileckas, 1996) for more details and an extended bibliography. In Section 4.16.2, to simplify the technicalities of the proof, Theorem 4.63 is formulated for domains with several axially symmetric conical or superconical exits. It remains true, however, for certain domains whose exits are not axially symmetric. More precisely, we can consider exits E = {x ∈ IR3 ; y3 > F (y ′ )},

(4.17.23)

where F is globally Lipschitz, i.e., there exists M > 0 such that |F (u′ ) − F (z ′ )| ≤ M |u′ − z ′ |, u′ , z ′ ∈ IR2 ,

(4.17.24)

276


where t → F (t, kt) is increasing on (0, ∞) for any k ∈ IR,

(4.17.25)

and where there exist constants 0 < k1 < k2 < ∞ and α ∈ (0, 1] such that k1 |x′ |α < F (x′ ) < k2 |x′ |α .

(4.17.26)

If Ω is a domain with a finite number of exists (4.17.23)–(4.17.26), then Theorem 4.63 remains true (see (Novo et al., 2003)). 4.17.4.2 Cylindrical or subconical exits Theorem 4.64 is due to Novo, Novotn´ y, Pokorn´ y (Novo et al., 2003). It was originally proved in (Novo and Novotn´ y, 2003a) for domains with only cylindrical exists. Theorem 4.64 on domains with cylindrical exits generalizes results of Novotn´ y, Padula, Penel (Novotn´ y et al., 1996) and of Lions (Lions, 1998) from locally smooth solutions to the classes of bounded energy weak solutions. In Section 4.16.2, we have formulated it for domains with several axially symmetric exits. However, the hypothesis about axial symmetry can be avoided. More precisely, suppose that the exits are defined by (4.17.23), where F is only locally Lipschitz. We set [E]SS = {x ∈ E, x3 = R}, S > 1 and we denote by dE S the infimum over all distances of two parallel lines lying in the plane {x ∈ IR3 ; x3 = S} such that ESS is between these two lines. Theorem 4.64 still holds for domains with several subconical exits (4.17.23), where F is only locally Lipschitz, if we replace the hypothesis (4.16.16) by δ i |[Ei ]SS | ≤ cS β , dE S ≤ cS , i = 1, . . . , M,

(4.17.27)

where | · | is the Lebesgue measure on IR2 and δ, β ≥ 0, δ +

β 1 ≤ . 2 2

(4.17.28)

This result is proved in (Novo et al., 2003). If the exits are axially symmetric, then in (4.17.27), (4.17.28) 2δ = β and in (4.16.16), α = 1δ , and we are in the situation of the hypothesis of Theorem 4.64. If α = 0, then β ≤ 1; this corresponds to layer-like domains contained in a cone. 4.17.4.3 Stability with respect to external data, solutions with extremal fluxes The following result can be found in (Novo et al., 2003): Theorem 4.67 Suppose that γ > 3. Let Ω be a domain with M > 1 conical or superconical exits as in Theorem 4.63. Let {ρki }M i=1 be a sequence of vectors of positive numbers converging in IRM to a vector {ρi }M i=1 of positive components and let f k → f and g k → g strongly in (L1 (Ω))3 and in (L∞ (Ω))3 . Let (ρk , uk )


277

be a sequence of renormalized bounded energy weak solutions of problem (4.1.1)– (4.1.5), (4.16.1), (4.16.2) corresponding to these external data, whose existence is guaranteed by Theorem 4.63 (as we do not have uniqueness, we choose any of the weak solutions guaranteed by this theorem). Then there exists a subsequence ′ ′ (ρk , uk ) such that ′

′

1,2 3 k ρk → ρ in L2γ loc (Ω), u → u in (D0 (Ω)) ,

where (ρ, u) is a renormalized bounded energy weak solution of problem (4.1.1)– (4.1.4), (4.16.1), (4.16.2) such that ρ − ρi ∈ L3 ([Ei ]1 ) ∩ L2γ ([Ei ]1 ) corresponding to the external data {ρi }M i=1 , f and g. The proof of this result follows the lines of the proof of Theorem 4.63. It is easy to show that the fluxes Φki are bounded by c∇uk 0,2,Ω (1 + (ρk − hk )1{|ρ−hk |≥1} 0,2γ,Ω ), where hk is defined by (4.16.17) with ρi replaced by ρki . After this, the energy inequality (4.16.12) yields the bound c∇uk 0,2,Ω (1+(ρk −hk )1{|ρ−hk |≥1} 0,2γ,Ω ) for ∇uk 20,2,Ω . Then, testing the momentum equation (4.16.9) by φ = BΩ (|ρk − hk |γ 1{|ρ−hk |≥1} ), where BΩ is the operator defined in Lemma 3.28, after similar calculation as that of Section 4.13.3.1, one gets the bound for (ρk − hk )1{|ρ−hk |≥1} 0,2γ,Ω and consequently the bound on ∇uk 0,2,Ω , both independent of k. Finally, testing (4.16.9) by φ = BΩ ((ρk − hk )2 1{|ρ−hk | 1 conical or superconical exits as in Theorem 4.63. Denote Φi = sup{Φi } and Φi = inf{Φi } where the supremum (infimum) is taken over all renormalized bounded energy weak solutions of problem (4.1.1)–(4.1.5), (4.16.1), (4.16.2) corresponding to fixed data {ρi }M i=1 , f , g and guaranteed by Theorem 4.63. Then in the set of these solutions there exists at least one whose flux through the i-th exit is equal to Φi (Φi ), i = 1, . . . , M . 4.17.4.4 Some open problems In the case of domains with at least one subconical exit given by (4.16.16), where 1 < α < 4, existence or nonexistence of weak solutions to problem (4.1.1)–(4.1.5), (4.16.1),(4.16.2) is an open question. Criteria for uniqueness of weak solutions is another very difficult open problem.

278


It is an important open problem whether system (4.1.1)–(4.1.5) admits a renormalized bounded energy weak solution provided condition (4.16.1) is replaced by m Φi = ci , i = 1, . . . , m, lim|x|→∞, x∈Ω u(x) → 0, i=1 ci = 0, ci constants.

4.17.5

Flow of mixtures

Another area of research is directed at compressible mixtures. Thus, e.g., in (Frehse et al., 2002) a system of compressible Navier–Stokes like equations which model flow of mixtures of fluids in IR3 is considered. The convective terms are neglected and the densities ρi of the species and their velocity fields ui are prescribed at x = ∞. The existence of weak solutions is obtained in the class ρi − ρi,∞ ∈ L2 (IR3 ) ∩ L2γ (IR3 ), ui ∈ (W01,2 (IR3 ))3 , ρi ∈ Lqloc (IR3 ), ∇ui ∈ (Lqloc (IR3 ))3×3 for all q ∈ [1, ∞). For physical background of mixtures we refer to (Rajagopal and Tao, 1995).

5 STRONG SOLUTIONS FOR STEADY NAVIER–STOKES EQUATIONS OF COMPRESSIBLE BAROTROPIC FLOW AND SMALL DATA In this chapter we investigate strong solutions near equilibria of steady barotropic Navier–Stokes equations in bounded and exterior domains (all results can be relatively easily extended to a general viscous compressible heat-conducting gas). The functional framework and main results are formulated in Section 5.1. The main ideas of the proofs via the Helmholtz decomposition are explained in Section 5.2. In Section 5.3, we investigate several auxiliary problems resulting from the decomposition, i.e. we deal with the Dirichlet and Neumann problems for the Laplacian, with the Stokes problem, with the transport equation in a bounded or in an exterior domains and with the Oseen problem in an exterior domain. The main existence theorems for the original system are proved in Sections 5.4– 5.5. Section 5.6 is devoted to comments and further results, as, e.g., asymptotic structure of solutions in an exterior domain and existence of the wake region. 5.1 Notation and main results 5.1.1 Formulation of the problem In this chapter, we shall deal with the existence and some qualitative properties of strong solutions to the problem ∂j (ρuuj ) − µ∆u − (µ + λ)∇div u + ∇p(ρ) = ρf in Ω, div (ρu) = 0 in Ω,

(5.1.1) (5.1.2)

1

p ∈ C ([0, ∞)),

(5.1.3)

u(x) = 0, x ∈ ∂Ω,

(5.1.4)

where λ and µ are constants satisfying (4.1.3), and Ω ⊂ IR3 is a bounded or an exterior domain. In the latter case, we complete the system by the following conditions at infinity u(x) → u∞ , ρ(x) → ρ∞ as |x| → ∞,

(5.1.5)

3

where ρ∞ is a given nonnegative constant and u∞ ∈ IR is a constant vector (compare with (4.1.1)–(4.1.6) and see Section 4.17.3.1). To simplify, without loss of generality, we suppose that u∞ = (u∞ , 0, 0), where u∞ ≥ 0. (5.1.6)

If Ω is a bounded domain, we denote by m = Ω ρ the total mass of gas in Ω. 1,s 2,s (Ω) × Wloc (Ω), 1 ≤ s < ∞, is called a strong solution to A couple (ρ, u) ∈ Wloc 279

280

STRONG SOLUTIONS FOR STEADY NAVIER–STOKES EQUATIONS

problem (5.1.1)–(5.1.5) if equations (5.1.1)–(5.1.2) are satisfied a.e. in Ω, equation (5.1.4) is satisfied in the sense of traces and conditions (5.1.5) hold pointwise. 5.1.2 Existence theorem in a bounded domain In the case of Ω a bounded domain, we have the following existence theorem: Theorem 5.1 Let m > 0, s > 3, k = 0, 1, . . . , f ∈ (W k,s (Ω))3 and let Ω ∈ C k+3 be a bounded domain. Then there exist positive constants α0 and α1 (depending only on k, s, ∂Ω, m and p) such that if f k,s ≤ α1 , then in the ball Bα0 = {(τ, w) ∈ W k+1,s × (W01,s (Ω) ∩ W k+2,s (Ω))3 ; τ k+1,s + wk+2,s ≤ α0 }

m +σ, u) is a strong there exists a unique couple (σ, u) such that the couple (ρ = |Ω| solution of problem (5.1.1)–(5.1.4). This solution satisfies the estimate

σk+1,s + uk+2,s ≤ cf k,s , where the constant c depends on k, s, ∂Ω, m and on the constitutive law p for pressure. We recall that W k,s (Ω) = {τ ∈ W k,s (Ω); Ω τ = 0}. Remark 5.2 The same statement holds if we put s = 2 and k = 1, 2, . . . in Theorem 5.1.

5.1.3 Functional spaces for exterior domains We recall that an exterior domain is defined in (3.3.46), (3.3.47). To simplify, we suppose without loss of generality that IR3 \ Ω ⊂ B1 .

(5.1.7)

Before formulation of the existence theorems in the case of Ω an exterior domain, we introduce some notation. For 1 < q < 3 < s < ∞, k = 0, 1, . . ., Wk,q,s (Ω) = W 1,q (Ω) ∩ W k,s (Ω)

is a Banach space with norm · Wk,q,s = · 1,q + · k,s , /  (D1,q (Ω))3 ∩ (D1,s (Ω))3    ∩{w; ∇w ∈ (W 1,q (Ω) ∩ (W k,s (Ω))3×3 } if a = 0 / Dak+1,q,s = (D1,q (Ω))3 ∩ (D1,s (Ω))3   4q if a > 0  ∩{w; ∇w ∈ (W 1,q (Ω) ∩ W k,s (Ω))3×3 } ∩ (L 4−q (Ω))3 1

= ∇·1,q +∇·k,s +a 4 ·0, 4q 11 is a Banach space space with norm ·Dk+1,q,s a 4−q and k+1,q,s Kk+1,q,s u∞ ,0 (Ω) = {w ∈ D|u∞ | (Ω); w|∂Ω = −u∞ , (div w)|∂Ω = 0}

is a Banach space (if u∞ = 0) or a closed convex subset of Dk+1,q,s (if u∞ = 0). |u∞ | 11 For

the definition of homogenous Sobolev spaces D1,r (Ω) see Section 1.3.6.

NOTATION AND MAIN RESULTS

281

For w ∈ Kk+1,q,s u∞ ,0 (Ω) we set k,q,s Wk,q,s (Ω); div (τ (w + u∞ )) ∈ Wk,q,s (Ω)}, w+u∞ (Ω) = {τ ∈ W

a Banach space with norm τ Wk,q,s = τ Wk,q,s + div (τ (w + u∞ )Wk,q,s . w+u∞ Last but not least, to measure the decay at infinity in the case of u∞ = 0, inspired by the decay of the fundamental solution to the Stokes problem (see Exercise 5.12), we set Sk+1,q,s = Kk+1,q,s ∩ {w; |x|w ∈ (L∞ (Ω))3 , |x|div w ∈ Lq (Ω) ∩ Ls (Ω)}. 0 0,0 It is a Banach space with norm wSk+1,q,s = wDk+1,q,s + |x|w0,∞ + 0 0 |x|div w0,q + |x|div w0,s .12 Finally, we write briefly Lk,q,s (Ω) = (D−1,q (Ω) ∩ Lq (Ω) ∩ W k,s (Ω))3 which is a Banach space endowed with the norm f Lk,q,s = |f |−1,q + f 0,q + f k,s . 5.1.4 Existence theorems in exterior domains If u∞ = 0, we have the following existence result:

Theorem 5.3 Let 23 < q < 3 < s < ∞, k = 0, 1, . . ., u∞ = 0, ρ∞ > 0, and let Ω ∈ C k+3 be an exterior domain. Suppose that f ∈ Lk,q,s (Ω) and that |x|f ∈ (L1 (Ω) ∩ L3 (Ω))3 . Then there exist positive constants α0 and α1 (depending only on k, s, ∂Ω, ρ∞ and p) such that if f Lk,q,s + |x|f 0,1 + |x|f 0,3 ≤ α1 , then in the closed convex subset Bα0 = {(τ, w) ∈ Wk+1,q,s × Sk+2,q,s (Ω); 0 τ Wk+1,q,s + wSk+2,q,s ≤ α0 }

(5.1.8)

0

2

(D01,2 (Ω))3

of L (Ω) × there exists a unique couple (σ, u) such that the couple (ρ = ρ∞ + σ, u) is a strong solution of problem (5.1.1)–(5.1.5). This solution satisfies the estimate σWk+1,q,s + uSk+2,q,s 0

≤ c[f Lk,q,s + |x|f 0,1 + |x|f 0,3 ], where the constant c depends on k, q, s, ∂Ω, ρ∞ and on the constitutive law p for pressure. 12 If u ∞ = 0, the pointwise behavior of solutions at infinity should be anisotropic and should be related to the fundamental solution of the Oseen problem. However, the existence result can be obtained without this information. The question of pointwise behavior is more involved than the simple existence problem. We shall discuss it briefly in Section 5.6.2.

282


If u∞ = 0, we have the following existence theorem:

Theorem 5.4 Let

3 2

< q < 3 < s < ∞, k = 0, 1, . . ., ρ∞ > 0, u∞ = 0 (see 3q

(5.1.6)), and let Ω ∈ C k+3 be an exterior domain. Suppose that f ∈ (L 3+q (Ω) ∩ W k,s (Ω))3 . Then there exists a positive constant κ (depending only on k, s, ∂Ω, ρ∞ and p) such that for any u∞ with 0 < |u∞ | ≤ κ there exist positive constants α0 , α1 (depending again only on k, s, ∂Ω, ρ∞ , u∞ and p) with the following property: If f 0,

3q 3+q

+ f k,s ≤ α1 ,

then in the closed convex subset Bα0 = {(τ, w) ∈ Wk+1,q,s × Kk+2,q,s u∞ ,0 (Ω); τ Wk+1,q,s + wDk+2,q,s ≤ α0 }

(5.1.9)

|u∞ |

of L2 (Ω) × (D1,2 (Ω) ∩ L4 (Ω))3 there exists a unique couple (σ, u) such that the couple (ρ = ρ∞ + σ, u) is a strong solution of problem (5.1.1)–(5.1.6). This solution satisfies the estimate σWk+1,q,s + uDk+2,q,s ≤ c[f 0, |u∞ |

3q 3+q

+ f k,s + |u∞ |],

where the constant c depends on k, q, s, ∂Ω, ρ∞ and p. 5.2 Heuristic approach 5.2.1 Perturbations and linearization of the problem Without loss of generality and to simplify, in subsequent sections, we shall consider problem (5.1.1)–(5.1.5) with p(ρ) = ρ (an isothermal motion). We shall concentrate only on the problems in exterior domains. The easier case of a bounded domain treated in Theorem 5.1 is left to the reader as an exercise. Since we deal with the small data problem, it is convenient to write the equivalent system for perturbations (σ, ℘), where ρ = ρ∞ + σ, u = u∞ + ℘. It reads −µ∆℘ − (µ + λ)∇div ℘ + ∇σ + u∞ ∂1 ℘ = F (σ, ℘) in Ω, div ℘ + div (σ(u∞ + ℘)) = 0 in Ω, ℘ = −u∞ in ∂Ω, ℘(x) → 0, σ(x) → 0 as |x| → ∞,

(5.2.1)

HEURISTIC APPROACH

283

where F (σ, ℘) = (1 + σ)f − div ((1 + σ)℘ ⊗ ℘) − u∞ ∂1 (σ℘),

(5.2.2)

cf. (5.1.6). To prove the existence of solutions for (5.2.1)–(5.2.2), we are thus naturally led to investigate first the linearized system −µ∆℘ − (µ + λ)∇div ℘ + ∇σ + u∞ ∂1 ℘ = F in Ω, div ℘ + div (σ(u∞ + w)) = 0 in Ω, ℘ = −u∞ in ∂Ω,

(5.2.3)

℘(x) → 0, σ(x) → 0 as |x| → ∞. In (5.2.3), the unknown functions are σ, ℘ while w, F are given functions such that w|∂Ω = −u∞ and div w|∂Ω = 0. Systems (5.2.3)u∞ =0 and (5.2.3)u∞ =0 are as important for compressible fluids as Stokes and Oseen equations are for incompressible fluids. From the physical point of view, such equations form a suitable linear approximation of the full system when in the momentum equation the whole (Stokes-like approximation) or a part (Oseen-like approximation) of the convective term is disregarded. This means that the inertial term is much smaller than that due to the friction (a slow motion). If we have a “good” existence theorem and estimates for linearized system (5.2.3), we show the existence of solutions for nonlinear problem (5.2.1)–(5.2.2) by applying a contraction principle to the composite map N : (τ, w) → F (τ, w) → (σ, ℘),

(5.2.4)

where (σ, ℘) is a solution of linear system (5.2.3) corresponding to F = F (τ, w). 5.2.2

Helmholtz decomposition and effective viscous flux

To show the existence of solutions for linear system (5.2.3), we proceed as follows.13 We look for ℘ in the form of its Helmholtz decomposition ℘ = v + ∇ϕ, div v = 0 in Ω, u · n = 0, ∂n ϕ = −u∞ · n in ∂Ω,

(5.2.5)

cf. Section 5.3.2. The divergence of (5.2.5)1 together with (5.2.1)2 , yields ∆ϕ = −div (σ(u∞ + w)), ∂n ϕ|∂Ω = −u∞ · n, ∇ϕ(x) → 0 as |x| → ∞.

(5.2.6)

Then from (5.2.3)1 , we get 13 Due to the presence of the term (u ∞ + w) · ∇σ, system (5.2.3) is not an elliptic system in the sense of Agmon, Douglis, Nirenberg unless u∞ + w vanishes identically on Ω, cf. (Agmon et al., 1959).

284


−µ∆v + ∇P + u∞ ∂1 v = F, div v = 0,

(5.2.7)

v|∂Ω = −u∞ − ∇ϕ|∂Ω , v(x), P (x) → 0 as |x| → ∞, where P = σ − (2µ + λ)div ℘ + u∞ ∂1 ϕ.

(5.2.8)

Due to (5.2.3)2 , the last identity is equivalent to σ + (2µ + λ)div (σ(u∞ + w)) = P − u∞ ∂1 ϕ, σ(x) → 0 as |x| → ∞.

(5.2.9)

Last but not least, we observe that ∆P = div F

(5.2.10)

(it is enough to calculate div of (5.2.7)) and that w|∂Ω = −u∞ , div w|∂Ω = 0 imply ℘|∂Ω = −u∞ , div ℘|∂Ω = 0,

(5.2.11)

cf. (5.2.3)2−3 . We are thus naturally led to define a formal linear map L : τ → σ as follows: i) For a given τ (and w) we solve the Neumann problem ∆ϕ = −div (τ (u∞ + w)), ∂n ϕ|∂Ω = −u∞ · n, ∇ϕ(x) → 0 as |x| → ∞

(5.2.12)

for an unknown function ϕ. ii) Once ϕ is known, we solve, for a given F, the nonhomogenous Stokes (u∞ = 0) or Oseen (u∞ = 0) problems (5.2.7) for unknown functions P , v. iii) When P is found, we solve the transport equation (5.2.9) for the unknown function σ. The chain of maps τ → ϕ → (P, v) → σ defines a formal linear map L : τ → σ.

(5.2.13)

If there exists a (“sufficiently regular”) fixed point σ of L, then σ and the corresponding ℘ = v + ∇ϕ satisfy the original linearized problem (5.2.3). The greatest difficulty in this scheme is the lack of regularity on the right-hand side of (5.2.12), caused by the presence of ∇τ . To overcome this difficulty, we essentially use the fact that by the fundamental properties of transport equation (5.2.9), σ and div (σ(w+u∞ )) must have the same regularity as the quantity P − u∞ ∂1 ϕ given by (5.2.7). Therefore, the functional spaces have to be taken so that

AUXILIARY LINEAR PROBLEMS

285

τ has the same regularity as div (τ (w + u∞ )). Another important ingredient of the proof is observation (5.2.11). It guarantees that the Ls -norm of ∇div (τ (w + u∞ )) can be estimated only by the D−1,s -norm of ∆div (τ (w + u∞ )) and that the Ls -norm of ∇div (σ(w + u∞ )) can be estimated only by D−1,s -norm of ∆div (σ(w + u∞ )). The key point which makes it possible to close the scheme for the fixed point is equation (5.2.10). This surprising property of the quantity P is due to the structure of equations (5.2.3). Indeed, the quantity P is nothing but the linearized effective viscous flux which has already played an essential role in the proof of the existence of weak solutions (cf. Sections 4.2.3 and 4.4.4). 5.2.3

Existence theorem for the linearized system

The theorem about the existence of strong solutions to linear problem (5.2.3) in the context of functional spaces introduced in Section 5.1.3 reads as follows: Theorem 5.5 Let 32 < q < 3 < s < ∞, k = 0, 1, . . ., ρ∞ > 0, u∞ ∈ IR3 (see (5.1.6)) and let Ω ∈ C k+3 be an exterior domain. Suppose that k,q,s w ∈ Kk+2,q,s (Ω). u∞ ,0 (Ω), F ∈ L

(5.2.14)

Then there exists a positive constant α (depending only on k, s, ∂Ω and ρ∞ ) such that if θ(w, u∞ ) := wDk+2,q,s + |u∞ | ≤ α, (5.2.15) 0

then there exists just one solution (σ, ℘) ∈ Wk+1,q,s (Ω) × Kk+2,q,s u∞ ,0 (Ω) of problem (5.2.3) satisfying the inequality ≤ c[FLk,q,s + |u∞ |]. σWk+1,q,s + ℘Dk+2,q,s u ∞

(5.2.16)

Here, c is a positive constant depending at most on k, q, s, ∂Ω and ρ∞ . 5.3

Auxiliary linear problems

In this section we recall well known results about weak solutions and regularity in Sobolev spaces of the problems which are elements of the decomposition presented in Section 5.2.2. We shall deal with the Neumann and Dirichlet problem for Laplacian, Stokes and Oseen problems, and the transport equation in a bounded and in an exterior domain. All these problems and results are broadly present in the mathematical literature. It is not our ambition to go into the fine details but to treat the issue with just enough generality for the applications in view. Thus, in what follows, the assumptions on the regularity of the domain are not optimal, and, in the Oseen problem, when we treat the regularity of solutions, we do it only for the problem with div v = 0.

286


5.3.1

Neumann problem for the Laplacian

The existence of weak solutions, uniqueness and regularity of the Neumann problem for the Laplacian in a bounded domain has already been investigated in Section 4.5. In this section, we shall investigate the same problem, i.e. −∆ϕ = g := div b, ∂n ϕ|∂Ω = −b · n, ∇ϕ(x) → 0 as |x| → ∞

(5.3.1)

in an exterior domain. The following lemma holds true: Lemma 5.6 Let 1 < s < ∞ and let Ω be an exterior domain. (i) If Ω ∈ C 2 and b ∈ (Ls (Ω))3 , then there exists a unique ϕ ∈ D1,s (Ω) satisfying

− Ω ∇ϕ · ∇η = Ω b · ∇η, η ∈ C0∞ (IR3 ) and the estimate

∇ϕ0,s ≤ c(s, Ω)b0,s .

(ii) If moreover Ω ∈ C k+2 , b ∈ E0q (Ω), div b ∈ W k,s (Ω), k = 0, 1, . . ., then the solution from (i) satisfies ∇ϕ ∈ W k+1,s (Ω) and ∇ϕk+1,s ≤ c(k, s, Ω)[b0,s + div bk,p ]. Proof See (Simader and Sohr, 1992a), Chapter III in (Galdi, 1994a) and the appendix in (Novotn´ y and Padula, 1994). 2 5.3.2

Helmholtz decomposition

Let Ω ∈ IR3 be a domain. We denote ·0,s

Hs (Ω) = {v ∈ (D(Ω))3 ; div v = 0}

,

1,s (Ω)}, G s (Ω) = {w ∈ (Ls (Ω))3 ; w = ∇ϕ for some ϕ ∈ Wloc

where 1 ≤ s < ∞. We have: Lemma 5.7 Let Ω ∈ C 2 be a bounded or an exterior domain and let 1 < s < ∞. Suppose that b ∈ (Lq (Ω))3 . Then there exists a unique couple (v, w) ∈ Hs (Ω) × G s (Ω) satisfying b = v + w. (5.3.2) In fact, w = ∇ϕ where ϕ is constructed in Lemma 4.27 (if Ω is a bounded domain) and in Lemma 5.6 (if Ω is an exterior domain). Formula (5.3.2) is called Helmholtz decomposition. Proof See Chapter III, Lemma 1.2 in (Galdi, 1994b) and Lemmas 4.27, 5.6. 2


5.3.3

287

Dirichlet problem for the Laplacian

In this section we consider the Dirichlet problem for the Laplacian in Ω, a bounded or an exterior domain: −∆φ = g, φ|∂Ω = 0, φ(x) → 0 as |x| → ∞ if Ω is an exterior domain.

(5.3.3)

The following lemma holds true: Lemma 5.8 (i) Let Ω ∈ C 2 be a bounded domain and g ∈ W −1,s (Ω), where 1 < s < ∞. Then there exists a unique φ ∈ W01,s (Ω) satisfying

∇φ · ∇η = g, η, η ∈ C0∞ (Ω) (5.3.4) Ω (where ·, · denotes the duality between W −1,s (Ω) and W01,s (Ω)) and the estimate φ1,s ≤ c(s, Ω)g−1,s . If moreover Ω ∈ C k+2 , g ∈ W k,s (Ω), k = 0, 1, . . ., then φ ∈ W k+2,s (Ω) and φk+2,s ≤ c(k, s, Ω)gk,s . (ii) Let Ω ∈ C 2 be an exterior domain and g ∈ D−1,q (Ω), where 32 < q < 3. Then there exists a unique φ ∈ D01,q (Ω) satisfying (5.3.4) (where, now, ·, · denotes the duality between D−1,s (Ω) and D01,s (Ω)) and the estimate |φ|1,q ≤ c(q, Ω)|g|−1,q . If moreover Ω ∈ C k+2 , g ∈ D−1,s ∩ W k,s (Ω), 1 < s < ∞, k = 0, 1, . . ., then ∇φ ∈ W k+1,s (Ω) and ∇φ0,q + ∇φk+1,s ≤ c(k, s, Ω)[|g|−1,q + |g|−1,s + gk,s ]. Proof See (Simader and Sohr, 1996), (Simader, 1990), (Simader and Sohr, 1992b) and the appendix in (Novotn´ y and Padula, 1994). 2 5.3.4

Stokes and Oseen problems

In the sequel, we shall investigate the problem −µ∆v + ∇Π + u∞ ∂1 v = g, div v = h,

(5.3.5)

v|∂Ω = ψ, v(x), P (x) → 0 as |x| → ∞ in Ω, a bounded domain (if u∞ = 0) and in Ω an exterior domain. If u∞ = 0, then the above system is called the Stokes system, and if u∞ = 0, it is called the Oseen system.

288

5.3.4.1


Stokes problem

Lemma 5.9 Let u∞ = 0. (i) Suppose that Ω ∈ C 2 is a bounded domain, 1 < s < ∞, g ∈ (W −1,s (Ω))3 ,

1 ψ ∈ (W 1− s ,s (∂Ω))3 , h ∈ Ls (Ω) such that Ω h = ∂Ω ψ · n. Then there exists a unique (Π, v) ∈ Ls (Ω) × (W 1,s (Ω))3 such that

µ Ω ∂i v · ∂i η − Ω Πdiv η = g, η, η ∈ (C0∞ (Ω))3 , div v = h,

(5.3.6)

v|∂Ω = ψ (where ·, · denotes the duality between (W −1,s (Ω))3 and (W01,s (Ω))3 ) which satisfies the estimate Π0,s + v1,s ≤ c[g−1,s + h0,s + ψ1− 1s ,s,∂Ω ]. 1

If moreover Ω ∈ C k+2 , g ∈ (W k,s (Ω))3 , h ∈ W k+1,s (Ω), ψ ∈ (W k+2− s ,s (∂Ω))3 , k = 0, 1, . . . , then (Π, v) ∈ W k+1,s (Ω) × (W k+2,s (Ω))3 and Πk+1,s + vk+2,s ≤ c[gk,s + hk+1,s + ψk+2− 1s ,s,∂Ω ]. (ii) Suppose that Ω ∈ C 2 is an exterior domain, 32 < q < 3, g ∈ (D−1,q (Ω))3 , 1 ψ ∈ (W 1− q ,q (∂Ω))3 , h ∈ Lq (Ω). Then there exists a unique (Π, v) ∈ Lq (Ω) × (D1,q (Ω))3 satisfying (5.3.6) (where, now, ·, · denotes the duality between (D−1,q (Ω))3 and (D01,q (Ω))3 ) and the estimate Π0,q + ∇v0,q ≤ c[|g|−1,q + h0,q + ψ1− q1 ,q,∂Ω ]. If moreover Ω ∈ C k+2 , g ∈ (D−1,s (Ω) ∩ Lq (Ω) ∩ W k,s (Ω))3 , h ∈ W 1,q (Ω) ∩ 1 1 W k+1,s (Ω), ψ ∈ (W 2− q ,q (∂Ω) ∩ W k+2− s ,s (∂Ω))3 , k = 0, 1, . . ., 1 < s < ∞, then (Π, ∇v) ∈ [W 1,q (Ω) ∩ W k+1,s (Ω)] × (W 1,q (Ω) ∩ W k+1,s (Ω))3×3 and Π1,q + Πk+1,s + ∇v1,q + ∇vk+1,s ≤ c[|g|−1,q + g0,q + |g|−1,s + gk,s +h1,q + hk+1,s + ψ2− q1 ,q,∂Ω + ψk+2− 1s ,s,∂Ω ]. Proof See Chapter V in (Galdi, 1994a) and the appendix in (Novotný and Padula, 1994). 2 5.3.4.2

Oseen problem

Lemma 5.10 Let u∞ = 0. Suppose that Ω ∈ C 2 is an exterior domain, 32 < 1 q < 4, g ∈ (D−1,q (Ω))3 , ψ ∈ (W 1− q ,q (∂Ω))3 , h ∈ D−1,q (Ω) ∩ Lq (Ω). Then there 4q exists a unique (Π, v) ∈ Lq (Ω) × (D1,q (Ω) ∩ L 4−q (Ω))3 satisfying


µ

Ω

∂i v · ∂i η −

Ω

Πdiv η + u∞

Ω

289

∂1 v · η = g, η, η ∈ (C0∞ (Ω))3 ,

div v = h,

(5.3.7)

v|∂Ω = ψ (where ·, · denotes the duality between (D−1,q (Ω))3 and (D01,q (Ω))3 ) and the estimate 1

Π0,q +|u∞ | 4 v0,

4q 4−q

+∇v0,q ≤ c[|g|−1,q +ψ1− q1 ,q,∂Ω +h0,q +|u∞ ||h|−1,q ].

If moreover Ω ∈ C k+2 , g ∈ (D−1,s (Ω) ∩ Lq (Ω) ∩ W k,s (Ω))3 , h = 0, ψ ∈ 1 1 (W 2− q ,q (∂Ω) ∩ W k+2− s ,s (∂Ω))3 , k = 0, 1, . . ., 1 < s < ∞, then (Π, ∇v) ∈ [W 1,q (Ω) ∩ W k+1,s (Ω)] × (W 1,q (Ω) ∩ W k+1,s (Ω))3×3 and 1

Π1,q + Πk+1,s + |u∞ | 4 v0,

4q 4−q

+ ∇v1,q + ∇vk+1,s

≤ c[|g|−1,q + g0,q + |g|−1,s + gk,s + ψ2− q1 ,q,∂Ω + ψk+2− 1s ,s,∂Ω ]. Proof See Chapter VII in (Galdi, 1994a) and the appendix in (Novotn´ y and Padula, 1994). 2 5.3.5

Steady transport equation

The last auxiliary problem to be investigated is the steady transport equation Λσ + div (w′ σ) = h, w′ · n|∂Ω = 0,

(5.3.8)

where Ω is a bounded or an exterior domain and Λ > 0, without loss of generality. From the various versions of existence theorems for (5.3.8) (see (Novotn´ y, 1998) for the detailed treatment), we shall need in the sequel the following formulation: Lemma 5.11 Let 1 < q < 3 < s < ∞, k = 0, 1, . . .. (i) Let Ω ∈ C k+2 be a bounded domain and h ∈ W k+1,s (Ω), w′ ∈ (W k+2,s (Ω))3 , (w′ · n)|∂Ω = 0. Then there exists α > 0 (depending only on Ω, k, s) such that if αw′ k+2,s < Λ, then there exists just one solution σ ∈ W k+1,s (Ω) with div (w′ σ) ∈ W k+1,s (Ω) of problem (5.3.8). This solution satisfies the estimate Λσk+1,s ≤ hk+1,s + αwk+2,s σk+1,s . Furthermore, there exists α′ (depending only on Ω, k, s) such that Λ∆σk−1,s ≤ ∆hk−1,s + α′ w′ k+2,s σk+1,s .

If Ω h = 0, then Ω σ = 0. If w′ ∈ W0l,s (Ω)3 and h ∈ W0l,s (Ω) with some l = 1, . . . , k + 1, then σ ∈ W0l,s (Ω).

290


(ii) Let Ω ∈ C k+2 be an exterior domain and let h ∈ Wk+1,q,s . Suppose that ′ , where w′∞ ∈ IR3 , w ′ ∈ Dk+2,q,s (Ω) and (w′ · n)|∂Ω = 0. Then w′ = w′∞ + w 0 there exists α > 0 (depending only on Ω, k, q, s) such that if ′ Dk+2,q,s + |w′∞ |) < Λ, α(w 0

then there exists just one solution σ ∈ Wk+1,q,s (Ω) with div (w′ σ) ∈ Wk+1,q,s (Ω) of problem (5.3.8). This solution satisfies the estimate ′ Dk+2,q,s + |w′∞ |]σWk+1,q,s . ΛσWk+1,q,s ≤ hWk+1,q,s + α[w

(5.3.9)

′ D2,q,s + |w′∞ |]σ1,t , Λ|∆σ|−1,t ≤ ∆h−1,t + α′ [w

(5.3.10)

′ Dk+2,q,s + |w′∞ |]σk+1,s . Λ∆σk−1,s ≤ ∆hk−1,s + α′ [w

(5.3.11)

0

Furthermore, there exists α′ (depending only on Ω, k, q, s) such that 0

where t stands for q or for s, and if k > 0, then also 0

If w′ ∈ (D01,s (Ω))3 and h ∈ D01,s (Ω), then σ ∈ D01,s (Ω).

Proof For all details and a more general formulation see (Novotn´ y, 1998). 2 5.4

The linearized system

In this section, we shall prove Theorem 5.5. To do so, we shall show that the map (5.2.13) is well defined from Wk+1,q,s (Ω) to Wk+1,q,s (Ω) and that, under assumptions (5.2.14), (5.2.15), it is a contraction. To start, we observe that for any w ∈ Kk+2,q,s u∞ ,0 (Ω), there holds (div (τ (w + u∞ )))|∂Ω = 0. Therefore, by virtue of Lemma 5.8, we have ∇div (τ (w + u∞ ))0,q + ∇div (τ (w + u∞ ))k,s ≤ c|τ |w+u∞ ,k−1,q,s , where |τ |w+u∞ ,k−1,q,s = |∆div (τ (w + u∞ ))|−1,q +|∆div (τ (w + u∞ ))|−1,s + κk ∆div (τ (w + u∞ ))k−1,s (κk = 0 if k = 0 and κk = 1 if k ≥ 1). By H¨ older’s inequality, Sobolev’s imbeddings and interpolation |div (τ (w + u∞ ))|−1,q + |div (τ (w + u∞ ))|−1,s +|div (τ (w + u∞ ))|−1, 4q ≤ cτ W1,q,s [wD2,q,s + |u∞ |]. 4−q

0

THE LINEARIZED SYSTEM

291

Here and in the sequel through this section c resp. c′ are generic constants which depend at most on k, q, s, Ω, ρ∞ and µ, λ. Therefore, in view of these auxiliary estimates, Lemma 5.6 (ii) applied to the Neumann problem (5.2.12) yields14 : ≤ c[|τ |w+u∞ ,k−1,q,s + θ(w, u∞ )τ W1,q,s + |u∞ |]. ∇ϕDk+2,q,s u ∞

(5.4.1)

Once ϕ is known, we apply Lemma 5.9 (ii) (if u∞ = 0) or Lemma 5.10 (ii) (if u∞ = 0) to system (5.2.7). After using the trace theorem to estimate ∇ϕ2− q1 ,q,∂Ω and ∇ϕk+2− 1s ,s,∂Ω , we obtain, in view of (5.4.1), the following bound for v and P : vDk+2,q,s + P Wk+1,q,s u ∞

≤ c[FLk,q,s + |τ |w+u∞ ,k−1,q,s + |u∞ | + θ(w, u∞ )τ Wk+1,q,s ].

(5.4.2)

In view of (5.2.10), |∆P |−1,q + |∆P |−1,s + ∆P k−1,s ≤ FLk,q,s .

(5.4.3)

Now, we apply Lemma 5.11 to transport equation (5.2.9) and employ estimates (5.4.1)–(5.4.3). Thus, the estimate for σ deduced from (5.3.9) by Lemma 5.11 reads: σWk+1,q,s ≤ c[FLk,q,s + |τ |w+u∞ ,k−1,q,s +|u∞ | + θ(w, u∞ )τ Wk+1,q,s ].

(5.4.4)

Once we deduce an estimate for ∆σ from (5.3.10) and (5.3.11), we find the same bound for ∆div (σ(w+u∞ )) directly from the equation ∆σ+(2µ+λ)∆div (σ(w+ u∞ )) = ∆P − u∞ ∂1 ∆ϕ. It reads: |σ|w+u∞ ,k−1,q,s ≤ c[FLk,q,s + |u∞ |(|τ |w+u∞ ,k−1,q,s + |u∞ |) +θ(w, u∞ )τ Wk+1,q,s ].

(5.4.5)

The last two inequalities imply σWk+1,q,s + 2c|σ|w+u∞ ,k−1,q,s ≤ c|τ |w+u∞ ,k−1,q,s + c′ [FLk,q,s +|u∞ | + θ(w, u∞ )(|τ |w+u∞ ,k−1,q,s + τ Wk+1,q,s + |u∞ |)].

(5.4.6)

Since the norms σWk+1,q,s +2c|σ|w+u∞ ,k−1,q,s and σWk+1,q,s are equivalent in w+u∞

Wk+1,q,s w+u∞ (Ω), and since L is a linear operator, from (5.4.6) we conclude that L is a contraction provided θ(w, u∞ ) is small, i.e. provided (5.2.15) holds. According to the Banach contraction principle (see Section 1.4.11.2), it possesses a unique

14 In the case of u ∞ = 0, Lemma 5.6 is not applicable to (5.2.12) directly. To overcome this difficulty, we search for the solution in the form ϕ = u∞ ·x+ϕ′ , where ∆ϕ′ = −div (τ (w+u∞ )), ∂n ϕ′ |∂Ω = 0 and we apply Lemma 5.6 to the latter problem.

292


fixed point τ = σ. The couple (σ, ℘ = v + ∇ϕ), where v and ϕ are solutions to problems (5.2.12) and (5.2.7) corresponding to this fixed point, is the unique solution of the original linearized system (5.2.3). Estimate (5.2.16) follows from (5.4.1)–(5.4.4) and (5.4.6) written at the fixed point. This completes the proof of Theorem 5.5. 5.5

The fully nonlinear system

5.5.1

The case of zero velocity at infinity

In this section, we shall prove the existence of solutions for small data for system (5.2.1)–(5.2.2) with u∞ = 0. We shall however start with some preparation. 5.5.1.1

Fundamental solutions of Stokes and Laplace operators

1 1 Exercise 5.12 (i) Prove that E(x) = − 4π |x| satisfies the equation ∆E = δ in ′ 3 ∞ D (IR ) and that ϕ = E ∗ g, where g ∈ C0 (IR3 ) satisfies ∆ϕ = g in IR3 .

(ii) Prove that 3 S = Sij

i,j=1

1 , Sij (x) = − 8π

δij |x|

3 , Pi (x) = ∂i E(x) = P = Pi i=1

+

xi xj |x|3

,

1 xi 4π |x|3

satisfy ∆Sij + ∂j P = δij , ∂j Sij = 0 and that vi = −Sij ∗ gj + Pi ∗ h, Π = Pj ∗ gj + h, where (g, h) ∈ (C0∞ (IR3 ))4 , satisfy −∆v + ∇Π = g, div v = h in IR3 . Hint: See, e.g., (Galdi, 1994a), Chapter IV. The function E is called the fundamental solution of the Laplace operator and the couple (P, S) is called the fundamental solution of the Stokes operator. Exercise 5.13 Prove that: (i) E and Sij are weakly singular kernels with λ = 1, cf. Section 1.3.4.5; (ii) ∂k E and ∂k Sij are weakly singular kernels with λ = 2, cf. again Section 1.3.4.5; on–Zygmund type, cf. Sec(iii) ∂k ∂s E and ∂k ∂s Sij are singular kernels of Calder´ tion 1.3.4.6. Hint: See, e.g., Chapter IV in (Galdi, 1994a).

THE FULLY NONLINEAR SYSTEM

h(y)

Exercise 5.14 Set I1 (h) = Ω |x−y| , I2 (h) = Ω that the following estimates hold true:

h(y) |x−y|2 ,

293

h ∈ C0∞ (IR3 ). Prove

|x|I1 (h)0,∞ ≤ c(q, s)(|x|h0,q + |x|h0,s ), 1 ≤ q
0 on Γ where Γ = {x ∈ ∂Ω; u∞ · n > 0} is the inflow portion of the boundary (nonzero inflow and outflow boundary conditions). Compare with the results available in the nonsteady situation discussed in Section 7.12.5 and with some remarks concerning weak solutions in Section 4.17.2.1. The list of references presented here is far from being exhaustive. There is a huge literature dealing with other boundary and free boundary value problems to steady compressible Navier–Stokes equations in the context of small data. For an interesting overview see (Padula, 1997). 5.6.2

Exterior domains

5.6.2.1 Existence theorems Theorems 5.3 and 5.4 and their proofs are taken over from (Novotn´ y and Padula, 1994). An existence and uniqueness result for small data and zero velocity at infinity in the Hilbert space setting was firstly formulated without proof in (Matsumura and Nishida, 1989). 5.6.2.2 Nonzero velocity at infinity – physically reasonable solutions and the wake region Exercises 5.16–5.19 deal with various properties of solutions to the Oseen problem −∆v + ∇Π + a∂1 v = g, div v = h, a > 0

(5.6.11)

in the whole of IR3 or in an exterior domain of IR3 related to the behavior at large distances. Exercise 5.16 (i) Prove that (P, O), where P is given in Exercise 5.12 and 3 , Oij (x) = (δij ∆ − ∂i ∂j )ϕO (x; a) O = Oij i,j=1

with

1 ψ ϕO (x; a) = − 4πa

satisfies

as(x) 2

, ψ(z) =

z 0

1−e−τ τ

dτ, s(x) = |x| − x1 ,

∆Oij + ∂j P − aOij = δij δ, ∂j Oij = 0.

(ii) Prove that O(x, a) = aO(ax; 1). (iii) Prove that

vi = −Oij ∗ gj + Pi ∗ h, Π = Pj ∗ gj + h − aP1 , where (g, h) ∈ (C0∞ (IR3 ))4 , satisfy the Oseen system (5.6.11) in IR3 . Hint: See, e.g., (Galdi, 1994a), Chapter VII or (Kraˇcmar et al., 2001). The couple (O, P) is called the fundamental solution of the Oseen operator.

298


Exercise 5.17 (i) Prove that 1 O(x; a) = S(x) + aO(1), ∇O(x; a) = ∇S(x) + a2 O( a|x| ) 1 as a|x| → 0. ∇2 O(x; a) = ∇2 S(x) + a3 O( a2 |x| 2)

(ii) Prove that for a|x| > 1 c |x|(1+s(ax) ,

O(x; a) ≤

1

∇O(x; a) ≤

3 |x| 2

∇2 O(x; a) ≤

ca 2

3

(1+s(ax)) 2

,

ca |x|2 (1+s(ax))2 .

Hint: See, e.g., (Galdi, 1994a), Chapter VII or (Kraˇcmar et al., 2001). Exercise 5.18 Prove that the following estimates hold true: |x|(1 + sγ (ax))O(x; a) ∗ g0,∞,IR3

≤ c|x|2 (1 + s2γ (ax))g0,∞,IR3 ; 0 ≤ γ ≤ 1, 0 < a ≤ 1, 3

3

5

|x| 2 (1 + s 2 γ (ax))O(x; a) ∗ g0,∞,IR3 5

≤ c|x| 2 (1 + s 2 γ (ax))g0,∞,IR3 ; α β

1 5

< γ ≤ 1, 0 < a ≤ 1,

|x| s (ax)T ∗ g0,r ≤ c(r, α, β)|x|α sβ (ax)g0,r ,

1 < r < ∞, 0 ≤ α, 0 ≤ β
n0 ,

| Ω (gn (t) − gm (t))η| ≤ gn (t) − gm (t)−1,s η1,s′ (6.1.5) ≤ ǫη1,s′ , η ∈ D(Ω), t ∈ I.

We thus deduce that for all η ∈ D(Ω), the maps t → Ω gn (t)η form a Cauchy sequence in C 0 (I) which converges in C 0 (I) to a limit lη . We have,

(6.1.6) |lη (t)| ≤ | lim supn→∞ Ω gn (t)η| ≤ cη0,q′ , η ∈ D(Ω), t ∈ I.

Hence the map η → lη (t) is a linear and densely defined bounded operator ′ ′ from Lq (Ω) to IR. Its closure is a continuous linear functional on Lq (Ω) (see Sections 1.4.5.1 and 1.4.7.6) which, due to the Riesz theorem (cf. Section 1.4.6), possesses the representation η → Ω g(t)η with g(t) ∈ Lq (Ω).

Due to (6.1.6), supt∈I g(t)0,q < ∞. Further, we know that the map t → Ω g(t)η belongs to C 0 (I) for all η ∈ D(Ω). The estimate

| Ω (g(t) − g(t′ ))η| ≤ | Ω (g(t) − g(t′ ))η|

′ +| Ω (g(t) − g(t′ ))(η − η)|, η ∈ Lq (Ω), η ∈ D(Ω),

302

SOME MATHEMATICAL TOOLS . . . ′

together with the density of D(Ω) in Lq (Ω) ensures that this map is continuous ′ for any η ∈ Lq (Ω) as well. From the estimate

supt∈I | Ω (gn (t) − g(t))η| ≤ supt∈I | Ω (gn (t) − g(t))η|

′ + supt∈I | Ω (gn (t) − g(t))(η − η)|, η ∈ Lq (Ω), η ∈ D(Ω),

it follows that

g η Ω n

→

′

Ω

gη in C 0 (I) for all η ∈ Lq (Ω).

This is statement (6.1.2). Now, we prove statement (ii). Under the assumptions on q and p, the imbed′ ′ ding W01,p (Ω) ֒→ Lq (Ω) is compact. By duality, the imbedding Lq (Ω) ֒→ W −1,p (Ω) is compact as well. Now, statement (6.1.3) follows again from the abstract Arzelà–Ascoli theorem. The proof of Lemma 6.2 is complete. 2 The proof of the next result is left to the reader as an exercise. Exercise 6.3 Let 1 < q < ∞ and a bounded domain of IRN . Suppose

let Ω be ∞ q 1 that f ∈ L (I, L (Ω)) and ∂t Ω f η ∈ L (I), η ∈ D(Ω). Then there exists g ∈ C 0 (I, Lqweak (Ω)) such that for a.e. t ∈ I, g(t) = f (t) a.e. in Ω. Hint: Use Lemma 1.7 and the Riesz theorem. The last result of this section follows directly from the properties of compact imbeddings (see Section 1.4.10.6). Lemma 6.4 Let Ω be a bounded Lipschitz domain of IRN , N ≥ 2 and 1 < q < ∞. If gn → g in C 0 (I, Lqweak (Ω)), then gn → g strongly in Lp (I, W −1,r (Ω)) for r < q < ∞.18 all 1 ≤ p < ∞, and all 1 ≤ r ≤ NN−1 or NN−1 < r < ∞ if NN+r Proof By assumption gn (t) → g(t) weakly in Lq (Ω), t ∈ I. Since the imbedding Lq (Ω) ֒→ W −1,r (Ω) is compact, the last formula yields gn (t) → g(t) strongly in W −1,r (Ω), t ∈ I. Moreover, in particular, gn → g weakly-∗ in L∞ (I, Lq (Ω)) and therefore gn is bounded in L∞ (I, Lq (Ω)). Due to Exercise 6.1, the sequence supt∈I gn (t)0,q is bounded as well and hence the sequence supt∈I gn (t)−1,r is bounded. We conclude by the Lebesgue dominated convergence theorem (see Theorem 1.16)

T 2 that 0 gn (t) − g(t)p−1,r dt → 0. The proof is thus complete.

q −1,r (Ω) 18 An equivalent formulation is: If g → g in C 0 (I, Lq n weak (Ω)) and if L (Ω) ֒→֒→ W then gn → g strongly in Lp (I, W −1,r (Ω)) for all 1 ≤ p < ∞.

SOME AUXILIARY RESULTS FROM FUNCTIONAL ANALYSIS

6.1.2

303

The time and space mollifiers

Lemma 6.5 (i) Let 1 ≤ p < ∞, 1 ≤ q ≤ ∞, N ≥ 1 and f ∈ Lp (IR, Lq (IRN )) or f ∈ C 0 (IR, Lq (IRN )). If {Sǫ }ǫ>0 is a family of one-dimensional mollifiers

Sǫ (f )(x, t) := IR ωǫ (t − τ )f (x, τ ) dτ, where ωǫ is defined in Section 1.3.4.4, then

Sǫ (f ) ∈ C ∞ (IR, Lq (IRN )),

Sǫ f → f strongly in Lp (IR, Lq (IRN )), f ∈ Lp (IR, Lq (IRN )), Sǫ f → f strongly in C 0 (I, Lq (IRN )), f ∈ C 0 (IR, Lq (IRN )),

where I is a bounded interval of IR. (ii) Let 1 ≤ p < ∞, 1 ≤ q < ∞, N ≥ 1 and f ∈ Lp (IR, Lq (IRN )). If {Sǫ }ǫ>0 is a family of N-dimensional mollifiers

Sǫ (f )(x, t) := IRN ωǫ (x − y)f (y, t) dy, where ωǫ is defined in Section 1.3.4.4, then

Sǫ (f ) ∈ Lp (IR, C ∞ (IRN )),

Sǫ f → f strongly in Lp (IR, Lq (IRN )), f ∈ Lp (IR, Lq (IRN )). Proof For part (i), see formulae (4.2.11) and (4.2.24) in Section III.4 of (Amann, 1995), or modify the classical proofs in Section 2.5 of (Kufner et al., 1977) to treat functions with values in Lq (IRN ). This is an easy exercise. The proof of part (ii). The fact that Sǫ (f ) ∈ Lp (IR, C ∞ (IRN )) is easily seen from general theorems about differentiation of integrals dependent on a parameter (see Theorem 1.11). To show convergence, we realize that [Sǫ (f )](t) → f (t) strongly in Lq (Ω), for a.e. t ∈ I (cf. Section 1.3.4.4) and that [Sǫ (f )](t)0,q,IRN ≤ f (t)0,q,IRN (a consequence of Young’s inequality for convolutions, see Section 1.3.4.3). With these two facts at hand, we can use the Lebesgue dominated convergence theorem (cf. Theorem 1.16) which implies [Sǫ (f )]0,q,IRN → f 0,q,IRN strongly in Lp (I). This yields the required result (cf. Section 1.4.5.22). The proof of part (ii) is thus complete. 2

304

6.1.3

SOME MATHEMATICAL TOOLS . . .

Local weak compactness in unbounded domains

The next lemma concerns the method of “invading domains”. We shall often use it when dealing with unbounded domains. Lemma 6.6 Let {fn }, fn ∈ Lp (I, Lqloc (IRN )) (1 < p, q ≤ ∞, N ≥ 1), be a sequence such that fn Lp (I,Lq (BM )) ≤ K(M ), M = M0 , M0 + 1, . . .. Then there exists a subsequence {n′ } ⊂ {n} such that fn′ → f weakly-∗ in Lp (I, Lq (BR )), R > 0. Proof By virtue of the Banach–Alaoglu theorem (cf. Sections 1.4.5.25 and 0 1.4.5.26), there exists {n0i }∞ weakly-∗ in i=1 ⊂ {n} such that fn0i →i→∞ f p q 1 ∞ 0 L (I, L (BM0 )), and there also exists {ni }i=1 ⊂ {ni } such that fn1i →i→∞ f 1 weakly-∗ in Lp (I, Lq (BM0 +1 )). Moreover f 1 = f 0 a.e. in BM0 × I. Further k we proceed by induction. If k ∈ IN , there exists {nk+1 }∞ i=1 ⊂ {ni } such that i k+1 p q weakly-∗ in L (I, L (BM0 +k+1 )), and moreover f k+1 = f k fnk+1 →i→∞ f i a.e. in BM0 +k × I. We set f (x, t) = f k (x, t) provided x ∈ BM0 +k . Choosing n′i = nii , we have fn′i →i→∞ f weakly-∗ in Lp (I, Lq (BM0 +k )) for any k ∈ IN . This completes the proof. 2 6.2

Renormalized solutions of the continuity equation

In this section we shall list several results concerning the continuity equation which will be needed to build the mathematical theory of nonstationary compressible barotropic Navier–Stokes equations. The topic is closely related to that treated in Section 3.1. The first part evokes a “nonstationary version” of DiPerna’s and Lions’ generalization to Friedrich’s lemma about commutators (see (DiPerna and Lions, 1989)) which is needed as a basic tool in proofs of the majority of results of this section. In the second part, we deal with the prolongation of the continuity equation from its domain to the whole space. The main result in this part stated in Lemma 6.8 is taken over from (Feireisl et al., 2001). In Section 6.2.3, we deal with the renormalized continuity equation, and the last section (Section 6.2.4) is concerned with the strong continuity of weakly continuous renormalized solutions to the continuity equation. These results follow from the DiPerna–Lions transport theory (DiPerna and Lions, 1989). We present their proofs only for the sake of completeness. The reader can consult Section 7.3 to get an overall idea of how these topics are needed and where they are used in the existence proof. 6.2.1

Friedrichs’ lemma about commutators

Lemma 6.7 Suppose that N ≥ 2. Let 1 ≤ q, β ≤ ∞, (q, β) = (1, ∞), Let 1 ≤ α ≤ ∞ and α1 + p1 ≤ 1. Suppose that 1,q ρ ∈ Lα (I, Lβloc (IRN )), u ∈ Lp (I, (Wloc (IRN ))N ).

1 q

+ β1 ≤ 1.

RENORMALIZED SOLUTIONS

Then

305

Sǫ (u · ∇ρ) − u · ∇Sǫ (ρ) → 0 strongly in Ls (I, Lrloc (IRN )), 1 s

=

1 α

+ p1 ,

r ∈ [1, q) if β = ∞, q ∈ (1, ∞] 1 1 1 + ≤ ≤ 1 otherwise, β q r

where Sǫ is the usual mollifier in IRN (see Section 1.3.4.4) and u · ∇ρ := div (ρu) − ρdiv u.

Proof (i) We have

Sǫ (u · ∇ρ) − u · ∇Sǫ (ρ), ϕ =

T

−

where

0

IRN

T

0

Iǫ (x, t)ϕ(x, t) dx dt

IRN

ρ(y, t)[u(y, t) − u(x, t)] · ∇ωǫ (x − y) dy,

Jǫ (x, t) = IRN ωǫ (x − y)[ρdiv u](y, t) dy,

Iǫ (x, t) =

(6.2.1)

Jǫ (x, t)ϕ(x, t) dx dt,

IRN

(6.2.2)

cf. the proof of Lemma 3.1. As in (3.1.4), we set   if β = q = ∞  ∈ [1, ∞) if β = q = ∞  r0 r0 q if β = ∞, q ∈ (1, ∞) β˜ = q−r if β = ∞, q ∈ (1, ∞) r0 : ∈ (1, q) 0  1 = 1 + 1 otherwise,  β if 1 ≤ β < ∞. r0 β q

By Lemma 6.5 it is evident that

Jǫ → ρdiv u strongly in Ls (I, Lr0 (BR )), R > 0.

(6.2.3)

In the sequel, in parts (ii) and (iii) we shall prove that Iǫ → ρdiv u strongly in Ls (I, Lr0 (BR )), R > 0.

(6.2.4)

If this is done the proof of Lemma 6.7 is finished. (ii) Repeating the argument of part (ii) of Lemma 3.1, we arrive at the inequality ˜ Iǫ (t)0,r ,B ≤ c(R, β)ρ(t) ∇u(t)0,q,B for a.a. t ∈ (0, T ). ˜ 0

R

0,β,BR+1

R+2

(6.2.5) (iii) Thanks to the last estimate, in the same way as in part (iii) of the proof of Lemma 3.1, we show that 0 Iǫ (t) → [ρdiv u](t) in Lrloc (IRN ) for a.a. t ∈ (0, T ),

see (3.1.6). Moreover, by H¨ older’s inequality, we obtain

s ∇u(t)s0,q,BR+2 dt ≤ ρsLα (I,Lβ˜ (B )) ∇usLp (I,Lq (BR+2 )) . ρ(t)0,β,B ˜ I R+1

R

Due to the last two facts, we can conclude by using the Lebesgue dominated convergence theorem that Iǫ → ρdiv u in Ls (I, Lr0 (BR ))

completing the proof of Lemma 6.7.

2

306

6.2.2


Continuity equation and its prolongation

Lemma 6.8 Let Ω be a bounded Lipschitz domain in IRN , N ≥ 2. Let ρ ∈ L2 (QT ), u ∈ L2 (I, (W01,2 (Ω))N ) and f ∈ L1 (QT ) satisfy ∂t ρ + div (ρu) = f in D′ (QT ). Then, prolonging (ρ, u, f ) by (0, 0, 0) outside Ω,

(6.2.6)

19

∂t ρ + div (ρu) = f in D′ (IRN × I).

(6.2.7)

Proof The proof of formula (6.2.7) is equivalent to showing − We have

T

0

IRN

ρ∂t η −

T

0

T

0

and

IRN

0

IRN

ρu · ∇η =

T

0

IRN

f η, η ∈ D(IRN × I).

T

T

f η = 0 IRN f ηΦm + 0 IRN f (1 − Φm )η,

T

T

ρ∂t η = 0 IRN ρ∂t (ηΦm ) + 0 IRN ρ(1 − Φm )∂t η

IRN

T

0

+

T

T

0

IRN

ρu · ∇η =

T

ρu · ∇(ηΦm )

T

ρu(1 − Φm ) · ∇η − 0 IRN ρu · ∇Φm η

IRN

0

IRN

for all m ∈ IN with Φm given by (3.1.10). By virtue of (3.1.11)

T

0

T

ρ(1 − Φm )∂t η → 0, 0 IRN f (1 − Φm )η → 0,

T

ρu(1 − Φm ) · ∇η → 0 as m → ∞. 0 IRN

IRN

It is therefore enough to prove that

T

0

IRN

ρu · ∇Φm η → 0 as m → ∞.

(6.2.8)

Integral (6.2.8) is bounded by 2 sup(x,t)∈QT |η(x, t)|

T 0

u ρ0,2,(supp∇Φm )×I dist(x,∂Ω) 0,2,QT

u ≤ cρ0,2,(supp∇Φm )×I dist(x,∂Ω) 0,2,QT .

In accordance with (3.1.12) and since ρ ∈ L2 (QT ), ρ0,2,(supp∇Φm )×I → 0 (cf. u Section 1.1.17). Since u ∈ L2 (I, (W01,2 (Ω))N ), we obtain, dist(x,∂Ω) ∈ (L2 (QT ))N by the Hardy inequality (cf. Section 1.3.5.9). Therefore formula (6.2.8) holds true. This completes the proof. 2 19 The

same statement holds true if we prolong (ρ, u, f ) by (c, 0, 0), where c ∈ IR, as well.


6.2.3 Renormalized continuity equation 6.2.3.1 Renormalized continuity equation with b ∈ C 1 set

307

As in Section 3.1.3, we

b ∈ C 0 ([0, ∞)) ∩ C 1 ((0, ∞)), |b′ (t)| ≤ ct−λ0 , t ∈ (0, 1], λ0 < 1

(6.2.9)

with growth conditions at infinity |b′ (t)| ≤ ctλ1 , t ≥ 1, where c > 0, −1 < λ1 < ∞.

(6.2.10)

We have the following nonsteady version of Lemma 3.3. Lemma 6.9 Let N ≥ 2. Let 2 ≤ β < ∞, and λ1 ≤

β 2

(6.2.11)

− 1.

1,2 (IRN ))N ) and Let ρ ∈ Lβ (I, Lβloc (IRN )), ρ ≥ 0 a.e. in IRN × I, u ∈ L2 (I, (Wloc ′ f ∈ Lz (I, Lzloc (IRN )), where z = λβ1 if λ1 > 0 and z = 1 if λ1 ≤ 0. Suppose that ∂t ρ + div (ρu) = f in D′ (IRN × I). (6.2.12)

(i) Then for any function b ∈ C 1 ([0, ∞)) satisfying (6.2.10) and (6.2.11)

∂t b(ρ) + div (b(ρ)u) + {ρb′ (ρ) − b(ρ)}div u = f b′ (ρ) in D′ (IRN × I). (6.2.13)

(ii) If f = 0, then (6.2.13) holds with any b satisfying (6.2.9)–(6.2.11). Proof To start, we shall suppose that λ1 = 0.

(6.2.14)

We shall explain in Exercise 6.12 how to relax this condition up to (6.2.11). Under the additional hypothesis (6.2.14), the proof is similar as that of Lemma 3.3. For the reasons explained in that proof, we shall concentrate on part (ii), leaving the proof of part (i) to the interested reader as an exercise. Regularizing (6.2.12) over the space variables by the usual mollifier (notation Sǫ , ǫ > 0), we get ∂t Sǫ (ρ) + div (Sǫ (ρ)u) = rǫ (ρ, u) a.e. in IRN × I,

(6.2.15)

where rǫ (ρ, u) = div (Sǫ (ρ)u) − div (Sǫ (ρu)).

Due to Lemma 6.7, rǫ → We multiply equation obtain

0 in L (I, Lrloc (IRN )), 1r = β1 + 21 . (6.2.15) by b′(h) (Sǫ (ρ)), where b(h) (·)

(6.2.16)

r

= b(h + ·) and we

∂t b(h) (Sǫ (ρ)) + div (b(h) (Sǫ (ρ))u) +{Sǫ (ρ)b′(h) (Sǫ (ρ)) − b(h) (Sǫ (ρ)}div u = rǫ b′(h) (Sǫ (ρ)) a.e. in IRN × I. (6.2.17) Now, we pass to the limit ǫ → 0+ . Due to the properties of mollifiers, Sǫ (ρ) → ρ in Lβ (I, (Lβloc (IRN )), and therefore a.e. in IRN ×I for an appropriately

308


chosen subsequence. Now, by Vitali’s convergence theorem (see Theorem 1.18), by virtue of growth conditions (6.2.10), (6.2.14), we have b(h) (Sǫ (ρ)) → b(h) (ρ) and Sǫ (ρ)b′(h) (Sǫ (ρ))−b(h) (Sǫ (ρ)) → ρb′(h) (ρ)−b(h) (ρ) in Lploc (IRN ×I), 1 ≤ p < 2. Since these sequences are bounded in L2 (Ω′ × I) with Ω′ any bounded domain in IRN , both the last mentioned limits are limits in L2weak (Ω′ × I) as well. This gives the convergence to the corresponding terms at the left-hand side of equation (6.2.17). The L1 -norm of the right-hand side over Ω′ × I is bounded by

T ′ [b(h) (Sǫ (ρ))0,r′ ,Ω′ rǫ 0,r,Ω′ ] ≤ crǫ Lr (I,Lr (Ω′ )) and therefore tends to zero. 0 Hence, we have ∂t b(h) (ρ)+div (b(h) (ρ)u)+{ρb′(h) (ρ)−b(h) (ρ)}div u = 0 in D′ (IRN ×I). (6.2.18) Now, we proceed to the limit h → 0+ . Similarly as in (3.1.26), we observe that for any k > 1 and for any Ω′ a bounded domain in IRN , |{(x, t); ρ ≥ k} ∩ Ω′ × I| ≤ k −β ρβ0,β,{ρ≥k}∩(Ω′ ×I) ≤ k −β ρβ0,β,Ω′ ×I . (6.2.19) We find as in the proof of Lemma 3.3 that, for example, the third term at the left-hand side of (6.2.18), namely IRN ×I {ρb′(h) (ρ) − b(h) (ρ)}div uψ (ψ ∈

D(IRN × I), suppψ ⊂ Ω′ × I), can be written as the sum of IRN ×I {ρb′(h) (ρ) −

b(h) (ρ)}div uψ1{ρ≤k} and IRN ×I {ρb′(h) (ρ) − b(h) (ρ)}div u ψ1{ρ>k} , k > 0. The

first integral tends to IRN ×I {ρb′ (ρ) − b(ρ)}div uψ1{ρ≤k} due to the Lebesgue β

dominated convergence theorem. The second one is majorized by ck 1− 2 (1+ β 2

div u0,2,Ω′ ×I ) ρ0,β,(Ω′ ×I)∩{ρ>k} supΩ′ ×I |ψ|. It tends to zero as k → ∞ (cf.

Section 1.1.17 and formula (6.2.19)). The sequence IRN ×I (ρb′ (ρ) − b(ρ))div uψ

1{ρ≤k} tends to IRN ×I (ρb′ (ρ) − b(ρ))div uψ by the same argument. The analysis of the convergence in the other terms is similar. 2 A nonsteady version of Exercise 3.4 reads: Exercise 6.10 (i) Let Ω, ρ, u and f satisfy the assumptions of Lemma 6.8. Suppose that ∂t ρ + div (ρu) ≥ f in D′ (Ω × I). Then, prolonging ρ, u and f by zero outside Ω, we have ∂t ρ + div (ρu) ≥ f in D′ (IRN × I).

(6.2.20)

(ii) Let β, Ω, ρ, u, f and b satisfy the assumptions of Lemma 6.9 and moreover let b be a nondecreasing function. Suppose that (6.2.20) holds. Then ∂t b(ρ) + div (b(ρ)u) + {ρb′ (ρ) − b(ρ)}div u ≥ f b′ (ρ) in D′ (IRN × I). (6.2.21)


309

6.2.3.2 Renormalized continuity equation with b′ having a jump We set as in (3.1.29) bk (s) =

b(s) if s ∈ [0, k) b(k) if s ∈ [k, ∞),

where b ∈ C 1 ([0, ∞)).

(6.2.22)

First, we observe that (bk )′+ (ρ(x, t)) =

b′k (ρ(x, t) if (x, t) ∈ {(y, s); ρ(y, s) = k} 0 if (x, t) ∈ {(y, s); ρ(y, s) = k},

(6.2.23)

where (bk )′+ is the right derivative of bk , see (3.1.31). Second, we observe that, similarly as in Auxiliary lemma 3.6, under the assumptions of Lemma 6.9, we have kdiv u = f a.e. in {(x, t); ρ(x, t) = k}. (6.2.24) With these observations at hand, copying the proof of Lemma 3.5, we show:

Lemma 6.11 Let β, ρ, u, satisfy the assumptions of Lemma 6.9 and f ∈ L1loc (IRN × I). Then (i) ∂t bk (ρ) + div (bk (ρ)u) + {ρ(bk )′+ (ρ) − bk (ρ)}div u (6.2.25) = f (bk )′+ (ρ) in D′ (IRN × I), k > 0

with any function bk satisfying (6.2.22). (ii) If f = 0, then the condition on b in (6.2.22) can be relaxed to (6.2.9).

Exercise 6.12 Let β > 1, and suppose that ρ, u, f and b satisfy the assumptions of Lemma 6.9. Then equation (6.2.25) yields (6.2.13) as k → ∞. Hint: Use the Lebesgue dominated convergence theorem. The statement of Exercise 6.12 completes the proof of Lemma 6.9 for λ1 satisfying (6.2.11). If the vector field u in Lemma 6.9 is smoother, the growth conditions imposed on b can be relaxed. The proof of the following statement is left to the reader as an exercise. 1,∞ (IR3 )) Exercise 6.13 Let β ∈ (1, ∞), ρ ∈ Lβ (I, Lβloc (IR3 )), u ∈ L∞ (I, Wloc and f = 0 (for simplicity). Then the statement of Lemma 6.9 holds with any function b satisfying (6.2.9) and (6.2.10), where λ1 + 1 ≤ β. Hint: Prove Lemma 6.9 with new β, ρ, u and with λ1 = 0. Prove Lemma 6.11 with new β, ρ, u. Use Exercise 6.12.

Remark 6.14 If (6.2.7) holds with f = 0 and some (ρ, u) ∈ (L1 (QT ))N +1 , ρu ∈ (L1 (QT ))N , then

ρ(t) = Ω ρ(s) for a.e. t, s ∈ I. Ω

Thus, the validity of the continuity equation in D′ (IRN × I) does imply the conservation of mass. In other words, Lemma 6.8 shows that the validity of

310


only (6.2.6) guarantees conservation of Ω ρ(t) provided ρ ∈ L2 (QT ) and u ∈ L2 (I, (W01,2 (Ω))N ). If ρ belongs only to Lp (QT ) with some 1 ≤ p < 2 and, of course, u ∈ 1,2 2 N L (I,

(W0 (Ω)) ), then the sole equation (6.2.6) does not imply conservation of Ω ρ(t). In other words, the hypothesis β ≥ 2 in Lemma 6.8 is essential also from the physical point of view: if it is violated, the validity of the continuity equation in D′ (Ω × I) may not imply the conservation of mass. This is seen from the following counterexample in one dimension, taken over from (Feireisl and Petzeltov´ a, 2000). Let Ω = (0, 1) and

x 1 u(x) = [x(1 − x)]α , ρ(x, t) = u(x) h(t − 0 u1 ), 12 < α < 1, where h ∈ C 1 (IR) is an arbitrary function satisfying h(s) = 0 for all s ≤ 0. One checks easily that u ∈ W01,2 ((0, 1)), ρ ∈ C([0, T ], Lp ((0, 1)), 1 ≤ p
0. We get ∂t Sǫ (˜ ρθ ) + div (Sǫ (˜ ρθ )u) = (1 − θ)Sǫ (˜ ρθ div u) + rǫ (˜ ρθ , u) in D′ (IRN × I), (6.2.31) where rǫ is defined in (6.2.16). By virtue of (6.2.30), by using classical results for integrals dependent on a parameter (cf. Theorem 1.11), we have Sǫ (˜ ρθ ) ∈ C 0 (Ω × I).

(6.2.32)

ρθ (t))0,q,IRN and By Young’s inequality for convolutions Sǫ (˜ ρθ (t))0,q,IRN ≤ ˜ therefore there exists ǫ0 > 0, such that ρθ (t)0,q < ∞, 1 ≤ q ≤ sup sup Sǫ (˜

ǫ∈(0,ǫ0 ) t∈I

β . θ

(6.2.33)

t∈I

(6.2.34)

By the standard properties of mollifiers Sǫ (˜ ρθ (t)) → ρ˜θ (t) strongly in Lq (Ω), 1 ≤ q ≤ and

β θ,

2β

Sǫ (˜ ρθ div u) → ρ˜θ div u strongly in L2 (I, L 2θ+β (Ω)).

(6.2.35)

We recall that, by Lemma 6.7, 2β

rǫ (˜ ρθ , u) → 0 strongly in L2 (I, L 2θ+β (Ω)).

(6.2.36)

We now apply Lemma 6.9 with b(s) = (s + 1)2 to equation (6.2.31) and obtain ρθ ) + 1]2 + div ([Sǫ (˜ ρθ ) + 1]2 u) + {[Sǫ (˜ ρθ )]2 − 1}div u ∂t [Sǫ (˜ ρθ ) + 1)Sǫ (˜ ρθ div u) = 2(1 − θ)(Sǫ (˜ θ

θ

′

(6.2.37) N

ρ ) + 1)rǫ (˜ ρ , u) in D (IR × I). +2(Sǫ (˜

ρθ )]2 η}ǫ>0 , for all η ∈ D(Ω), is uniBy virtue of (6.2.33), the sequence { Ω [Sǫ (˜ formly bounded on I, and, thanks to (6.2.37) together with (6.2.26), (6.2.33), (6.2.35), (6.2.36), it is also uniformly continuous on I. Hence, due to the Arzelà– Ascoli theorem (see Theorem 1.70) and in accordance with (6.2.34),

θ2

(6.2.38) [S (˜ ρθ )]2 η → Ω [˜ ρ ] η in C 0 (I), η ∈ D(Ω). Ω ǫ Now, relations (6.2.30) and (6.2.38) combined with a density argument yield ρ˜θ ∈ C 0 (I,L2 (Ω)) (cf. Section 1.4.5.22). This means by the interpolation of Lebesgue spaces (see Theorem 1.49) that ρ˜∈C 0 (I, Lp (Ω)) for all 1 ≤ p < β. Now, due to (6.2.26), identity (6.2.29) holds, in fact, for all t ∈ I. This finishes the proof. 2

7 WEAK SOLUTIONS FOR NONSTEADY NAVIER–STOKES EQUATIONS OF COMPRESSIBLE BAROTROPIC FLOW This chapter is devoted to the weak solvability of nonstationary compressible Navier–Stokes equations in the barotropic regime, and to the investigation of some of their qualitative properties. In Section 7.1 we formulate the problem in bounded and exterior domains with homogenous Dirichlet boundary conditions and state the main results. In the next section (Section 7.2), we study the continuity with respect to time of the linear momentum and of the total energy. The main ideas to prove existence and the main difficulties of the proof are explained in Section 7.3. Then, in Section 7.4 we introduce the chain of approximations which we use to solve the original problem. Section 7.5 is devoted to the investigation of the weak continuity of the effective viscous flux, the property which plays a key role in the existence proof. In the next section (Section 7.6), we recall several results for the continuity equation with dissipation which are needed throughout the proofs. Then, in Sections 7.7–7.9, the systems on different levels of approximations are solved and estimates independent of the parameters of the approximations are derived. In Section 7.10, the complete system of the viscous barotropic gas in a bounded domain with homogenous Dirichlet boundary conditions is solved. Section 7.11 is devoted to the same problem in an exterior domain. Finally, in Section 7.12, other problems are discussed such as nonmonotone pressure laws, nonsmooth boundaries, nonhomogenous Dirichlet boundary conditions and nonzero outflow/inflow conditions in a bounded domain, flows past an obstacle in an exterior domain, slip boundary conditions.

7.1

Formulation of problems and main results

In this section we formulate the problems which we are going to solve in the sequel. We define weak solutions, renormalized weak solutions, finite energy weak solutions and bounded energy weak solutions in bounded and unbounded domains and explain how these definitions are motivated. Then, we investigate relations between bounded energy and finite energy weak solutions. Finally, we formulate two existence theorems. The first one is concerned with the homogenous Dirichlet problem in bounded domains and the second one deals with the homogenous Dirichlet problem in exterior domains. They will be proved in Sections 7.4 and 7.7–7.11. We shall discuss possible generalizations to domains with nonsmooth boundaries, to other unbounded domains, to nonmonotone pressure laws and to other boundary conditions in Section 7.12. 312


7.1.1

313

Definition of weak solutions

7.1.1.1 Homogenous Dirichlet problem We shall study the following system of equations: ∂t (ρu) + ∂j (ρuuj ) − µ∆u − (µ + λ)∇div u + ∇p(ρ) = ρf + g in Ω × I, (7.1.1) (7.1.2) ∂t ρ + div (ρu) = 0 in Ω × I, where Ω is a domain of IR3 and I = (0, T ), T > 0, ρ(x, t), u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t)), x ∈ Ω, t ∈ I are unknown quantities and f (x, t) = (f 1 (x, t), f 2 (x, t), f 3 (x, t)), g(x, t) = (g 1 (x, t), g 2 (x, t), g 3 (x, t)), x ∈ Ω, t ∈ I are given vector fields. The function p ∈ C 0 ([0, ∞))

(7.1.3)

which appears in (7.1.1) is a given function (the so-called constitutive law for pressure). In the sequel, we suppose that p(s) = sγ , γ > We also assume that20

3 . 2

(7.1.4)

2 µ > 0, λ + µ ≥ 0. 3

(7.1.5)

u(x, t) = 0, (x, t) ∈ ∂Ω × I

(7.1.6)

ρ(x, 0) = ρ0 (x), (ρu)(x, 0) = q 0 (x),

(7.1.7)

Further, we suppose and where q 0 has to be at least such that q 0 (x) = 0 whenever ρ0 (x) = 0. If Ω is unbounded, one reasonable possibility is to impose the conditions u(x, t) → 0, ρ(x, t) → ρ∞ as |x| → ∞, (x, t) ∈ Ω × I,

(7.1.8)

where ρ∞ is a given nonnegative constant. Of course, for the initial condition ρ0 , we must assume that the compatibility condition of type ρ0 (x) → ρ∞ , q 0 (x) → 0 as |x| → ∞, x ∈ Ω is satisfied. 20 See

footnote at formula (4.1.3)

314

WEAK SOLUTIONS FOR NONSTEADY NAVIER–STOKES EQUATIONS

For the physical background to these conditions and equations see Section 1.2. In Section 7.12, we consider also more general situations: more general (possibly nonmonotone) pressure, other boundary conditions than boundary conditions (7.1.6) (such as, e.g., slip boundary conditions or nonhomogenous Dirichlet boundary conditions) and, in the case of unbounded domains, nonzero velocity at infinity instead of (7.1.8), namely u(x, t) → a∞ ∈ IR3 , ρ(x, t) → ρ∞ as |x| → ∞, (x, t) ∈ Ω × I.

(7.1.9)

7.1.1.2 Renormalized continuity equation and the energy inequality In this section, we perform some formal calculations. To this end, we suppose that Ω is “sufficiently” smooth and (ρ, u) are smooth solutions satisfying equations (7.1.1), (7.1.2), boundary conditions (7.1.6) and ρ > 0 throughout QT . If Ω is unbounded, we suppose that (7.1.8) holds with “sufficient” rate of decay. Multiplying equation (7.1.2) by b′ (ρ), where b ∈ C 1 ((0, ∞)), we obtain the so-called renormalized continuity equation ∂t b(ρ) + div (b(ρ)u) + (ρb′ (ρ) − b(ρ))div u = 0. Energy equation (1.2.89) integrated over Ω gives21

d 2 2 dt [E(ρ, q)] + µ Ω |∇u| + (µ + λ) Ω |div u| = Ω (ρf + g) · u in I,

where

E(ρ, q) = Ω [ 12 ρ|u|2 + P (ρ)], q = ρu,  sγ  γ−1 / 1 γ P (s) = [sγ − γsργ−1 ∞ + (γ − 1)ρ∞ ] γ−1  . if Ω is unbounded and ρ∞ > 0

(7.1.10)

(7.1.11)

(7.1.12)

Due to the lower weak semicontinuity of norms at the right-hand side of (7.1.11), for weak solutions one expects an inequality rather than an identity, namely

d 2 2 dt [E(ρ, q)] + µ Ω |∇u| + (µ + λ) Ω |div u| dx ≤ Ω (ρf + g) · u in I. (7.1.13) Integrating (7.1.13) over (0, t), t ∈ I, we get

t

t

E(ρ(t), q(t)) + µ 0 Ω |∇u|2 + (µ + λ) 0 Ω |divu|2 (7.1.14)

t

≤ E(ρ(0), q(0)) + 0 Ω (ρf + g) · u. 21 As we already observed in Section 1.2.18, equation (7.1.11) is not an independent identity for barotropic flows. It can also be obtained by scalar multiplying equation (7.1.1) by u and integrating it over Ω. In order to arrive at (7.1.11), one has to employ integration by parts several times and to use identity (7.1.2). If Ω is unbounded, one should integrate over Ω ∩ BR and discuss how integrals over ∂BR ∩ Ω, coming from integration by parts, disappear as R → ∞ (see (4.1.10)). One also has to add to γ ργ d the usual P (ρ) = γ−1 in (1.2.89) the term a = −γργ−1 ∞ ρ + (γ − 1)ρ∞ (of course, dt Ω a = 0), to force the new P to be integrable and nonnegative.


315

In Section 7.12, in different situations, we shall deal with different forms of the renormalized continuity equation and with more general forms of the energy inequality. Remark 7.1 In fact, if Ω is a bounded domain, P can be determined up to a linear function cρ, c ∈ IR, see (1.2.84), (1.2.85). In the case of general pressure p(ρ), p ∈ C 0 ([0, ∞)), P takes the form

ρ (7.1.15) P (ρ) = ρ 1 p(s) s2 + c0 ρ,

where the constant c0 is chosen in such a way that P (ρ) ≥ 0 (for example, 1 in (7.1.12)). Of course, with this P and in the case of smooth solutions, c0 = γ−1 the “pointwise” energy equation (1.2.89) and consequently the “integral” energy equation (7.1.11) hold. In the case of an unbounded domain and again for smooth solutions, the function P from (7.1.12) guarantees P (ρ) ≥ 0 and the “integral” energy equation (7.1.11). Due to the presence of the term ργ∞ in P , the “pointwise” energy equation (1.2.89) with this P may be violated.

Remark 7.2 In physics, Epot (ρ) = Ω P (ρ) is called the potential energy of the fluid in the domain Ω, Ekin (ρ, q) = 12 Ω ρ|u|2 (where q = ρu) is called the kinetic energy of the fluid in the domain Ω and E(ρ, q) is the total energy of the fluid in the domain Ω (cf. Section 1.2.7). For smooth solutions with positive density,

2 the kinetic energy is equal to 21 Ω |q|ρ . We shall see in Section 7.2 that for weak solutions, the latter formula written in the form

2 Ekin (ρ, q) = 21 Ω |q|ρ 1{x; ρ(t)>0} is more adequate. From now on, we set

E(ρ(t), q(t)) = Ekin (ρ(t), q(t)) + Epot (ρ(t))

2 = Ω 12 |q(t)| ρ(t) 1{x; ρ(t)>0} + Ω P (ρ(t)), q = ρu,

(7.1.16)

where P is given in (7.1.12).

These considerations motivate the definitions of different types of weak solutions. They will be introduced in the next section. 7.1.1.3 Definition of weak solutions In the sequel we explain what we mean by weak solution of problem (7.1.1)–(7.1.8). Definition 7.3 Let the couple (ρ0 , u0 ) satisfy ρ0 ∈ L1loc (Ω), P (ρ0 ) ∈ L1 (Ω), ρ0 ≥ 0 a.e. in Ω , q 0 ∈ (L1loc (Ω))3 such that

|q 0 |2 ρ0 1{ρ0 >0}

∈ L1loc (Ω)

and such that q 0 (x) = 0 whenever x ∈ {ρ0 = 0},

(7.1.17)

316


where the quantity P is defined in (7.1.12). Further, suppose that ′

6γ

f ∈ L1 (I, (Lγloc (Ω))3 ) ∩ L2 (I, (L 5γ−6 (Ω))3 )∩ 6

6

L2 (I, (L 5 (Ω))3 ), g ∈ L2 (I, (L 5 (Ω))3 ).

(7.1.18)

(i) A couple (ρ, u) is called a weak solution of problem (7.1.1)–(7.1.8) if: ρ ∈ Ls (I, Lsloc (Ω)) for some γ ≤ s ≤ ∞,

P (ρ) ∈ L∞ (I, L1 (Ω)), ρ ≥ 0 a.e. in Ω × I, u∈

L2 (I, (D01,2 (Ω))3 ),

(7.1.19)

ρ|u|2 ∈ L∞ (I, L1 (Ω)).

Equation (7.1.1) holds in (D′ (QT ))3 .

(7.1.20)

Equation (7.1.2) holds in D′ (QT ).

limt→0

+ Ω ρ(t)ψ = Ω ρ0 ψ, ψ ∈ D(Ω), 3 limt→0+ Ω ρu(t) · φ = Ω q 0 · φ, φ ∈ (D(Ω)) .

(7.1.21) (7.1.22)

Equation (7.1.2) holds in D′ (IR3 × I) provided (ρ, u) is prolonged by zero outside Ω.

(7.1.23)

Equation (7.1.10) holds in D′ (IR3 × I) provided (ρ, u) is prolonged by zero outside Ω for any function b belonging to (6.2.9), (6.2.10), (6.2.11), where β = γ.

(7.1.24)

(ii) A couple (ρ, u) is called a renormalized weak solution of problem (7.1.1)– (7.1.8) if in addition to (7.1.19)–(7.1.22) we have:

(iii) A couple (ρ, u) is called a finite energy weak solution of problem (7.1.1)– (7.1.8) if in addition to (7.1.19)–(7.1.22), we have: Inequality (7.1.13) with E(ρ, q) defined in (7.1.16) holds in D′ (I).

(7.1.25)

(iv) A couple (ρ, u) is called a bounded energy weak solution of problem (7.1.1)– (7.1.8) if in addition to (7.1.19)–(7.1.22) we have:

2 The quantity E0 = Ω [ 12 |qρ00| 1{ρ0 >0} + P (ρ0 )] is finite and inequality (7.1.14) with E defined by (7.1.16) and with E0 in place of E(ρ(0), q(0)) holds a.e. in I.

(7.1.26)

7.1.1.4 Finite energy and bounded energy weak solutions In this section, we investigate the relation between finite energy and bounded energy weak solutions. We also explain one way to define the energy of weak solutions for all t ∈ I (and not only for almost all t ∈ I as one would expect under the regularity of weak solutions). We start with the following auxiliary statement which is left to the reader as an exercise.


317

Exercise 7.4 Let E be a nonnegative function belonging to L∞ (I) and let A, D ∈ L1 (I). Then d ′ (7.1.27) dt E + D ≤ A in D (I) if and only if E(t) − E(s) +

t s

D≤

t s

A for a.a. t, s ∈ I, t ≥ s.

(7.1.28)

If E ∈ L∞ (I) and E ≥ 0, we have 0≤

1 h

t+h t

E ≤ ess sups∈I E(s) < ∞, t ∈ [0, T ), h > 0, t + h ∈ I.

From the theorem about the Lebesgue points (cf. Section 1.1.15), we know that lim inf h→0+

1 h

t+h t

E = E(t) for a.a. t ∈ I.

Taking into account this fact and Exercise 7.4, the reader can prove the following statement. Exercise 7.5 (i) Let E ∈ L∞ (I), E ≥ 0. Then the quantity = lim inf h→0+ E(t)

1 h

t+h t

(7.1.29)

E

exists for all t ∈ I, it is nonnegative, bounded from above by ess sups∈I E(s) and equal almost everywhere in I to E. (ii) If A, D ∈ L1 (I) and if (7.1.27) holds, then d dt E +

D ≤ A in D′ (I).

(7.1.30)

(iii) Under the same assumptions on A, D, if (7.1.28) holds, then

− E(s) + t D ≤ t A, s, t ∈ [0, T ), s ≤ t. E(t) s s

(7.1.31)

(iv) E is a left lower semicontinuous function on (0, T ] and a right upper semicontinuous function on [0, T ), cf. Section 1.4.2.7. Remark 7.6 In view of Exercises 7.4 and 7.5, any finite energy weak solution defined in point (iii) of Definition 7.3, satisfies

+ t D ≤ t A + lim inf s→0+ E(s), E(t) t ∈ I. 0 0 In this formula

= lim inf h→0+ E(t)

1 h

t+h "

with P being given in (7.1.12),

t

2 [ 1 |q(τρ )| (τ )1{ρ(τ )>0} Ω 2

# + P (τ )] dτ

318


D(t) = µ and

Ω

|∇u(t)|2 + (µ + λ)

A(t) =

Ω

Ω

|div u(t)|2 ,

(ρ(t)f (t) + g(t)) · u(t).

is bounded by a constant throughout the interval [0, T ). However, Hence, E(t) it is not clear whether ≤ E0 . lim inf E(s) + s→0

In other words, it is not clear whether (ρ, u) is a bounded energy weak solution in the sense of point (iv) of Definition 7.3. On the other hand, the question whether and under what conditions, any bounded energy weak solution is a finite energy weak solution, is an open problem. It is intimately related to the question of uniqueness and regularity of solutions. 7.1.2

Existence in bounded domains

The first main theorem of this section concerns the existence of weak solutions of problem (7.1.1), (7.1.2), (7.1.6), (7.1.7) in a bounded domain with sufficiently smooth boundary and homogeneous Dirichlet boundary condition for the velocity. It reads: Theorem 7.7 Assume that Ω is a bounded domain of class C 2,θ , θ ∈ (0, 1],

(7.1.32)

f ∈ (L∞ (QT ))3 , g ∈ (L∞ (QT ))3 ,

(7.1.33)

1

q 0 ∈ (L

2γ γ+1

P (ρ0 ) ∈ L (Ω), ρ0 ≥ 0 a.e. in Ω, 2 (Ω)) , q 0 1{ρ0 =0} = 0 a.e. in Ω, |qρ00| 1{ρ0 >0} ∈ L1 (Ω) 3

(7.1.34)

and suppose that (7.1.4), (7.1.5) hold. Then there exists a renormalized finite as well as bounded energy weak solution (ρ, u) of problem (7.1.1), (7.1.2), (7.1.6), (7.1.7) which is such that ρ ∈ L∞ (I, Lγ (IR3 )) ∩ Ls(γ) (IR3 × I)), s(γ) =

5γ−3 3 ,

ρ ∈ C 0 (I, Lγweak (Ω)) ∩ C 0 (I, Lp (Ω)), 1 ≤ p < γ, ρ ≥ 0 a.e. in Ω × I, ρ = 0 in (IR3 \ Ω) × I,

u ∈ L2 (I, (W 1,2 (IR3 ))3 ), u = 0 in (IR3 \ Ω) × I,

(7.1.35)

2γ γ+1

ρu ∈ C 0 (I, (Lweak (Ω))3 ),

6γ

ρ|u|2 ∈ L∞ (I, L1 (IR3 )) ∩ L2 (I, L 4γ+3 (Ω)). The renormalized continuity equation (7.1.10) in D′ (IR3 × I) holds with any b satisfying (6.2.9), (6.2.10), (6.2.11), where β = s(γ).


319

Proof The proof of Theorem 7.7 will be carried out through Sections 7.4–7.10 via several levels of approximations which are described in Section 7.4. 2 Exercise 7.8 In general, the statement u = 0 in (IR3 \ Ω) × I and u ∈ L2 (I, (W 1,2 (IR3 ))3 ) is weaker than u ∈ L2 (I, (W01,2 (Ω))3 ). However, if Ω is a Lipschitz domain, i.e. in particular in our situation, both statements are equivalent. The proof of this fact relies on a direct application of the Stokes formula, cf. Exercise 4.2. Remark 7.9 One easily verifies by a density argument and (7.1.35) that equation (7.1.20) holds with any test function φ ∈ L[ 2γ L1 (I, (L γ−1 (Ω))3 ).

s(γ) ′ γ ]

6γ 1, 2γ−3

(I, (W0

(Ω))3 ), ∂t φ ∈

Remark 7.10 Given c > 0, we set E (c) (ρ, u) = where P (c) (ρ) =

Ω

|q|2 ρ 1{ρ>0}

1 γ γ−1 [ρ

+

Ω

P (c) (ρ),

+ (γ − 1)cγ − γcγ−1 ρ].

Then, in addition to the energy inequality (7.1.13), we have

+ µ Ω |∇u|2 + (µ + λ) Ω |div u|2

≤ Ω (ρf + g) · u in D′ (I),

d (c) (ρ, u) dt E

(7.1.36)

and, in addition to (7.1.14), we have

t

t E (c) (ρ(t), u(t)) + µ 0 Ω |∇u|2 + (µ + λ) 0 |div u|2

t

≤ E (c) (ρ0 , u0 ) + 0 Ω (ρf + g) · u for a.a. t ∈ I.

(7.1.37)

These versions of the energy inequalities are proved in Section 7.10.6. We need them in Section 7.11 when we investigate flows in unbounded domains. Remark 7.11 Hypothesis (7.1.33) are not optimal. Since their optimization is not our goal, we leave it to the reader as an exercise. Remark 7.12 The renormalized continuity equation holds also with any bk , k > 0 defined by (6.2.22) with b satisfying (6.2.9), i.e. we have ∂t bk (ρ) + div (bk (ρ)u) + [ρ(bk )′+ (ρ) − bk (ρ)]div u = 0 in D′ (I).

320


Remark 7.13 By virtue of Lemmas 6.8 and 6.9, any weak solution of problem (7.1.1), (7.1.2), (7.1.6), (7.1.7) is automatically a renormalized one, provided γ ≥ 95 (i.e. s(γ) ≥ 2). If γ < 95 , the question whether any weak solution is a renormalized one is an open problem. In this case, it is not even clear whether any weak solution, which satisfies the continuity equation in D′ (IR3 × I), is a renormalized one. The condition γ > 23 marks the limit of the present theory. If γ < 23 , the term ρu ⊗ u may not belong to L1 (I, (L1 (Ω))3×3 ). Due to this fact, its interpretation in the momentum equation is questionable. Remark 7.14 If equation (7.1.2) holds in D′ (IR3 × I), one can take in (7.1.23) a test function η ∈ D(IR3 ), η = 1 in Ω. One obtains the conservation of mass in the integral form

ρ(t) = Ω ρ0 , for a.e. t ∈ I. (7.1.38) Ω

Of course, the weak solutions constructed in Theorem 7.7 satisfy (7.1.38) for all t ∈ I. If equation (7.1.2) holds only in D′ (QT ), conservation of mass may be violated, cf. Remark 6.14. 7.1.3

Existence in exterior domains

The second main theorem of this section concerns existence of weak solutions of problem (7.1.1), (7.1.2), (7.1.6), (7.1.7) and (7.1.8) in an exterior domain with prescribed zero velocity at infinity and density equal to a nonnegative constant. (The case with prescribed nonzero velocity at infinity will be discussed in Section 7.12.6.) In (3.3.46) and (3.3.47), it is defined what we mean under an exterior domain and what is an exterior domain of class C k,θ , k ∈ IN ∪ {0}, θ ∈ [0, 1]. We shall prove the following theorem. Theorem 7.15 (i) Assume that ρ∞ > 0 and Ω is an exterior domain of class C 2,θ , θ ∈ (0, 1],

(7.1.39)

f ∈ (L∞ (QT ))3 ∩L∞ (I, (L1 (Ω))3 ), g ∈ (L∞ (QT ))3 ∩L∞ (I, (L1 (Ω))3 ), (7.1.40) 2γ γ+1

P (ρ0 ) ∈ L1 (Ω), ρ0 ≥ 0 a.e. in Ω,

q 0 ∈ (Lloc (Ω))3 , q 0 1{ρ0 =0} = 0 a.e. in Ω,

|q 0 |2 ρ0 1{ρ0 >0}

∈ L1 (Ω)

(7.1.41)

(P is defined in (7.1.12)) and suppose that (7.1.4), (7.1.5) hold. Then there exists a bounded energy renormalized weak solution22 (ρ, u) of problem (7.1.1), (7.1.2), (7.1.6), (7.1.7), (7.1.8) which is such that 22 The existence of a finite energy weak solution is an open problem. See Section 7.12 for more details.

LINEAR MOMENTUM AND TOTAL ENERGY s(γ)

ρ ∈ Lloc (IR3 × I), s(γ) =

5γ−3 3 ,

321

P (ρ) ∈ L∞ (I, L1 (IR3 )),

ρ ∈ C 0 (I, Lγweak (Ω′ )) ∩ C 0 (I, Lp (Ω′ )), 1 ≤ p < γ,

Ω′ a bounded subdomain of Ω, ρ ≥ 0 a.e. in Ω × I, ρ = 0 in (IR3 \ Ω) × I, u ∈ L2 (I, (D1,2 (IR3 ))3 ),

(7.1.42)

2γ

γ+1 (Ω′ ))3 ), u = 0 in (IR3 \ Ω) × I, ρu ∈ C 0 (I, (Lweak 6γ

4γ+3 (IR3 )). ρ|u|2 ∈ L∞ (I, L1 (IR3 )) ∩ L2 (I, Lloc

The renormalized continuity equation (7.1.10) in D′ (IR3 × I) holds with any b satisfying (6.2.9), (6.2.10), (6.2.11), where β = s(γ). (ii) Let ρ∞ = 0 and suppose that in addition to the assumptions of part (i) we have also ρ0 ∈ L 1 (Ω). Then statement (i) remains valid and in addition to

(7.1.42) there holds Ω ρ = Ω ρ0 in I.

Proof The proof of Theorem 7.15 will be carried out in Section 7.11. It is based on Theorem 7.7 and on the so-called method of invading domains. 2

Exercise 7.16 If Ω is a Lipschitz exterior domain, then u ∈ L2 (I, (D01,2 (IR3 ))3 ) and u = 0 in IR3 \ Ω is equivalent to u ∈ L2 (I, (D01,2 (Ω))3 ). 7.2 Linear momentum and total energy In this section, we deal with the linear momentum and with the kinetic, potential and total energies. In the first part we explain how it is convenient to understand the statement ρu ∈ C 0 (I, Lpweak (Ω)), for a weak solution (ρ, u) satisfying ρ ∈ C 0 (I, L1weak (Ω)) and ρ(t)u(t) = q(t) a.e. in Ω for a.a. t ∈ I, where q ∈ C 0 (I, (Lpweak (Ω))3 ). In the second part we show that the kinetic, potential and total energies of the fluid defined in Remark 7.2 (see especially formula (7.1.16)) are defined for all t ∈ I and that they are lower semicontinuous on I. 7.2.1 Linear momentum Lemma 7.17 Let Ω be a domain in IR3 and 1 ≤ p < ∞. Assume that ρ ∈ C 0 (I, L1weak (Ω)), ρ ≥ 0 a.e. in QT , q ∈ C 0 (I, (Lpweak (Ω))3 ), u ∈L2 (I,(D01,2 (Ω))3 ) and ρ|u|2 ∈ L∞ (I, L1 (Ω)). Set 23 V (t) = {x ∈ Ω; ρ(x, t) = 0} ⊂ Ω, t ∈ I. Suppose that for a.e. t ∈ I, ρ(t)u(t) = q(t) a.e. in Ω. Then q(t) = 0 a.e. in V (t) and there exists a representant ˜ (t) = u(t) a.e. in Ω for a.a. t ∈ I ˜ ∈ L2 (I, (D01,2 (Ω))3 ), u u such that ρ(t)˜ u(t) = q(t) a.e. in Ω for all t ∈ I. 23 For any representant ρ(t) of the equivalence class {z; ρ(t) = z a.e. in Ω}, the set V (t) is uniquely determined.

322


Proof We first show that q(t) = 0 a.e. in V (t), t ∈ I. The statement is trivial if |V (t)| = 0. Suppose that |V (t)| = 0. We introduce J = {τ ∈ I; ρu(τ ) = q(τ ) a.e. in Ω

and ρ(τ )|u(τ )|2 0,1,Ω ≤ ρ|u|2 L∞ (I,L1 (Ω)) } ⊂ I. By the assumptions, |I \ J| = 0. Therefore there exists a sequence tn ∈ J such that tn → t. By virtue of the lower weak continuity of the L1 -norm and due to the L1weak -continuity of ρ,

V (t)

|q i (tn )| = lim inf tn →t V (t) |ρ(tn )ui (tn )| )

≤ lim inf tn →t { V (t) ρ(tn ) V (t) ρ(tn )|u(tn )|2 } = 0.

|q i (t)| ≤ lim inf tn →t

V (t)

This shows the first part. Now we define

  u(x, t) if x ∈ Ω and t ∈ J, ˜ (x, t) = u(x, t) if x ∈ V (t) and t ∈ I \ J, u  q(x,t) if x ∈ Ω \ V (t) and t ∈ I \ J. ρ(x,t)

(7.2.1)

˜ (t) = u(t) which means that u ˜ ∈ L2 (I, (D01,2 (Ω))3 ) We see that for a.e. t ∈ I, u 1,2 2 3 2 provided u ∈ L (I, (D0 (Ω)) ). This finishes the proof. 7.2.2 7.2.2.1

Total energy Two expressions for the kinetic energy

Lemma 7.18 Suppose that Ω, ρ, u and q satisfy assumptions of Lemma 7.17 and that ρ(t) ∈ Lqloc (Ω) for a.e. t ∈ I, where 23 ≤ q < ∞. Then

Ω

ρ(t)|u(t)|2 =

Proof By Hölder’s inequality

V

Ω

|q(t)|2 ρ(t) 1{ρ(t)>0}

for a.a. t ∈ I.

ρ(t)|u(t)|2 ≤ ρ(t)0,q,V u(t)20,2q′ ,V for a.a. t ∈ I

with V any bounded measurable subset of Ω. The last inequality

implies, via the Sobolev imbedding (cf. Section 1.3.6.4), that for a.a. t ∈ I, V ′ ρ(t)|u(t)|2 = 0 with any bounded and measurable V ′ ⊂ V (t), where V (t) is defined in Lemma 7.17. Therefore: for a.a. t ∈ I, ρ(t)|u(t)|2 = 0 a.e. in V (t). Moreover, according to Lemma 7.17, we have ρ(t) > 0 in Ω \ V (t) and q(t) = ρ(t)u(t) a.e. in Ω \ V (t). From the last three observations we easily deduce the statement of the lemma. 2

LINEAR MOMENTUM AND TOTAL ENERGY

323

7.2.2.2 Lower semicontinuity of the kinetic energy Lemma 7.19 Let Ω be a domain in IR3 and 1 0} ∈ L∞ (I, L1 (Ω)). Then the function

2 t → Ω |q(t)| ρ(t) 1{ρ(t)>0} is lower semicontinuous from [0, T ) to IR ∪ {∞}.

Proof We set

√ ǫ (ρ(t)) , ǫ > 0, a(t) = q(t)h ρ(t)+ǫ

(7.2.2)

where hǫ ∈ C 0 (IR), 0 ≤ hǫ ≤ 1, hǫ (z) = 0, z ≤ 0, for any z ∈ IR, hǫ (z) is nonincreasing in IR+ as a function of ǫ, limǫ→0+ hǫ (z) = 1, z > 0. We have

|q(t)|2

2 2 |a(t)|2 ≤ Ω |q(t)| ρ(t) 1{ρ(t)>0} , limǫ→0+ Ω |a(t)| = Ω ρ(t) 1{ρ(t)>0} , t ∈ I, Ω

cf. Lemma 1.15 about monotone convergence of integrals. It is therefore enough to investigate the lower semicontinuity of the map

[0, T ) → IR ∪ {∞}, t → Ω |a(t)|2 . (7.2.3)

Due to the assumptions, we have

a ∈ C 0 (I, (L1weak (Ω))3 ) and a ∈ L∞ (I, (L2 (Ω))3 ). There also holds

t+h

t+h

|a(t)η| ≤ t [a(t)0,2,Ω η0,2,Ω ] Ω t

≤ haL∞ (I,L2 (Ω)) η0,2,Ω , t, t + h ∈ I, h > 0, η ∈ L2 (Ω).

(7.2.4)

(7.2.5)

˜ (t) ∈ (L2 (Ω))3 , By the Riesz theorem, this formula implies the existence of a t ∈ [0, T ) such that

t+h

(7.2.6) ˜ (t)η = lim inf h→0+ h1 t a(τ )η dτ, η ∈ L2 (Ω). a Ω Ω

By virtue of (7.2.4) and due to the lemma about Lebesgue’s points (cf. Section 1.1.15), we have

˜ (t)η = Ω a(t)η, t ∈ [0, T ), η ∈ L2 (Ω), a (7.2.7) Ω

so that

˜ ∈ C 0 (I, (L1weak (Ω))3 ). a

(7.2.8)

Formula (7.2.5) implies ˜ a(t)0,2,Ω ≤ aL∞ (I,L2 (Ω)) , t ∈ [0, T ). Now, (7.2.8) ˜ ∈ C 0 (I, together with the last mentioned bound, by a density argument, yields a 3 2 ˜ (t) = a(t) a.e. in Ω, t ∈ [0, T ) which means (Lweak (Ω)) ). Due to (7.2.7), we have a that a ∈ C 0 (I, (L2weak (Ω))3 ) as well. Now, we use Corollary 3.33 to show that the function (7.2.2) is lower semicontinuous on [0, T ). This completes the proof. 2

324


7.2.2.3 Lower semicontinuity of the potential and of the total energy The statement of the following exercise is a direct consequence of Corollary 3.33. Exercise 7.20 Let ρ ∈ C 0 (I, Lγweak (Ω)), where Ω is a domain and 1 < γ < ∞. Then the function t → Ω P (ρ(t)), where P is defined in (7.1.12) is a lower semicontinuous function on [0, T ). Due to Lemma 7.19 and Exercise 7.20, the following theorem holds true. Proposition 7.21 Let all assumptions of Lemma 7.19 and of Exercise 7.20 be satisfied. Then the function t → E(ρ(t), q(t)), where E is defined in (7.1.16), is a lower semicontinuous function on [0, T ). Remark 7.22 The total energy E(ρ(t), q(t)), kinetic energy Ekin (ρ(t), q(t)) and potential energy Epot (ρ(t)), introduced in Remark 7.2, which correspond to the weak solution (ρ, u) constructed in Theorem 7.7, are lower semicontinuous on [0, T ). The same is true for weak solutions constructed in Theorem 7.15. In

2 this case, one firstly shows that functions t → Ω |q(t)| ρ(t) 1{ρ(t)>0} 1Bn (0) , t →

P (ρ(t))1Bn (0) are lower semicontinuous on [0, T ) for any fixed n ∈ IN , and Ω then one applies to these sequences the monotone convergence lemma. 7.3

Heuristic approach

This section is intended to be a brief guide to the proofs of existence of nonsteady weak solutions to compressible barotropic Navier–Stokes equations. In particular, we explain what we call the compactness arguments of Lions and of Feireisl. They play a crucial role throughout the following parts of this chapter. In Sections 7.4–7.11 we shall give a detailed proof of Theorems 7.7 and 7.15 via several levels of approximations and several passages to limits. These proofs are quite involved and it may happen that the main ideas remain hidden in the “jungle” of technicalities. On any level of approximations, there are several underlying ideas which lead to the existence of a weak solution. We shall explain them in this section by investigating more or less formally the weak compactness of the set of renormalized bounded energy weak solutions to system (7.1.1)–(7.1.7). More precisely, we shall describe Lions’ approach (Lions, 1998)(which applies to values γ ≥ 95 ) developed and generalized later in (Feireisl, 2001) to treat the case γ > 23 . 7.3.1

Compactness of weak solutions

Theorem 7.23 Let Ω be a bounded domain with a Lipschitz continuous boundary. Let (ρn , un ) be a sequence of bounded energy renormalized weak solutions to problem (7.1.1)–(7.1.7) with f = 0, g = 0 satisfying (7.1.35) and ρn (0) → ρ0 strongly in L1 (Ω), q n (0) → q 0 strongly in (L1 (Ω))3 ,

HEURISTIC APPROACH

325

E(ρn (0), q n (0)) → E(ρ0 , q 0 ) < ∞ in IR.

Then passing to subsequences if necessary, we have

ρn → ρ in C 0 (I, Lγweak (Ω)),

un → u weakly in L2 (I, (W01,2 (Ω))3 ),

(7.3.1)

where (ρ, u) is a bounded energy renormalized weak solution of the same problem with initial conditions ρ(0) = ρ0 and q(0) = q 0 again satisfying (7.1.35). In the sequel, we explain the main points of the proof of the above theorem. 7.3.2

Estimates due to the energy inequality

Inequality (7.1.14) yields ρn L∞ (I,Lγ (Ω)) ≤ L(ρ0 , q 0 ), un L2 (I,(W 1,2 (Ω))3 ) ≤ L(ρ0 , q 0 ), 0

(7.3.2)

2

ρn |un | L∞ (I,L1 (Ω)) ≤ L(ρ0 , q 0 ). We wish to emphasize, that these bounds do not depend on Ω. They are well known and standard and do not mean anything new. 7.3.3

Improved estimate of the density

If we test equation (7.1.1) written for (ρn , un ) by the test function

φ(x, t) = BΩ (ρθn − Ω ρθn ),

(7.3.3)

where BΩ is the Bogovskii operator (see Lemma 3.17) and θ > 0, we obtain after a long calculation

γ+θ ρ ≤ L(ρ0 , q 0 , Ω), γ > 23 , θ(γ) = 23 γ − 1. (7.3.4) I Ω n

To get the above estimate, one needs that (ρn , un ) satisfies the renormalized continuity equation (7.1.10) in IR3 with b(s) = sθ (including the property that the continuity equation holds in IR3 ) when (ρn , un ) are prolonged by 0 outside Ω. Notice that φ is well defined provided Ω is at least a Lipschitz domain and that, due to the properties of BΩ , the constant L in estimate (7.3.4) depends also on Ω (cf. again Lemma 3.17). For all details of this proof, see Section 7.9.5. This is an important estimate ensuring that the limiting pressure cannot be a mere measure. Notice that a weaker, only local estimate of the same type, namely

γ+θ ρ ≤ L(K), K any compact subset of Ω, (7.3.5) I K n

would provide the same service. Both estimates were discovered, for the purpose of weak solutions, by Lions, in local form in (Lions, 1998) and in the global form in (Lions, 1999) (see also (Lions, 1993b), (Lions, 1993a)), by using tools other than the Bogovskii operator.

326


7.3.4 Limit passage Estimates (7.3.2), (7.3.4) together with (7.1.20), (7.1.23) and together with the results presented in Section 6.1.1, yield ρn → ρ weakly in Lγ+θ (IR3 × I)

and in C 0 (I, Lγweak (Ω)), ρ = 0 in (IR3 \ Ω) × I,

un → u weakly in L2 (I, (W 1,2 (IR3 ))3 ), u = 0 in (IR3 \ Ω) × I, 2γ

γ+1 (Ω))3 ), ρn un → ρu weakly in C 0 (I, (Lweak

(7.3.6)

ρn uin ujn → ρui uj in D′ (QT ),

ργn → ργ weakly in L

γ+θ γ

(QT ).

Consequently, by virtue of (7.1.20), (7.1.23), (ρ, u, ργ ) satisfies ∂t ρ + div (ρu) = 0 in D′ (IR3 × I), ∂t (ρu) + div (ρu ⊗ u) − µ∆u −(µ + λ)∇div u +

∇ργ

(7.3.7) ′

3

= ρf + g in (D (QT )) .

Moreover, due to (7.1.24) written with (ρn , un ), we have ∂t b(ρ) + div [b(ρ)u] + [ρb′ (ρ) − b(ρ)]div u = 0 in D′ (IR3 × I),

(7.3.8)

where b satisfies (6.2.9)–(6.2.11) with β = γ and b(ρn ) → b(ρ) in L

2(γ+θ) γ

(QT ),

[ρn b′ (ρn ) − b(ρn )]div un → [ρb′ (ρ) − b(ρ)]div u weakly in L

2(γ+θ) 2γ+θ

(QT ).

In virtue of the first formula in (7.3.7), equation (7.3.8) holds if we replace b(ρ) by cρ + b(ρ), c ∈ IR. Finally, due to Lemma 6.15, ρ belongs to C 0 (I, Lp (Ω)), 1 ≤ p < γ. The results presented in this section rely on standard analysis and do not mean anything new. For their detailed proof, see Section 7.10.1. 7.3.5 Effective viscous flux Now, the main difficulty in the proof consists in showing that ργ = ργ . This requires some sort of compactness of the sequence ρn which is not available from the estimates (7.3.2), (7.3.4). As in the steady case (cf. Section 4.2.3), this missing information will be replaced by the observation that the quantity P (ρ) = ργ − (2µ + λ)div u usually called effective viscous flux, possesses this property. This is the crucial observation of Lions in (Lions, 1998). By now, it is “cantus firmus” of all proofs dealing with the existence theory of compressible flows. One of its versions is formulated in the following lemma.

HEURISTIC APPROACH

Lemma 7.24 If γ > 32 , then 0 0 η[ργn − (2µ + λ)div un ]Tk (ρn ) = lim n→∞

QT

QT

327

η[ργ Tk (ρ) − (2µ + λ)Tk (ρ) div u],

where Tk (s) = s if s ∈ [0, k), Tk (s) = k if s ∈ [k, ∞), k > 0 and η ∈ D(QT ).

A detailed proof of this identity in the general situation is in Section 7.5; its application to the present situation is explained in Section 7.10.2. Notice that the proof of this identity does not need the global estimate of the type (7.3.4). The sole local estimate of type (7.3.5) (which does not require Ω to be a Lipschitz domain) is sufficient. Lions has used this identity with Tk replaced by the function s → sϑ , ϑ > 0. The choice Tk due to Feireisl seems to be more appropriate for treating γ’s up to 32 . In the following argument, the Lions and Feireisl approaches differ in some points. In the present book, we shall follow Feireisl’s method from (Feireisl, 2001), since it seems to be more general. It, however, uses a lot of Lions’ ideas. In order to catch the difference, we make a short detour and explain, how the Lions proof continues.

7.3.6 Strong convergence of density – Lions’ approach 7.3.6.1 Renormalized continuity equation In this section, we explain schematically Lions’ approach after Lemma 7.24. Thanks to (7.3.4) (we emphasize that at this point, the sole local estimate (7.3.5) is not enough), ρ belongs to L2 (QT ) provided γ ≥ 59 . In this case, the DiPerna–Lions transport theory applies to the continuity equation in (7.3.7). It implies that the renormalized continuity equation ∂t (b(ρ)) + div (b(ρ)u) + (ρb′ (ρ) − b(ρ))div u = 0 in D′ (IR3 × I)

(7.3.9)

holds with any b satisfying (6.2.9)–(6.2.11), where β = γ, see Lemma 6.9 for all details. Notice that this (more or less technical) point is the only one in the Lions proof, where the limitation γ ≥ 95 is really needed. 7.3.6.2 Propagation of oscillations and strong convergence of density The continuity equation is an equation of hyperbolic type which, in general, allows the creation of oscillations in the sequence of densities as time runs on. However, in our situation, the renormalized continuity equation causes the effect that oscillations cannot be created during the evolution of the system. This is another of Lions’ key observations. In the sequel Lions uses (7.3.8) and (7.3.9) with b(s) = s ln s, to get an equation for the evolution of oscillations of the density measured by ρ ln ρ−ρ ln ρ. This equation reads 24 24 This is only a formal argument which works rigorously only provided γ + θ ≥ γ + 1, i.e., provided γ ≥ 3. If 59 ≤ γ < 3, one has to use b(s) = sϑ , ϑ ∈ (0, 1); compare with Section 4.2.4. We refer to Chapter 5 in (Lions, 1998) for all details.

328


∂t [ρ ln ρ − ρ ln ρ] + div [(ρ ln ρ − ρ ln ρ)u] =

1 [ρ ργ − ργ+1 ], 2µ + λ

(7.3.10)

where the overlined quantities denote corresponding weak limits. In getting this identity, Lemma 7.24 was used in an essential way. Formally, integrating this equality over Ω×I, using monotonicity of s → sγ and convexity of s → s ln s, one obtains ρ ln ρ = ρ ln ρ. The last identity means strong convergence of the sequence ρn , cf. Section 3.4. This finishes the sketch of the proof by Lions’ approach. 7.3.7

Strong convergence of density – Feireisl’s approach

7.3.7.1 Amplitude of oscillations If γ < 59 , ρ may not belong to L2 (QT ), and it is not clear whether the renormalized continuity equation (7.3.9) holds. This missing information is replaced by the following crucial observation due to Feireisl (Feireisl, 2001). Lemma 7.25 If γ > 32 , then supk>0 lim supn→∞ Tk (ρn ) − Tk (ρ)γ+1 Lγ+1 (Ω×I)

≤ supk>0 limn→∞ QT (ργn Tk (ρn ) − ργ Tk (ρ)) ≤ L(ρ0 , q 0 , Ω, T ).

This formula is an immediate consequence of Lemma 7.24. For its detailed proof see Section 7.10.3. Notice that in this proof, neither the global estimate of type (7.3.4) nor the local estimate (7.3.5) are needed. The sole bounds (7.3.2) and Lemma 7.24 are enough to get the result. This opens new possibilities for generalizations, see Section 7.12. The meaning of Lemma 7.3.7.1 can be translated as follows: Although the sequence ρn is bounded only in Lγ+θ (QT ), its oscillations, measured by the left-hand side of the above formula, are always bounded in “better” space than L2 (QT ). This is the property which will replace the missing condition ρ ∈ L2 (QT ) in the DiPerna–Lions transport theory. 7.3.7.2 Renormalized continuity equation The following assertion of Feireisl states that the couple (ρ, u) defined by (7.3.6) satisfies the renormalized continuity equation. In fact, the estimate given in Lemma 7.25 is enough for the renormalized continuity equation to hold. More precisely: Lemma 7.26 If γ > 32 , then the equation ∂t (b(ρ)) + div (b(ρ)u) + (ρb′ (ρ) − b(ρ))div u = 0 in D′ (IR3 × I) holds with any b satisfying (6.2.9)–(6.2.11), where we have taken β = γ. For a detailed proof of this statement, see Section 7.10.4. Of course, due to the first equation in (7.3.7), the renormalized continuity equation holds with any function s → cs + b(s), where c ∈ IR and b is the same as in Lemma 7.26.

HEURISTIC APPROACH

329

7.3.7.3 Strong convergence of the density This part is just a modification of Lions’ idea about the propagation of oscillations. We take Lk (s) =

s ln s, s ∈ [0, k), s ln k + s − k, s ∈ [k, ∞).

By virtue of (7.3.8) with b = Lk and due to Lemma 7.26, we have ∂t (Lk (ρ) − Lk (ρ)) +div [(Lk (ρ) − Lk (ρ))u] +[Tk (ρ)div u − Tk (ρ)div u] = 0.

(7.3.11) (7.3.12)

This identity, when manipulated conveniently (integrating over Ω × I, letting the second term diseppear, using convexity of s → s ln s in the first term, and Lemma 7.25 in the third term), gives lim lim sup Tk (ρn ) − Tk (ρ)Lγ+1 (QT ) ≤ 0.

k→∞ n→∞

This and (7.3.4) imply ρn → ρ strongly in Lp (QT ), 1 ≤ p < γ + θ.

(7.3.13)

We have presented a rather formal sketch of the last part of the proof. For all details see Section 7.10.5. Both Lions’ and Feireisl’s methods can be applied to other situations. The reader can consult Section 7.12 to learn more about this issue. 7.3.8

Remarks on approximations

7.3.8.1 Proof by using Lions’ existence theorem One point of view would be to apply Theorems 7.1, 7.2 and the results of Section 7.5 of Lions’ book (Lions, 1998) to system (7.4.1), (7.4.2), (7.1.6), (7.1.7) with γ > 23 and, e.g., β ≥ 2. The limit passage δ → 0, using Lions’ and Feireisl’s compactness arguments 7.3.2–7.3.5, 7.3.7.1–7.3.7.3, would then yield the statement of Theorem 7.7. This is, however, not the path we want to follow. Our goal is to present a consistent theory starting with basic approximations and ending with the weak solutions of the original system. 7.3.8.2 Parabolic regularization of the continuity equation It is natural to regularize the continuity equation by adding to its left-hand side the term −ǫ∆ρ, ǫ > 0 (see (7.4.23)). The originally hyperbolic equation then becomes a parabolic equation. We shall complete it with the homogenous Neumann boundary conditions. This is suggested by the fact that conservation of mass in the form

∂t Ω ρ = 0 should hold. The new problem can be treated by standard methods and it yields a nonnegative density, provided the initial density is also nonnegative. See Section 7.6 for all details.

330


Once the term −ǫ∆ρ is added to the continuity equation, we have to modify the momentum equation by adding to the left-hand side the term ǫ∇ρ · ∇u (see (7.4.22)), in order to conserve a reasonable form of the energy inequality. See Section 7.7.4.2 for more details. In this way, we obtain system (7.4.22), (7.4.23) with δ = 0 and with boundary and initial conditions (7.1.6), (7.1.7) and (7.4.24). We want to solve it by the Galerkin method (see Section 7.7). When passing to the limit in the Galerkin approximation, one encounters some technical difficulties for small values of γ. The most restrictive one appears when one wants to pass to the limit in the term ǫ∇ρ·∇u (see Section 7.8.2). At this stage, this is the main reason why we modify the pressure term ∇ργ by adding the artificial pressure δ∇ρβ with, e.g., β ≥ 8. The choice β ≥ 8 is certainly not optimal (in (Feireisl et al., 2001)), in the same situation, only β > 4 is required). It is however not important from the point of view of the final goal and it simplifies some of the technical arguments. After this modification, system (7.4.22), (7.4.23), (7.1.6), (7.1.7) and (7.4.24) is easily solvable by the Galerkin method, see Propositions 7.31, 7.34 and Sections 7.7, 7.8. With this choice of β, Lions’ approach based on Sections 7.3.2–7.3.6 can be used without difficulty in passing to the limit ǫ → 0+ (see Propositions 7.27, 7.31 and Section 7.9). (What is most important here is the fact that, since ρ ∈ L∞ (I, Lβ (Ω)), the renormalized continuity equation holds simply as a consequence of the DiPerna–Lions transport theory.) The last limit passage δ → 0+ uses Lions’ and Feireisl’s compactness arguments described in Sections 7.3.2–7.3.5 and 7.3.7.1–7.3.7.3. The reader can consult Proposition 7.27 and Section 7.10 for full details. 7.4

Approximations in bounded domains

In this section we introduce the chain of approximations which we use to solve the original problem (7.1.1)–(7.1.7). In Section 7.3.8 we have explained the motivation for this choice. At any level of approximations we formulate the statements about the existence of weak solutions and their properties which are needed to carry out the proof of existence for the original system. These statements will be proved in Sections 7.7–7.9. 7.4.1

First level approximations – artificial pressure

We consider the following system of equations ∂t (ρu) + ∂j (ρuuj ) − µ∆u − (µ + λ)∇div u

(7.4.1)

∂t ρ + div (ρu) = 0 in QT

(7.4.2)

+∇ργ + δ∇ρβ = ρf + g in QT ,

with initial conditions (7.1.7), with boundary conditions (7.1.6) and with δ > 0, β ≥ max{γ, 8}.

(7.4.3)


331

A weak solution of the original problem (7.1.1)–(7.1.7) will be obtained as a weak limit (ρ, u) as δ → 0+ to the sequence (ρδ , uδ ) of weak solutions of system (7.4.1), (7.4.2), (7.1.6), (7.1.7). This limit process will be carried out in Section 7.10; its main ingredient will be the following proposition which concerns existence of solutions and estimates independent of δ. It will be proved in Section 7.9. Proposition 7.27 Let µ, λ satisfy (7.1.5), γ satisfy (7.1.4) and let δ, β satisfy (7.4.3). Let Ω belong to the class (7.1.32), f , g satisfy (7.1.33), and suppose that ρ0 ∈ Lβ (Ω), ρ0 ≥ 0 a.e. in Ω, 2β

q 0 ∈ (L β+1 (Ω))3 , q 0 1{ρ0 =0} = 0 a.e. in Ω,

|q 0 |2 ρ0 1{ρ0 >0}

(7.4.4)

∈ L1 (Ω).

Then there exists a couple (ρδ , uδ ) with the following properties: (i) ρδ ∈ Lβ+1 (IR3 × I), ρδ ∈ C 0 (I, Lβweak (Ω)) ∩ C 0 (I, Lp (Ω)), 1 ≤ p < β, ρδ ≥ 0 a.e. in Ω × I, ρδ = 0 in (IR3 \ Ω) × I, uδ ∈ L2 (I, (W 1,2 (IR3 ))3 ), uδ = 0 in (IR3 \ Ω) × I,

q δ := ρδ uδ ∈ L2 (I, (L

6β 6+β

(7.4.5)

2β β+1

(IR3 ))3 ) ∩ C 0 (I, (Lweak (Ω))3 ), 6β

3β

ρδ |uδ |2 ∈ L∞ (I, L1 (IR3 )) ∩ L2 (I, L 4β+3 (IR3 )) ∩ L1 (I, L β+3 (IR3 )). (ii) ∂t (ρδ uδ ) + ∂j (ρδ uδ ujδ ) − µ∆uδ − (µ + λ)∇div uδ

(7.4.6)

+∇ργδ + δ∇ρβδ = ρf + g in (D′ (QT ))3 .

(iii) ∂t ρδ + div (ρδ uδ ) = 0 in D′ (IR3 × I).

(7.4.7) β

λ1 +1 (iv) For any b from (6.2.9)–(6.2.11), the function b(ρδ ) is in C 0 (I, Lweak (Ω)) ∩ β C 0 (I, Lp (Ω)), 1 ≤ p < λ1 +1 . Moreover,

∂t b(ρδ ) + div (b(ρδ )uδ ) +{ρδ b′ (ρδ ) − b(ρδ )}div uδ = 0 in D′ (IR3 × I).

(7.4.8)

For any bk , k > 0 defined by (6.2.22) with b belonging to (6.2.9), bk (ρδ ) belongs to C 0 (I, Lp (Ω)), 1 ≤ p < ∞ and moreover, ∂t bk (ρδ ) + div (bk (ρδ )uδ ) +{ρδ (bk )′+ (ρδ ) − bk (ρδ )}div uδ = 0 in D′ (IR3 × I).

(7.4.9)

332


(v)

limt→0+ Ω ρδ (t)η → Ω ρ0 η, η ∈ D(Ω),

limt→0+ Ω ρδ uδ (t) · φ → Ω q 0 · φ, φ ∈ (D(Ω))3 .

(vi)

(7.4.10)

+ µ Ω |∇uδ |2 + (µ + λ) Ω |div uδ |2

≤ Ω (ρδ f + g) · uδ in D′ (I),

d dt Eδ (ρδ , q δ )

where

Eδ (ρ, q) =

" 1 |q|2 Ω

2 ρ

1{ρ>0} +

1 γ γ−1 ρ

+

δ β β−1 ρ

The integral version of the energy inequality in the form

#

.

t

t

Eδ (ρδ (t), q δ (t)) + µ 0 Ω |∇uδ |2 + (µ + λ) 0 Ω |div uδ |2

t

≤ Eδ (ρ0 , q 0 ) + 0 Ω (ρδ f + g) · uδ for a.a. t ∈ I

(7.4.11)

(7.4.12)

(7.4.13)

is satisfied as well.

(vii) For δ ∈ (0, 1), the following estimates hold uδ L2 (I,W 1,2 (Ω)) ≤ L(E 0 , f , g, T ), ρδ L∞ (I,Lγ (Ω)) ≤ L(E 0 , f , g, T ),

δ 1/β ρδ L∞ (I,Lβ (Ω)) ≤ L(E 0 , f , g, T ), ρδ |uδ |2 L∞ (I,L1 (Ω)) ≤ L(E 0 , f , g), δ

ρδ 0,s,QT ≤ L(E 0 , f , g, Ω, T ), ρδ 0,β+θ,QT ≤ L(E 0 , f , g, Ω, T )

1 β+θ

ρδ uδ

2γ

L∞ (I,L γ+1 (Ω))

ρδ |uδ |2

3γ

L1 (I,L γ+3 (Ω))

+ ρδ uδ

/

, θ=

6γ

L2 (I,L γ+6 (Ω))

+ ρδ |uδ |2

6γ

2 γ − 1, s = γ + θ, 3 ≤ L(E 0 , f , g, T ),

L2 (I,L 4γ+3 (Ω))

≤ L(E 0 , f , g, T ),

(7.4.14)

(7.4.15) (7.4.16) (7.4.17) (7.4.18) (7.4.19)

|q |2 1 1 where E 0 = supδ∈(0,1) Eδ (ρ0 , q 0 ) = Ω { 21 ρ00 1{ρ0 >0} + γ−1 ργ0 + β−1 ρβ0 } and L is a positive constant, which is, in particular, independent of δ. Remark 7.28 By a density argument and (7.4.5) one easily verifies that equaβ+θ tion (7.4.6) holds with any test function φ ∈ L θ (I, (W01,r (Ω))3 ) such that 6γ , β+θ ∂t φ ∈ (L2 (QT ))3 , where r = max{ 2γ−3 θ }.


333

Remark 7.29 Given c > 0, we set (c)

Eδ (ρ, q) =

Then

and

2 1 |q | 1{ρ>0} Ω 2 ρ

δ + β−1

Ω

+

1 γ−1

Ω

[ργ + (γ − 1)cγ − γcγ−1 ρ]

[ρβ + (β − 1)cβ − βcβ−1 ρ].

+ µ Ω |∇uδ |2 + (µ + λ) Ω |div uδ |2

≤ Ω (ρδ f + g) · uδ in D′ (I)

d (c) dt Eδ (ρδ , q δ )

t

t

(c) Eδ (ρδ (t), q δ (t)) + µ 0 Ω |∇uδ |2 + (µ + λ) 0 Ω |div uδ |2

t

(c) ≤ Eδ (ρ0 , u0 ) + 0 Ω (ρδ f + g) · uδ for a.a. t ∈ I.

(7.4.20)

(7.4.21)

These versions of energy inequalities will be proved in Section 7.7.4.2. Remark 7.30 In agreement with Exercise 7.8, in our situation, the condition uδ ∈ L2 (I, (W 1,2 (IR3 ))3 ), uδ = 0 in (IR3 \ Ω) × I is equivalent to uδ ∈L2 (I, (W01,2 (Ω))3 ). 7.4.2

Second level approximation – continuity equation with dissipation

The sequence (ρδ , uδ ) satisfying Proposition 7.27 will be obtained as a weak limit as ǫ → 0+ of the sequence (ρδ,ǫ , uδ,ǫ ) of weak solutions to the following system of partial differential equations ∂t (ρu) + ∂j (ρuuj ) − µ∆u − (µ + λ)∇div u

+∇ργ + δ∇ρβ + ǫ(∇ρ · ∇)u = ρf + g in QT , ∂t ρ + div (ρu) − ǫ∆ρ = 0 in QT

(7.4.22) (7.4.23)

with initial conditions (7.1.7), boundary conditions (7.1.6) and ∂n ρ = 0 in ∂Ω.

(7.4.24)

This limit process will be carried out in Section 7.9 by using the following proposition about the solvability and estimates independent of ǫ for system (7.4.22), (7.4.23), (7.1.6),(7.1.7), (7.4.24) (we write here ρǫ rather than ρδ,ǫ , etc.). This proposition will be proved in Section 7.8. It reads: Proposition 7.31 Let µ, λ satisfy (7.1.5), let γ satisfy (7.1.4), let δ, β be given by (7.4.3) and let (7.4.25) ǫ > 0, 0 < ρ < ρ < ∞. Let Ω belong to the class (7.1.32), f , g satisfy (7.1.33) and ρ ≤ ρ0 ≤ ρ, ρ0 ∈ W 1,∞ (Ω), q 0 ∈ (L2 (Ω))3 .

(7.4.26)

334


Then there exists a couple (ρǫ , uǫ ) with the following properties: (i) ρǫ ∈ C 0 (I, Lβweak (Ω)) ∩ C 0 (I, Lp (Ω)) ∩ Lβ+1 (QT ), 1 ≤ p < β, β

ρǫ ≥ 0 a.e. in QT , ρǫ2 ∈ L2 (I, W 1,2 (Ω)), ∂t ρǫ ∈ L ∇2 ρǫ ∈ (L

5β−3 4β

5β−3 4β

(QT ),

(QT ))3×3 , uǫ ∈ L2 (I, (W01,2 (Ω))3 ), 2β

6β

β+1 q ǫ := ρǫ uǫ ∈ C 0 (I, (Lweak (Ω))3 ) ∩ L2 (I, (L β+6 (Ω))3 ), 6β

ρǫ |uǫ |2 ∈ L∞ (I, L1 (Ω)) ∩ L2 (I, L 4β+3 (Ω)), ∇uǫ · ∇ρǫ ∈ (L ∇ρǫ , ρǫ uǫ ∈ L

5β−3 4β

10β−6 5β−3 , 4β

(E03β+3

(ii)

(Ω)),

Ω

ρǫ =

5β−3 4β

(QT ))3 ,

ρ0 .

∂t (ρǫ uǫ ) + ∂j (ρǫ uǫ ujǫ ) − µ∆uǫ − (µ + λ)∇div uǫ

+∇ργǫ + δ∇ρβǫ + ǫ(∇ρǫ · ∇)uǫ = ρǫ f + g in (D′ (QT ))3 .

(iii) d dt

(iv)

Ω

ρǫ η −

ρǫ uǫ · ∇η

+ǫ Ω ∇ρǫ · ∇η = 0 in D′ (I), η ∈ C ∞ (IR3 ). Ω

limt→0+ Ω ρǫ (t)η → Ω ρ0 η, η ∈ D(Ω),

limt→0+ Ω ρǫ uǫ (t) · φ → Ω q 0 · φ, φ ∈ (D(Ω))3 .

(v) The energy inequality holds in the differential form

d 2 dt Eδ (ρǫ , q ǫ ) + µ Ω |∇uǫ |

+(µ + λ) Ω |div uǫ |2 + ǫδβ Ω ρβ−2 |∇ρǫ |2 ǫ

≤ Ω (ρǫ f + g) · uǫ in D′ (I),

as well as in the integral form

t

Eδ (ρǫ (t), q ǫ (t)) + µ 0 Ω |∇uǫ |2

t

t

+(µ + λ) 0 Ω |div uǫ |2 + ǫδβ 0 Ω ρβ−2 |∇ρǫ |2 ǫ

t

≤ Eδ (ρ0 , q 0 ) + 0 Ω (ρǫ f + g) · uǫ for a.a. t in I.

(7.4.27)

(7.4.28)

(7.4.29)

(7.4.30)

(7.4.31)

(7.4.32)

(vi) If δ ∈ (0, 1), the following estimates hold

uǫ L2 (I,W 1,2 (Ω)) ≤ L(E 0 , f , g, T ),

(7.4.33)


ρǫ L∞ (I,Lγ (Ω)) ≤ L(E 0 , f , g, T )

(7.4.34)

ρǫ |uǫ |2 L∞ (I,L1 (Ω)) ≤ L(E 0 , f , g, T ),

(7.4.35)

δ ρǫ L∞ (I,Lβ (Ω)) ≤ L(E 0 , f , g, T )

(7.4.36)

1 β

ρǫ uǫ

335

2β

L∞ (I,L β+1 (Ω))

+ ρǫ uǫ

ρǫ uǫ

10β−6 L 3β+3

2

6β

L2 (I,L β+6 (Ω))

ρǫ |uǫ |

(QT )

≤ L(E 0 , δ, f , g, T )

≤ L(E 0 , δ, f , g, Ω, T ),

6β

L2 (I,L 4β+3 (Ω))

≤ L(E 0 , δ, f , g, Ω, T ),

ρǫ 0,β+1,QT ≤ L(E 0 , δ, f , g, Ω, T ),

ǫ∇ρǫ 0, 10β−6 ,QT ≤ L(E 0 , δ, f , g, Ω, T ), 3β+3 √ ǫ∇ρǫ 0,2,QT ≤ L(E 0 , δ, f , g, Ω, T ),

ǫ∇ρǫ · ∇uǫ 0, 5β−3 ,QT ≤ L(E 0 , δ, f , g, Ω, T ). 4β

(7.4.37) (7.4.38) (7.4.39) (7.4.40) (7.4.41) (7.4.42)

Here L is a positive constant, which is, in particular, independent of ǫ. Moreover, if δ is not mentioned explicitly in the argument of L, then L is independent of δ as well. Remark 7.32 In accordance with (7.4.27) and due to the density argument, equation (7.4.28) holds with any test function φ ∈ L4 (I, (W01,4 (Ω))3 ) such that ∂t φ ∈ (L2 (QT ))3 and equation (7.4.29) holds with any test function η ∈ L2 (I, 2β ′ W 1, β−1 (IR3 ), ∂t η ∈L1 (I,Lβ (Ω)). 7.4.3

Third level approximation – Galerkin method

We know from Lemmas 4.32 and 4.33 that there exist countable sets {λi }∞ i=1 , 0 < λ1 ≤ λ2 ≤ . . . , and

1,p 3 2,p (Ω))3 , 1 ≤ p < ∞ {φi }∞ i=1 ⊂ (W0 (Ω)) ∩ (W

(7.4.43)

such that −µ∆φi − (µ + λ)∇div φi = λi φi (no summation over i) 2 3 {φi }∞ i=1 is an orthonormal basis in (L (Ω))

and an orthogonal basis in (W01,2 (Ω))3 with respect

to the scalar product Ω [µ∂j u · ∂j v + (µ + λ)div u div v].

(7.4.44)

We define an n-dimensional Euclidean space Xn with scalar product ·, · by

Xn = span{φi }ni=1 , u, v = Ω u · v, u, v ∈ Xn (7.4.45)

and denote by Pn the orthogonal projection of L2 (Ω) onto Xn . Some of its evident properties needed in the sequel are collected in the following exercise.

336


Exercise 7.33 Prove that:

P (u) · v = Ω u · Pn (v), u, v ∈ (L2 (Ω))3 , Pn L(L2 (Ω),L2 (Ω)) = 1, Ω n limn→∞ (Pn − I)z0,2 = 0, z ∈ L2 (Ω),

Pn zk,2 ≤ czk,2 , z ∈ (W01,2 (Ω))3 ∩ (W k,2 (Ω))3 , k = 1, 2 limn→∞ (Pn − I)z1,2 = 0, z ∈ W01,2 (Ω),

(7.4.46)

Pn z−1,2 ≤ cz−1,2 , z ∈ L2 (Ω), (I−Pn )z −1,2,Ω limn→∞ supz ∈L2 (Ω) = 0. z 0,2,Ω

Hint for the last statement: Suppose that the statement is not true. In this case there exists z n′ , z n′ 0,2 = 1 such that Pn′ z n′ − z n′ −1,2 ≥ ǫ0 > 0. Then use the compact imbedding L2 (Ω) ֒→ W −1,2 (Ω) to get a contradiction. In Section 7.7, we shall prove the following proposition: Proposition 7.34 Let µ, λ satisfy (7.1.5), γ satisfy (7.1.4), δ, β satisfy (7.4.3) and ǫ, ρ, ρ satisfy (7.4.25). Assume that Ω belong to the class (7.1.32), f , g satisfy (7.1.33) and 0 < ρ ≤ ρ0 ≤ ρ, ρ0 ∈ W 1,∞ (Ω), u0 ∈ Xn .

(7.4.47)

Then there exists a unique couple (ρn , un ) with the following properties: (i) ρn ∈ C 0 (I, W 1,p (Ω)) ∩ L2 (I, W 2,p (Ω)), ∂t ρn ∈ L2 (I, Lp (Ω)),

1 0 in Ω × I, un ∈ C 0 (I, Xn ), ∂t un ∈ L2 (I, Xn ), 2

∇ρn ∈ L

(I, E0p (Ω)),

ρn un ∈ C

0

(7.4.48)

(I, E0p (Ω).

(ii)

Ω

∂t (ρn un ) · φ +

Ω

[∂j (ρn un ujn ) − µ∆un

−(µ + λ)∇div un + ∇ργn + δ∇ρβn + ǫ∇ρn · ∇un ] · φ =

Ω

(ρn f + g) · φ,

t ∈ I, φ ∈ Xn .

(7.4.49)

(iii) (iv)

∂t ρn + div (ρn un ) − ǫ∆ρn = 0 a.e. in QT .

(7.4.50)

ρn (0) = ρ0 , un (0) = u0 .

(7.4.51)

(v) With q n = ρn un ,

+ µ Ω |∇un |2 + (µ + λ) Ω |div un |2

+ǫδβ Ω ρβ−2 |∇ρn |2 ≤ Ω (ρn f + g) · un a.e. in I n

d dt Eδ (ρn , q n )

(7.4.52)


and Eδ (ρn (t), q n (t)) + µ

t

0

+(µ + λ)

Ω

t

0

≤ Eδ (ρ0 , q 0 ) +

337

|∇un |2

t

|div un |2 + ǫδβ 0 Ω ρβ−2 |∇ρn |2 n Ω

t

(ρ f + g) · un for all t ∈ I. 0 Ω n

(7.4.53)

(vi) If δ ∈ (0, 1), the following estimates hold

un L2 (I,W 1,2 (Ω)) ≤ L(E 0 , f , g, T ),

(7.4.54)

ρn L∞ (I,Lγ (Ω)) ≤ L(E 0 , f , g, T ),

(7.4.55)

ρn |un |2 L∞ (I,L1 (Ω)) ≤ L(E 0 , f , g, T ),

(7.4.56)

δ ρn L∞ (I,Lβ (Ω)) ≤ L(E 0 , f , g, T ),

(7.4.57)

ρn L2 (I,W 1,2 (Ω)) ≤ L(E 0 , ǫ, δ, f , g, T ),

(7.4.58)

ρn 34 β ≤ L(E 0 , ǫ, δ, f , g, Ω, T ), (QT ) L √ ǫ∇ρn L2 (QT ) ≤ L(E 0 , δ, f , g, Ω, T ).

(7.4.59)

1 β

β 2

(7.4.60)

Here L is a positive constant which is, in particular, independent of n. Moreover, if δ resp. ǫ is not explicitly written in the argument of L, then L is independent of δ resp. ǫ as well. The couple (ρǫ , uǫ ) of Proposition 7.31 will be constructed as a weak limit of the sequence (ρǫ,n , uǫ,n ) guaranteed by Proposition 7.34. This limit process will be carried out in Section 7.8. Remark 7.35 To treat unbounded domains, we shall need a slightly different form of energy inequalities, namely

d (c) 2 dt Eδ (ρn , q n ) + µ Ω |∇un |

(7.4.61) +(µ + λ) Ω |div un |2 + ǫδβ Ω ρβ−2 |∇ρn |2 n

′ ≤ Ω (ρn f + g) · un in D (I)

and

(c)

t

|∇un |2

t

t

+(µ + λ) 0 Ω |div un |2 + ǫδβ 0 Ω ρβ−2 |∇ρn |2 n

t

(c) ≤ Eδ (ρ0 , q 0 ) + 0 Ω (ρn f + g) · un for all t ∈ I,

Eδ (ρn (t), q n (t)) + µ

(c)

0

Ω

(7.4.62)

where c > 0 and Eδ is defined in Remark 7.29. d (c) Eδ (ρn , q n ) = The first inequality follows immediately from the fact that dt d ′ dt Eδ (ρn , q n ) in D (I). The second inequality is obtained by integrating the first (c) one, taking into account Lemma 1.7 and the fact that Eδ (ρn , q n ) ∈ C 0 (I).

338


We briefly summarize the whole procedure to prove Theorem 7.7. In Section 7.7 we prove the existence and estimates for the Galerkin approximations stated in Proposition 7.34. In Section 7.8, we carry out the limit as n → ∞ for solutions guaranteed by Proposition 7.34 thus proving Proposition 7.31. Afterwards, we pass to the limit ǫ → 0+ in the sequence of solutions guaranteed by Proposition 7.31 and prove Proposition 7.27. This will be done in Section 7.9. The last step will be to carry out the limit δ → 0+ to obtain Theorem 7.7. This is performed in Section 7.10. 7.5

Effective viscous flux

Weak compactness of the effective viscous flux for steady equations was studied in Section 4.4. The main goal of this section is to investigate weak compactness of this quantity in the case of nonsteady equations. We have already emphasized that this property is essential in all existence proofs. For system (7.1.1) and (7.1.2), this property was formulated in Section 7.3.5. Here we deal with a more general formulation which applies to the first and second levels of approximating equations described in Section 7.4 (the last, third level of approximating equations can be solved by means of classical analysis without using the effective viscous flux). The proof relies on Feireisl’s modification (Feireisl, 2001) of original Lions’ approach (Lions, 1998). The precise formulation of the result is given in Proposition 7.36. It reads: Proposition 7.36 Let Ω be a domain of IR3 , and 1 < q, s, r, z < ∞, max{2, r′ } ≤ w ≤ ∞, z1 + 1q < 56 , 1r + q1∗ < 1, 1s + q1∗ < 1. Assume that for any bounded subdomain Ω′ such that Ω′ ⊂ Ω, q n → q in C 0 (I, (Lzweak (Ω′ ))3 ), un → u weakly in (L2 (Ω′ × I))3 ,

∇un → ∇u weakly in (L2 (Ω′ × I))3×3 , pn → p weakly in Lr (Ω′ × I), s

′

(7.5.1) (7.5.2) (7.5.3)

3

F n → F weakly in (L (Ω × I)) ,

gn → g in C 0 (I, Lqweak (Ω′ )) and weakly-∗ in Lw (Ω′ × I), ηfn → ηf weakly in L2 (I, W −1,2 (Ω)) and ′

A(ηfn ) → A(ηf ) strongly in L2 (I, Lz (Ω′ )), η ∈ D(Ω).

(7.5.4) (7.5.5) (7.5.6)

Here A stands for Ai , i = 1, 2, 3, and the operators Ai are defined in Section 4.4.1. Further suppose that ∂t qni + ∂j (qni ujn ) − µ∆uin

−(µ + λ)∂i div un + ∂i pn = Fni in D′ (Ω × I)

(7.5.7)


339

and ∂t gn + div (gn un ) = fn in D′ (Ω × I).

(7.5.8)

(i) Then

limn→∞ I ψ Ω ηgn [pn − (2µ + λ)div un ]

= I ψ Ω ηg[p − (2µ + λ)div u], η ∈ D(Ω), ψ ∈ D(I).

(7.5.9)

(ii) If q > 2, then there exists a subsequence such that gn div un → gdiv u 2q wr weakly in L2 (I, L 2+q (Ω′ )) and pn gn → pg in D′ (QT ) with pg ∈ Lmin{2, w+r } (I, 2q wr min{ 2+q , w+r }

Lloc

(Ω)). Moreover

pg − (2µ + λ)gdiv u = pg − (2µ + λ)gdiv u a.e. in QT .

(7.5.10)

Proof By virtue of (7.5.8) and in accordance with (4.4.5), (4.4.7) and (4.4.9), we have ∂t Ai (gn η˜) + Rij (gn ujn η˜) = Ai (fn η˜) + Ai (gn ujn ∂j η˜) in D′ (Ω × I), η˜ ∈ D(Ω),

(7.5.11)

where Ai and Rij are linear operators defined respectively by (4.4.1) and (4.4.6). Taking into account (4.4.2), (4.4.3), we find that Ai (gn η˜) ∈ Lw (I, W 1,w (Ω′ )) ∩ L∞ (I, W 1,q (Ω′ )), and having in mind (7.5.2), (7.5.5), (7.5.6) and (7.5.11), we ′ observe that ∂t Ai (gn η˜) ∈ L2 (I, Lz (Ω′ )). By density and by (7.5.1)–(7.5.6), we show that the vector field ψηAi (gn η˜), where ψ ∈ D(I) and η ∈ D(Ω), is an admissible test function for system (7.5.7). After a long but straightforward calculation which uses integration by parts several times, identity (7.5.11) and properties (4.4.5) of the operators Ai , (4.4.7)–(4.4.9) of the operators Rij , one obtains the following equality

ψ Ω η η˜gn [pn − (2µ + λ)div un ] I

= − I ψ Ω ∂i ηpn Ai (gn η˜) + (µ + λ) I ψ Ω ∂i ηdiv un Ai (gn η˜)

+µ I ψ Ω ∂j η∂j uin Ai (gn η˜) − µ ψ Ω ∂j ηuin ∂i Aj (gn η˜) + µ I ψ Ω ∂i η η˜uin gn

− I ψ Ω ηFni Ai (gn η˜) − I ψ Ω ∂j ηqni ujn Ai (gn η˜) − I ψ Ω ηqni Ai (fn η˜)

− I ∂t ψ Ω ηqni Ai (gn η˜) + I ψ Ω Ai (ηqni )gn ujn ∂j η˜

+ I ψ Ω ujn [gn η˜Rij (qni η) − qni ηRij (gn η˜)]. (7.5.12) Our next task is to pass to the limit n → ∞ in (7.5.12)25 . From now on, in this proof, in order to simplify the notation, whenever there is no danger of confusion, we shall write A instead of Ai and R instead of Rij . 25 We shall see in the sequel that all terms at the right-hand side, except for the last term, converge to corresponding expected limits as a consequence of classical results of functional

340


We start by listing several consequences of convergences (7.5.1)–(7.5.6). By virtue of (7.5.2)–(7.5.5), gn is bounded in L∞ (I, Lq (Ω′ )), gn un is bounded in 6q L2 (I, L 6+q (Ω′ )) and fn in L2 (I, W −1,2 (Ω′ )). Therefore by (7.5.8), ∂t gn is a 6q }. Since the imbedding bounded sequence in L2 (I, W −1,p (Ω′ )), p = min{2, 6+q q ′ −1,2 ′ (Ω ) is compact, we can use the Aubin–Lions lemma (cf. TheoL (Ω ) ֒→ W rem 1.71) to get gn → g strongly in Lp (I, W −1,2 (Ω′ )), 1 ≤ p < ∞

(7.5.13)

at least for an appropriately chosen subsequence. This in turn with (7.5.2) yields 6q

gn un → gu weakly in L2 (I, (L 6+q (Ω′ ))3 )

(7.5.14)

for the same subsequence. By virtue of (4.4.2), (4.4.3), (4.4.5) and due to (7.5.1), A(q n (t)η) → A(q(t)η) weakly in (W 1,z (Ω′ ))3×3 , t ∈ I.

(7.5.15)

For 1 ≤ p < z ∗ the imbedding W 1,z (Ω′ ) ֒→ Lp (Ω′ ) is compact and therefore A(q n (t)η) → A(q(t)η) strongly in (Lp (Ω′ ))3×3 , t ∈ I (cf. Section 1.4.10.6). It is a consequence of (7.5.1) and of Exercise 6.1 that the sequence A(q n η) is uniformly bounded in (Lp (Ω′ ))3×3 ; therefore by Vitali’s convergence theorem we finally obtain A(q n η) → A(qη) strongly in Lp1 (I, (Lp2 (Ω′ ))3×3 ), 1 ≤ p1 < ∞, 1 ≤ p2 < z ∗ (7.5.16) (cf. Theorem 1.18). Due to (7.5.1) and the compact imbedding of Lz (Ω′ ) into W −1,2 (Ω′ ) (recall that z > 65 ), we obtain q n (t) → q(t) strongly in (W −1,2 (Ω′ ))3 , t ∈ I. Moreover, the sequence q n is uniformly bounded in (Lz (Ω′ ))3 , and therefore by Vitali’s convergence theorem q n → q strongly in Lp (I, (W −1,2 (Ω′ ))3 ), 1 ≤ p < ∞ (see Lemma 6.4). This in turn with (7.5.2) yields 6z

q n un → qu weakly in L2 (I, (L 6+z (Ω′ ))3×3 ).

(7.5.17)

By virtue of (4.4.2), (4.4.3), (4.4.5) and (7.5.5), A(gn (t)˜ η ) → A(g(t)˜ η ) weakly in (W 1,q (Ω′ ))3 , t ∈ I.

(7.5.18)

Since the imbedding W 1,q (Ω′ ) ֒→ Lp (Ω′ ) is compact for any 1 ≤ p < q ∗ , we have, for these values of p, A(gn (t)˜ η ) → A(g(t)˜ η ) strongly in (Lp (Ω′ ))3 , t ∈ I. Similarly as before, we therefore obtain A(gn η˜) → A(g η˜) strongly in Lp1 (I, (Lp2 (Ω′ ))3 ), 1 ≤ p1 < ∞, 1 ≤ p2 < q ∗ . (7.5.19) analysis. The compactness of the last term follows from Feireisl’s commutator lemma (see Lemma 4.25) or from similar results of modern harmonic analysis, see e.g. (Coifman and Meyer, 1975). It is therefore important to write equation (7.5.12) in the form letting the commutator appear. This crucial observation is due to Lions, and it makes the proof possible.


341

Since the imbedding Lq (Ω′ ) ֒→ W −1,2 (Ω′ ) is compact, by virtue of (7.5.18) we also have R(gn (t)˜ η ) → R(g(t)˜ η ) strongly in (W −1,2 (Ω′ ))3×3 , t ∈ I and consequently R(gn η˜) → R(g η˜) strongly in Lp (I, (W −1,2 (Ω′ ))3×3 ), 1 ≤ p < ∞.

(7.5.20)

On the other hand, Lemma 4.25, together with (7.5.1) and (7.5.5) yields gn (t)˜ η Rij (qni (t)η) − qni (t)ηRij (gn (t)˜ η ) → g(t)˜ η Rij (q i (t)η) − q i (t)ηRij (g(t)˜ η) zq

weakly in L z+q (Ω′ ), t ∈ I. Since

zq z+q

>

6 5,

the imbedding L

zq z+q

′

(Ω ) ֒→ W

−1,2

(7.5.21) (Ω ) is compact and therefore ′

η Rij (qni (t)η) − qni (t)ηRij (gn (t)˜ η) gn (t)˜

→ g(t)˜ η Rij (q i (t)η) − q i (t)ηRij (g(t)˜ η ) strongly in W −1,2 (Ω′ ), t ∈ I.

(7.5.22)

In addition to this, the sequence gn η˜Rij (qni η) − qni ηRij (gn η˜) zq

is uniformly bounded in L z+q (Ω′ ) as seen from (7.5.1), (7.5.5), (4.4.2) and the uniform boundedness of gn and q n in Lq (Ω′ ) and Lz (Ω′ ), respectively. This zq fact and the compact imbedding L z+q (Ω′ ) ֒→֒→ W −1,2 (Ω′ ) along with Vitali’s convergence theorem, yield gn η˜Rij (qni η) − qni ηRij (gn η˜) → g η˜Rij (q i η) − q i ηRij (g η˜) strongly in Lp (I, W −1,2 (Ω′ )), 1 ≤ p < ∞.

(7.5.23)

Knowing this and (7.5.2), one easily concludes that

ujn [gn (t)˜ η Rij (qni (t)η) − qni (t)ηRij (gn (t)˜ η )]

j → I ψ Ω u [g(t)˜ η Rij (q i (t)η) − q i (t)ηRij (g(t)˜ η )]. I

ψ

Ω

(7.5.24)

Now, everything is ready to pass to the limit in equation (7.5.12). At the right-hand side, we use (7.5.3) and (7.5.19) to pass to the limit in the first term, (7.5.2) and (7.5.19) to let n → ∞ in the second and third terms, (7.5.2) and (7.5.20) to treat the fourth term, (7.5.14) to deal with the fifth term, (7.5.4) and (7.5.19) to find the limit in the sixth term, (7.5.17) and (7.5.19) to handle the seventh term, (7.5.1) and (7.5.6) to pass to the limit in the eigth

term and (7.5.1) and (7.5.19) in the ninth term. The tenth term tends to I ψ Ω Ai (ηq i )guj ∂j η˜ by virtue of (7.5.14) and (7.5.16). Finally, the limit of the last term is given by (7.5.24). We thus obtain

342


limn→∞ I ψ Ω η η˜gn [pn − (2µ + λ)div un ]

= − I ψ Ω ∂i ηpAi (g η˜) + (µ + λ) I ψ Ω ∂i ηdiv uAi (g η˜)

+µ I ψ Ω ∂j η∂j ui Ai (g η˜) − µ ψ Ω ∂j ηui ∂i Aj (g η˜) + µ I ψ Ω ∂i η η˜ui g

− I ψ Ω ηF i Ai (g η˜) − I ψ Ω ∂j ηq i uj Ai (g η˜) − I ψ Ω ηq i Ai (f η˜)

− I ∂t ψ Ω ηq i Ai (g η˜) + I ψ Ω Ai (ηq i )guj ∂j η˜

+ I ψ Ω uj [g η˜Rij (q i η) − q i ηRij (g η˜)]. (7.5.25) Passing to the limit n → ∞ in equations (7.5.7), (7.5.8) and (7.5.11), we obtain ∂t q i + ∂j (q i uj ) − µ∆ui − (µ + λ)∂i div u + ∂i p = F i in D′ (Ω × I), ∂t g + div (gu) = f in D′ (Ω × I)

(7.5.26) (7.5.27)

and ∂t Ai (g η˜) + Rij (guj η˜) = Ai (f η˜) + Ai (guj ∂j η˜) in D′ (Ω × I), η˜ ∈ D(Ω).

(7.5.28)

Taking into account (4.4.2) and (4.4.3), we find that Ai (g η˜)∈L∞ (I,W 1,q (Ω′ ))∩ Lw (I,W 1,w (Ω′ )), and with the help of (7.5.2), (7.5.5), (7.5.6) and (7.5.11), we ob′ serve that ∂t Ai (g η˜) ∈ L2 (I, Lz (Ω′ )). By a density argument and (7.5.1)–(7.5.6) one verifies that the vector field ψηAi (g η˜) is an admissible test function for system (7.5.26). After a long but straightforward calculation which uses integration by parts several times, identity (7.5.28) and properties (4.4.5), (4.4.7)–(4.4.9) of operators Rij , one obtains, in the same way as that which led to (7.5.12), the identity

η η˜g[p − (2µ + λ)div u]

= − I ψ Ω ∂i ηpAi (g η˜) + (µ + λ) I ψ Ω ∂i ηdiv uAi (g η˜)

+µ I ψ Ω ∂j η∂j ui Ai (g η˜) − µ ψ Ω ∂j ηui ∂i Aj (g η˜) + µ I ψ Ω ∂i η η˜ui g

− I ψ Ω ηF i Ai (g η˜) − I ψ Ω ∂j ηq i uj Ai (g η˜) − I ψ Ω ηq i Ai (f η˜)

− I ∂t ψ Ω Ω ηq i Ai (g η˜) + I ψ Ω Ai (ηq i )guj ∂j η˜

+ I ψ Ω uj [g η˜Rij (q i η) − q i ηRij (g η˜)]. (7.5.29) Comparing (7.5.25) and (7.5.29), we arrive at the identity I

ψ

Ω

limn→∞

I

ψ

Ω

η η˜gn [pn − (2µ + λ)div un ] =

I

ψ

Ω

η η˜g[p − (2µ + λ)div u].

This is nothing but the identity (7.5.9). Part (i) is thus proved.

CONTINUITY EQUATION WITH DISSIPATION

343

2q

If q > 2, then gn div un is a bounded sequence in L2 (I, L 2+q (Ω′ ) and there is a weakly convergent subsequence with the limit gdiv u. We thus get 0 0 0 0 ψ ηpn gn = ψ η{(2µ + λ)gdiv u + g[p − (2µ + λ)div u]}. (7.5.30) lim n→∞

I

Ω

I

Ω

This means that pn gn converges in D′ (QT ) to a limit (denoted pg) which is equal to pg + (2µ + λ)(gdiv u − gdiv u). We conclude the argument by using Lemma 1.33, completing the proof of Proposition 7.36. 2 7.6

Continuity equation with dissipation

In the first part of this section, we deal with the heat equation complemented with the Neumann boundary conditions. We recall the classical results about existence, uniqueness and regularity in the Lp -setting. These results will be needed as auxiliary tools for the proof of similar regularity properties for solutions of the continuity equation with dissipation. The continuity equation with dissipation forms one part of the systems at the second and third levels of approximations (see systems (7.4.22)–(7.4.24), (7.1.6), (7.1.7) and the Galerkin approximation in Section 7.4.3). In order to solve these systems, we need several properties of this equation such as, e.g., existence, uniqueness and regularity of solutions, pointwise lower and upper bounds, etc. For our purpose all useful properties are formulated in Proposition 7.39. The subsequent sections are devoted to its proof. Throughout this section, if not stated explicitly otherwise, c, c′ denote generic positive constants independent of the parameter ǫ. If we judge it useful for better understanding of the subject, or if it is needed in the sequel, the dependence on other parameters of studied problems is sometimes explicitly written in the argument of these constants or it is stressed out on corresponding places in the text. All results presented in this section are well known and their proofs rely on classical analysis. 7.6.1

Regularity for the parabolic Neumann problem

We recall that Ω is a bounded domain of IR3 and that I = (0, T ), where T > 0. We consider the following parabolic initial and boundary value problem ∂t ρ − ǫ∆ρ = h in QT ,

(7.6.1)

ρ(x, 0) = ρ0 (x), x ∈ Ω,

(7.6.2)

∂n ρ = 0 in ∂Ω × I,

(7.6.3)

where ǫ > 0, ρ0 and h are given functions on Ω and Ω × I, respectively and ρ is an unknown function on Ω × I. The following result rephrases the well known statements about the regularity of parabolic systems.

344


Lemma 7.37 Let 0 < θ ≤ 1, 1 < p, q < ∞ and Ω be a bounded domain. If & 2− p2 ,q (Ω), h ∈ Lp (I, Lq (Ω)) (7.6.4) Ω ∈ C 2,θ , ρ0 ∈ W

& 2− p2 ,q (Ω) is a completion of the space {z ∈ C ∞ (Ω); ∂n z|∂Ω = 0} in (where W 2 W 2− p ,q (Ω), see Section 1.3.5.10), then there exists a unique ρ ∈ Lp (I, W 2,q (Ω))∩ 2 C 0 (I, W 2− p ,q (Ω)), ∂t ρ ∈ Lp (I, Lq (Ω)) satisfying equation (7.6.1) a.e. in QT , equation (7.6.2) a.e. in Ω, equation (7.6.3) in the sense of normal traces a.e. in I, and which verifies estimate 1

ǫ1− p ρ

L∞ (I,W

2− 2 ,q p (Ω))

+ ∂t ρLp (I,Lq (Ω)) + ǫρLp (I,W 2,q (Ω)) 1

≤ c(p, q, Ω)[ǫ1− p ρ0 2− p2 ,q + hLp (I,Lq (Ω)) ].

(7.6.5)

Lemma 7.37 follows, e.g., from Chapter III in (Amann, 1995).26 In section 7.8 we shall need the following existence and uniqueness result for the Neumann problem (7.6.1)–(7.6.3) with the right-hand side in divergence form: Lemma 7.38 Let 0 < θ ≤ 1, 1 < p, q < ∞ and Ω be a bounded domain. Suppose that Ω ∈ C 2,θ , ρ0 ∈ Lq (Ω), b ∈ Lp (I, Lq (Ω)).

Then there exists a unique ρ ∈ Lp (I, W 1,q (Ω)) ∩ C 0 (I, Lq (Ω)) which satisfies (7.6.2) a.e. in Ω,

d ′ ∞ dt Ω ρη + ǫ Ω ∇ρ · ∇η = − Ω b · ∇η, η ∈ C (Ω), in D (I) and

1

1

ǫ1− p ρL∞ (I,Lq (Ω)) + ǫ∇ρLp (I,Lq (Ω)) ≤ c(p, q, Ω)[ǫ1− p ρ0 0,q + bLp (I,Lq (Ω)) ]. (7.6.8) 26 For example, the statements of Theorem 4.10.8 and Remark 4.10.9 in (Amann, 1995) applied to the equation

∂t ξ − ∆ξ = h in Ω × (0, ∞),

(7.6.6)

ξ(0) = ρ0 in Ω, ∂n ξ = 0 in ∂Ω × (0, ∞),

where ρ0 ∈ C ∞ (Ω), ∂n ρ0 |∂Ω = 0, h ∈ C0∞ (Ω × (0, ∞)), yield existence of a unique solution ξ ∈ Lp ((0, ∞), W 2,q (Ω)) ∩ W 1,p ((0, ∞), Lq (Ω)) ∩ C 0 ([0, ∞), W

2 ,q 2− p

(Ω))

(7.6.7)

which satisfies the estimate ξ

L∞ (It ,W

≤ c(p, Ω)[ρ0

2− 2 ,q p (Ω))

W

+ ξLp (It ,W 2,q (Ω)) + ∂t ξLp (It ,Lq (Ω))

2− 2 ,q p (Ω)

+ hLp (It ,Lq (Ω)) ], It = (0, t), t ∈ (0, ∞).

Using the rescaling t → ǫt in equation (7.6.6) and in the above estimate, we easily verify the statement of Lemma 7.37.


345

Also this lemma follows from Theorem 4.10.8 and Remark 4.10.9 in (Amann, 1995). 7.6.2

Continuity equation with dissipation

Next we investigate the equation ∂t ρ + div (ρu) − ǫ∆ρ = 0 in QT ,

(7.6.9)

completed with initial conditions ρ(0) = ρ0 in Ω

(7.6.10)

∂n ρ = 0 in ∂Ω × I.

(7.6.11)

and boundary conditions The unknown function is ρ(x, t), t ∈ I, x ∈ Ω ⊂ IR3 , Ω is a bounded domain, ǫ > 0 is a given constant, ρ0 (x) is a given function and u(x, t) is a given vector field vanishing on the boundary of Ω.27 We shall prove the following theorem about the solvability of (7.6.9)–(7.6.11). Proposition 7.39 Let 0 < θ ≤ 1, Ω be a bounded domain of class C 2,θ , 0 < ρ ≤ ρ < ∞, and ρ0 ∈ W 1,∞ (Ω), ρ ≤ ρ0 ≤ ρ. Then there exists a unique mapping Sρ0 : L∞ (I, (W01,∞ (Ω))3 ) → C 0 (I, W 1,2 (Ω)),

(7.6.12)

where W01,∞ (Ω) is defined in (4.6.45), such that: (i) Sρ0 (u) ∈ RT := {ρ; ρ ∈ L2 (I, W 2,p (Ω)) (7.6.13) 1,p 2 p 0 ∩C (I, W (Ω)), ∂t ρ ∈ L (I, L (Ω))}, 1 < p < ∞. (ii) The function ρ = Sρ0 (u) satisfies (7.6.9) a.e. in QT , (7.6.10) a.e. in Ω and (7.6.11) in the sense of traces a.e. in I. (iii) t t ρe− 0 u(τ )1,∞ dτ ≤ [Sρ0 (u)](x, t) ≤ ρe 0 u(τ )1,∞ dτ, (7.6.14) t ∈ I, for a.a. x ∈ Ω. (iv) If uL∞ (I,W 1,∞ (Ω)) ≤ K, where K > 0, then c

Sρ0 (u)L∞ (It ,W 1,2 (Ω)) ≤ cρ0 1,2 e 2ǫ (K+K 27 The

2

)t

, t ∈ I,

(7.6.15)

condition u · n|∂Ω = 0 would be enough for the conclusions of this section to hold.

346


∇2 Sρ0 (u)L2 (Qt ) ≤ ∂t Sρ0 (u)L2 (Qt )

c ǫ

√

c

tρ0 1,2 Ke 2ǫ (K+K

2

)t

t ∈ I, √ 2 c ≤ c tρ0 1,2 Ke 2ǫ (K+K )t , t ∈ I.

(7.6.16)

(v) [Sρ0 (u1 ) − Sρ0 (u2 )](t)0,2,Ω ≤ c(K, ǫ, T )tρ0 1,2 u1 − u2 L∞ (It ,W 1,∞ (Ω)) , t ∈ I.

(7.6.17)

The constant c in estimates (7.6.15), (7.6.16) depends at most on Ω; in particular, it is independent of ǫ, K, T, ρ0 , u. Proof The proof of this theorem will be carried out in Sections (7.6.3)–(7.6.7). 2 7.6.3

Construction of a solution – Galerkin method

In this section, we shall construct a strong solution of problem (7.6.9)–(7.6.11), where (7.6.18) u ∈ L2 (I, (W01,∞ (Ω))3 ), ρ0 ∈ W 1,2 (Ω). We shall look for ρ ∈ C 0 (I, W 1,2 (Ω)) ∩ L2 (I, W 2,2 (Ω)), ∂t ρ ∈ L2 (QT )

(7.6.19)

satisfying (7.6.9) a.e. in QT , (7.6.10) a.e. in Ω and (7.6.11) in the sense of normal traces a.e. in I. We shall prove the following statement: Auxiliary lemma 7.40 If Ω ∈ C 2 and if assumptions (7.6.18) hold, then in class (7.6.19), there exists a unique strong solution to problem (7.6.9)–(7.6.11). Proof To start, we recall some elements needed for the Galerkin method. In accordance with Lemma 4.28, there exist countable sets ∞ √1 , {λi }∞ i=0 , λ0 = 0 < λ1 ≤ . . . , {Φi }i=0 , Φ0 = |Ω|

Φi ∈ W 2,p (Ω), ∇Φi ∈

E0p (Ω),

1 ≤ p < ∞, ∆Φi = λi Φi , i = 0, 1, . . .

(in the last formula, there is no summation over i), such that {Φi }∞ i=0 is an 1,2 orthonormal basis in L2 (Ω) and an orthogonal basis in W (Ω). We denote En :=

span {Φi }ni=0 , (ρ, η) = Ω ρη the scalar product on En and Pn the orthogonal projection of L2 (Ω) onto En . Now, we are ready to begin the proof. Since it is standard and since similar ideas are used later in more general situations, we present it in the form of several exercises. 1) In the exercises 7.41–7.44 below we assume that u ∈ C 0 (I, (W01,∞ (Ω))3 ).


347

Exercise 7.41 We define the operator Au : I → L(En , En ) by setting [Au (t)]ξ = −ǫ∆ξ + Pn (div (ξu)), ξ ∈ En (i) Prove that the Au ∈ C 0 (I, L(En , En )). (ii) Prove that the system of ordinary differential equations ρ′n + Au ρn = 0, ρn (0) = Pn (ρ0 )

(7.6.20)

with t → ρn (t) ∈ En , t ≥ 0, admits a unique solution ρn ∈ C 0 (I, En )∩C 1 (I, En ). (iii) Prove that ρn satisfies

∂ ρ η + ǫ QT ∇ρn · ∇η + QT div (ρn u)η = 0, η ∈ L2 (I, En ). (7.6.21) QT t n

Hint for (ii): Use Proposition 2.17.

Exercise 7.42 Prove that the functions ρn constructed in Exercise 7.41 satisfy the following estimates ρn (t)1,2 ≤ L(T, uL2 (I,W 1,∞ (Ω)) , ǫ), t ∈ I, ∇ρn L2 (I,W 1,2 (Ω)) ≤ L(T, uL2 (I,W 1,∞ (Ω)) , ǫ), ∂t ρn L2 (QT ) ≤ L(T, uL2 (I,W 1,∞ (Ω)) , ǫ). Hint: Scalar multiply (7.6.20) in En by ρn , then by −∆ρn and finally by ∂t ρn , use standard properties of orthogonal projection Pn (see e.g. Exercise 7.33) and effectuate estimates in the same spirit as is done in Section 7.6.6 later. Exercise 7.43 Prove that in class (7.6.19) there exists ρ such that ρn → ρ weakly in L2 (I, W 2,2 (Ω)) and in C 0 (I, L2 (Ω)),

ρn → ρ weakly-∗ in L∞ (I, W 1,2 (Ω)), ∂t ρn → ∂t ρ weakly in L2 (QT )

(7.6.22)

at least for an appropriately chosen subsequence. Hint: Use the Banach–Alaoglu and Arzel` a-Ascoli theorems, cf. Sections 1.4.5.25, 1.4.5.26, 1.3.12. Exercise 7.44 (i) First prove that

∂ ρη + ǫ QT ∇ρ · ∇η + QT div (ρu)η = 0, η ∈ L2 (I, W 1,2 (Ω)) QT t

(7.6.23)

and then show that ρ satisfies (7.6.9) a.e. in QT , (7.6.10) a.e. in Ω and γn (∇ρ) = 0 a.e. in I. (ii) Prove that ρ ∈ C 0 (I, W 1,2 (Ω)). Hint for (i): Use (7.6.21), Exercise 7.43, a density argument and integration by parts with convenient test functions. Hint for (ii): Use Theorem 1.67 or apply Lemma 7.37 to (7.6.9)–(7.6.11) with h = −div (ρu) and p = q = 2.

348


Exercise 7.44 completes the proof of Auxiliary Lemma 7.40 under assumption u ∈ C 0 (I, (W01,∞ (Ω))3 ). 2) If u ∈ L2 (I, (W01,∞ (Ω))3 ), we take a sequence un ∈ C 0 (I, (W01,∞ (Ω))3 ), un → u strongly in L2 (I, (W01,∞ (Ω))3 )

(7.6.24)

and the corresponding strong solutions ρn ∈ C 0 (I, W 1,2 (Ω)) ∩ L2 (I, W 2,2 (Ω)), ∂t ρn ∈ L2 (QT ) of problem (7.6.9)–(7.6.11) with u replaced by un . Due to (7.6.23),

∂ ρ η − ǫ Ω ∆ρn η + Ω div (ρn un )η = 0 a.e. in I, η ∈ L2 (Ω). (7.6.25) Ω t n

For a.a. t ∈ I, functions η = ρn (t), η = −∆ρn (t), η = ∂t ρn (t) are admissible test functions for (7.6.25). Taking into account strong convergence (7.6.24) and repeating with minor changes the argument of Exercises 7.42 and 7.43, we easily obtain that also for this sequence, formulae (7.6.22) hold. Now, we finish by passing to the limit n → ∞ in (7.6.25). It remains to show that ρ ∈ C 0 (I, W 1,2 (Ω)). To do this, we observe that div (ρu) ∈ L2 (QT ) and apply Lemma 7.37 to (7.6.9)–(7.6.10) with p = q = 2. Another possibility is to use Theorem 1.67. This completes the proof of Auxiliary Lemma 7.40. 2 Now, we set Sρ (u) = ρ completing the proof of (7.6.12) and of statement (ii) of Proposition 7.39. 7.6.4

Regularity of solutions

To prove that ρ possesses regularity (7.6.13), we use Lemma 7.37 with h = −div (ρu), and a bootstrap argument starting with (7.6.19). The details are left to the interested reader. 7.6.5

Boundedness from below and from above

The function

t

R(t) = ρe

0

div u(τ )0,∞ dτ

(7.6.26)

solves the ordinary differential equation R′ − div u0,∞,Ω R = 0, R(0) = ρ.

(7.6.27)

R′ + div (Ru) ≥ 0 a.e. in QT .

(7.6.28)

ω(0) = ρ0 − ρ ≤ 0, ∂n ω = 0 in ∂Ω.

(7.6.30)

Therefore, By virtue of (7.6.9)–(7.6.11) and (7.6.28), the difference ω(x, t) = ρ(x, t) − R(t), satisfies (7.6.29) ∂t ω + div (ωu) − ǫ∆ω ≤ 0 a.e. in QT , +

+

Multiplying (7.6.29) by ω (ω is an admissible test function, see Section 1.3.5.5) and integrating over Ω, we get

CONTINUITY EQUATION WITH DISSIPATION 1 d + 2 2 dt ω 0,2

+ǫ

Ω

|∇ω + |2 ≤ − 12

Ω

349

|ω + |2 div u ≤ 21 div u0,∞ ω + 20,2

(cf. Theorems 1.67 and 1.68). Therefore d + 2 dt ω 0,2

≤ div u0,∞ ω + 20,2 .

Hence, by Gronwall’s lemma (cf. Section 1.1.4), one obtains 1

ω + (t)0,2 ≤ ω + (0)0,2 e 2

t 0

div u(τ )0,∞ dτ

= 0,

i.e. ρ(x, t) − R(t) ≤ 0 a.e. in QT . Due to regularity (7.6.13), this is nothing but the second inequality in (7.6.14). The first inequality in (7.6.14) can be deduced similarly by using the function ω = ρ − r, where r = ρe−

t 0

div u(τ )0,∞ dτ

.

The new function satisfies ∂t ω + div (ωu) − ǫ∆ω ≥ 0

(7.6.31)

with initial and boundary conditions ω(0) = ρ0 − ρ ≥ 0, ∂n ω = 0 on ∂Ω. Testing (7.6.31) with ω − leads to ω − (t)0,2 = 0, i.e. ρ(x, t) − r(t) ≥ 0 a.e. in QT . This is the desired result. 7.6.6

L2 -estimates

We test (7.6.9) by ρ to get

d dt

Ω

ρ2 + 2ǫ

Ω

|∇ρ|2 = −

where the right-hand side is majorized, e.g., by

Ω

ρ2 div u,

cu1,∞ ρ20.2 . We multiply (7.6.9) by −∆ρ and integrate over Ω, to obtain similarly

d 2 2 dt Ω |∇ρ| + 2ǫ Ω |∆ρ| = 2 Ω ρdiv u∆ρ + 2 Ω u · ∇ρ∆ρ,

(7.6.32)

(7.6.33)

(7.6.34)

cf. again Theorems 1.67, 1.68; the right-hand side is majorized e.g. by cu1,∞ ρ1,2 ∆ρ0,2 ≤

ǫ c′ u21,∞ ρ21,2 + ∆ρ20,2 . ǫ 2

Putting (7.6.32) and (7.6.35) together, we arrive at

(7.6.35)

350

WEAK SOLUTIONS FOR NONSTEADY NAVIER–STOKES EQUATIONS d 2 dt ρ1,2

≤ cǫ (u1,∞ + u21,∞ )ρ21,2 .

Hence, by Gronwall’s lemma

≤ ρ0 1,2 exp

ρ(t)L∞ (It ,W 1,2 (Ω))

c ∞ 1,∞ (Ω)) 2ǫ (uL (I,W

+ u2L∞ (I,W 1,∞ (Ω)) )t , t ∈ I.

(7.6.36)

This completes the proof of the bound (7.6.15). Integrating (7.6.34) (this is possible, see Lemma 1.7 and Theorem 1.67), while using (7.6.36), we obtain

t

t (7.6.37) ǫ 0 ∆ρ20,2 ≤ cuL∞ (I,W 1,∞ (Ω)) ρL∞ (It ,W 1,2 (Ω)) 0 ∆ρ0,2 .

After an evident application of Young’s inequality and of the estimate ∇2 ρ0,2 ≤ c∆ρ0,2 which follows from Lemma 4.27, estimate (7.6.37) gives the first inequality in (7.6.16). If one takes into account the last two previous bounds, the second inequality (7.6.16) follows directly from identity (7.6.9). Part (iv) of Proposition 7.39 is thus proved. 7.6.7

L2 -estimate of differences

Let ρ1 and ρ2 be two solutions (7.6.13) of problem (7.6.9)–(7.6.11) corresponding to u1 and u2 , respectively. Their difference satisfies ∂t (ρ1 − ρ2 ) − ǫ∆(ρ1 − ρ2 ) = −ρ1 div (u1 − u2 ) − ∇ρ1 · (u1 − u2 ) −(ρ1 − ρ2 )div u2 − ∇(ρ1 − ρ2 ) · u2 a.e. in QT . Testing this equation by (ρ1 − ρ2 ), one gets

d 2 2 dt Ω (ρ1 − ρ2 ) + 2ǫ Ω |∇(ρ1 − ρ2 )|

= −2 Ω [ρ1 div (u1 − u2 ) + ∇ρ1 · (u1 − u2 )

(7.6.38)

(7.6.39)

+(ρ1 − ρ2 )div u2 + ∇(ρ1 − ρ2 ) · u2 ](ρ1 − ρ2 ).

Estimating the right-hand side by c[ρ1 1,2 ρ1 − ρ2 0,2 u1 − u2 1,∞ + u2 1,∞ ρ1 − ρ2 20,2 ], one gets, in particular, d dt ρ1

− ρ2 0,2 ≤ cρ1 1,2 u1 − u2 1,∞ + cu2 1,∞ ρ1 − ρ2 0,2 . c

Therefore, by Gronwall’s lemma, ρ1 − ρ2 0,2 (t) ≤ c

t

t cu2 1,∞ (s)ds τ ρ dτ, t ∈ I. (τ ) (u − u )(τ ) e 1 1,2 1 2 1,∞ 0

The last inequality and (7.6.15) yield (7.6.17).


351

7.6.8 A renormalized inequality with dissipation Proposition 7.45 Assume that Ω is a domain in IR3 . Let 2 ≤ β < ∞ and let 1 ≤ p < ∞. Suppose that a couple (ρ, u) satisfies ρ ∈ L∞ (I, Lβloc (Ω)), ∆ρ ∈ Lploc (Ω × I),

1,2 (Ω))3 ) ρ ≥ 0 a.e. in Ω × I, u ∈ L2 (I, (Wloc

and ∂t ρ + div (ρu) − ǫ∆ρ = 0 in D′ (Ω × I).

(7.6.40)

Then for any convex function

b ∈ C 1 ([0, ∞)) ∩ C 2 ((0, ∞)) satisfying growth conditions (6.2.10) and (6.2.11), ∂t b(ρ) + div (b(ρ)u) + (ρb′ (ρ) − b(ρ))div u − ǫ∆b(ρ) ≤ 0 in D′ (Ω × I). (7.6.41)

Proof Let Ω′ , Ω′ ⊂ Ω be a bounded domain. Then for any sufficiently small α > 0, we have ∂t Sα (ρ) + div (Sα (ρ)u) − ǫ∆Sα (ρ) = rα (ρ, u) a.e. in Ω′ × I,

(7.6.42)

where Sα is the mollifying operator over space variables (cf. Section 1.3.4.4) and rα is defined in (6.2.16). Notice that ∂t Sα (ρ) ∈ Lmin{2,p} (I, C ∞ (Ω′ )). We multiply (7.6.42) by b′ (Sα (ρ)) to obtain, similarly as in (6.2.17), ∂t b(Sα (ρ)) + div [b(Sα (ρ))u] +[Sα (ρ)b′ (Sα (ρ)) − b(Sα (ρ))]div u − ǫb′ (Sα (ρ))∆Sα (ρ)

(7.6.43)

= b′ (Sα (ρ))rα (ρ, u) a.e. in Ω′ × I. The identity

∆b(Sα (ρ)) = b′′ (Sα (ρ))|∇Sα (ρ)|2 + b′ (Sα (ρ))∆Sα (ρ) together with the convexity of b implies ∆b(Sα (ρ)) ≥ b′ (Sα (ρ))∆Sα (ρ) a.e. in Ω′ × I. If we use the last inequality in equation (7.6.43), we obtain ∂t b(Sα (ρ)) + div [b(Sα (ρ))u] + [Sα (ρ)b′ (Sα (ρ)) − b(Sα (ρ))]div u −ǫ∆b(Sα (ρ)) ≤ b′ (Sα (ρ))rα (ρ, u) a.e. in Ω′ × I.

(7.6.44)

Letting α → 0+ , by using Lemma 6.5, the Lebesgue dominated convergence theorem, Vitali’s convergence theorem (here we have to take into account the growth conditions of b) and Lemma 6.7, we finally arrive at ∂t b(ρ) + div (b(ρ)u) + (ρb′ (ρ) − b(ρ))div u − ǫ∆b(ρ) ≤ 0 in D′ (Ω′ × I).

Since Ω′ , Ω′ ⊂ Ω, was an arbitrary bounded subdomain of Ω, the last inequality implies the desired result. 2

352


7.7

Galerkin approximation of the system with dissipation in the continuity equation and with artificial pressure

This section is devoted to the existence of the Galerkin approximation of the full system with dissipation in the continuity equation and with artificial pressure in the momentum equation. We will also derive estimates independent of the dimension of approximation. In other words, we will prove Proposition 7.34. The reader can consult Section 7.4.3 for the corresponding formulation and notation. 7.7.1

Preparatory calculations

In Sections 7.7.1–7.7.4, if not stated explicitly otherwise, c, c, c′ , c, d, di and di are generic positive constants independent of the initial conditions ρ0 , u0 (but which may, in particular, depend on n and on other parameters of the problem) while b, b are generic positive constants independent of ρ0 , u0 and n. Finally, L is a generic positive constant which is, in particular, independent of n (and which may depend on other parameters including the initial data ρ0 , u0 ). If we judge it useful for understanding the proofs, or if it is needed in the sequel, we shall indicate the dependence (or independence) of these constants on some of other parameters of the problem (β, γ, δ, ǫ, λ, µ, Ω, T , f and g) directly in the corresponding places in the text or explicitly in the argument of these constants. Given g ∈ C 0 (I, L1 (Ω)), ∂t g ∈ L1 (QT ), ess inf (x,t)∈QT g(x, t) ≥ a > 0, we define for all t ∈ I Mg(t) : Xn → Xn , Mg(t) v, w ≡ Realizing that

Ω

g(t)v · w, v, w ∈ Xn .

W k,p (Ω)-norms, k = 0, 1, . . ., 1 ≤ p ≤ ∞ are equivalent on Xn , we obtain Mg(t) L(Xn ,Xn ) ≤ c(n)

Ω

g(t), t ∈ I.

(7.7.1)

(7.7.2)

(7.7.3) (7.7.4)

It is easy to observe that M−1 g(t) exists for all t ∈ I and that 1 M−1 g(t) L(Xn ,Xn ) ≤ a .

(7.7.5)

By virtue of (7.7.4) and (7.7.5), we also have −1 M−1 g(t) Mg1 (t) Mg(t) L(Xn ,Xn ) ≤

c(n) a2

g1 (t)0,1 , t ∈ I.

(7.7.6)

For the differences, the following formula Mg2 (t) − Mg1 (t) L(Xn ,Xn ) ≤ c(n)(g2 − g1 )(t)0,1 , t ∈ I

(7.7.7)

−1 −1 −1 holds true. Due to the identity M−1 g2 − Mg1 = Mg2 (Mg1 − Mg2 )Mg1 , we get

GALERKIN APPROXIMATION −1 M−1 g2 (t) − Mg1 (t) L(Xn ,Xn ) ≤

c(n) a2 (g2

− g1 )(t)0,1 t ∈ I

353

(7.7.8)

for any g1 , g2 belonging to (7.7.1). We observe that −1 Mg v, w = v, Mg w, M−1 g v, w = v, Mg w,

∂t Mg v, w = Ω ∂t gv · w, v, w ∈ Xn .

(7.7.9)

Once we know these facts, we can calculate ∂t M−1 g v, w. This is the subject of the following exercise. Exercise 7.46 Let g ∈ W 1,1 (QT ), ess inf (x,t)∈QT g(x, t) ≥ a > 0. Prove that −1 −1 ′ ∂t M−1 g v, w = −Mg M∂t g Mg v, w in D (I), v, w ∈ Xn

and

−1 −1 −1 ∂t M−1 g v, w = Mg M∂t g Mg v, w + Mg ∂t v, w

in D′ (I), v ∈ C 1 (I, Xn ), w ∈ Xn .

(7.7.10)

(7.7.11)

Hint for (7.7.10): Use the identity 0 = ∂t v, φ = ∂t z, Mg φ + z, M∂t g φ in ∞ D′ (I) for all φ ∈ Xn , where z = M−1 g v, v ∈ Xn and g ∈ C (QT ), g > 0. Then −1 −1 verify that ∂t z, ψ = Mg M∂t g Mg v, ψ, where ψ = Mg φ. If g ∈ W 1,1 (QT ), g ≥ a a.e. in QT , use regularization and a density argument. 7.7.2

Galerkin approximation

We shall look for T ′ ∈ (0, T ] and u ∈ C 0 (I ′ , Xn ), I ′ = (0, T ′ ),

(7.7.12)

satisfying

Ω

ρ(t)u(t) · φ −

Ω

q0 · φ =

t

0

Ω

[µ∆u + (µ + λ)∇div u − ∇ργ − δ∇ρβ

−div (ρu ⊗ u) − ǫ(∇ρ · ∇)u + ρf + g] · φ,

(7.7.13)

t ∈ I ′ , φ ∈ Xn ,

where ρ(t) = [Sρ0 (u)](t) is the solution of problem (7.6.9)–(7.6.11) constructed in Proposition 7.39 and where q 0 = ρ0 u0 . Using Proposition 7.39, and the operator Mρ(t) , equation (7.7.13) can be rephrased as " #

t u(t) = M−1 [Sρ0 (u)](t) P q 0 + 0 P [N (Sρ0 (u), u)] ,

(7.7.14)

where P = Pn is the orthogonal projection of L2 (Ω) onto Xn introduced in Section 7.4.3 and N (ρ, u) = µ∆u + (µ + λ)∇div u − ∇ργ − δ∇ρβ −div (ρu ⊗ u) − ǫ∇ρ · ∇u + ρf + g.

(7.7.15)

354


We denote Aρ0 ,q0 = {t ∈ (0, T ]; there exists a unique (ρ, u) ∈

Rt × C 0 (It , Xn ) satisfying (7.7.13) and (7.6.9)–(7.6.11)},

(7.7.16)

where Rt is defined in (7.6.13) and It = (0, t). In the sequel we prove the existence and uniqueness part of Proposition 7.34 (see statements (i)–(iv)) by showing two auxiliary results. Auxiliary lemma 7.47 Aρ0 ,q0 is not empty. Auxiliary lemma 7.48 Aρ0 ,q0 = (0, T ]. The first statement is equivalent to the local existence and uniqueness in time of a solution (ρ, u) to (7.7.13), (7.6.9)–(7.6.11). The second statement claims that there is a global solution which is unique. Auxiliary lemma 7.47 is proved in the next section; the proof of Auxiliary lemma 7.48 is the subject of Section 7.7.4. 7.7.3

Local existence of solutions

In this section we prove the first auxiliary lemma. 7.7.3.1 Some auxiliary estimates First, we derive some auxiliary estimates which concern couples (ρ = Sρ0 (v), v), resp. (ρk = Sρ0 (v k ), v k ), k = 1, 2, where v resp. v k belong to the class C 0 (I, Xn ) and vC 0 (I,Xn ) ≤ K, resp. v k C 0 (I,Xn ) ≤ K, where K is a positive constant. From (7.7.15), by using equivalence (see (7.7.3)), we get P N (ρ, v)Xn ≤ c(n)[vXn + ρ0,∞ (vXn + v2Xn + f 0,∞ ) +g0,1 + ργ0,∞ + ρβ0,∞ ].

(7.7.17)

From (7.7.17) and (7.6.14) we deduce P [N (Sρ0 (v), v)](t)Xn ≤ d(K, ρ, T, f , g, n), t ∈ I,

(7.7.18)

where d is nondecreasing in the second variable. Using (7.7.15) along with formula

z F (z1 ) − F (z2 ) = z12 F ′ (s) ds, where F (s) = sγ and F (s) = sβ , we obtain N (ρ1 , v 1 ) − N (ρ2 , v 2 ), φ

= Ω [µ∆(v 1 − v 2 ) + (µ + λ)∇div (v 1 − v 2 )] · φ

ρ + Ω [ ρ12 (γsγ−1 + δβsβ−1 ) ds]div φ

+ Ω (ρ1 − ρ2 )ui1 uj1 ∂j φi + Ω ρ2 (ui1 − ui2 )uj1 ∂j φi

+ Ω ρ2 ui2 (uj1 − uj2 )∂j φi + ǫ Ω (ρ1 − ρ2 )(∆v 1 · φ + ∂j ui1 ∂j ψ i )

+ǫ Ω ρ2 [∆(v 1 − v 2 ) · φ + ∂j (ui1 − ui2 )∂j φi ] + Ω (ρ1 − ρ2 )f · φ.

(7.7.19)

GALERKIN APPROXIMATION

355

By virtue of (7.7.19), (7.6.14), due to elementary properties of the projections P = Pn (see Exercise 7.33) and using the equivalence of W k,p -norms on Xn , we get P [N (ρ1 , v 1 ) − N (ρ2 , v 2 )](t)Xn ≤ d(K, ρ, T, f , g, n){(v 1 − v 2 )(t)Xn + (ρ1 − ρ2 )(t)0,1 }, t ∈ I,

(7.7.20)

where d is again nondecreasing in the second variable. Due to (7.7.8) and (7.6.14), (7.6.17), we have −1 M[S −M−1 [Sρ0 (v 1 )](t) L(Xn ,Xn ) ρ0 (v 2 )](t) d(K, T, n) ρ0 1,2 t v 1 − v 2 C 0 (It ,Xn ) , ≤ ρ2

(7.7.21)

v 1 , v 2 ∈ C 0 (I, Xn ), t ∈ I, It = (0, t). Since

t2 P [N (Sρ0 (v), v)] − M−1 [Sρ0 (v )](t2 ) 0 P [N (Sρ0 (v), v)]

t1 = M−1 [Sρ0 (v )](t1 ) t2 P [N (Sρ0 (v), v)]

t2 −1 +{M−1 [Sρ0 (v )](t1 ) − M[Sρ0 (v )](t2 ) } 0 P [N (Sρ0 (v), v)],

M−1 [Sρ0 (v )](t1 )

t1 0

thanks to (7.7.5), (7.7.18) on one hand, and (7.6.13) on

· due to (7.7.8), (7.7.18), 0 Xn ). Even P [N (S (v), v)] ∈ C (I, the other hand, we find that M−1 ρ0 [Sρ0 (v )](·) 0 more simply, we observe that M−1 Sρ (v )

t

P [N (Sρ0 (v), v)]C 0 (I t ,Xn ) ≤ d1 (K, ρ, ρ, T, f , g, n) t, t ∈ I, (7.7.22) where d1 is nonincreasing in the second and nondecreasing in the third variable. 0 Similarly, but more easily, M−1 Sρ (v ) (P q 0 ) ∈ C (I, Xn ) and 0

0

0

−1 Kt M−1 e P q 0 Xn , t ∈ I. Sρ (v ) (P q 0 )C 0 (I t ,Xn ) ≤ ρ 0

7.7.3.2

Fixed point argument, existence of a local solution If 4 max

we take

(7.7.23)

" P q

0 Xn

ρ

# , u0 Xn < K,

T0 = T0 (K, ρ, ρ, T, f , g, n) = min

" ln 2 K

K ,T , 2d 1

(7.7.24) #

(7.7.25)

so that T0 is nondecreasing in the second and nonincreasing in the third variable. K With this choice we have ρ−1 eKT0 P q 0 Xn < K 2 and d1 T0 < 2 . Therefore, by virtue of (7.7.14), (7.7.22), (7.7.23), the mapping

356


Tρ0 ,q0 : C 0 (I τ0 , Xn ) → C 0 (I τ0 , Xn ), " #

t Tρ0 ,q0 (w) := M−1 Sρ (w) P q 0 + 0 P [N (Sρ0 (w), w)]

(7.7.26)

0

maps

" BK,τ0 = w ∈ C 0 (I τ0 , Xn ); wC 0 (I τ

0

,W 1,∞ (Ω))

≤K

#

(7.7.27)

into itself, for any 0 < τ0 ≤ T0 . In the next step we prove that Tρ0 ,q 0 is a contraction. Due to the formula −1 −1 −1 −1 M−1 ρ1 (w 1 ) − Mρ2 (w 2 ) = (Mρ1 − Mρ2 )(w 1 ) + Mρ2 (w 1 − w 2 ) and due to (7.7.26), we get the identity Tρ0 ,q0 (w1 ) − Tρ0 ,q0 (w2 )

t −1 = (M−1 Sρ0 (w1 ) − MSρ0 (w2 ) ){P q 0 + 0 P [N (Sρ0 (w 1 ), w 1 )]}

t +M−1 Sρ (w2 ) 0 {P [N (Sρ0 (w 1 ), w 1 )] − P [N (Sρ0 (w 2 ), w 2 )]}.

(7.7.28)

0

We apply (7.7.24),(7.7.21), (7.7.18) to bound the first term and (7.7.5), (7.7.20), (7.6.14), (7.6.17) to majorize the second term. We finally get Tρ0 ,q0 (w1 ) − Tρ0 ,q0 (w2 )Xn (t) ≤ d2 (K, ρ, ρ, f , g, n)(1 + ρ0 1,2 ) t w1 − w2 C 0 (I t ,Xn ) ,

(7.7.29)

t ∈ Iτ0 , w1 , w2 ∈ BK,τ0 , where d2 is nonincreasing in the second and nondecreasing in the third variable. If we take 0 ,τ1 } T = min{τ where τ1 < d12 , 1+ρ0 1,2 ,

then Tρ0 ,q0 maps BK,T ⊂ C 0 (I T , Xn ) into itself and it is a contraction. It therefore possesses in BK,T a unique fixed point u, which satisfies (7.7.14). The couple (ρ = Sρ0 (u), u) fulfils (7.7.13) and (7.6.9)–(7.6.11), or equivalently (7.7.14) and (7.6.9)–(7.6.11). Existence is thus proved. We observe that T has the form T =

d3 (K,ρ,ρ,f ,g ,T,n) , 1+ρ0 1,2

(7.7.30)

where d3 is nondecreasing in ρ and nonincreasing in ρ. 7.7.3.3 Local uniqueness If (ω = Sρ0 (v), v) ∈ RT × C 0 (I T , Xn ) is another solution to problem (7.7.13) and (7.6.9)–(7.6.11) with the same initial data, then a similar calculation to that leading to (7.7.29) suggests v(t) − u(t)Xn ≤ κv − uC 0 (It ,Xn ) , t ∈ IT , where 0 < κ < 1. We conclude that (ρ, u) = (ω, v) on QT thus completing the proof of Auxiliary lemma 7.47.


7.7.4

357

Existence of maximal solutions

In this section we prove the second Auxiliary lemma. In the sequel, T ′ = sup Aρ0 ,q0 and I ′ = (0, T ′ ).

(7.7.31)

The solution (ρ, u) ∈ RT ′ × C 0 (I ′ , Xn ) is called a maximal solution. 7.7.4.1 Further regularity of u We firstly need to improve the regularity of u in the variable t. Due to (7.7.14) and by virtue of (7.7.11) " #

t −1 (7.7.32) ∂t u = M−1 P q 0 + 0 P [N (ρ, u)] + M−1 ρ P [N (ρ, u)]. ρ M∂t ρ Mρ

Using (7.7.5), (7.7.6) and (7.7.18) together with (7.6.13), (7.6.14) to evaluate the last expression, one sees that ∂t u ∈ L2 (I ′ , Xn ).

(7.7.33)

7.7.4.2 Energy inequality for maximal solution Now we are in a position to derive the energy inequality on I ′ . Differentiating (7.7.13) with respect to t and integrating by parts, one gets

∂ ρu · φ + Ω ρ∂t u · φ − Ω ρui uj ∂j φi + µ Ω ∂i uj ∂i φj Ω t

+(µ + λ) Ω div u div φ + Ω ∇ργ · φ + δ Ω ∇ρβ · φ + ǫ Ω (∇ρ · ∇)u · φ

= Ω (ρf + g) · φ, φ ∈ Xn , a. e. in I ′ . (7.7.34) Using integration by parts, employing (7.6.9), thanks to regularity (7.6.13), (7.7.12) and (7.7.33) for (ρ, u), we easily justify the following identities

∂ (ρu) · u − Ω ρui uj ∂j ui = Ω ∂t ρ|u|2 + 21 Ω ρ∂t |u|2 − 12 Ω ρui ∂i |u|2 Ω t

= 21 Ω ρ∂t |u|2 + 12 Ω ∂t ρ|u|2 + 2ǫ Ω ∆ρ|u|2

= 21 ∂t Ω ρ|u|2 − 2ǫ Ω ∇ρ · ∇|u|2 a.e. in I ′ .

Similarly as in (4.8.12), since Ω ργ−1 ∆ρ ≤ 0 in I ′ , we have

γ−1 γ γ ∇ργ · u = γ−1 ρu · ∇ργ−1 = − γ−1 ρ div (ρu) Ω Ω Ω

γ

γ−1 ǫγ 1 1 ∆ρ ≥ γ−1 ∂t Ω ργ a e. in I ′ . = γ−1 ∂t Ω ρ − γ−1 Ω ρ Adapting slightly the reasoning of (4.8.11), we find

β−1

ǫβ 1 ∇ρβ · u = β−1 ∂t Ω ρβ − β−1 ρ ∆ρ Ω Ω

1 ∂t Ω ρβ + ǫβ Ω ρβ−2 |∇ρ|2 a e. in I ′ . = β−1

Last but not least,

358


ǫ

Ω

ǫ 2

∂i ρ∂i u · u =

Ω

(∇ρ · ∇)|u|2 .

Thus, using φ = u in (7.7.34), we arrive at d dt Eδ (ρ, q)

≤

+µ

Ω

Ω

|∇u|2 + (µ + λ)

(ρf + g) · u a.e. in I ′ ,

Ω

|div u|2 + ǫδβ

Ω

ρβ−2 |∇ρ|2

(7.7.35)

where Eδ is the total energy defined in (7.4.12). This is nothing but the energy inequality (7.4.52) on I ′ . 7.7.4.3 Global estimates of maximal solution The goal of this section is to provide uniform bounds in (0, T ′ ) for u and q = ρu. We majorize the right-hand side of (7.7.35) by 1

1

2 2 f 0, ρ|u|2 0,1 ρ0,γ

2γ γ−1

+ u0,6 g0, 56 ,

(7.7.36)

and apply the Young inequality to it several times to get the estimate 1 2 2 ρ|u| 0,1

+ 12 ρ0,γ f 20,

≤ 12 ρ|u|2 0,1 + +ϑu20,6 +

+ ϑu20,6 +

2γ γ−1

γ 1 γ−1 ρ0,γ

1 2 4ϑ g0, 65 ,

+

1 γ−1 2γ ′ ( 2γ )

1 2 4ϑ g0, 56

γ′ γ

′

f 2γ 0, 2γ

(7.7.37)

γ−1

ϑ > 0.

Then we choose suitable ϑ(λ, µ) and use the Poincaré type inequality 2b2 u21,2 ≤ µ∇u20,2 +(µ + λ)div u20,2 , in order to transform (7.7.35) to the form β−2

+ b2 u21,2 + ǫδβρ 2 ∇ρ20,2 ′ ≤ Eδ (ρ, q) + b g20, 6 + f 2γ , 2γ 0,

d dt Eδ (ρ, q)

5

(7.7.38)

γ−1

where b(λ, µ, Ω) and b(γ, λ, µ, Ω) > 0. By virtue of (7.7.16), Eδ (ρ, q) ∈ C 0 (I) and, of course, [Eδ (ρ, q)](0) = Eδ (ρ0 , q 0 ). Hence, by Gronwall’s lemma, we obtain

t

t β−2 u21,2 + ǫδβ 0 ρ 2 ∇ρ20,2

t ′ ≤ E δ,0 et + b 0 g20,2 + f 2γ (s)e(t−s) ds 0, 2γ

Eδ (ρ, q)(t) + b2

0

(7.7.39)

γ−1

≡ G(t, E δ,0 )

for all t ∈ I ′ and for (ρ0 , u0 ) such that Eδ (ρ0 , q 0 ) ≤ E δ,0 , where E δ,0 is a given positive constant. (G depends also on f , g but this dependence is irrelevant for


359

the purpose of our proof.) We observe that G = G(t, s) is increasing in both variables. Therefore, inequality (7.7.39), in particular, implies

and

Ω

ρ|u|2 (t) ≤ 2G(T, E δ,0 ), t ∈ I ′

uL2 (I ′ ,W 1,2 (Ω)) ≤

1 b

)

G(T, E δ,0 ).

(7.7.40)

(7.7.41)

Using inequality u1,∞ ≤ c(n)u1,2 , u ∈ Xn (see (7.7.3)), from (7.6.14), we obtain √ T′ ρ ≥ ρe−c 0 u(s)1,2 ds ≥ ρe−c T uL2 (I ′ ,W 1,2 (Ω)) (7.7.42) √ √ c ≥ ρe− c T G(T,E δ,0 ) ≡ d4 (ρ, E δ,0 , T, n). By (7.7.40) and (7.7.42), we have u2L∞ (I ′ ,L2 (Ω)) ≤

2 G(T, E δ,0 ). d4

(7.7.43)

From this and due to the inequality u0,2 ≥ cu1,∞ on Xn , one arrives at uC 0 (I ′ ,W 1,∞ (Ω))

1 ≤ c

4

2G(T, E δ,0 ) ≡ d5 (ρ, E δ,0 , T, n). d4

(7.7.44)

Moreover, since u1,∞ ≥ c′ uXn on Xn , one also gets uC 0 (I ′ ,Xn ) ≤ d5 (ρ, E δ,0 , T, n).

On the other hand, by the second inequality (7.6.14), ρ ≤ ρec

T′ 0

≤ ρe(c/b)

u(s)1,2 ds

√ T uL2 (I ′ ,W 1,2 (Ω))

≤ ρec

√ √ T G(T,E δ,0 )

≡ d4 (ρ, E δ,0 , T, n).

(7.7.45)

Therefore, by (7.7.40), (7.7.43) and (7.7.45), one gets ρuC 0 (I ′ ,L2 (Ω)) ≤

4

G(T, 2E δ,0 ) c √T ρe b d4

√

G(T,E δ,0 )

≡ d6 (ρ, ρ, E δ,0 , T, n) (7.7.46)

and Pn (ρu)C 0 (I ′ ,Xn ) ≤ d6 (ρ, ρ, E δ,0 , T, n).

Now, everything is ready for proving that T ′ = T . This is done in the next section.

360


7.7.4.4 Global existence and uniqueness of solutions Let us fix (ρ0 , u0 ) in class (7.4.47) and fix a constant K satisfying (7.7.24). We set # " > 4 max d5 , d6 . K d4

(7.7.47)

Suppose that T ′ < T . Let Tk be an increasing sequence converging to T ′ . We can take subsequently ρ0 = ρ(Tk ), q 0 = ρ(Tk )u(Tk ) as initial conditions for problem (7.7.13) and (7.6.9)–(7.6.11). By virtue of (7.7.44), (7.7.46) and (7.7.47), in place of K. condition (7.7.24) holds with these new initial values and with K We can thus repeat the reasoning of Sections 7.7.3.2 and 7.7.3.3. In this way we construct couples (ρk , uk ) = (Sρ(Tk ) (uk ), uk ) ∈ RTk × C 0 (ITk , Xn ), which solve (on ITk ) system (7.6.9)–(7.6.11) and (7.7.13) with ρ0 replaced by ρ(Tk ) and q 0 replaced by ρ(Tk )u(Tk ). According to (7.7.30) and by virtue of (7.6.14), (7.6.15), we have min ρ(Tk ), max ρ(Tk ), T ) d4 , d4 , T ) d3 (K, d3 (K, Ω Ω ≥ T0 = Tk = . c 2 1 + ρ(Tk )1,2 1 + cρ0 1,2 e 2ǫ (d5 +d5 )T Notice that T0 is independent of k. Now, we define a sequence of couples (ωk , v k ) by putting

(ρ(t), u(t)), t ∈ [0, Tk ), (ρk (t − Tk ), uk (t − Tk )), t ∈ [Tk , min{T, Tk + T0 }). (7.7.48) Of course (ωk , uk ) ∈ Rmin{T,Tk +T0 } × C 0 (Imin{T,Tk +T0 } , Xn ). Moreover, due to local uniqueness, (ωk , v k ) coincides with (ρ, u) on (Tk , min{T ′ , Tk + T0 }), and it solves (7.6.9)–(7.6.11) and (7.7.13) with the initial conditions ρ0 , q 0 on [0, T ′ ) and with the initial conditions ρ(Tk ), q(Tk ) on (Tk , min{T, Tk + T0 }). Therefore, it solves system (7.6.9)–(7.6.11) and (7.7.13) on (0, min{T, Tk + T0 }). Since there exists k0 such that Tk0 + T0 > T ′ , there exists a solution belonging to Rmin{T,Tk +T0 } × C 0 (Imin{T,Tk +T0 } , Xn ). This contradicts assumption (7.7.31). We have thus proved that T ′ = T . Due to the local uniqueness, this solution is unique. Auxiliary Lemma 7.48 is thus proved. By now, we have proved statements (i)–(iv) of Proposition 7.34. It remains to prove the energy inequalities and derive estimates independent of n. This will be done in the next section. (ωk (t), v k (t)) =

7.7.5 Energy inequalities and estimates 7.7.5.1 Differential and integral form of energy inequalities Since T ′ = T , the differential form (7.4.52) of the energy inequality is nothing but (7.7.35). If we integrate it by using Lemma 1.7, we get, similarly as in (7.7.46),

t

t

t

[Eδ (ρ, q)](t) + µ 0 Ω |∇u|2 + (µ + λ) 0 Ω |div u|2 + ǫδβ 0 Ω ρβ−2 |∇ρ|2

t

≤ Eδ (ρ0 , q 0 ) + 0 Ω (ρf + g) · u, t ∈ I. (7.7.49)

COMPLETE SYSTEM WITH DISSIPATION

361

This finishes the proof of (7.4.53). 7.7.5.2 Estimates independent of the dimension of the Galerkin approximation Our goal in this section is to derive estimates independent of n. We come back to (7.7.38) and realize that the coefficients b and b really depend at most on γ, λ, µ and Ω. In particular, they are independent of n. Thus, estimate (7.7.39) yields bounds (7.4.54)–(7.4.58). From (7.4.58), by imbedding W 1,2 (Ω) ֒→ L6 (Ω), we deduce ρβn L1 (I,L3 (Ω)) ≤ L(E 0 , ǫ, δ, f , g, T ), where E 0 = supδ∈(0,1) E(ρ0 , q 0 ). By interpolation (see Theorem 1.49), ρβn 0,2 ≤ 1/4

3/4

ρβn 0,1 ρβn 0,3 , therefore, by virtue of (7.4.57),

ρβn L4/3 (I,L2 (Ω)) ≤ L(E 0 , ǫ, δ, f , g, T ), which implies, in particular, (7.4.59). Testing

(7.6.9), where we write

(ρn , un ) on place of (ρ, u), by ρ , and using the identity div (ρ u )ρ = − ρ u ·∇ρn = n n n n Ω Ω n n

2 1 ρ div u , we obtain n 2 Ω n 1 2 2 ρn (t)0,2

+ǫ

t 0

∇ρn 20,2 = 21 ρ0 20,2 −

1 2

t

0

Ω

ρ2n div un , t ∈ I.

(7.7.50)

√ The right-hand side is majorized by ρn L∞ (I,L4 (Ω)) T div un L2 (QT ) . By virtue of (7.4.54) and (7.4.57), it is bounded on condition β ≥ 4. This implies estimate (7.4.60) and completes the proof of Proposition 7.34. 7.8

Complete system with dissipation in the continuity equation and with artificial pressure

In this section we prove Proposition 7.31 by letting n → ∞ in Proposition 7.34. The first section deals with the modified continuity equation (statement (iii) of Proposition 7.31). The second one concerns the momentum equation (statement (ii) of Proposition 7.31). In these two parts we also prove statements (i) and (iv)). The third section is devoted to energy inequalities and estimates independent of the dissipation parameter ǫ, except estimate (7.4.39) which is proved in the last section. We recall that I = (0, T ) where T > 0. If not stated explicitly otherwise, in this section, c > 0 denote generic constants which depend on λ, µ, Ω and eventually on the coefficients of Sobolev spaces. In particular, they are independent of δ, ǫ (and of ρ0 , q 0 , f , g, T ). Generic positive constants which depend on ρ0 , q 0 , f , g, T , δ, ǫ (and possibly on β, γ, λ, µ, Ω) are denoted by L. The dependence of constants L on one of parameters ρ0 , q 0 , f , g, T , δ, ǫ is always written explicitly in their argument. We underline, that both c’s and L’s are independent of n.

362


7.8.1

Limit in the modified continuity equation

By (7.4.54), (7.4.57) and (7.4.60), at least for some chosen subsequences un → u weakly in L2 (I, (W01,2 (Ω))3 ),

(7.8.1)

ρn → ρ weakly-∗ in L∞ (I, Lβ (Ω)),

(7.8.2)

∇ρn → ∇ρ weakly in (L (QT )) .

(7.8.3)

2

Since

1/2

Ω

1/2

ρn un · φ ≤ ρn 0,β ρn |un |2 0,1 φ0,

2β β−1

3

(on condition β > 1), one gets

by (7.4.56), (7.4.57)

ρn un

2β

L∞ (I,L β+1 (Ω))

≤ L(E 0 , δ, f , g, T ),

(7.8.4)

where E 0 = supδ∈(0,1) Eδ (ρ0 , q 0 ). From (7.4.50), we obtain ∂t

Ω

ρn η −

Ω

ρn uin ∂i η + ǫ

Ω

∇ρn · ∇η = 0 in D′ (I), η ∈ C ∞ (Ω).

(7.8.5)

Therefore, by virtue of (7.8.4) and (7.4.60), the sequence ρn is uniformly contin2β uous in W −1, β+1 (Ω) on I. Since it is also uniformly bounded in Lβ (Ω), we have by Lemma 6.2 ρn → ρ in C 0 (I, Lβweak (Ω)), (7.8.6) at least for a chosen subsequence. This gives sense to the initial conditions for ρ and implies, in particular, the first formula (7.4.30) for initial data. By virtue of (7.8.4), (7.8.5) and (7.4.60), ∂t ρn

L2 (I,[W

1,

2β β−1

(Ω)]∗ )

≤ L(E 0 , δ, f , g, T ).

(7.8.7)

Using (7.4.60) and the compact imbedding W 1,2 (Ω) ֒→ L2 (Ω), we conclude by the Aubin–Lions lemma (see Theorem 1.71), that, after eventual selection, ρn → ρ strongly in L2 (QT ). After this, by interpolation and (7.4.59), we obtain

By the inequality

ρn → ρ strongly in Lp (QT ), 1 ≤ p < 43 β. Ω

ρn un ·φ ≤ ρn 0,β |un 0,6 φ0,

6β 5β−6

(7.8.8)

(on condition β > 6/5),

and by (7.4.54), (7.4.57)

ρn un

6β

L2 (I,L β+6 (Ω))

≤ L(E 0 , δ, f , g, T ).

(7.8.9)

By virtue of (7.8.1), (7.8.4), (7.8.8) and (7.8.9), 6β

2β

ρn un → ρu weakly in L2 (I, (L β+6 (Ω))3 ), weakly-∗ in L∞ (I, (L β+1 (Ω))3 ). (7.8.10) Formulae (7.8.2), (7.8.3) and (7.8.10) make it possible to pass to the limit in (7.4.50) and obtain the modified continuity equation (7.4.29).


7.8.2

363

Limit in the momentum equation

Due to interpolation, for any β ≥ 4 there exists 2 < r(β) < ρn un θ0, ρn un 0,r(β) ≤ ρn un 1−θ 0, 2β β+1

(in fact

34 15

≤ r(β) =

10β−6 3β+3

3/2),

ρ ∈ C 0 (I, Lp (Ω))3 , 1 ≤ p < β. Ω

ρn un uin · φ ≤ ρn un 0,

ρn un un

6β

L2 (I,L 4β+3 (Ω))

2β β+1

un 0,6 φ0,

(7.8.16) 6β 2β−3

≤ L(E 0 , δ, f , g, T ).

(on condition (7.8.17)

By virtue of the identity ∂t (P a) = P ∂t a a.e. in QT , which holds for all a ∈ (L2 (QT ))3 , ∂t a ∈ (L2 (QT ))3 , equation (7.4.49) can be rephrased in the form

364

∂t


Ω

Pn (ρn un ) · φ =

Ω

[ρn un ujn · ∂j (Pn φ) − µ∇un · ∇(Pn φ)

−(µ + λ)div un div (Pn φ) + (ργn + δρβn )div (Pn φ) − ǫ(∇ρn · ∇)un · (Pn φ)]

+ Ω (ρn f + g) · (Pn φ), t ∈ I, φ ∈ D(Ω). (7.8.18) From (7.8.18) when using the definition of the W −2,2 -norm, Exercise 7.33, the H¨ older inequality, estimates (7.4.54), (7.4.57), (7.4.59), (7.8.13), (7.8.17) and Sobolev’s imbeddings, we obtain ∂t Pn (ρn un )Lt(β) (I,W −2,2 (Ω)) ≤ L(E 0 , δ, ǫ, f , g, T ).

(7.8.19)

Since Pn (ρn un )−1,2 ≤ Pn (ρn un )0,2 ≤ ρn un 0,2 ≤ cρn un 0,r(β) , we have Pn (ρn un )Lr(β) (I,W −1,2 (Ω)) ≤ Pn (ρn un )Lr(β) (I,L2 (Ω)) ≤ L(E 0 , δ, f , g, T ). (7.8.20) Due to the compact imbedding L2 (Ω) ֒→ W −1,2 (Ω), (7.8.19), (7.8.20) and the Aubin–Lions lemma, the sequence Pn (ρn un ) converges strongly in L2 (I,(W −1,2 (Ω))3 ) to a limit which, by virtue of (7.8.10), has to be ρu. After this we write ρu − ρn un = (ρu − Pn (ρn un )) +(Pn (ρn un ) − ρn un ) and apply to the second term the last statement of Exercise 7.33. We thus get ρn un → ρu strongly in L2 (I, (W −1,2 (Ω))3 ).

(7.8.21)

This convergence, (7.8.1) and (7.8.17) imply 6β

ρn uin ujn → ρui uj weakly in L2 (I, L 4β+3 (Ω)).

(7.8.22)

In accordance with Remark 7.32, we observe that η = ρ = ρǫ , is an admis2β which sible test function in (7.4.29) (on conditions t(β) ≥ β ′ and r(β) ≥ β−1 are certainly satisfied for β ≥ 8). Using Theorem 1.67 we obtain, after some calculation,

t

2 1 1 t 1 2 2 2 2 ρ(t)0,2 + ǫ 0 ∇ρ0,2 = 2 ρ0 0,2 − 2 0 Ω ρ div u, t ∈ I. Integrating this identity over t ∈ I, one gets

T

(T − t)∇ρ(t)20,2,Ω ) dt

T = T2 ρ0 20,2,Ω − 12 0 (T − t) Ω ρ2 (t)div u(t) dt. 1 2 2 ρ0,2,QT

+ǫ

0

(7.8.23)

On the other hand, integrating (7.7.50) over I and passing to the limit n → ∞, one gets thanks to (7.8.1), (7.8.8),

T + ǫ limn→∞ 0 (T − t)∇ρn (t)20,2,Ω ) dt

T = T2 ρ0 20,2 − 21 0 (T − t) Ω ρ2 (t)div u(t) dt.

1 2 2 ρ0,2,QT

(7.8.24)


365

Comparing (7.8.23) and (7.8.24) yields limn→∞

T 0

T (T − t)∇ρn (t)20,2 dt = 0 (T − t)∇ρ(t)20,2 dt.

This implies |∇ρn |2 → |∇ρ|2 in D′ (QT ) which in turn with (7.8.3) and (7.8.12) yields (7.8.25) ∇ρn → ∇ρ strongly in L2 (QT ). By (7.8.1), (7.8.25) and (7.8.13), we finally obtain ∇ρn · ∇un → ∇ρ · ∇u in Lt(β) (QT ).

(7.8.26)

We are now ready to prove that (ρ, u) satisfies the momentum equation (7.4.28). Indeed, (7.4.28) is obtained as n → ∞ in (7.4.49), as follows: In the first term, we use (7.8.10); to get the limit of the second term, one employs (7.8.22); and the limit in the third and fourth terms follows from (7.8.1). In the fifth and sixth terms, one uses (7.8.8). Finally, the limit of the last term at the left-hand side is obtained from (7.8.26). The limit of the right-hand side is obtained with the help of (7.8.2). Finally, we are in a position to show the continuity in time of ρu. From (7.8.10), (7.4.28), (7.4.54)–(7.4.60) and (7.8.26) we easily see that 2β

ρu ∈ L∞ (I, (L β+1 (Ω))3 ), ∂t

Ω

ρu · φ ∈ L1 (I), φ ∈ D(Ω).

In accordance with Exercise 6.3 (or see directly Lemma 1.7), there exists q ∈ 2β

β+1 (Ω))3 ), such that for a.e. t ∈ I, q(t) = ρ(t)u(t) a.e. in Ω. Therefore, C 0 (I, (Lweak in agreement with Lemma 7.17, changing u eventually on a set of measure 0 in 2β

β+1 (Ω))3 ). This yields the remaining information I, we find that ρu ∈ C 0 (I, (Lweak in (7.4.27) and completes the proof of the second formula (7.4.30). So far, we have proved statements (i)–(iv) of Proposition 7.31. It remains to prove energy inequalities and estimates independent of ǫ. This is the subject of the next section, except for estimate (7.4.39) whose proof is more involved. It is performed in Section 7.8.4.

7.8.3

Limit in the energy inequality and estimates independent of vanishing dissipation

By virtue of (7.8.8) and (7.8.22),

E (ρ , q n ) → Ω Eδ (ρ, q) in D′ (I). Ω δ n

Due to the lower weak semicontinuity of convex functionals (see Lemma 3.32), we have

T

T

ψ Ω |∇u|2 ≤ lim inf n→∞ 0 ψ Ω |∇un |2 , 0

T

T

ψ Ω |div u|2 ≤ lim inf n→∞ 0 ψ Ω |div un |2 , 0

366


where ψ ∈ C 0 (I) is a nonnegative function. By virtue of (7.8.25) and (7.8.8), |∇ρn |2 → ρβ−2 |∇ρ|2 a.e. in QT (at least for a chosen subsequence), cf. ρβ−2 n Lemma 1.13. Therefore by Fatou’s lemma (see Lemma 1.14), we get

T β−2

T

ψ Ωρ |∇ρ|2 ≤ lim inf n→∞ 0 ψ Ω ρβ−2 |∇ρn |2 . n 0

Using these facts and (7.8.1), (7.8.8) in (7.4.52) we obtain (7.4.31). With the same argument applied to (7.4.53), we obtain (7.4.32) in D′ (I). However, since Eδ (ρ, q) ∈ L∞ (I), this inequality holds a.e. in I as well. Estimates (7.4.33)–(7.4.36) and (7.4.41) follow directly from (7.4.54)–(7.4.57) and (7.4.60) by the lower weak semicontinuity of norms. Estimates (7.4.37), (7.4.38), (7.4.40) and (7.4.42) follow from (7.8.4), (7.8.9), (7.8.11), (7.8.17), (7.8.12) and (7.8.13) by the same means. In the next section, we will concentrate on (7.4.39). 7.8.4 Improved estimate of density Due to (7.4.36) and Lemma 3.17, the function (7.8.27) ϕ(t, x) = ψ(t)φ(t, x), φ = B(ρǫ − m), ψ ∈ D(I),

1,4 m where m = |Ω| and m = Ω ρ0 = Ω ρǫ , belongs to L4 (I, W0 (Ω)). We easily find that ∂t ϕ = ψ ′ B(ρǫ − m) + ψB(∂t ρǫ ) (7.8.28) = ψ ′ B(ρǫ − m) + ψB(div (ǫ∇ρǫ − ρǫ uǫ )). 2,t(β)

(Ω) for a.a. t ∈ I. Therefore, By virtue of (7.8.15), ǫ∇ρǫ (t) − ρǫ (t)uǫ (t) ∈ E0 again by Lemma 3.17, ∂t φ ∈ L2 (QT ). In accordance with Remark 7.32, ϕ is an admissible test function for the momentum equation (7.4.28). By Lemma 3.17 (see also Remark 3.19), we have φ0,p,Ω + ∇φ0,p,Ω ≤ c(p, β, Ω)ρǫ 0,p,Ω , 1 < p ≤ β, ∂t φ0,p,Ω ≤ c(p, β, Ω)[ρǫ 0,p,Ω + ǫ∇ρǫ 0,p,Ω

(7.8.29)

+ρǫ uǫ 0,p,Ω ], 1 < p ≤ r(β), where p is defined in (1.3.64). When using ϕ in (7.4.28) as a test function, after some elementary calculations which consist mostly of integration by parts, we obtain

T

ψ Ω (ργǫ + δρβǫ )ρǫ 0

T

= m 0 ψ Ω (ργǫ + δρβǫ )

T

T

+(µ + λ) 0 ψ Ω ρǫ div uǫ − (µ + λ)m 0 ψ Ω div uǫ (7.8.30)

T

T

T

+µ 0 ψ Ω ∂j uiǫ ∂j φi − 0 ψ ′ Ω ρǫ uiǫ φi − 0 ψ Ω ρǫ uiǫ ujǫ ∂j φi

T

T

− 0 ψ Ω ρǫ uiǫ ∂t φi + ǫ 0 ψ Ω ∂i ρǫ ∂i ujǫ φj

T

9 − 0 ψ Ω (ρǫ f + g) · φ = k=1 Ik .


367

The integrals at the right-hand side are handled as follows: (i) By using (7.4.34) and (7.4.36) we get |I1 | = |m

T 0

ψ

Ω

(ργǫ + δρβǫ )| ≤ |ψ|C 0 (I) L(E 0 , f , g, T ).

(ii) By H¨ older’s inequality and by virtue of (7.4.33) and (7.4.36),

T

|I2 | = |(µ + λ) 0 ψ Ω ρǫ div uǫ |

T ≤ 0 |ψ|ρǫ 0,2 div uǫ 0,2 ≤ |ψ|C 0 (I) L(E0 , δ, f , g, T ).

(iii) In virtue of (7.4.33), there holds

T

|I3 | = |(µ + λ)m 0 ψ Ω div uǫ | ≤ |ψ|C 0 (I) L(E 0 , f , g, T ).

(iv) By H¨ older’s inequality, (7.4.33), (7.4.36) and (7.8.29), we have

T

|I4 | = |µ 0 ψ Ω ∂j uiǫ ∂j φi | ≤ c|ψ|C 0 (I) ∇uǫ 0,2,QT ∇φ0,2,QT

≤ c|ψ|C 0 (I) ∇uǫ 0,2,QT ρǫ 0,2,QT ≤ |ψ|C 0 (I) L(E0 , δ, f , g, T ).

(v) |I5 | = | ≤c

T 0

T 0

ψ′

Ω

ρǫ uiǫ φi | ≤ 1

T 0

√ √ |ψ ′ | ρǫ uǫ 0,2 ρǫ 0,2β φ0,

2β β−1

1

2 2 ρǫ 0,β ρǫ 0, |ψ ′ | ρǫ |uǫ |2 0,1 1

2β β−1

3/2

≤ cψ ′ L1 (I) ρǫ |uǫ |2 L2 ∞ (I,L1 (Ω)) ρǫ L∞ (I,Lβ (Ω)) ≤ ψ ′ L1 (I) L(E 0 , δ, f , g, T ).

In (v), the first inequality is H¨ older’s inequality, and the second one holds due to 2β (7.8.29). Then we use the imbedding Lβ (Ω) ֒→ L β−1 (Ω). The last bound follows from (7.4.35) and (7.4.36). (vi)

T

T

|I6 | = | 0 ψ Ω ρǫ uiǫ ujǫ ∂j φi | ≤ c 0 |ψ| ρǫ |uǫ |2 0, 6β ∇φ0, 6β 4β+3 2β−3

T ≤ c 0 |ψ| ρǫ |uǫ |2 0, 6β ρǫ 0, 6β ≤ |ψ|C 0 (I) L(E 0 , δ, f , g, T ). 4β+3

2β−3

Here we used Hölder’s inequality, then estimate (7.8.29); the final bound holds by virtue of (7.4.38) and (7.4.36). (vii)

T

T |I7 | = | 0 ψ Ω ρǫ uiǫ ∂t φi | ≤ c 0 |ψ| ρǫ uǫ 0, 6β ∂t φ0, 6β β+6 5β−6

T ≤ c 0 |ψ| ρǫ uǫ 0, 6β ρǫ 0, 6β + ǫ∇ρǫ 0, 6β + ρǫ uǫ 0, 6β β+6

≤ |ψ|C 0 (I) L(E 0 , δ, f , g, T ).

5β−6

5β−6

5β−6

368


The first inequality is H¨ older’s inequality, in the next inequality we used (7.8.29), 6β and the final bound holds due to the imbeddings L2 (Ω) ֒→ L 5β−6 (Ω), Lβ (Ω) ֒→ 6β 6β 6β L 5β−6 (Ω), L β+6 (Ω) ֒→ L 5β−6 (Ω) and estimates (7.4.36), (7.4.37), (7.4.41). (viii) T

T |I8 | = ǫ 0 ψ Ω ∂i ρǫ ∂i ujǫ φj ≤ cǫ 0 |ψ| ∇ρǫ 0,2 ∇uǫ 0,2 φ0,∞

T ≤ cǫρǫ L∞ (I,Lβ (Ω)) 0 |ψ| ∇uǫ 0,2 ∇ρǫ 0,2 ≤ |ψ|C 0 (I) L(E 0 , δ, f , g, T ).

In (viii), the first inequality is H¨ older’s inequality, the bound φL∞ (QT )) ≤ c ρǫ L∞ (I,Lβ (Ω)) holds due to (7.8.29) and yields the second inequality. The final bound is a consequence of (7.4.33), (7.4.36) and (7.4.41). (ix) The last integral can be estimated by H¨ older’s inequality, (7.8.29), employing ′ the imbedding Lβ (Ω) ֒→ Lβ (Ω) and (7.4.36), as follows T

T |I9 | = 0 ψ Ω (ρǫ f + g) · φ ≤ c 0 |ψ| ρǫ 0,β f 0,∞ φ0,β ′ + g0,∞ φ0,β ≤ c|ψ|C 0 (I) (1 + ρǫ L∞ (I,Lβ (Ω)) )ρǫ L∞ (I,Lβ (Ω)) (f 0,∞ + g0,∞ ).

Estimates (i)–(ix) hold in particular with any ψ = ψm , where ψm ∈ D(I), m ∈ IN, 0 ≤ ψm ≤ 1, 1 1 ψm (t) = 1, t ∈ [ m ,1 − m ], |ψ ′ | ≤ 2m.

(7.8.31)

If we put them together and if we realize that ψm → 1 pointwise in I and ′ that ψm L1 (I) (which appears in estimate (v)) is bounded by 4T , we arrive at (7.4.39). The proof of Proposition 7.31 is thus complete. 28 7.9

Complete system with artificial pressure

In this section, we prove Proposition 7.27 by letting ǫ → 0+ in Proposition 7.31. We are for the first time confronted with a missing estimate for the sequence of densities which would guarantee strong convergence. This problem is closely related to the problem of identification the limiting pressure with the pressure corresponding to the limiting density. To overcome these difficulties, we shall apply to system (7.4.22)–(7.4.24) Lions’ approach described in Section 7.3. In the first section, we deal with weak limits of modified continuity and momentum equations. Using results of Section 6.2, we prove that the continuity 28 So

far, we have proved Proposition 7.31 for fixed initial conditions satisfying (7.4.47). If they satisfy only assumptions (7.4.26), we can approximate q 0 by a sequence q 0n which satisfies (7.4.47) and which is such that Eδ (ρ0 , q 0n ) → Eδ (ρ0 , q 0 ). q0

It is sufficient to take u0n = Pn ( ρ ). Of course, Proposition 7.34 holds with these new initial 0 conditions, and due to the last formula, estimates (7.4.54)–(7.4.60) are independent of n. Therefore, by the arguments explained in Sections 7.8.1-7.8.4, we get Proposition 7.31 as well.

COMPLETE SYSTEM WITH ARTIFICIAL PRESSURE

369

equation can be prolonged to the whole space and that weak limits (ρ, u) prolonged by 0 outside Ω satisfy the renormalized continuity equation everywhere. In the next Section 7.9.2, we investigate weak compactness of the effective viscous flux. To do this, we apply to our system general results of Section 7.5. In Section 7.9.3, we prove the strong convergence of the sequence of densities by using the renormalized continuity equation with a convenient renormalization function and the weak compactness of the effective viscous flux. This completes the limit in the momentum equation. The next section is devoted to energy inequalities and to estimates independent of δ except for estimate (7.4.17) whose proof is more involved. It is given in Section 7.9.5. If not stated explicitly otherwise, throughout this section, c denotes a generic positive constant which may depend on β, γ, λ, µ, Ω, T , which is, in particular, independent of ǫ, δ, of the parameter k introduced in part (iv) of assertion 7.49 (and of ρ0 , u0 , f , g). Generic positive constants which may depend on ρ0 , u0 , f , g, T (and possibly on β, γ, λ, µ, Ω) are denoted by L. The dependence on parameters ρ0 , u0 , f , g, T is usually written explicitly in the argument of L, as well as dependence on other quantities, if we judge it useful. Again, constants L are always independent of ǫ, δ and k. 7.9.1 Weak limits as dissipation tends to zero From estimates (7.4.33), (7.4.36), (7.4.39), (7.4.41), at least for a chosen subsequence, we obtain uǫ → u weakly in L2 (I, (W01,2 (Ω))3 ),

(7.9.1)

ρǫ → ρ weakly in Lβ+1 (QT ), weakly-∗ in L∞ (I, Lβ (Ω)), ρ ≥ 0 a.e. in QT , ργǫ → ργ weakly in L ρβǫ → ρβ weakly in L

β+1 γ

β+1 β

(7.9.2)

(QT ), ργ ≥ 0 a.e. in QT ,

(7.9.3)

(QT ), ρβ ≥ 0 a.e. in QT ,

(7.9.4)

2

ǫ∇ρǫ → 0 strongly in L (QT ).

(7.9.5)

(ρ, u, ρβ , ργ ) are extended by (0, 0, 0, 0) outside Ω.

(7.9.6)

In the sequel, we shall suppose that

These new quantities are denoted again by ρ, u, ρβ , ργ . d ρ η ∈ L1 (Ω), we deduce from Since in (7.4.29) Ω ρǫ η ∈ C 0 (I) and dt Ω ǫ Lemma 1.7 that

t′

t′

(ρ (t′ ) − ρǫ (t))η = t Ω ρǫ uǫ · ∇η + ǫ t Ω ∇ρǫ · ∇η, η ∈ D(Ω), t, t′ ∈ I. Ω ǫ The right-hand side of the last equality is bounded e.g. by c

ǫ∇ρǫ 0,2,Ω )∇η0,

2β β−1 ,Ω

t′ t

(ρǫ uǫ 0,

2β β+1 ,Ω

+

. Thus, by virtue of estimates (7.4.37), (7.4.41), the se2β

quence ρǫ is uniformly continuous in W −1, β+1 (Ω). Since it belongs to C 0 (I, Lβweak

370


(Ω)), and since it is uniformly bounded on I in Lβ (Ω) (cf. Exercise 6.1), we obtain by Lemma 6.2 ρǫ → ρ in C 0 (I, Lβweak (Ω)).

(7.9.7)

This gives sense to the initial condition for ρ and in turn with the first formula in (7.4.30) justifies the first formula in (7.4.10). Since the imbedding Lβ (Ω) ֒→ W −1,2 (Ω) is compact, we get by Lemma 6.4 ρǫ → ρ in Lp (I, W −1,2 (Ω)), 1 ≤ p < ∞.

(7.9.8)

This, together with (7.9.1) and (7.4.37), yields 2β

ρǫ uǫ → ρu weakly-∗ in L∞ (I, (L β+1 (Ω))3 ) 6β

and weakly in L2 (I, (L β+6 (Ω))3 ) and (L

10β−6 3β+3

(7.9.9)

(QT ))3 .

On the other hand, equation (7.4.28) implies d dt

Since

Ω

Ω

Ω

ρǫ uiǫ ujǫ ∂j φi − µ Ω ∂j uiǫ ∂j φi

−(µ + λ) Ω div u div φ + Ω (ργǫ + δρβǫ )div φ − ǫ Ω ∇ρǫ · ∇uǫ · φ

+ Ω (ρǫ f + g) · φ, t, t′ ∈ I, φ ∈ D(Ω).

ρǫ uǫ · φ =

Ω

ρǫ uǫ · φ ∈ C 0 (I) and

d dt

Ω

ρǫ uǫ · φ ∈ L1 (I), Lemma 1.7 yields

t′

t′

(ρǫ (t′ )uǫ (t′ ) − ρǫ (t)uǫ (t)) · φ = t Ω ρǫ uiǫ ujǫ ∂j φi − µ t Ω ∂j uiǫ ∂j φi

t′

t′

−(µ + λ) t Ω div u div φ + t Ω (ργǫ + δρβǫ )div φ

t′

t′

−ǫ t Ω (∇ρǫ · ∇)uǫ · φ + t Ω (ρǫ f + g) · φ, t, t′ ∈ I, φ ∈ D(Ω).

Using H¨ older’s inequality several times, we find that the right-hand side is bounded by

t′ t

[ρǫ uǫ uǫ 0,

6β 4β+3

∇φ0,

+ρǫ γ0,β+1 ∇φ0,

β+1 β+1−γ

6β 2β−3

+ (2µ + λ)∇uǫ 0,2 ∇φ0,2

+ δρǫ β0,β+1 ∇φ0,β+1

+ǫ∇ρǫ · ∇uǫ 0, 5β−3 φ0, 5β−3 + ρǫ 0,β+1 f 0,∞ φ0, β+1 + g0,∞ φ0,1 ]. 4β

β−3

β

Taking into account estimates (7.4.33)–(7.4.42), we conclude by H¨ older’s inequalβ+1 ′ ity over the interval (t, t ) that ρǫ uǫ is uniformly continuous in (W −1, β (Ω))3 . 2β

β+1 (Ω)) and that it is We already know that this sequence belongs to C 0 (I, Lweak


371

2β

uniformly bounded in L β+1 (Ω), cf. (7.4.27), (7.4.37) and Exercise 6.1. Lemma 6.2 thus yields 2β

β+1 ρǫ uǫ → q in C 0 (I, (Lweak (Ω))3 ), where q = ρu a.e. in QT .

Finally, relabelling u on a set of measure 0 in I in the spirit of Lemma 7.17 and denoting this new quantity again by u, we can write 2β

β+1 (Ω))3 ). ρǫ uǫ → ρu in C 0 (I, (Lweak

(7.9.10)

This gives sense to the initial conditions for ρu, and in turn with the second formula in (7.4.30), justifies the second formula in (7.4.10). 2β Since the imbedding L β+1 (Ω) ֒→ W −1,2 (Ω) is compact, we have ρǫ uǫ → ρu strongly in Lp (I, (W −1,2 (Ω))3 ), 1 ≤ p < ∞,

(7.9.11)

cf. again Lemma 6.4. This in turn with (7.9.1) yields ρǫ uiǫ ujǫ → ρui uj in D′ (QT ) and thanks to (7.4.38), one gets 6β

ρǫ uiǫ ujǫ → ρui uj weakly in L2 (I, L 4β+3 (Ω)).

(7.9.12)

Finally, (7.9.1) and (7.9.5) imply ǫ(∇ρǫ · ∇)uǫ → 0 in (D′ (QT ))3 and due to (7.4.42), we even have ǫ(∇ρǫ · ∇)uǫ → 0 weakly in (L

5β−3 4β

(QT ))3 .

(7.9.13)

We are now in a position to pass to the limit in the momentum equation (7.4.28) and in the modified continuity equation (7.4.29). The result and some of its consequences are summarized in the following assertion. Auxiliary lemma 7.49 Let the assumptions of Proposition 7.27 be satisfied and let ρ, u, ργ and ρβ be defined by (7.9.1)–(7.9.4). Then under convention (7.9.6), we have (i) (7.9.14) ∂t ρ + div (ρu) = 0 in D′ (IR3 × I). (ii) ∂t (ρu) + ∂j (ρuuj ) − µ∆u

(7.9.15)

−(µ + λ)∇div u + ∇ργ + δ∇ρβ = ρf + g in (D′ (QT ))3 . (iii) With any function b satisfying (6.2.9)–(6.2.10), where λ1 ≤ β 1+λ1

belongs to C 0 (I, Lweak (Ω)) ∩ C 0 (I, Lp (Ω)), 1 ≤ p
0 defined by (6.2.22), where b satisfies (6.2.9), bk (ρ) belongs to C 0 (I, Lp (Ω)), 1 ≤ p < ∞ and equation ∂t bk (ρ) + div (bk (ρ)u) +(ρ(bk )′+ (ρ) − bk (ρ))div u = 0 in D′ (IR3 × I)

(7.9.17)

holds true. (v) We have ρ ∈ C 0 (I, Lp (Ω)), 1 ≤ p < β.

(7.9.18)

Proof Equation (7.9.14) in D′ (QT ) follows from (7.4.29) when using (7.9.2), (7.9.5) and (7.9.9). By virtue of Lemma 6.8 (in order to satisfy the hypothesis, β must be greater than or equal to 2), it is valid also on D′ (IR3 × I). Similarly, (7.9.15) follows from (7.4.28) when using (7.9.1)–(7.9.4), (7.9.9), (7.9.12) and (7.9.13). The renormalized continuity equations (7.9.16) and (7.9.17) are obtained from continuity equation (7.9.14) by direct application of Lemmas 6.9 and 6.11 (their assumptions are satisfied provided β ≥ 2). We know that (7.9.16) holds, e.g., with any b(s) = sθ , where θ ∈ (0, 1). After this observation, statement (v) follows directly from Lemma 6.15. It immediately implies the continuity in time of b(ρ) and bk (ρ) claimed in statements (iii) and 2 (iv).29 This completes the proof of Auxiliary lemma 7.49. 7.9.2


We take in Proposition 7.36 z= and

2β β+1 ,

r=

β+1 β ,

s=

5β−3 4β ,

q = β, w = β + 1

q n = ρǫ uǫ , un = uǫ , pn = ργǫ + δρβǫ F n = ρǫ f + g − ǫ(∇ρǫ · ∇)uǫ , gn = ρǫ , fn = ǫ∆ρǫ

(7.9.19)

(7.9.20)

(one can put, e.g., ǫ = n1 ). With this choice, by virtue of (7.4.28), (7.4.29), equations (7.5.7) and (7.5.8) are satisfied. Moreover, due to (7.9.1)–(7.9.4), (7.9.7), (7.9.10), (7.9.12) and (7.9.13), assumptions (7.5.1)-(7.5.5) are verified with q = ρu, u p = ργ + δρβ , F = ρf + g and g = ρ. It remains to verify assumption (7.5.6). The first formula (with f = 0) follows immediately from (7.9.5). It suffices to write the weak limit in L2 (I, W −1,2 (Ω)) by using its definition. To prove 29 One can use, e.g., Vitali’s convergence theorem in combination with growth conditions (6.2.9)–(6.2.10) to show that b(ρ(tn)) → b(ρ(t)) resp. bk (ρ(tn )) → bk (ρ(t)) in Lp (Ω) provided tn , t ∈ I, tn → t. Due to (7.9.18), Ω b(ρ)η, ∈ C 0 (I), η ∈ D(Ω). This fact and density imply β λ1 +1 b(ρ) ∈ C 0 I, Lweak (Ω) .


373

the second formula, we use properties (4.4.2), (4.4.3) (4.4.4), (4.4.6) of operator A and H¨ older’s inequality, to get

| I Ai (η∆ρǫ ), φ| ≤ | I Ω η∂j ρǫ Rij (φ)| + | I Ω ∂j η∂j ρǫ Ai (φ)|

≤ c|η|C 1 (Ω) I ∇ρǫ 0,z′ ,Ω φ0,z,Ω , where η ∈ D(Ω), for all φ in C0∞ (QT ). Thus, if

ǫ∇ρǫ L2 (I,Lz′ (Ω)) → 0,

(7.9.21)

then Ai (η∆ρǫ )L2 (I,Lz′ (Ω)) → 0 and the proof of the second formula in (7.5.6) 2β 2β and we have 10β−6 is finished. In our case, z ′ = β−1 3β+3 > β−1 > 2. By using interpolation formula ∇ρǫ

2β

L2 (I,L β−1 (Ω))

1−α ≤ ∇ρǫ α L2 (QT ) ∇ρǫ

L2 (I,L

10β−6 3β+3

, (Ω))

where α ∈ (0, 1) satisfies α 2

+

(1−α)(3β+3) 10β−6

=

β−1 2β ,

we easily see that (7.9.21) holds true, cf. (7.9.5) and (7.4.40). This completes the verification of the assumptions. Due to (7.4.33), (7.4.36), (7.4.39), the sequences ρǫ div uǫ , ργ+1 and ρβ+1 are ǫ ǫ β+1 2β 1 2 β+2 γ+1 (Ω)), L (QT ) and L (QT ), respectively. Therefore, we bounded in L (I, L 2β can define weak limits ρdiv u ∈ L2 (I, L β+1 (Ω)) and ργ+1 , ρβ+1 in [C 0 (QT )]∗ by setting 2β

gn div un = ρǫ div uǫ → gdiv u = ρdiv u weakly in L2 (I, L 2+β (Ω))

(7.9.22)

and β+1

ργ+1 → ργ+1 weakly-∗ in [C 0 (QT )]∗ (weakly in L γ+1 (QT ) provided β > γ), ǫ ρβ+1 → ρβ+1 weakly in [C 0 (QT )]∗ , ǫ

(7.9.23) where the limiting process runs over a conveniently chosen subsequence. With these definitions at hand, we easily see that the weak-∗ limit pg of the sequence pn gn = ργ+1 + δρβ+1 in [C 0 (QT )]∗ is ργ+1 + δρβ+1 . ǫ ǫ After this long preparation, Proposition 7.36 yields the following statement: Lemma 7.50 Let ρ, u, ργ , ρβ , ργ+1 , ρβ+1 , ρdiv u be defined in (7.9.1), (7.9.2), (7.9.3), (7.9.4) and (7.9.22), (7.9.23). Then under the hypothesis of Proposition 7.27, β+1 ργ+1 ∈ L γ+1 (QT ), ρβ+1 ∈ L1 (QT ) (7.9.24) and

ργ+1 + δρβ+1 − (2µ + λ)ρdiv u = ργ ρ + δρβ ρ − (2µ + λ)ρdiv u a.e. in QT .

(7.9.25)

374


7.9.3

Renormalized equation of continuity and strong convergence of density

Equation (7.9.16) yields in particular

d ′ ′ dt Ω b(ρ) + Ω (ρb (ρ) − b(ρ))div u = 0 in D (I).

(7.9.26)

It is an easy exercise to show that b(s) = s ln s satisfies assumption (6.2.9)– (6.2.11). We can therefore use it in (7.9.26) in order to obtain

d ′ (7.9.27) dt Ω ρ ln ρ + Ω ρdiv u = 0 in D (I).

Due to (7.9.1), (7.9.2), (7.9.18) and (7.9.27), we have Ω ρ ln ρ ∈ C 0 (I) and

d 2 dt Ω ρ ln ρ ∈ L (I). We can thus integrate (7.9.27) over (0, T ) by using Lemma 1.7 in order to get finally

ρ(T ) ln ρ(T ) − Ω ρ(0) ln ρ(0) + I Ω ρdiv u = 0. (7.9.28) Ω

On the other hand, (ρǫ , uǫ ) and the function b(s) = s ln(s + h), where h > 0, satisfy the assumptions of Proposition 7.45 and so the renormalized inequality with dissipation (7.6.41) holds. In fact, by virtue of (7.4.27), for a.e. t ∈ I, ln(ρǫ (t) + h) belongs to Lp (Ω), 1 ≤ p < ∞, ∇ ln(ρǫ (t) + h) = 10β−6 1 p 3 ρǫ (t)+h ∇ρǫ (t) belongs to (L (Ω)) , 1 ≤ p ≤ 3β+3 , ∇[ρǫ (t) ln(ρǫ (t) + h)] =

ǫ (t) [ ρǫρ(t)+h +ln(ρǫ (t)+h)]∇ρǫ (t) ∈ (Lp (Ω))3 , 1 ≤ p < ǫ (t) + ln(ρǫ (t) + h)]∇2 ρǫ (t) [ ρǫρ(t)+h

(Lp (Ω))3×3 with any 1 ≤ p
0 (see Lemma 6.5). For any fixed open interval I ′ , I ′ ⊂ I and for any 0 < α < α0 (I ′ ), where α0 is sufficiently small (we can choose, e.g., α0 = min{sup I − sup I ′ , inf I ′ − inf I}), we get ∂t Sα (bk (ρδ )) + div Sα [bk (ρδ )uδ ] +Sα {[ρδ (bk )′+ (ρδ ) − bk (ρδ )]div uδ } = 0 in D′ (IR3 × I ′ ).

(7.9.37)


377

By virtue of Lemma 6.5 and (7.4.5), we have Sα [bk (ρδ )uδ ] ∈ C ∞ (I ′ , L6 (IR3 )),

Sα (bk (ρδ )), ∂t Sα (bk (ρδ )) ∈ C ∞ (I ′ , L∞ (IR3 )),

Sα {[ρδ (bk )′+ (ρδ ) − bk (ρδ )]div uδ } ∈ C ∞ (I ′ , L2 (IR3 )),

(7.9.38)

div Sα [bk (ρδ )uδ ] ∈ C ∞ (I ′ , L2 (IR3 )),

so that, on one hand, equation (7.9.37) can be rewritten in the form

ψ IR3 Sα [bk (ρδ )uδ ] · ∇η I

= I ψ Ω [∂t Sα (bk (ρδ )) + Sα {[ρδ (bk )′+ (ρδ ) − bk (ρδ )]div uδ }η, ′

(7.9.39)

3

ψ ∈ D(I ), η ∈ D(IR ),

and, on the other hand, it holds a.e. in QT . Employing these facts and Stokes formula (3.2.3) in (7.9.39), we find

1 ,2 1 ,2 ψγn (Sα [bk (ρδ )uδ ]), γ0 (η) = 0. ∗ I 2 2 {[W

(∂Ω)] ,W

(∂Ω)}

In view of Lemma 3.12, we conclude that

Sα [bk (ρδ )uδ ] ∈ C ∞ (I ′ , E02 (Ω)).

(7.9.40)

Now, we shall investigate the function ϕ(x, t) = ψ(t)φ(t, x), φ = B(Sα [bk (ρδ )] −

Ω

Sα [bk (ρδ )]), ψ ∈ D(I ′ ), (7.9.41)

where B is the Bogovskii operator defined in Lemma 3.17. By virtue of (7.9.37), we have

∂t ϕ = ψ ′ B Sα [bk (ρδ )] − Sα [bk (ρδ )] + ψB ∂t Sα [bk (ρδ )] − ∂t Sα [bk (ρδ )]

= ψ ′ B Sα [bk (ρδ )] − Ω Sα [bk (ρδ )] − ψB Sα [(ρδ (bk )′+ (ρδ ) − bk (ρδ ))div uδ ]

− Ω Sα [(ρδ (bk )′+ (ρδ ) − bk (ρδ ))div uδ ] − ψB div Sα [bk (ρδ )uδ ] . (7.9.42) We apply the conclusions of Lemma 3.17 and Remark 3.19 to these formulae. In view of (7.9.38) and (7.9.40), we therefore obtain φ(t)0,p,Ω + ∇φ(t)0,p,Ω ≤ c(p, Ω) ×{Sα [bk (ρδ )]}(t)0,p,Ω , t ∈ I, 1 < p < ∞, ∂t φ(t)0,p,Ω ≤ c(p, p, Ω) {Sα [(ρδ (bk )′+ (ρδ ) − bk (ρδ ))

×div uδ ]}(t)0,p,Ω + {Sα [bk (ρδ )uδ ]}(t)0,p,Ω , t ∈ I, 1 < p ≤ 2 (7.9.43) (for the definition of p, p, see (1.3.64)). In agreement with Remark 7.28, ϕ is an admissible test function for (7.4.6).

378


7.9.5.2 Underlying estimates In view of Remark 7.28, equation (7.4.6) with test function ϕ reads

Jleft := I ψ Ω (ργδ + δρβδ )Sα [bk (ρδ )]

= I ψ{ Ω Sα [bk (ρδ )] × Ω (ργδ + δρβδ )} + µ I Ω ∂j uiδ ∂j ϕi

+(µ + λ) I Ω div uδ div ϕ − I ψ Ω ρδ uiδ ∂t φi − I ψ ′ Ω ρδ uiδ φi

7 − I Ω ρδ uiδ ujδ ∂j ϕi − I Ω (ρδ f + g) · ϕ = k=1 Jk . (7.9.44) The integrals at the right-hand side are handled as follows: (i) In view of (7.4.15), we have |J1 | = |

I

ψ{ Ω Sα [bk (ρδ )] × Ω (ργδ + δρβδ )}|

(7.9.45)

≤ L(E 0 , f , g, T )|ψ|C 0 (I) Sα [bk (ρδ )]L1 (Ω×I ′ ) .

(ii) By the Cauchy–Schwartz inequality, (7.4.14), and by virtue of the first inequality in (7.9.43), one obtains

|J2 | + |J3 | = |(µ + λ) I Ω div uδ div ϕ| + µ| I Ω ∂j uiδ ∂j ϕi |

≤ c I |ψ| ∇uδ 0,2,Ω ∇φ0,2,Ω (7.9.46) ≤ L(E 0 , f , g, T )|ψ|C 0 (I) Sα [bk (ρδ )]L2 (Ω×I ′ ) .

(iii) Using H¨ older’s inequality, we get

|J4 | = | I ψ Ω ρδ uiδ ∂t φi | ≤ I |ψ|uδ 0,6,Ω ρδ 0,γ,Ω ∂t φ0,

6γ 5γ−6 ,Ω

, γ > 65 .

Then, we use the second formula (7.9.43) and bounds (7.4.14), (7.4.15) to obtain |J4 | ≤ L(E 0 , f , g, T )|ψ|C 0 (I) Sα [(ρδ (bk )′+ (ρδ ) − bk (ρδ ))div uδ ]L2 (I ′ ,Lp (Ω))

6γ +Sα [bk (ρδ )uδ ]L2 (I ′ ,Lp (Ω)) , p = 5γ−6 , γ > 65 . (7.9.47) (iv) |J5 | = | ≤

I

I

ψ′

Ω

ρδ uiδ φi | ≤ 1

1

1

I

2 2 ρδ |uδ |2 0,1,Ω φ0, |ψ ′ | ρδ 0,γ,Ω 1

2 2 ρδ |uδ |2 0,1,Ω ∇φ0, |ψ ′ | ρδ 0,γ,Ω

≤ L(E 0 , f , g, T )ψ ′ L1 (I) Sα [bk (ρδ )]

2γ γ−1 ,Ω

(7.9.48)

6γ 5γ−3 ,Ω 6γ

C 0 (I ′ ,L 5γ−3 (Ω))

.

In (iv), the first inequality is H¨ older’s inequality, then we have used Sobolev’s imbedding theorem and finally bounds (7.4.15), (7.4.16) and the first inequality in (7.9.43).


379

(v) If we apply H¨ older’s inequality, estimates (7.4.14), (7.4.15) and the first formula (7.9.43) to estimate the sixth integral, we get

|J6 | = | I Ω ρδ uiδ ujδ ∂j ϕi |

≤ I |ψ| ρδ 0,γ,Ω uδ |20,6,Ω ∇φ0, 3γ ,Ω (7.9.49) 2γ−3 ≤ L(E 0 , f , g, T )|ψ|C 0 (I) Sα [bk (ρδ )]

3γ

C 0 (I ′ ,L 2γ−3 (Ω))

, γ > 23 .

(vi) Last but not least, using H¨ older’s inequality, (7.4.15) and the first formula (7.9.43), we obtain

|J7 | = | I Ω (ρδ f + g)φ|

≤ I |ψ|(ρδ 0,γ,Ω f 0,∞,Ω + g0,γ,Ω )ϕ0,γ ′ ,Ω (7.9.50)

≤ L(E 0 , f , g, T )|ψ|C 0 (I) I ′ Sα [bk (ρδ )]0,[γ ′ ],Ω . 7.9.5.3 Limit α → 0+ ticular,

From Lemma 6.5 and from (7.4.5), we deduce, in par-

Sα (bk (ρδ )uδ ) → bk (ρδ )uδ strongly in Lp (I ′ , (L6 (IR3 ))3 ), 1 ≤ p ≤ 2, Sα (bk (ρδ )) → bk (ρδ ) in C 0 (I ′ , Lq (IR3 )), 1 ≤ q < ∞ and strongly in Lp (I ′ , L∞ (IR3 )), 1 ≤ p < ∞,

(7.9.51)

Sα [(ρδ (bk )′+ (ρδ ) − bk (ρδ ))div uδ ]

→ (ρδ (bk )′+ (ρδ ) − bk (ρδ ))div uδ in Lp (I ′ , L2 (IR3 )), 1 ≤ p ≤ 2.

Therefore, if we let α → 0+ in (7.9.45)–(7.9.50), we transform these inequalities to |J1 | ≤ L|ψ|C 0 (I) bk (ρδ )0,1,QT , |J2 | + |J3 | ≤ L|ψ|C 0 (I) bk (ρδ )0,2,QT , |J4 | ≤ J41 + J42 , where   1 J4 = L|ψ|C 0 (I) (ρδ (bk )′+ (ρδ ) − bk (ρδ ))×       ×div uδ L2 (I,Lp (Ω)) , p ≤ 2 ,p=      2  J4 = L|ψ|C 0 (I) bk (ρδ )uδ L2 (I,Lp (Ω)) , p ≤ 6 |J5 | ≤ Lψ ′ L1 (I) bk (ρδ )

6γ

L∞ (I,L 5γ−3 (Ω))

6γ 5γ−6 ,

γ > 56 , (7.9.52)

,

3γ , γ > 23 , |J6 | ≤ L|ψ|C 0 (I) bk (ρδ ) ∞ L (I,L 2γ−3 (Ω))

|J7 | ≤ L|ψ|C 0 (I) I bk (ρδ )0,[γ ′ ],Ω ,

where L = L(E 0 , f , g, T ) is a positive constant which is, in particular, independent of δ. Finally, at the left-hand side of inequality (7.9.44), one gets

(7.9.53) lim supα→0+ Jleft = I ψ Ω (ργδ + δρβδ )bk (ρδ ).

380


7.9.5.4 Limit k → ∞ Since we have

Now, we take b(s) = sθ , where θ > 0 will be fixed later. / sθ if s ∈ [0, k) bk (s) = ≤ sθ k θ if s ∈ [k, ∞)

and |s(bk )′+ (s) − bk (s)| =

|1 − θ|sθ if s ∈ [0, k) k θ if s ∈ [k, ∞)

/

≤ max{1, |1 − θ|}sθ ,

we obtain from (7.9.52), by majorazing all right-hand sides pointwise in k, the following bounds: |J1 | ≤ L|ψ|C 0 (I) ρθδ 0,1,QT , |J2 | + |J3 | ≤ L|ψ|C 0 (I) ρθδ 0,2,QT , |J4 | ≤ J41 + J42 , where   J41 = max{1, |1 − θ|}L|ψ|C 0 (I) ρθδ ×       ×div uδ L2 (I,Lp (Ω)) , p ≤ 2 , p=      2  J4 = L|ψ|C 0 (I) ρθδ uδ L2 (I,Lp (Ω)) , p ≤ 6 |J5 | ≤ Lψ ′ L1 (I) ρθδ

|J6 | ≤ L|ψ|C 0 (I) ρθδ

6γ

L∞ (I,L 5γ−3 (Ω)) 3γ

L∞ (I,L 2γ−3 (Ω))

|J7 | ≤ L|ψ|C 0 (I) By virtue of (7.4.14) and (7.4.15),

I

6γ 5γ−6 ,

(7.9.54)

,

, γ > 23 ,

ρθδ 0,[γ ′ ],Ω .

|J1 | + |J2 | + |J3 | ≤ L(E 0 , f , g, T ), 0 < θ ≤ We realize that [

6γ ]= 5γ − 6

γ > 56 ,

γ . 2

6γ 7γ−6

(γ < 6) arbitrary > 1 (γ ≥ 6).

Thus, J41 ≤ L(E 0 , f , g, T ) provided

θ ≤ 32 γ − 1 (γ < 6) (γ ≥ 6), θ < γ2

J42 ≤ L(E 0 , f , g, T ) provided

θ ≤ γ − 1 (γ < 6) θ < 65 γ (γ ≥ 6),

|J5 | ≤ L(E 0 , f , g, T )ψ ′ L1 (I) provided θ ≤ |J6 | ≤ L(E 0 , f , g, T ) provided θ ≤

5 1 γ− , 6 2

2 γ − 1, 3

COMPLETE SYSTEM OF ISENTROPIC EQUATIONS

|J7 | ≤ L(E 0 , f , g, T ) provided

381

θ ≤ 34 γ − 1 (γ < 3) θ 32 , 0 < θ ≤ 32 γ − 1.

(7.9.55)

Inequality (7.9.55) is valid with any function ψ = ψm , m ∈ IN, defined by (7.8.31) in Section 7.8.4. Inserting ψm into (7.9.55) and letting m → ∞, in a similar way as presented in that section, we get

(ργδ + δρβδ )bk (ρδ ) ≤ L, γ > 23 , 0 < θ ≤ 23 γ − 1. (7.9.56) QT Finally, as k → ∞, by the monotone convergence theorem, we obtain: Lemma 7.52 Under the assumptions of Proposition 7.27,

(ρδγ+θ + δρδβ+θ ) ≤ L(E 0 , f , g, T ), γ > 32 , θ(γ) = 32 γ − 1, QT

(7.9.57)

where ρδ is the weak limit defined by (7.9.2).

Estimate (7.4.17) is verified. This completes the proof of Proposition 7.27.31 7.10

Complete system of isentropic Navier–Stokes equations

In this section, we prove Theorem 7.7 by letting δ → 0+ in Proposition 7.27. We are again confronted with a missing estimate for the sequence of densities which would guarantee strong convergence. Additional problems arise from the fact that a priori bounds for the density do not allow us to apply the DiPerna– Lions transport theory. To overcome these difficulties, we shall apply to system (7.4.1)–(7.4.2) Feireisl’s approach described in Section 7.3. In the first section, we deal with weak limits of continuity, renormalized continuity and momentum equations. In the next Section 7.10.2, we investigate weak compactness of the effective viscous flux. To do this, we apply to our system the general results of Section 7.5. In Section 7.10.3, by using weak compactness of the effective viscous flux, we prove the boundedness of a quantity which measures amplitude of oscillations in the sequence of densities. This result implies that the renormalized continuity equation with convenient renormalization functions still 31 So far, we have proved Proposition 7.27 for initial conditions satisfying (7.4.26). If they satisfy only assumptions (7.4.4), we can approximate them by a sequence (ρ0ǫ , q 0,ǫ ) which satisfies (7.4.26) and is such that

Eδ (ρ0ǫ , q 0ǫ ) → Eδ (ρ0 , q 0 ). The way to do this is explained in Section 7.10.7. Clearly, Proposition 7.31 holds with these new initial conditions and due to the above formula, estimates (7.4.33)–(7.4.42) are independent of ǫ. By the same procedure as described in Sections 7.9.1-7.9.5, we therefore get Proposition 7.27 as well.

382


holds. We shall see this in Section 7.10.4. Then, via a renormalized continuity equation, we prove that oscillations (measured in an appropriate way) do not develop in the sequence of densities during the evolution. Strong convergence of densities follows from this fact. This is the subject of Section 7.10.5. The next section is devoted to energy inequalities. We finish this section with a paragraph devoted to general initial conditions and their relaxation. If not stated explicitly otherwise, throughout this section, c denotes a generic positive constant which may depend on γ, λ, µ, Ω, T , and which is, in particular, independent of δ, of parameter k introduced in part (iii) of Auxiliary lemma 7.53 (and of ρ0 , u0 , f , g) . Generic positive constants which may depend on ρ0 , u0 , f , g, T (and possibly on γ, λ, µ, Ω) are denoted by L. They are, in particular, independent of δ and k. The dependence of L on parameters ρ0 , u0 , f , g, T is usually written explicitly in its argument, as well as the dependence on other quantities, if we judge it useful. 7.10.1

Weak limits at vanishing artificial pressure

From estimates (7.4.14), (7.4.15), (7.4.17), we have uδ → u weakly in L2 (I, (W01,2 (Ω))3 )

and in L2 (I, (W 1,2 (IR3 ))3 ), u = 0 in (IR3 \ Ω) × I,

(7.10.1)

ρδ → ρ weakly in Lγ+θ (IR3 × I), θ = 32 γ − 1 and weakly-∗ in L∞ (I, Lγ (IR3 )),

(7.10.2)

3

ρ ≥ 0 a.e. in QT , ρ = 0 in (IR \ Ω) × I. ργδ → ργ weakly in L ργ

3

≥ 0 a.e. in IR × I,

ργ

γ+θ γ

(QT ),

= 0 in (IR3 \ Ω) × I,

δρβδ → 0 weakly in L

β+θ β

(QT ),

(7.10.3) (7.10.4)

at least for a chosen subsequence. Due to (7.4.7), we have

d ′ dt Ω ρδ η = Ω ρδ uδ · ∇η in D (I), η ∈ D(Ω). 2γ

Consequently, ρδ is uniformly continuous in W −1, γ+1 (Ω). Since it belongs to C 0 (I, Lγweak (Ω)) and since it is uniformly bounded in Lγ (Ω) as well (this last fact follows from (7.4.15) and Exercise 6.1), we use Lemma 6.2, in order to get at least for a chosen subsequence ρδ → ρ in C 0 (I, Lγweak (Ω)).

(7.10.5)

Formula (7.10.5) gives sense to the initial condition for ρ and in turn with the first formula in (7.4.10) justifies the first formula in (7.1.22). Once we realize


that the imbedding Ls (Ω) ֒→ W −1,2 (Ω), s > to ρδ and we obtain

6 5

383

is compact, we apply Lemma 6.4

ρδ → ρ strongly in Lp (I, W −1,2 (Ω)), 1 ≤ p < ∞.

(7.10.6)

Since (ρδ , uδ ) is zero outside Ω, this formula and (7.10.1) together with (7.4.18), yield 2γ

ρδ uδ → ρu weakly-∗ in L∞ (I, (L γ+1 (IR3 ))3 ) 6γ

(7.10.7)

and weakly in L2 (I, (L γ+6 (IR3 ))3 ). On the other hand, equation (7.4.6) implies d dt

Ω

ρδ u δ · φ =

ρδ uiδ ujδ ∂j φi − µ Ω ∂j uiδ ∂j φi

−(µ + λ) Ω div uδ div φ + Ω (ργδ + δρβδ )div φ

+ Ω (ρδ f + g) · φ in D′ (I), φ ∈ D(Ω). Ω

Consequently, Lemma 1.7 yields

Ω

t′

t′

(ρδ (t′ )uδ (t′ ) − ρδ (t)uδ (t)) · φ = t Ω ρδ uiδ ujδ ∂j φi − µ t Ω ∂j uiǫ ∂j φi

t′

t′

−(µ + λ) t Ω div uδ div φ + t Ω (ργδ + δρβδ )div φ

t′

+ t Ω (ρδ f + g) · φ, t, t′ ∈ I, φ ∈ (D(Ω))3 .

For γ > 32 , the right-hand side is bounded by

t′ t

[ρδ uδ uδ 0,

6γ 4γ+3

∇φ0,

6γ 2γ−3

+ (2µ + λ)∇uδ 0,2 ∇φ0,2

+ρδ γ0,γ+θ ∇φ0, γ+θ + δρδ β0,β+θ ∇φ0, β+θ θ

θ

+ρδ 0,β+θ f 0,∞ φ0, β+θ + g0,∞ φ0,1 ]. θ

Taking into account (7.4.14)–(7.4.19) we conclude that ρδ uδ is uniformly continuous in W −1,s (Ω), where s =

β+θ β .

2γ

γ+1 (Ω)) and since Since it belongs to C 0 (I, Lweak

2γ

it is uniformly bounded in L γ+1 (Ω) (see (7.4.18) and Exercise 6.1), Lemma 6.2 together with Lemma 7.17 yield 2γ

γ+1 (Ω))3 ). ρδ uδ → ρu in C 0 (I, (Lweak

(7.10.8)

This gives sense to the initial conditions for ρu and in turn with the second formula in (7.4.10) justifies the second formula in (7.1.22).

384

WEAK SOLUTIONS FOR NONSTEADY NAVIER–STOKES EQUATIONS 2γ

If γ > 23 , then L γ+1 ֒→֒→ W −1,2 (Ω) and we easily deduce from Lemma 6.4 ρδ uδ → ρu strongly in Lp (I, (W −1,2 (Ω))3 ), 1 ≤ p < ∞.

(7.10.9)

This in turn with (7.10.1) and (7.4.19) finally gives 6γ

ρδ uiδ ujδ → ρui uj weakly in L2 (I, L 4γ+3 (Ω)).

(7.10.10)

γ 2

Let b belong to class (6.2.9)–(6.2.10), where we take λ1 < − 1, and let bk , k > 0 belong to class (6.2.22), where b satisfies (6.2.9). The number k being fixed, in agreement with (7.4.14) and (7.4.15), we shall define weak limits bk (ρδ ) → bk (ρ) weakly-∗ in L∞ (IR3 × I), (ρδ (bk )′+ (ρδ ) − bk (ρδ ))div uδ

→ (ρ(bk )′+ (ρ) − bk (ρ))div u weakly in L2 (IR3 × I),

(7.10.11)

γ

b(ρδ ) → b(ρ) weakly-∗ in L∞ (I, L 1+λ1 (IR3 )), (ρδ b′ (ρδ ) − b(ρδ ))div uδ → ρb′ (ρ) − b(ρ))div u 2γ

weakly in L2 (I, L 2+2λ1 +γ (IR3 )), where the limit process is effectuated over a conveniently chosen subsequence. We are now in a position to pass to the limit in equations (7.4.6),(7.4.7), (7.4.8) and (7.4.9). The result and some of its consequences are summarized in the following assertion. Auxiliary lemma 7.53 Let the assumptions of Proposition 7.27 be satisfied and let ρ, u and ργ be defined by (7.10.1)–(7.10.3). Then (i) (7.10.12) ∂t ρ + div (ρu) = 0 in D′ (IR3 × I). (ii)

∂t (ρu) + ∂j (ρuuj ) − µ∆u

(7.10.13)

−(µ + λ)∇div u + ∇ργ = ρf + g in D′ (QT ).

(iii) For any function bk , k > 0 defined by (6.2.22) with b belonging to (6.2.9), ∂t bk (ρ) + div (bk (ρ)u) + (ρ(bk )′+ (ρ) − bk (ρ))div u = 0 in D′ (IR3 × I). (7.10.14) Moreover bk (ρ) ∈ C 0 (I, Lpweak (Ω)), 1 ≤ p < ∞. (iv) For any function b satisfying (6.2.9) and (6.2.10) with λ1
0 is uniformly continuous in W −1,2 (Ω) and {b(ρδ )}δ>0 has the same property in W −1,s (Ω), s = min{ 6λ1 6γ +6+γ , 2}. We therefore get by Lemma 6.2 bk (ρδ ) → bk (ρ) in C 0 (I, Lpweak (Ω)), 1 ≤ p < ∞, (7.10.16) γ 1+λ1 (Ω)), b(ρδ ) → b(ρ) in C 0 (I, Lweak and then by Lemma 6.4, bk (ρδ ) → bk (ρ) strongly in Lp (I, W −1,2 (Ω)), 1 ≤ p < ∞, b(ρδ ) → b(ρ) strongly in Lp (I, W −1,2 (Ω)), 1 ≤ p < ∞.

(7.10.17)

These limits along with (7.10.1) and the fact that uδ = 0 outside Ω, yield bk (ρδ )uδ → bk (ρ)u weakly in L2 (I, (L6 (IR3 ))3 ), 6γ

b(ρδ )uδ → b(ρ)u weakly in L2 (I, (L 6+6λ1 +γ (IR3 ))3 ).

(7.10.18)

After these observations, we let δ → 0+ in (7.4.9), in order to get (7.10.14) and in (7.4.8), in order to get (7.10.15). This completes the proof of Auxiliary lemma 7.53. 2 Remark 7.54 Equation (7.10.15) is only a weak limit of equation (7.4.8). We do not have ρ ∈ L2 (QT ) as was the case in Auxiliary lemma 7.49. So, Lemma 6.9, in general, does not apply. As a consequence, we do not know whether the renormalized continuity equation ∂t b(ρ) + div (b(ρ)u) + [ρb′ (ρ) − b(ρ)]div u = 0 in D′ (IR3 × I) holds. The proof of the strong convergence for the sequence of densities, however, uses this equation as an important tool (see Sections 7.9.3 and 7.10.5). In Section 7.10.4, we shall prove by other means that the solutions we have constructed through Propositions 7.27–7.34, satisfy the renormalized continuity equation. We emphasize that it need not be the case for any weak solution defined in Definition 7.3. The question of whether any weak solution is a renormalized one, remains an open problem. Note that this difficulty does not exist provided γ ≥ 59 . In this case, ρ ∈ Lγ+θ(γ) (QT ) ⊂ L2 (QT )) and so, by Lemma 6.9, the renormalized continuity equation automatically holds.

386


7.10.2 Effective viscous flux We set, as in (4.11.17), Tk (s) = bk (s), where b(s) = s.

(7.10.19)

Weak limits (7.10.11) of sequences Tk (ρδ ) and (ρδ (Tk )′+ (ρδ ) − Tk (ρδ ))div uδ are denoted Tk (ρ) and (ρ(Tk )′+ (ρ) − Tk (ρ))div u, respectively. Further, in accordance with (7.4.14), (7.4.15), we define weak limits Tk (ρδ )div uδ → Tk (ρ)div u weakly in L2 (QT ), ργδ Tk (ρδ ) → ργ Tk (ρ) weakly in L

γ+θ γ

(7.10.20)

(QT ).

We take in Proposition 7.36 z=

2γ γ+1 ,

r=

β+θ(γ) , β

s = γ + θ(γ),

6γ 2γ−3

< q < ∞, w = ∞

(7.10.21)

and q n = ρδ uδ , un = uδ , pn = ργδ + δρβδ , F n = ρδ f + g, gn = Tk (ρδ ), fn = −(ρδ (Tk )′+ (ρδ ) − Tk (ρδ ))div uδ

(7.10.22)

(one can take, e.g., δ = n1 ). With this choice, due to (7.4.6) and (7.4.9), equations (7.5.7) and (7.5.8) are satisfied. Moreover, due to (7.10.1)–(7.10.4), (7.10.8), (7.10.10), (7.10.16) and the properties of operator A recalled in Section 4.1.1, assumptions (7.5.1)–(7.5.6) are satisfied with q = ρu, u, p = ργ , F = ρf + g, g = Tk (ρ) and f = −(ρ(Tk )′+ (ρ) − Tk (ρ))div u). Finally, we realize that pn gn → β+θ

ργ Tk (ρ) weakly in L β (QT ), at least for a conveniently chosen subsequence. Thus, in the present situation, Proposition 7.36 can be rephrased as follows:

Lemma 7.55 Let ρ, u, ργ , Tk (ρ), Tk (ρ)div u, ργ Tk (ρ) be defined in (7.10.1), (7.10.2), (7.10.3), (7.10.11) and (7.10.20). Then, under the hypothesis of Proposition 7.27, ργ Tk (ρ) − (2µ + λ)Tk (ρ)div u = ργ Tk (ρ) − (2µ + λ)Tk (ρ)div u a.e. in QT . (7.10.23) 7.10.3 Amplitude of oscillations The main achievement of this section is the following lemma. Lemma 7.56 Suppose that ρ, Tk (ρ), ργ , ργ Tk (ρ), where Tk , k > 0 is defined in (7.10.19), are the same weak limits as in Lemma 7.55. Then, under the assumptions of Proposition 7.27, we have

lim supδ→0+ QT |Tk (ρδ ) − Tk (ρ)|γ+1 ≤ QT [ργ Tk (ρ) − ργ Tk (ρ)] (7.10.24)

and there exists a constant L which may depend on initial conditions, the spacetime cylinder and external forces, such that supk>1 lim supδ→0+ Tk (ρδ ) − Tk (ρ)0,γ+1,QT ≤ L(E 0 , f , g, T, Ω).

(7.10.25)


387

In the nonsteady case, this lemma plays the same role as Lemma 4.49 in the steady case. The proof of Lemma 7.56 follows the lines of the proof of Lemma 4.49 so that the attentive reader can do this as an exercise. We present it here only for the sake of completeness. Proof Taking into account (7.10.3), the second formula in (7.10.20) and first formula in (7.10.11), we can write the identity

ργ Tk (ρ) − ργ Tk (ρ) = limδ→0+ QT (ργδ − ργ )(Tk (ρδ ) − Tk (ρ)) QT (7.10.26)

+ QT (ργ − ργ )(Tk (ρ) − Tk (ρ)).

Since t → tγ is convex and t → Tk (t) concave on [0, ∞), the second term at the right-hand side is nonnegative (see statement (iii) of Corollary 3.33). According to (4.11.19), which says that (Tk (t) − Tk (s))(tγ − sγ ) ≥ |Tk (t) − Tk (s)|γ+1 , t, s ≥ 0, the first term

at the right-hand side is greater than or equal to lim supδ→0+ QT |Tk (ρδ ) − Tk (ρ)|γ+1 . We thus get (7.10.24). Using Lemma 7.55, we obtain

lim supδ→0+ QT |Tk (ρδ ) − Tk (ρ)|γ+1

≤ (2µ + λ) lim supδ→0+ QT div uδ (Tk (ρδ ) − Tk (ρ)).

(7.10.27)

The right-hand side of this inequality can be written in the form

(2µ + λ) lim supδ→0+ QT div uδ {(Tk (ρδ ) − Tk (ρ)) + (Tk (ρ) − Tk (ρ))}, and majorized, due to the Schwartz inequality, by

c lim sup[div uδ L2 (QT ) (Tk (ρδ ) − Tk (ρ)L2 (QT ) + Tk (ρ) − Tk (ρ)L2 (QT ) )]. δ→0+

Now, we apply lower weak semicontinuity of norms in the form Tk (ρ) − Tk (ρδ )L2 (QT ) Tk (ρ) − Tk (ρ)L2 (QT ) ≤ lim inf + δ→0

to get majoration of the right-hand side of (7.10.27) by c lim sup[div uδ L2 (QT ) Tk (ρδ ) − Tk (ρ)L2 (QT ) ]. δ→0+

Finally, we use the inclusion Lγ+1 (QT ) ⊂ L2 (QT ), to obtain the upper bound c(γ, QT ) lim sup[div uδ L2 (QT ) Tk (ρδ ) − Tk (ρ)Lγ+1 (QT ) ].

(7.10.28)

δ→0+

Comparing (7.10.27) with (7.10.28) and taking into account the bound (7.4.14), we obtain

lim supδ→0+ QT |Tk (ρδ ) − Tk (ρ)|γ+1 ≤ L(E 0 , f , g, T, Ω) lim supδ→0+ Tk (ρδ ) − Tk (ρ)Lγ+1 (QT ) .

This yields the desired conclusion and completes the proof.

2

388


7.10.4 Renormalized continuity equation We have the following result. Lemma 7.57 Let b satisfy (6.2.9), (6.2.10), where λ1 + 1 ≤ γ+θ(γ) and let ρ, u 2 be defined by (7.10.1) and (7.10.2). Then, under the assumptions of Proposition 7.27,32 ∂t b(ρ) + div (b(ρ)u) + {ρb′ (ρ) − b(ρ)}div u = 0 in D′ (IR3 × I).

(7.10.29)

Remark 7.58 If ρ ∈ L2loc (IR3 × I), equation (7.10.29) follows from equation (7.10.12) by using Lemma 6.9. (This was the case, e.g., in (7.9.16).) In our situation, the condition ρ ∈ L2loc (IR3 × I) is, in general, violated. The validity of (7.10.29) is therefore far from obvious. Proof The proof of identity (7.10.29) is similar to the proof of the steady version of the same identity (see Section 4.11.4 and Lemma 4.50). We start with an exercise. Exercise 7.59 Prove that for a sequence ρδ which satisfies (7.4.17) and for its weak limit ρ then 5 ! sup{δ>0} ρδ 1{ρδ ≥k} Lp (QT ) sup{δ>0} Tk (ρδ ) − ρδ Lp (QT ) ≤ L(E 0 , f , g, T, Ω)k and !

Tk (ρ) − ρLp (QT ) Tk (ρ) − ρLp (QT )

(7.10.30)

1 1 ( γ+θ(γ) −p )(γ+θ(γ))

, k > 0, 1 ≤ p < γ + θ(γ)

5

≤ L(E 0 , f , g, T, Ω)k

1 1 )(γ+θ(γ)) ( γ+θ(γ) −p

(7.10.31) , k > 0, 1 ≤ p < γ + θ(γ).

Hint: To start, use the evident inequality |{(x, t) ∈ QT ; ρδ (x, t) ≥ k}| ≤ ρδ 1{ρδ ≥k} and apply H¨ older’s inequality to its right-hand side.

1 k

QT

If the lemma holds with any b ∈ C 1 ([0, ∞)) which satisfies conditions (6.2.10) with λ1 ≤ γ+θ 2 − 1, then it is true also with b satisfying (6.2.9), (6.2.10) with the same λ1 . To see this, it suffices to repeat the reasoning at the end of the proof of Lemma 6.9. In agreement with this observation, we may suppose, without loss of generality, that b ∈ C 1 ([0, ∞)). Applying Lemma 6.11 to equation (7.10.14) with bk = Tk , we get ∂t bM (Tk (ρ)) + div (bM (Tk (ρ))u) + {Tk (ρ)[bM ]′+ (Tk (ρ)) − bM (Tk (ρ))}div u ′ = −{ρ(Tk )′+ (ρ) − Tk (ρ)}div u [bM ]+ (Tk (ρ)) in D′ (IR3 × I), M > 0, (7.10.32) 32 Since ∂ ρ + div (ρu) = 0, the renormalized equation (7.10.29) with any function s → t γ+θ(γ) , holds as well. cs + b(s), c ∈ IR, where b satisfies (6.2.9), (6.2.10) with λ1 + 1 ≤ 2


389

where [bM ]′+ is defined by (3.1.31). When k → ∞, the Lebesgue dominated convergence theorem together with (7.10.31) gives convergence in D′ (IR3 × I) of the left-hand side to ∂t bM (ρ) + div (bM (ρ)u) + {ρ[bM ]′+ (ρ) − bM (ρ)}div u. The L1 (Ω)-norm of the right-hand side can be estimated by maxs∈[0,M ] |b′ (s)| where

QT ,k,M

|{ρ(Tk )′+ (ρ) − Tk (ρ)}div u|,

QT,k,M = {(x, t) ∈ QT ; Tk (ρ)(x, t) ≤ M }. Due to the weak semicontinuity of norms, {ρ(Tk )′+ (ρ) − Tk (ρ)}div u0,1,QT ,k,M

≤ lim inf δ→0+ [ρδ (Tk )′+ (ρδ ) − Tk (ρδ )]div uδ 0,1,QT ,k,M . By the Schwartz inequality and the second inequality in (4.11.18), namely |t(Tk )′+ (t) − Tk (t)| ≤ Tk (t)1{t≥k} , the right-hand side of the last formula is bounded by cTk (ρδ )1{ρδ ≥k} 0,2,QT ,k,M div uδ 0,2,QT γ−1

γ+1

2γ 2γ Tk (ρδ )1{ρδ ≥k} 0,γ+1,Q . ≤ LTk (ρδ )1{ρδ ≥k} 0,1,Q T ,k,M T ,k,M

To get the last inequality, we have used the interpolation of L2 (QT,k,M ) between L1 (QT,k,M ) and Lγ+1 (QT,k,M ) (see Section 1.3.5.12) and the bound (7.4.14) for div uδ 0,2,QT . Since Tk (ρδ ) ≤ ρδ , it follows from (7.10.30) with p = 1 that lim supδ→0+ Tk (ρδ )1{ρδ ≥k} 0,1,QT → 0 as k → ∞. In accordance with Lemma 7.56, Tk (ρδ )1{ρδ ≥k} 0,γ+1,QT ,k,M ≤ Tk (ρδ ) − Tk (ρ)0,γ+1,QT ,k,M

1

+Tk (ρ)0,γ+1,QT ,k,M ≤ L + M |QT | γ+1 . Due to these facts {ρTk′ (ρ) − Tk (ρ)}div u → 0 in L1 (QT ) as k → ∞. Equation (7.10.32) thus yields ∂t bM (ρ) + div (bM (ρ)u) +{ρ[bM ]′+ (ρ) − bM (ρ)}div u = 0 in D′ (IR3 ), M > 0.

(7.10.33)

Now, Exercise 6.12 yields the statement of Lemma 7.57 as M → ∞. The proof is thus complete. 2

390


An immediate consequence of Lemma 7.57 is ρ ∈ C 0 (I, Lp (Ω)), 1 ≤ p < γ.

(7.10.34)

To see this, we just use Lemma 6.15 with β = γ. Finally, (7.10.34) implies γ

λ1 +1 b(ρ) ∈ C 0 (I, Lweak (Ω)) ∩ C 0 (I, Lp (Ω)), 1 ≤ p
1.

(7.10.36)

Clearly, Lk can be written in the form (4.11.30), (4.11.31), namely Lk (s) = (ln k + 1)s + l(s), l(s) = s(ln s − ln k)1{s≤k} − s1{s≤k} − k1{s>k} . (7.10.37) We observe that l satisfies (6.2.9), (6.2.10) with any λ1 > −1. Further we check that sL′k (s) − Lk (s) = Tk (s). By virtue of (7.10.5) and due to (7.10.16), we can write Lk (ρδ ) → Lk (ρ) = ρ + l(ρ) in C 0 (I, Lγweak (Ω)),

(7.10.38)

and we have, in particular Lk (ρ)(t) = Lk (ρ(t)), where Lk (ρ(t)) is a weak limit of Lk (ρδ (t)) in Lγ (Ω). Moreover, due to (7.10.35), Lk (ρ) ∈ C 0 (I, Lp (Ω), 1 ≤ p < γ. The continuity equation (7.10.12) and equation (7.10.15) with b = l yield

(7.10.39) ∂t Ω Lk (ρ) + Ω Tk (ρ)div u = 0 in D′ (I), k > 1.

On the other hand, again the continuity equation (7.10.12) and Lemma 7.57 with b = l give

∂t Ω Lk (ρ) + Ω Tk (ρ)div u = 0 in D′ (I), k > 1. (7.10.40)


391

We can integrate equation (7.10.39) by using Lemma 1.7. Since Lk (ρ)(0) = Lk (ρ(0)) = Lk ((ρ(0)), we obtain

L (ρ(T )) − Ω Lk (ρ(0)) + QT Tk (ρ)div u = 0. (7.10.41) Ω k On the other hand, integrating equation (7.10.40), we get

L (ρ(T )) − Ω Lk (ρ(0)) + QT Tk (ρ)div u = 0. Ω k

The last two equations imply

[L (ρ(T )) − Lk (ρ(T ))] = QT [Tk (ρ)div u − Tk (ρ)div u]. Ω k

(7.10.42)

(7.10.43)

Since Lk is convex on [0, ∞), the left-hand side of (7.10.43) is nonnegative (see Corollary 3.33). We therefore get

(Tk (ρ) − Tk (ρ))div u + QT (Tk (ρ)div u − Tk (ρ)div u) ≥ 0. QT Hence, due to (7.10.23) and (7.10.24), we have 1 2µ+λ

lim supδ→0+ Tk (ρδ ) − Tk (ρ)γ+1 0,γ+1,QT ≤

The right-hand side is majorized by

QT

(Tk (ρ) − Tk (ρ))div u. (7.10.44)

Tk (ρ) − Tk (ρ)0,2,QT div u0,2,QT ; after using interpolation between L1 (QT ) and Lγ+1 (QT ), we find that it is bounded by γ−1

γ+1

2γ 2γ Tk (ρ) − Tk (ρ)0,γ+1 . c(γ, QT )Tk (ρ) − Tk (ρ)0,1

According to Lemma 7.56 and due to (7.10.31), it tends to zero as k → ∞. Using all these facts in (7.10.44), we arrive at lim lim sup Tk (ρδ ) − Tk (ρ)0,γ+1 ≤ 0.

k→∞ δ→0+

(7.10.45)

Now, we write ρδ − ρ0,1,QT ≤ ρδ − Tk (ρδ )0,1,QT + Tk (ρδ ) − Tk (ρ)0,1 + Tk (ρ) − ρ0,1,QT , and employing (7.10.30), (7.10.31), (7.10.45), we obtain the strong convergence of at least a conveniently chosen subsequence ρδ in L1 (QT ). After using interpolation of Lebesgue spaces and the bound (7.4.17), we arrive at the following result. Lemma 7.60 Let ρδ be the sequence and ρ its weak limit from (7.10.2). Then under the assumptions of Proposition 7.27, at least for a chosen subsequence, ρδ → ρ strongly in Lp (QT ), 1 ≤ p < γ + θ(γ). By virtue of this result, ργ = ργ in Auxiliary lemma 7.53. This completes the proof of (7.1.20).

392


7.10.6 Energy inequalities Due to (7.10.3), (7.10.4), (7.10.10) and Lemma 7.60, we have

ρ |u |2 → Ω ρ|u|2 in L2 (I) Ω δ δ and

Ω

(ργδ + δρβδ ) →

Ω

ργ in L

β+θ β

(I).

So, similarly as in (7.9.35) and (7.9.36), we get Eδ (ρδ , uδ ) → E(ρ, u) in L

β+θ β

(I).

By Corollary 3.33, we obtain,

|∇uδ |2 ≥ Ω |∇u|2 in [C 0 (I)]∗ . Ω

In this way, the energy inequality (7.1.25) and its integral counterpart (7.1.26) follow from (7.4.11) and (7.4.13), respectively. Similarly, inequalities (7.1.36) and (7.1.37) in Remark 7.10 follow from Remark 7.29 letting δ → 0+ in (7.4.20) and in (7.4.21), respectively. 7.10.7 General initial conditions So far, we have proved Theorem 7.7 with initial conditions satisfying (7.4.26). If (ρ0 , q 0 ) belongs to class (7.1.34), we proceed as follows. We extend ρ0 by 0 outside Ω and put (7.10.46) ρ0δ = Sδ (ρ0 ) + δ, where Sδ is the regularizing operator over space variables, cf. Section 1.3.4.4. We evidently have (7.10.47) ρ0δ → ρ0 strongly in Lγ (Ω). Then we define

0δ = q

!

q0 0

)

ρ0δ ρ0

in {ρ0 > 0} otherwise.

(7.10.48)

q q Since √ρ0δ0δ = √ρ00 1{ρ0 >0} ∈ (L2 (Ω))3 , due to a density argument, there exists hδ ∈ (W 1,∞ (Ω))3 such that q √ 0δ − hδ ≤ δ. ρ0δ 0,2

We put

q 0δ =

√

ρ0δ hδ .

(7.10.49)

The reader easily verifies that 2γ

q 0δ → q 0 in (L γ+1 (Ω))3

(7.10.50)

Eδ (ρ0δ , q 0δ ) → E(ρ0 , q 0 ).

(7.10.51)

and The initial conditions (7.10.46) and (7.10.49) satisfy assumptions (7.4.26) of Proposition 7.31 as well as assumptions (7.4.4) of Proposition 7.27. Of course,

EXISTENCE OF SOLUTIONS IN EXTERIOR DOMAINS

393

Proposition 7.27 with these new conditions guarantees the existence of a couple (ρδ , uδ ). Due to (7.10.51), this couple satisfies estimates (7.4.14)–(7.4.19) uniformly with respect to δ. We therefore get Theorem 7.7 in the same way as described in Sections 7.10.1–7.10.6. 7.11

Existence of solutions in exterior domains

This section is devoted to the weak solvability of problem (7.1.1)–(7.1.8) in an exterior domain Ω. We prove Theorem 7.15 by using Theorem 7.7 on bounded invading domains ΩR = Ω ∩ BR and letting R → ∞. In fact, the limit process is a “localized version” of that presented in Sections 7.9 and 7.10. The only particularities arise in a priori estimates (see Sections 7.11.3, 7.11.4) and in the proof of strong convergence of density (see Section 7.11.7). The same proof applies to other sufficiently smooth unbounded domains with the same boundary conditions and conditions at infinity. Precise formulation and details are left to the reader as an exercise. In this section, we limit ourselves to the case ρ∞ > 0 which is more involved. It concerns statement (i) of Theorem 7.15. Without loss of generality, we shall suppose throughout the proof that ρ∞ ∈ (0, 1). The case ρ∞ = 0 (i.e. part (ii) of Theorem 7.15) is left to the reader as an exercise. Its proof (which is even slightly easier than that of part (i)) follows lines of the present proof. It differs from these lines only in the energy estimates in the sequence of densities (we obtain ρ ∈ L∞ (I, Lγ (Ω))) and in the fact that the mass conserves (this implies ρ ∈ L∞ (I, L1 (Ω))). We also remark that the part dealing with the bounded energy weak solution of statement (ii) in Theorem 7.15, can be considered as a consequence of Theorem 7.72 (see Section 7.12.4). Later, in Section 7.12.6, we shall discuss similar problems with nonzero velocity at infinity. c, c denote If not stated explicitly otherwise, throughout this section c, c′ , generic positive constants which may depend on γ, λ, µ, Ω, T and which are, in particular, independent of R, truncation parameter k introduced in (6.2.22), see also (7.11.7), and of ρ0 , u0 , f , g, ρ∞ . Constants denoted by ci , c′i may depend on γ, λ, µ, Ω, T , ρ∞ and they are independent of other parameters. Generic positive constants depending on ρ0 , u0 , T , f , g, ρ∞ (and possibly on γ, λ, µ, Ω) are denoted by L. The dependence on ρ0 , u0 , T , f , g, ρ∞ is usually indicated in the argument of L, as well as dependence on other quantities, if it is useful. In particular, L is independent of R and k. 7.11.1

Solutions on invading domains

Let (ρR , uR ) be a sequence of weak solutions to problem (7.1.1), (7.1.2), (7.1.4), (7.1.6), (7.1.7) in QR,T = ΩR ×I, where ΩR = Ω∩BR , R > 1 and I = (0, T ), T > 0 (without loss of generality, we may suppose that IR3 \ Ω ⊂ B1 ), with initial conditions (7.11.1) ρ0R (x) = ρ0 (x), q 0R (x) = q 0 (x), x ∈ ΩR .

394


The existence of such a sequence is guaranteed by Theorem 7.7. We recall that, by virtue of this theorem, the footnote of Lemma 6.8 and Remarks 7.10, 7.12, these sequences possess the following properties: (i) ρR ≥ 0, a.e. in QR,T , ρR = ρ∞ in (IR3 \ ΩR ) × I, ρR − ρ∞ ∈ L∞ (Lγ (IR3 )),

ρR ∈ C 0 (I, Lγweak (Bn )) ∩ C 0 (I, Lp (Bn )), 1 ≤ p < γ, ρR ∈ Ls(γ) (ΩR × I), s(γ) =

5γ−3 3 ,

uR ∈ L2 (I, (D1,2 (IR3 ))3 ),

(7.11.2)

2γ

γ+1 (Bn ))3 ), uR = 0 in (IR3 \ ΩR ) × I, ρR uR ∈ C 0 (I, (Lweak 6γ

ρR |uR |2 ∈ L∞ (I, L1 (IR3 )) ∩ L2 (I, L 4γ+3 (IR3 )), n ∈ IN. (ii) Initial conditions are verified, i.e.,

limt→0+ Ω ρR (t)η = Ω ρ0 η, η ∈ D(ΩR ),

limt→0+ Ω ρR (t)uR (t) · φ = Ω q 0 · φ, φ ∈ (D(ΩR ))3 .

(7.11.3)

(iii) The momentum equation holds in the sense of distributions, i.e., ∂t (ρR uiR ) + ∂j (ρR uiR ujR ) − µ∆uR i

−(µ + λ)∂i div uR + ∂i ργR = ρR f + g in (D′ (ΩR × I))3 .

(7.11.4)

(iv) The continuity equation holds, i.e., ∂t ρR + div (ρR uR ) = 0 in D′ (IR3 × I).

(7.11.5)

(v) The renormalized continuity equations in the form ∂t b(ρR ) + div (b(ρR )uR ) +[ρR b′ (ρR ) − b(ρR )]div uR = 0 in D′ (IR3 × I) with any b defined by (6.2.9), (6.2.10), where λ1 + 1 ≤

s(γ) 2

(7.11.6)

and in the form

∂t bk (ρR ) + div (bk (ρR )uR ) +[ρR (bk )′+ (ρR ) − bk (ρR )]div uR = 0 in D′ (IR3 × I)

(7.11.7)

with any bk , k > 0 defined by (6.2.22), where b satisfies (6.2.9), hold. In particular, (7.11.6) holds true with b = Lk , k > 0, where Lk is defined in (7.10.36), see


395

also (7.10.37), and (7.11.7) holds true with Tk defined in (7.10.19). (vi) Energy inequalities in the differential form d (ρ∞ ) (ρR , q R ) dt ER

≤

+µ

Ω

Ω

|∇uR |2 + (µ + λ)

(ρR f + g) · uR in D′ (I)

Ω

|div uR |2

(7.11.8)

and in the integral form

t

t

(ρ ) ER ∞ (ρR (t), q R (t)) + µ 0 Ω |∇uR |2 + (µ + λ) 0 Ω |div uR |2

t

(ρ ) ≤ ER ∞ (ρ0 , q 0 ) + 0 Ω (ρR f + g) · uR for a. a. t in I

(7.11.9)

are valid, with (ρ

)

ER ∞ (ρ, q) =

|q |2 [1 1{ρ>0} ΩR 2 ρ

P (ρ∞ ) (s) = 7.11.2

+ P (ρ∞ ) (ρ)], P (ρ∞ ) : [0, ∞) → [0, ∞),

1 γ γ−1 [s

+ (γ − 1)ργ∞ − γsργ−1 ∞ ].

(7.11.10)

Orlicz spaces Lpq (Ω)

In the sequel, we shall need the following special type of Orlicz spaces. Let 1 < q, p < ∞ and let

s

s Φδ1 ,δ2 (s) = 0 φ and Ψδ1 ,δ2 (s) = 0 ψ,

where φ is a continuous and increasing function satisfying (1.3.4) such that / sq−1 , 0 ≤ s ≤ δ1 , δ1 > 0, δ1q−1 < δ2p−1 , and ψ(s) = φ−1 (s), φ(s) = sp−1 , s > δ2

are complementary Young’s functions, cf. Section 1.3.2. The reader may verify that Φδ1 ,δ2 , Φδ1 ,δ2 , (δ1 , δ2 ) = (δ1 , δ2 ) are equivalent Young’s functions and that they satisfy the ∆2 condition. We introduce the Orlicz space

Lpq (Ω) = {u ∈ L1loc (Ω); Ω Φδ1 ,δ2 (|u|) < ∞} (7.11.11) and corresponding norms

uLpq (Ω) = sup{

Ω

uv;

Ω

Ψδ1 ,δ2 (|v|) ≤ 1}.

By virtue of Theorems 1.28 and 1.29, Lpq (Ω) does not depend of δ1 , δ2 and the corresponding norms are finite on Lpq (Ω) and equivalent with respect to δ1 , δ2 . By virtue of Theorems 1.27, 1.29, 1.30, 1.31, Lpq (Ω) endowed with the norm ·Lpq (Ω) , are separable (for example C0∞ (Ω) is a dense linear subspace), reflexive Banach ′ spaces and their duals are isometrically isomorphic to Lpq′ (Ω).

396


The reader may verify by direct calculation (it is convenient to consider cases q < p and q ≥ p separately) that Lpq (Ω) = {u ∈ L1loc (Ω); u1{|u|0} + P together with the Gronwall lemma, gives uniform bounds ER ∞ (ρR , q R ) +

Ω

I

ρR |uR |2 L∞ (I,L1 (IR3 )) ≤ L(E0 , f , g, T ), ∇uR L2 (IR3 ×I) ≤ L(E0 , f , g, T ), 1{|ρ−ρ∞ | 1 such that if R > RK , then K ⊂ B ′ ⊂ BR for some ball B ′ (centered at 0). We consider the vector fields ϕ(x, t) = ψ(t)φ(x, t), ψ ∈ D(I ′ ),

BΩ′ {Sα [bk (ρR )η] − Ω′ Sα [bk (ρR )η]}, x ∈ Ω′ := Ω ∩ B ′ φ= 0, x ∈ Ω \ Ω′ ,

(7.11.19)

where I ′ is a subinterval of I such that I ′ ⊂ I, Sα with 0 < α < α0 (I ′ ), α0 “sufficiently small”, is a one-dimensional regularizing operator, η ∈ D(B ′ ), 0 ≤ η ≤ 1, η = 1 in K, is a cut-off function,33 and BΩ′ is the Bogovskii operator defined in Lemma 3.17. In the same manner as in Section 7.9.5, in agreement with Remark 7.28, we verify that ϕ is an admissible test function for the momentum equation (7.11.4). In fact, we obtain

ψ Ω′ ηργR Sα [bk (ρR )] = I ψ{ Ω′ ηSα [bk (ρR )] × Ω′ ργR } I

+(µ + λ) I Ω′ div uR div ϕ + µ I Ω′ ∂j uiR ∂j ϕi − I ψ Ω′ ρR uiR ∂t φi

− I ψ ′ Ω′ ρR uiR φi − I Ω′ ρR uiR ujR ∂j ϕi

− I Ω′ (ρR f + g) · ϕ 7 = j=1 Jj . (7.11.20) If we estimate the right-hand side of (7.11.20) in a similar way as was done in Section 7.9.5, we arrive, after a long and cumbersome calculation, at

s(γ) (7.11.21) ρ ≤ L(E0 , f , g, T, K, Ω′ , ρ∞ ), γ > 32 , s(γ) = 5γ−3 3 , K ∈ K. K×I R 7.11.5

Weak limits at growing invading domains

From estimates (7.11.15)–(7.11.17), (7.11.21), recalling that (ρR , uR ) = (ρ∞ , 0) outside ΩR , and using Lemma 6.6, we obtain the existence of a triplet (ρ, u, ργ ) ∈ L∞ (I, Lγloc (IR3 )) ×L2 (I, (D1,2 (IR3 ))3 ) × L

s(γ) γ

s(γ)

(7.11.22)

(I, Llocγ (IR3 )),

such that34 uR → u weakly in L2 (I, (L6 (IR3 ))3 ), u = 0 in (IR3 \ Ω) × I, ∇uR → ∇u weakly in (L2 (IR3 × I))3×3 ,

(7.11.23)

33 The presence of η in (7.11.19) is inevitable. Indeed, in order to estimate one of the terms Ω′ (ρuη) is required to be zero. This would not in ∂t ϕ by using Lemma 3.17, the normal trace γn be true without the presence of η. See Section 7.9.5 for technical details. 34 For the definition of homogenous Sobolev space D 1,2 (IR3 ), see Section 1.3.6.


ρR − ρ∞ → ρ − ρ∞ weakly in L∞ (I, Lγ2 (IR3 )), ∞

γ

s(γ)

ρR → ρ weakly-∗ in L (I, L (Bn )), weakly in L and

(Bn × I)),

n ∈ IN, ρ ≥ 0 a.e. in QT , ρ = ρ∞ in (IR3 \ Ω) × I, ργR → ργ weakly in L

s(γ) γ

(Bn × I), n ∈ IN ,

399

(7.11.24) (7.11.25)

(7.11.26)

at least for a conveniently chosen subsequence. As in Section 7.10.1, we show that, 2γ for any fixed n ∈ IN , the sequence ρR is uniformly continuous in W −1, γ+1 (Bn ). Since it belongs to C 0 (I, Lγweak (Bn )), and since it is uniformly bounded in Lγ (Bn ), we get by Lemma 6.2, ρR → ρ|Bn in C 0 (I, Lγweak (Bn )), for a selected subsequence, if necessary. Using the usual diagonalization process, we thus obtain ρR → ρ in C 0 (I, Lγweak (Bn )), n ∈ IN ,

(7.11.27)

at least for a subsequence. Formula (7.11.27) gives sense to the initial condition for ρ and in turn with the first formula in (7.11.3) justifies the first formula in (7.1.22). Lemma 6.4 implies ρR → ρ strongly in Lp (I, W −1,2 (Bn )), 1 ≤ p < ∞, n ∈ IN .

(7.11.28)

This formula and (7.11.23) combined with (7.11.17) and Lemma 6.6, yield 2γ

ρR uR → ρu weakly-∗ in L∞ (I, (L γ+1 (Bn ))3 ) 6γ

(7.11.29)

and weakly in L2 (I, (L γ+6 (Bn ))3 ), n ∈ IN , at least for a subsequence. From equation (7.11.4) with the test function φ ∈ (D(Ωn ))3 , we get, again as in Section 7.10.1, that ρR uR is uniformly contin6γ , s(γ) uous in (W −1,s (Ωn ))3 , s = min{ 4γ+3 γ }. Since it is uniformly bounded in 2γ

(L γ+1 (Ωn ))3 as well, we obtain by Lemma 6.2 and Lemma 7.17, ρR uR → (ρu)|Ωn 2γ

γ+1 in C 0 (I, (Lweak (Ωn ))3 ), at least for a subsequence. Applying once more the diagonalization procedure and recalling that (ρR , uR ) = (ρ∞ , 0) outside ΩR , we arrive at 2γ

γ+1 (Bn ))3 ), n ∈ IN . ρR uR → ρu in C 0 (I, (Lweak

(7.11.30)

Now, by Lemma 6.4, we obtain ρR uR → ρu strongly in (Lp (I, W −1,2 (Bn ))3 ), 1 ≤ p < ∞, n ∈ IN

(7.11.31)

and then, by virtue of (7.11.17), (7.11.23), 6γ

ρR uiR ujR → ρui uj weakly in L2 (I, L 4γ+3 (Bn )), n ∈ IN again for a conveniently chosen subsequence.

(7.11.32)

400


The number k > 0 being fixed, by virtue of (7.11.15), (7.11.17) and Lemma 6.6, there exists Tk (ρ) ∈ L∞ (IR3 × I), Lk (ρ) ∈ L∞ (I, Lγloc (IR3 )), Tk (ρ)div u, (ρ(Tk )′+ (ρ) − Tk (ρ))div u ∈ L2 (IR3 × I), ργ Tk (ρ) ∈ L that

s(γ) γ

s(γ)

(I, Llocγ (IR3 )), such

Tk (ρR ) → Tk (ρ) weakly-∗ in L∞ (IR3 × I), Lk (ρR ) → Lk (ρ) weakly-∗ in L∞ (I, Lγ (Bn )), n ∈ IN, Tk (ρR )div uR → Tk (ρ)div u weakly in L2 (IR3 × I), (ρR (Tk )′+ (ρR ) − Tk (ρR ))div uR

(7.11.33)

→ (ρ(Tk )′+ (ρ) − Tk (ρ))div u weakly in L2 (IR3 × I), ργR Tk (ρR ) → ργ Tk (ρ) weakly in L

s(γ) γ

(I, L

s(γ) γ

(Bn )), n ∈ IN .

We are now in a position to let R → ∞ in equations (7.11.4), (7.11.5), (7.11.6) with b = Lk and in (7.11.7) with bk = Tk . The result, whose proof is similar to the proof of Auxiliary lemma 7.53, is summarized in the following statement. Auxiliary lemma 7.61 Let (ρR , uR ) be the sequence from Section 7.11.1 and let ρ, u, ργ , Tk (ρ), Lk (ρ), Tk (ρ)div u,(ρ(Tk )′+ (ρ) − Tk (ρ))div u be weak limits defined by (7.11.22)–(7.11.26) and (7.11.33). Then (i) (7.11.34) ∂t ρ + div (ρu) = 0 in D′ (IR3 × I). (ii) ∂t (ρu) + ∂j (ρuuj ) − µ∆u − (µ + λ)∇div u +∇ργ = ρf + g in D′ (QT ).

(7.11.35)

(iii) Tk (ρR ) → Tk (ρ) in C 0 (I, Lpweak (Bn )), n ∈ IN , 1 ≤ p < ∞ and ∂t Tk (ρ) + div (Tk (ρ)u) + (ρ(Tk )′+ (ρ) − Tk (ρ))div u = 0 in D′ (IR3 × I). (7.11.36) (iv) Lk (ρR ) → Lk (ρ) in C 0 (I, Lγweak (Bn )), n ∈ IN and ∂t Lk (ρ) + div (Lk (ρ)u) + Tk (ρ)div u = 0 in D′ (IR3 × I). 7.11.6

(7.11.37)

Effective viscous flux and renormalized continuity equation

7.11.6.1 Effective viscous flux Repeating the argument of Section 7.10.2 with minor changes, we easily get the following lemma. Lemma 7.62 Let ρ, u, ργ , Tk (ρ), Tk (ρ)div u, ργ Tk (ρ) be defined in (7.11.22)– (7.11.26) and (7.11.33). Then, under the hypothesis of Theorem 7.15, ργ Tk (ρ) − (2µ + λ)Tk (ρ)div u = ργ Tk (ρ) − (2µ + λ)Tk (ρ)div u a.e. in QT .

(7.11.38)


401

7.11.6.2 Amplitude of oscillations Repeating the reasoning of Section 7.10.3 with small modifications, we obtain the following result. Lemma 7.63 Suppose that ρ, Tk (ρ), ργ , ργ Tk (ρ), where Tk , k > 0 is defined in (7.10.19), are the same weak limits as in Lemma 7.62. Then, under the assumptions of Theorem 7.15, we have

lim supR→∞ QT |Tk (ρR ) − Tk (ρ)|γ+1 Φ (7.11.39)

≤ QT (ργ Tk (ρ) − ργ Tk (ρ))Φ, Φ ≥ 0, Φ ∈ D(IR3 ), and there exists a constant L which depends on E0 , Ω, T , f , g and K (and which is in particular independent of R, k), such that

sup lim sup Tk (ρR ) − Tk (ρ)0,γ+1,K×I ≤ L(E0 , f , g, T, K), K ∈ K, (7.11.40) k>1 R→∞

where K is defined in (7.11.18). 7.11.6.3 Renormalized solutions of the continuity equation Repeating the argument of Section 7.10.4 we get the following lemma. s(γ) 2

and let ρ, u

∂t b(ρ) + div (b(ρ)u) + {ρb′ (ρ) − b(ρ)}div u = 0 in D′ (IR3 × I).

(7.11.41)

Lemma 7.64 Let b satisfy (6.2.9), (6.2.10), where λ1 + 1 ≤ be defined by (7.11.23) and (7.11.25). Then

An immediate consequence of Lemmas 7.64 and 6.15 is ρ ∈ C 0 (I, Lploc (Ω)), 1 ≤ p < γ.

(7.11.42)

7.11.7 Strong convergence of the density We start with three technical exercises. Exercise 7.65 Let g ∈ L1loc (IR3 ), g − ρ∞ ∈ Lγ2 (IR3 ). Then Lk (g) − Lk (ρ∞ )Lp2 (IR3 ) ≤ c(ρ∞ , p, γ)g − ρ∞ Lγ2 (IR3 ) , k ≥ 1 + ρ∞ , where 1 ≤ p < γ. The constant c in the above estimate is independent of k. s−ρ∞ ln ρ∞ ∞ ln ρ∞ = 1 + ln ρ∞ , lims→∞ s ln |s−ρ Hint: First show that lims→ρ∞ s ln s−ρ q s−ρ∞ ∞| = 0, 1 < q < ∞. Then show that |Lk (s) − Lk (ρ∞ )|1{|s−ρ∞ |≤1} ≤ c(ρ∞ )|s − ρ∞ |1{|s−ρ∞ |≤1} , k ≥ 1 + ρ∞ and |Lk (s) − Lk (ρ∞ )|1{|s−ρ∞ |≥1} ≤ c(ρ∞ , p)|s − ρ∞ |q 1{|s−ρ∞ |≥1} , k ≥ 1 + ρ∞ , 1 < q < ∞. Finally use the equivalence of norms Lp2 evoked in Section 7.11.2. In the sequel we need cut-off functions defined in (4.14.12), namely x Φm (x) = Φ( m ), m ∈ IN, Φ ∈ (D(IR3 ))3 ,

0 ≤ Φ(x) ≤ 1, Φ(x) =

1, x ∈ B1 , 0, x ∈ B 2 .

(7.11.43)

We recall in the following exercise some of their evident properties which will be useful in the sequel.

402


Exercise 7.66 Prove that

limm→∞

supx∈IR3 |∇Φm (x)| ≤

IR3

c(Φ) m ,

|u · ∇Φm |p = 0, u ∈ (D1,2 (IR3 ))3 , 2 ≤ p < 6.

(7.11.44)

Hint: See Sections 1.3.6.2, 1.3.6.3, 1.3.6.4 and use interpolation. Exercise 7.67 Let ρR be a sequence defined in Section 7.11.1 satisfying bound (7.11.21) and let ρ be its weak limit (7.11.25). Prove that    supR>1 Tk (ρR ) − ρR 0,p,K×I  1 1 ≤ L(E0 , f , g, T, K)k ( s(γ) − p )s(γ) , Tk (ρ) − ρ0,p,K×I   Tk (ρ) − ρ0,p,K×I

where k > 0, 1 ≤ p < s(γ), K ∈ K. Hint: See Exercise 7.59.

We use Φm as test functions in (7.11.37) and in (7.11.41), where we take35 b = Lk . We obtain

d ′ (7.11.45) dt Ω Lk (ρ)Φm + Ω Tk (ρ)div uΦm − Ω Lk (ρ)u · ∇Φm = 0 in D (I)

and

d dt

Lk (ρ)u · ∇Φm = 0 in D′ (I). (7.11.46)

Since both Ω Lk (ρ)Φm and Ω Lk (ρ)Φm belong to C 0 (I), integrating (7.11.45) and (7.11.46) over I by using Lemma 1.7, we obtain

[L (ρ(T )) − Lk (ρ(0))]Φm + QT Tk (ρ)div u Φm − QT Lk (ρ)u · ∇Φm = 0 Ω k

and

Ω

Ω

Lk (ρ)Φm +

Ω

Tk (ρ)div u Φm −

[Lk (ρ(T )) − Lk (ρ(0))]Φm +

QT

Ω

Tk (ρ)div u Φm −

QT

Lk (ρ)u · ∇Φm = 0.

Subtracting both equations, we arrive at

[L (ρ(T )) − Lk (ρ(T ))]Φm Ω k

+ QT [Tk (ρ)div u − Tk (ρ)div u]Φm = QT [Lk (ρ) − Lk (ρ)]u · ∇Φm .

(7.11.47)

Since Lk is convex, the first term at the left-hand side is nonnegative. The second term at the left-hand side can be written as

[T (ρ) − Tk (ρ)]div u Φm + QT [Tk (ρ)div u − Tk (ρ)div u]Φm QT k

35 By virtue of (7.10.37), L is an admissible renormalization function, cf. footnote in Lemma k 7.57.


403

and, by virtue of (7.11.38), (7.11.39), it is greater than or equal to

1 [T (ρ) − Tk (ρ)]div u Φm + 2µ+λ lim supR→∞ QT |Tk (ρR ) − Tk (ρ)|γ+1 Φm . QT k

Equation (7.11.47) thus yields 1 2µ+λ

|Tk (ρR ) − Tk (ρ)|γ+1 Φm

≤ QT [Tk (ρ) − Tk (ρ)]div u Φm + QT [Lk (ρ) − Lk (ρ)]u · ∇Φm = 0. lim supR→∞

QT

(7.11.48)

The first term at the right-hand side possesses the bound

Tk (ρ) − Tk (ρ)0,2,Ω2m ×I div u0,2,QT . After using interpolation of Lebesgue spaces, we find that it is bounded by γ−1

γ+1

2γ 2γ c(γ, m, T )Tk (ρ) − Tk (ρ)0,1,Ω Tk (ρ) − Tk (ρ)0,γ+1,Ω div u0,2,QT . 2m ×I 2m ×I

Finally, employing (7.11.40) and Exercise 7.67 we observe that it tends to 0 as k → ∞. The second term at the right-hand side is majorized by ×

Lk (ρ) − Lk (ρ)L∞ (I,Lp2 (Ω)) I

[u · ∇Φm 0,2,Ω + u · ∇Φm 0,p′ ,Ω ],

3 2

< p < min{γ, 2}.

(7.11.49)

By virtue of Exercises 7.65 and 7.66 this tends to 0 as m → ∞. Using all these observations in (7.11.48), we arrive at lim lim sup Tk (ρR ) − Tk (ρ)0,γ+1,K×I ≤ 0, K ∈ K.

k→∞ R→∞

(7.11.50)

Now, we write ρR − ρ0,1,K×I ≤ ρR − Tk (ρR )0,1,K×I +Tk (ρR ) − Tk (ρ)0,1,K×I + Tk (ρ) − ρ0,1,K×I , and we obtain by using Exercise 7.67 and (7.11.50), the strong convergence of ρR in L1 (K × I). After using interpolation and bound (7.11.21), we can claim: Lemma 7.68 Let ρR be the sequence defined in Section 7.11.1 and ρ its weak limit from (7.11.25). Then, at least for a subsequence, ρR → ρ strongly in Lp (K × I), 1 ≤ p < s(γ), K ∈ K.

(7.11.51)

By virtue of this result, we have ργ = ργ in Auxiliary lemma 7.61. This completes the limit in momentum equation (7.1.20). To complete the proof of Theorem 7.15, it remains to show the energy inequality in the integral form (7.1.26).36 This is the subject of the following section. 36 It is an open problem whether the energy inequality in differential form (7.1.25) is still valid. See Section 7.12.4 for more details.

404


7.11.8

Energy inequality √ By virtue of (7.11.15), ρR uR is a bounded sequence in (L2 (QT ))3 . Due to this fact and (7.11.23), (7.11.51), √ √ ρR uR → ρu weakly in (L2 (QT ))3 , at least for a chosen subsequence. By lower weak semicontinuity, we therefore obtain

ρ|u|2 ≤ lim inf R→∞ Ω ρR |uR |2 in D′ (I). Ω Thanks to (7.11.13) and (7.11.15) P (ρ∞ ) (ρR ) is a bounded sequence in L2 (QT ). Thus, Lemma 7.68 and again lower weak semicontinuity imply

P (ρ∞ ) (ρ) ≤ lim inf R→∞ Ω P (ρ∞ ) (ρR ) in D′ (I). Ω Of course, by virtue of (7.11.23)

|∇u|2 ≤ lim inf R→∞ Ω |∇uR |2 in D′ (I). Ω

Finally, by (7.11.15), (ρR f +g)·uR is a bounded sequence in L2 (I, Lp (Ω)), where 1 1 1 γ + 6 ≤ p . Therefore, with help of (7.11.29), we have

Ω

(ρR f + g) · uR →

Ω

(ρf + g) · u in D′ (I).

Using all these facts in energy inequality (7.11.9) we obtain that (7.1.14) holds in D′ (I). Examining the summability of all terms, we conclude that it holds, in fact, a.e. in I. The proof of Theorem 7.15 is thus complete. 7.12

Other problems and bibliographic remarks

The first section is devoted to bibliographic remarks to Theorems 7.7 and 7.15. In the next two sections we discuss generalizations of existence theorems to slip boundary conditions and to nonmonotone pressure laws. Then we address the problem of domain dependence of weak solutions. In Section 7.12.5, we investigate the existence of weak solutions for the problem with nonhomogenous Dirichlet boundary conditions. Section 7.12.6 is devoted to flows past an obstacle. The last section is a brief review of existence results in domains with nonsmooth boundaries. 7.12.1

Bibliographic remarks on basic theorems

7.12.1.1 Underlying ideas In Sections 7.3.1–7.3.6 we outlined the underlying ideas and main tools of the existence theory for system (7.1.1)–(7.1.7) whose details are published in Lions’ pioneering book (Lions, 1998). Here we shall briefly discuss their origins. The first tool is the concept of renormalized weak solutions to the transport equation applied to the continuity equation (see Section 6.2). This concept was introduced in (DiPerna and Lions, 1989).

OTHER PROBLEMS AND BIBLIOGRAPHIC REMARKS

405

The second tool is the weak compactness of the so-called effective viscous flux (see Lemma 7.24 in Section 7.3 and Proposition 7.36 in Section 7.5). This type of identity was discovered by Lions in (Lions, 1998), Chapter 5 (see also (Lions, 1993b) and (Lions, 1993a)). He obtained them by testing the momentum equation with functions of type “∇∆−1 (b(ρ))” with convenient b. The main point was to introduce the commutator b(ρ)[Rij (ρui uj ) − ui Rij (ρuj )],

(7.12.1)

where Rij is a Riesz operator, cf. Section 4.4.1, and to observe that it was weakly compact. The latter observation was deduced from general results of harmonic analysis from (Coifman and Meyer, 1975) and (Coifman et al., 1976). For more details about these arguments, see (Lions, 1998), Chapters 5 and 7. The importance of the commutator (7.12.1) was already known to Serre (Serre, 1991), but he failed to prove its weak compactness in higher space dimensions. An alternative approach to prove the identities for the effective viscous flux was developed later in (Feireisl, 2001). Here, instead of (7.12.1), another commutator appears in the form uj [b(ρ)Rij (ρui ) − ρui Rij (b(ρ))]

(7.12.2)

and its weak compactness is proved with the help of the div–curl lemma, see (Murat, 1978), (Tartar, 1975) or (Yi, 1992). This is the way we followed in the present book (see Sections 4.4.1–4.4.4 and 7.5). In the context of small initial data, importance of effective viscous flux for the existence theory was also observed by D. Hoff in (Hoff, 1992a), (Hoff, 1992b). The third tool are improved estimates of the density. In fact, in order to guarantee that the limiting pressure is not a mere measure, there is a need for better estimates than that provided by the energy inequality (see estimates (7.3.4), (7.3.5) in Section 7.3 and their proofs in Sections 7.8.4, 7.9.5). In the context of weak solutions, estimates of this type are again due to Lions. At first, they were only local of type (7.3.5); see Theorem 7.1 in (Lions, 1998). The final goal is to prove the strong convergence of the sequence of densities (see Section 7.3.6). It was noticed in (Lions, 1993b), (Lions, 1993a) that it is closely related to the time propagation of oscillations of the density which are governed by a renormalized continuity equation (see, e.g., equation (7.3.10) in Section 7.3). It is worth observing that any reasonable solution operator we may associate with the finite (or bounded) energy weak solution may not be compact in the variable ρ. This is due to the hyperbolic character of the continuity resp. of the renormalized continuity equations. In accordance with the observations of (Lions, 1993b), (Lions, 1993a), oscillations in the density should propagate in time. Serre in (Serre, 1991) studied these phenomena and showed that the amplitude of the Young measure associated to the density oscillations is a nondecreasing function of time. His proof is complete for one-dimensional flow. It holds in several space variables, provided one takes the identity of the

406


effective viscous flux for granted. Having proved this identity, Lions completed the existence proof in higher dimensions. There is still a backlog which we are going to explain in what follows. The DiPerna–Lions transport theory can be applied to the continuity equation and the validity of the renormalized continuity equation can be rigorously proved, only provided ρ ∈ L2 (QT ), i.e., provided γ ≥ 2 (see estimates (7.3.2) in Section 7.3 and Lemma 6.9 in Section 6.2). Physically relevant cases, however, concern the values of γ less than 2 (e.g. γ = 53 for the monatomic gas). In order to overcome this difficulty, two possibilities were investigated: 1) To prove estimates of the density up to the boundary. 2) To prove that the renormalized continuity equation holds for ρ ∈ Lα (QT ) with α < 2. To do this, one has necessarily to use tools other than the DiPerna–Lions transport theory. The first possibility was worked out by Lions. By using known regularity results for the Stokes problem as an auxiliary tool, he proved the bound (7.3.4) for density, up to the boundary, for the Dirichlet problem with vanishing velocity at the boundary (see (Lions, 1999)). With this new information, ρ ∈ L2 (QT ), provided γ ≥ 95 . For these adiabatic exponents, the DiPerna–Lions transport theory can again be used. In this way, the existence theory was completed for γ ≥ 59 . Estimate (7.3.4) was obtained independently and for other purposes, by using the so-called Bogovskii operator, in (Novotn´ y and Straˇskraba, 2000) and in (Feireisl and Petzeltov´ a, 2000). Notice that all existing approaches require sufficiently smooth domains Ω (at least a Lipschitz domain). In the present book we have adopted the approach from (Novotn´ y and Straˇskraba, 2000), see Section 7.9.5. The second possibility was worked out in (Feireisl, 2001). The condition ρ ∈ L2 (QT ) necessary for the application of the transport theory is replaced by the condition of boundedness of the amplitude of oscillations in the density sequence, measured in an appropriate way (see Lemma 7.25). This new information allows us to show that the renormalized continuity equation with convenient renormalization functions holds, even if the density is not square integrable up to the boundary. Then, Lions’ argument about nonpropagation of oscillations again applies, giving the weak compactness of the set of finite energy renormalized weak solutions under the sole hypothesis γ > 32 (see Sections 7.3.7.1–7.3.7.3 and 7.10.3–7.10.5). This argument was worked out for approximations in (Feireisl et al., 2001) giving thus the existence theory for γ > 23 . This is the way we followed in this book (see Sections 7.4–7.10). Notice that the above result is the best possible, meaning that the convective term ρui uj may not belong to L1loc (Ω) provided γ < 23 . It is also worth remarking that Feireisl’s approach does not need an estimate of type (7.3.4) up to the boundary. (Indeed, the sole local estimate of type (7.3.5) is enough to guarantee p ∈ L1 (QT ) after the limit process.) This opens ways to relax the assumptions on the regularity of the boundary. So far, the main difficulty of the whole existence theory was connected with


407

the nonlinear pressure law. If γ = 1, we are concerned with isothermal flows. In this situation, the mathematical nature of the problem drastically changes. The main difficulty is connected with the convective term div (ρuui ) and more precisely, with the lack of “sufficiently strong” a priori estimate for the density. The paper (Padula, 1986) clears up the nature of these difficulties, but contains a gap that has not yet been removed (cf. (Padula, 1988a)). Even in two dimensions, the existence of solutions of compressible Navier–Stokes equations in the isothermal regime is a very difficult open problem. 7.12.1.2 Remarks on Theorems 7.7, 7.15 The first breakthrough into the existence of weak solutions to steady compressible Navier–Stokes equations in the isentropic regime was Lions’ book (Lions, 1998). It introduces and puts together all the principal ideas in the subject and represents the main source of further development of the theory. The first theorem about the existence of finite energy renormalized weak solutions is Theorem 7.2 in (Lions, 1998). Lions provides proofs under the following hypothesis: (i) Ω ⊂ IRN , N ≥ 2 is a periodic cell, γ ≥ 95 (N = 3), γ ≥ 23 (N = 2) and solutions (ρ, u) are periodic in x; (ii) Ω = IRN , N ≥ 2, ρ∞ = 0 and γ ≥ 59 (N = 3), γ ≥ 32 (N = 2); (iii) Ω is a bounded domain with smooth boundary and γ ≥ 59 (N = 3), γ ≥ 32 (N = 2); (iv) Ω is an exterior domain with smooth boundary, ρ∞ = 0 and γ ≥ 59 (N = 3), γ ≥ 23 (N = 2). In all theses situations the pressure is supposed to satisfy the constitutive relation p(ρ) = ργ . Besides these cases which are treated in detail, other boundary conditions and pressure laws are considered. Here we present an incomplete list. (v) In Section 7.5 it is explained how the above results can be extended to more general pressure laws: p is supposed to be a nondecreasing continuous function on [0, ∞) and such that lim inf t→∞ p(t)t−γ > 0. (vi) In Section 7.6 it is explained how the nonhomogenous boundary value problem with nonzero inflow and outflow can be well posed, and a result about existence of weak solutions to this problem is formulated along with some hints for proofs when γ ≥ 95 (N = 3), γ ≥ 32 (N = 2). (vii) In Section 7.6 it is also explained how the case of an unbounded domain (say for simplicity of an exterior domain) with constant density ρ∞ ≥ 0 and velocity a∞ ∈ IRN at infinity, can be well posed. Some elements of proofs are given, provided γ ≥ 59 (N = 3), γ ≥ 23 (N = 2). Feireisl’s observation about the boundedness of the amplitude of oscillations and its consequences, namely the facts that (i) the renormalized continuity equation can be proved without using DiPerna– Lions transport theory for γ > 23 (N = 3), γ > 1 (N = 2), and that

408


(ii) there is no need for the improved estimate of the density up to the boundary, pushed Lions’ existence theory a step further in the following directions: (i) The sole assumption γ > 23 (N = 3), γ > 1 (N = 2) is needed. (ii) Nonmonotone pressures can be considered. (iii) No smoothness of the boundary is required. For example, assumptions on the smoothness of the boundary in Theorems 7.7 and 7.15 can be considerably relaxed, see Sections 7.12.4 and 7.12.7. The hypothesis on pressure can be equally weakened so that a certain type of nonmonotonicity is allowed, see Section 7.12.3. Now, it is also known, in some cases, that the nonhomogenous Dirichlet problem and problems in unbounded domains with ρ∞ ≥ 0, a∞ ∈ IR3 are well posed under the sole hypothesis γ > 23 . We shall formulate these results precisely in Sections 7.12.5 and 7.12.6. Concerning Theorem 7.15 a natural question arises about the nature of limit ρ∞ → 0+ . This is one of open problems left to the reader. 7.12.2

Slip boundary conditions

As in Section 4.17.2.4, we suppose that µ, λ, Ω satisfy conditions (4.17.12) and (4.17.13). We consider problem (7.1.1), (7.1.2), (7.1.4), (7.1.7) with slip boundary conditions (7.12.3) u · n|∂Ω = 0, [τi dij (u)nj ]|∂Ω = 0, where dij (u) = 12 (∂j ui + ∂i uj ). In this case, we have to change the weak formulation of the momentum equation (7.1.20) in Definition 7.3 to

− I Ω ρui ∂t φi − I Ω ρui uj ∂j φi + λ I Ω div u div φ

+2µ I Ω dij (u)dij (φ) − I Ω p(ρ)div φ = I Ω (ρf + g) · φ, (7.12.4) φ ∈ (C ∞ (Ω × I))3 , φ · n|∂Ω×I = 0, φ(x, 0) = φ(x, T ) = 0, x ∈ Ω

and the energy inequalities (7.1.13) resp. (7.1.14) to

d ′ 2 dt E(ρ, q) + λ Ω |div u| + 2µ Ω dij (u)dij (u) ≤ Ω (ρf + g) · u in D (I) (7.12.5) and to

t

t

E(ρ, q) + λ 0 Ω |div u|2 + 2µ 0 Ω dij (u)dij (u) (7.12.6)

t

≤ E(ρ0 , q 0 ) + 0 Ω (ρf + g) · u a.e. I,

respectively. In (7.1.19), the assumptions for ρ are maintained while the condition u ∈ L2 (I, (D01,2 (Ω))3 ) is replaced by u ∈ L2 (I, (W 1,2 (Ω))3 ∩ E02 (Ω))). Other requirements are kept unchanged. After these arrangements we can claim: Theorem 7.69 Problem (7.1.1), (7.1.2), (7.1.4), (7.1.6), (7.12.3), where f , g satisfy (7.1.33), Ω is a bounded domain satisfying (7.1.32), (4.17.13) and λ, µ satisfy (4.17.12), admits at least one renormalized finite as well as bounded energy weak solution.


409

Repeating the argumentation (4.17.18), (4.17.19) and (4.17.20) of Section 4.17.2.4, we transform energy inequality (7.12.6) to

E(ρ, q) + cu21,2,QT ≤ E(ρ0 , q 0 ) + I Ω (ρf + g) · u a.e. I, (7.12.7)

with some c > 0. Inequality (7.12.7) will give later a priori estimates. With this observation at hand, the proof will follow the lines of the proof of Theorem 7.7, however, with a lot of new technicalities. The details are left to the reader. 7.12.3

Nonmonotone pressure

It was shown in (Feireisl, 2002) that a renormalized weak solution exists even for a large class of nonmonotone pressure laws. A version of this result is stated in the following theorem. Theorem 7.70 Let the assumptions of Theorem 7.7 be satisfied with the exception of hypothesis (7.1.4) which is replaced by p ∈ C 1 ([0, ∞)), p(0) = 0,

p(s) = asγ + r(s) + z(s), s > 0, where a > 0, γ > 23 ,

(7.12.8)

r ∈ C 0 ([0, ∞)) ∩ L∞ ((0, ∞)), z ∈ C 0 ([0, ∞)) nondecreasing. Then there exists a renormalized finite as well as bounded energy weak solution to problem (7.1.1), (7.1.2), (7.1.6), (7.1.7) in the sense of Definition 7.3, where P is given by (7.1.15). In order to remain in the same functional framework fixed by (7.1.35) and to avoid useless technical complexity, we suppose in addition to (7.12.8) that lims→∞

z(s) sγ

∈ [0, ∞).

In this situation, the proof of Theorem 7.70 is based on the same compactness argument as that described in Section 7.3 and rigorously given in Section 7.10. If we simplify it as much as possible, we find out, that there are essentially two observations which allow us to adapt this argument to the present situation: (1) The identity for the effective viscous flux reads p(ρ)Tk (ρ) − (2µ + λ)Tk (ρ)div u = p(ρ) Tk (ρ) − (2µ + λ)Tk (ρ)div u a.e. in QT . (2) Similarly as in (7.10.26), one can write

[p(ρ)Tk (ρ) − p(ρ) Tk (ρ)] QT

= a QT [ργ Tk (ρ) − ργ Tk (ρ)] + limn→∞ QT [r(ρn ) − r(ρ)][Tk (ρn ) − Tk (ρ)]

+ QT [r(ρ) − r(ρ)][Tk (ρ) − Tk (ρ)] + QT [z(ρ)Tk (ρ) − z(ρ) Tk (ρ)].

(In statements (1), (2), as usual, overlined quantities denote weak limits of corresponding sequences.) Since s → sγ is convex and Tk concave on [0, ∞), and

410


by virtue of the algebraic inequality (4.11.19), the first term on the

right-hand side of the above formula is bounded from below by a lim supn→∞ QT |Tk (ρn ) − and third terms are evidently bounded from below by Tk (ρ)|γ+1 . The second

−c(r) lim supn→∞ QT |Tk (ρn ) − Tk (ρ)| (c(r) = 4rL∞ ((0,∞)) ). Finally, since z and Tk are nondecreasing, the fourth term is nonnegative, cf. Exercise 3.37. We thus get the inequality

QT

[p(ρ)Tk (ρ) − p(ρ) Tk (ρ)]

≥ a lim supn→∞ QT |Tk (ρn ) − Tk (ρ)|γ+1

−c(r) lim supn→∞ QT |Tk (ρn ) − Tk (ρ)|

which replaces inequality (7.10.24) in Lemma 7.56. All remaining argumentation of Section 7.10 can be performed with minor changes only, which are induced by this modification. 7.12.4

Domain dependence

The details about the following issue can be found in (Feireisl et al., 2002). In that paper, the following concept of convergence of open sets is adopted:37 Definition 7.71 Let Ωn be a sequence of open sets in IR3 . We shall say that Ωn converges to an open set Ω (notation Ωn → Ω) if (i) For any compact K ⊂ Ω there exists n0 (K) ∈ IN such that K ⊂ Ωn for all n ≥ n0 ; (ii) Sets Ωn \ Ω are bounded and cap2 (Ωn \ Ω) → 0, where

cap2 (M ) = inf{ IR3 |∇φ|2 ; φ ∈ D(IR3 ), φ ≥ 1 on M }.

The dependence of weak solutions on spatial domains is described in the following theorem.

Theorem 7.72 Let Ωn be a sequence of open sets in IR3 such that Ωn → Ω, where Ω is an open nonempty set. Let (ρn , un ) be a bounded energy renormalized weak solution (prolonged by (0, 0) outside Ωn ) of problem (7.1.1)–(7.1.7) on the set Ωn × I satisfying ρn ∈ L∞ (I, L1 (Ωn ))38 with initial conditions (ρn0 , q n0 ) and driving forces f n , g n which are such that ρn0 → ρ0 , P (ρn0 ) → P (ρ0 ) in L1 (IR3 ), E(ρn0 , q n0 ) → E(ρ0 , q 0 ), f n → f , g n → g weakly-∗ in L∞ (IR3 × I).

Then, passing to subsequences if necessary, we have 37 Condition (ii) is satified if, e.g., Ω \ Ω are bounded, |∂Ω | = |∂Ω| = 0 and |Ω \ Ω| → 0 n n n as n → ∞. 38 This information is useless when Ω are bounded domains. n


411

ρn → ρ in C 0 (I, L1 (IR3 )), un → u weakly in L2 (I, (D01,2 (IR))3 ), ρn un → ρu in C 0 (I, L1weak (IR3 )),

where (ρ, u) is a bounded energy weak solution of the same problem on Ω × I for initial conditions (ρ0 , q 0 ) and driving forces f , g. Due to the standard observations of Section 7.10 (see the argument leading to (7.10.5), (7.10.8), (7.10.34) and Lemma 7.17), one can suppose without loss of 2γ/(γ+1) (Bm ))3 ), generality that ρn ∈ C 0 (I, Lp (Ω)), 1 ≤ p < γ; ρn un ∈ C 0 (I, (Lweak m ∈ IN , so that the statement of theorem makes sense. Of course, due to the same reasons, the limiting couple (ρ, u) satisfies ρ ∈ C 0 (I, Lp (Ω)) and ρu ∈ 2γ/(γ+1) (Bm ))3 ) as well. C 0 (I, (Lweak The proof of this result follows the lines of Section 7.3 (see especially Sections 7.3.5, 7.3.7.1–7.3.7.3). One essential difference lies in the fact that, due to assumption (i) of Definition 7.71, the sole local estimate of type (7.3.5) is available. This estimate reads

γ+θ (7.12.9) ρn ≤ L(K), θ = 2 γ − 1, K ∈ K, I

K

3

where L is, in particular, independent of n and = {K ⊂ Ω; K a compact set}. K

(7.12.10)

It is obtained by testing the momentum equation (7.1.1) for (ρn , un ) written on Ωn × I by the test function

ψBΩ′ (ρθn η − [ρθn η]) (7.12.11)

extended by 0 outside Ω′ (and conveniently relaxed, see (7.9.41)), in a similar way as described in Section 7.9.5. In (7.12.11), BΩ′ is the Bogovskii operator (cf. and Ω′ is a bounded Lemma 3.17), η ∈ D(Ω′ ), 0 ≤ η ≤ 1, η = 1 in K, K ∈ K ′ Lipschitz domain such that K ⊂ Ω ⊂ Ω. Therefore, instead of (7.3.13), we shall have only 1 ≤ p < 5γ−3 . (7.12.12) ρn → ρ strongly in Lp (I × K), K ∈ K, 3

This is, of course, enough to pass to the limit in the pressure term (and in all terms of the momentum and continuity equations). Another important technical point of the proof is to show that Ωn → Ω, un → u in L2 (I, (D01,2 (IR3 ))3 ), where un ∈ L2 (I, (D01,2 (Ω))3 ) implies that u ∈ L2 (I, (D01,2 (Ω))3 ). A further problem arises when one deals with the energy inequalities. In fact, formula (7.12.12) does not guarantee

P (ρn ) → Ω P (ρ) in D′ (I). Ω Instead of this, by lower weak semicontinuity, we have only

P (ρ) ≤ lim inf n→∞ Ω P (ρn ) in D′ (I). Ω

As a consequence, we do not know how to pass to the limit in the differential form of the energy inequality. This is the main reason why we are concerned only

412


with the bounded energy weak solutions. It is not known whether Theorem 7.72 is true for the finite energy weak solutions. Theorem 7.72 remains true for nonmonotone pressure (7.12.8). For the precise formulation, see (Feireisl et al., 2002). Any bounded or exterior domain with Lipschitz boundary can be approximated by smooth bounded or exterior domains Ωn in the sense of Definition 7.71 (see Section 4.15 for more details). Therefore, the existence of renormalized bounded energy weak solutions in Lipschitz bounded or in Lipschitz exterior domains with ρ∞ = 0 is a consequence of Theorem 7.72. More precisely, we have: Corollary 7.73 (i) Let Ω be a bounded Lipschitz domain. Suppose that all other assumptions of Theorem 7.7 are satisfied. Then there exists a renormalized bounded energy weak solution of problem (7.1.1)–(7.1.7) which satisfies (7.1.35). (ii) Let Ω be an exterior Lipschitz domain and let ρ∞ = 0. Suppose that all other assumptions of Theorem 7.15 are satisfied. Then statement (ii) of Theorem 7.15 remains valid. In both situations it is, however, not clear whether there exists a finite energy weak solution. By using techniques of local estimates of the density described above, the assumptions on the boundary of the domain can be weakened even more, see Section 7.12.7. For an adaptation of Theorem 7.72 to a problem of shape optimization in a viscous compressible flow see (Feireisl, 2003d) Another application of similar arguments is again due to (Feireisl, 2003b) who deals with a system of differential equations describing motion of rigid bodies in a viscous compressible fluid and obtains global existence of weak solutions. 7.12.5

Nonhomogeneous boundary conditions

In this section, we shall consider problem (7.1.1)–(7.1.5), (7.1.7) in a bounded domain Ω ⊂ IR3 completed with the nonhomogenous Dirichlet boundary conditions (7.12.13) u(x, t) = u∞ (x, t), (x, t) ∈ ∂Ω × I, ρ(x, t) = ρ∞ (x, t), t ∈ I, x ∈ Γt ,

(7.12.14)

Γt := {x ∈ ∂Ω; u∞ (x, t) · n(x) < 0}

(7.12.15)

where is the inflow portion of the boundary and ρ∞ ≥ 0, u∞ are given functions. Recall that I = (0, T ), where T > 0. This problem was discussed on the formal level and for values γ ≥ 95 in Section 7.6 of (Lions, 1998). Here we give a rigorous definition of renormalized weak solutions inspired by Lions’ considerations, and the papers (Feireisl, 2003d) and (Novo, 2003).


413

To start with and to simplify, we shall suppose that ρ∞ ≥ 0, ρ∞ = ξ|{(Γt ,t); t∈I} , u∞ = z|∂Ω×I , where ξ ∈ C 0 (IR3 × I), z ∈ (C 1 (IR3 × I))3 , ∇z ∈ C 0 (I, (C00 (IR3 ))3×3 ).

(7.12.16)

In the sequel, we identify (ρ∞ , u∞ ) with (ξ, z) so that ρ∞ ≥ 0, (ρ∞ , u∞ ) ∈ C 0 (IR3 × I) × (C 1 (IR3 × I))3 , ∇u∞ ∈ C 0 (I, (C00 (IR3 ))3×3 ).

(7.12.17)

If (ρ∞ , u∞ ), belonging to (7.12.17), also satisfies ∂t ρ∞ + div (ρ∞ u∞ ) = 0 in (IR3 \ Ω) × I,

(7.12.18)

then we say that the boundary conditions are admissibly prolongable to IR3 . With these notions at hand, we are ready to define weak solutions. Definition 7.74 Suppose that ρ0 , q 0 , f , g satisfy (7.1.17)–(7.1.18), Ω is a bounded domain and that (ρ∞ , u∞ ) fulfils (7.12.17). We say that a couple (ρ, u) is a weak solution of problem (7.1.1)–(7.1.5), (7.1.7), (7.12.13)–(7.12.15) if (i) P (ρ) =

1 γ γ−1 ρ

∈ L∞ (I, L1 (Ω)), ρ ≥ 0 a.e. in QT = Ω × I,

u − u∞ ∈ L2 (I, (W01,2 (Ω))3 ), ρ|u|2 ∈ L∞ (I, L1 (Ω)); Equation (7.1.1) holds in (D′ (QT ))3 ; Equation (7.1.2) holds in the form

(ρ∂t η + ρu · ∇η) = I Γt ρ∞ u∞ · nη, QT η ∈ D(QΓ ), where QΓ = t∈I [(Ω ∪ Γt ) × {t}];

limt→0+ Ω ρ(t)ψ → Ω ρ0 ψ, ψ ∈ D(Ω),

limt→0+ ρ(t)u(t) · φ → Ω q 0 φ, φ ∈ (D(Ω))3 .

(7.12.19) (7.12.20)

(7.12.21)

(7.12.22)

(ii) A couple (ρ, u) is called a renormalized weak solution of problem (7.1.1)– (7.1.5), (7.1.7), (7.12.13)–(7.12.15) if in addition to (7.12.19)–(7.12.22),

[b(ρ)∂t η + b(ρ)u · ∇η + (b(ρ) − ρb′ (ρ))div uη] QT

(7.12.23) = I Γt b(ρ∞ )u∞ · nη, η ∈ D(QΓ ) with any function b satisfying (6.2.9)–(6.2.11), where β = γ.

(iii) A couple (ρ, u) is called a finite energy weak solution of problem (7.1.1)– (7.1.5), (7.1.7), (7.12.13)–(7.12.15) if in addition to (7.12.19)–(7.12.22),

414


|∇(u − u∞ )|2 + (µ + λ) Ω |div (u − u∞ )|2

≤ Ω (ρf + g) · (u − u∞ ) − Γ P (ρ∞ )u∞ · n + Ω ρui (u∞ − u) · ∂i u∞

− Ω p(ρ)div u∞ − µ Ω ∂i u∞ · ∂i (u − u∞ )

−(µ + λ) Ω div u∞ div (u − u∞ ) in D′ (I), (7.12.24)

d dt E(ρ, q)

+µ

Ω

where39

q = ρu, E(ρ, q) = E (Ω) (ρ, q) =

2 ∞| [ 1 |q−ρu 1{ρ>0} ρ Ω 2

+ P (ρ)].

(7.12.25)

(iv) A couple (ρ, u) is called a bounded energy weak solution of problem (7.1.1)– (7.1.4), (7.1.7), (7.12.13)–(7.12.15) if in addition to (7.12.19)–(7.12.22),

t

t

E(ρ, q) + µ 0 Ω |∇(u − u∞ )|2 + (µ + λ) 0 Ω |div (u − u∞ )|2

t

t

t

≤ E0 + 0 Ω (ρf + g) · (u − u∞ ) − 0 Γ P (ρ∞ )u∞ · n − 0 Ω p(ρ)div u∞

t

t

+ 0 Ω ρui (u∞ − u) · ∂i u∞ − µ 0 Ω ∂i u∞ · ∂i (u − u∞ )

t

−(µ + λ) 0 Ω div u∞ div (u − u∞ ) a.e. in I, (7.12.26) where E0 = E (Ω) (ρ0 , q 0 ) =

2 [ 1 |q0 −ρρ00u∞ | 1{ρ0 >0} Ω 2

+ P (ρ0 )].

(7.12.27)

Remark 7.75 Suppose that Ω is a bounded Lipschitz domain. Let (ρ∞ , u∞ ) be admissibly prolongable to IR3 . Let us extend (ρ, u) to IR3 × I by an arbitrary admissible prolongation (ρ∞ , u∞ ), see (7.12.17), (7.12.18), and denote the new couple again by (ρ, u). Then for any b from (7.12.23) as well as for any b(s) = cs, where c > 0, we have

{b(ρ)∂t η + b(ρ)u · ∇η + [b(ρ) − ρb′ (ρ)]div u η} QT (7.12.28)

= I Γt b(ρ∞ )u∞ · nη, η ∈ D(QΓ ) = D( t∈I [(Ω ∪ Γt ) × {t}])

if and only if

∂t b(ρ) + div (b(ρ)u) + [ρb′ (ρ) − b(ρ)]div u = 0 in D′ ( t∈I [(IR3 \ (∂Ω \ Γt )) × {t}]).

(7.12.29)

If (ρ∞ , u∞ ) is admissibly prolongable to IR3 it might be useful to consider (7.12.29) rather than (7.12.23), to define the renormalized weak solutions. An expected result in this situation is formulated in the following conjecture. 39 E(ρ, q)

is equal to

1 Ω [ 2 ρ|u

− u∞ |2 + P (ρ)] a.e. in I.


415

Theorem 7.76 (Conjecture) Let Ω be a bounded Lipschitz domain, f , g satisfy (7.1.33). Suppose that (ρ∞ , u∞ ) satisfy (7.12.17). Suppose further that 2γ

ρ0 ∈ Lγ (Ω), ρ0 ≥ 0 a.e. in Ω,

q 0 ∈ (L γ+1 (Ω))3 , q 0 1{ρ0 =0} = 0 a.e.in Ω,

|q 0 |2 ρ0 1{ρ>0}

∈ L1 (Ω).

(7.12.30)

Then problem (7.1.1)–(7.1.5), (7.1.7), (7.12.13)–(7.12.15) admits at least one renormalized finite as well as bounded energy weak solution (ρ, u) which is such that40 s(γ)

ρ − ρ∞ ∈ L∞ (I, Lγ (IR3 )), ρ ∈ Ls(γ) (I, Lloc (Ω)), s(γ) =

5γ−3 3 ,

ρ ∈ C 0 (I, Lγweak (Ω)) ∩ C 0 (I, Lp (Ω)), 1 ≤ p < γ, ρ ≥ 0 a.e. in Ω × I, ρ = ρ∞ in (IR3 \ Ω) × I,

u − u∞ ∈ L2 (I, (W 1,2 (IR3 ))3 ), u = u∞ in (IR3 \ Ω) × I,

(7.12.31)

2γ

γ+1 (Ω))3 ), ρu ∈ C 0 (I, (Lweak

6γ

ρ|u|2 ∈ L∞ (I, L1 (Ω)) ∩ L2 (I, L 4γ+3 (Ω)). So far, no exhaustive proof of Theorem 7.76 in its full generality exists in the mathematical literature. Only some particular cases have been treated rigorously. For example, it is known (see (Novo, 2003)) that a version of Theorem 7.76 holds if we take Ω = BR \ S, where S ⊂ B1/2 is a domain, R ≥ 1 and ρ∞ = const ≥ 0, u∞ =

a∞ ∈ IR3 on ∂BR , 0 on ∂S.

(7.12.32) (7.12.33)

Before we give a precise formulation of this existence result and outline its proof, we observe that the above boundary conditions are admissibly prolongable to IR3 . We take a vector field U ∞ ∈ D(IR3 ), supp U ∞ ⊂ B2R′ , ! 0 in B 34 3 3 ′ div U ∞ = 0 ∈ IR \ BR , U ∞ = R a∞ in B12 ,

(7.12.34)

where R′ is chosen sufficiently large, e.g. R′ = R + |a∞ |T.

(7.12.35)

Such a vector field can be constructed relatively easily with the help of convenient cut-off functions, Lemma 3.17 and mollifiers. Notice that (ρ∞ , U ∞ ) is 40 Since Ω is Lipschitz, the information u−u 2 1,2 (IR3 ))3 ), u = u 3 ∞ ∈ L (I, (W ∞ in (IR \Ω)×I means that u − u∞ ∈ L2 (I, (W01,2 (Ω))3 ).

416


an admissible prolongation of (ρ∞ , u∞ ) to IR3 . In the sequel we shall therefore identify u∞ with U ∞ . We have the following theorem. Theorem 7.77 Let Ω be a domain satisfying (7.12.32), let f , g verify (7.1.33), let ρ∞ ≥ 0, let u∞ satisfy (7.12.34) and let (ρ0 , u0 ) satisfy (7.12.30). (i) If Ω is a Lipschitz domain, then problem (7.1.1)–(7.1.5), (7.1.7), (7.12.13)– (7.12.15) admits at least one renormalized bounded energy weak solution41 which satisfies (7.12.31). (ii) If Ω is a domain (no regularity assumption), then statement (i) remains true but u − u∞ may not belong to L2 (I, (W01,2 (Ω))3 ). (We have only u − u∞ ∈ L2 (I, (W 1,2 (IR3 ))3 ), u = u∞ in (IR3 \ Ω) × I.) (iii) In both cases (i), (ii), in addition to (7.12.26),

t

t

E (ρ∞ ) (ρ, q) + µ 0 Ω |∇(u − u∞ )|2 + (µ + λ) 0 Ω |div (u − u∞ )|2

t

t

(ρ ) ≤ E0 ∞ + 0 Ω (ρf + g) · (u − u∞ ) + 0 Ω ρui (u∞ − u) · ∂i u∞ (7.12.36)

t

t

− 0 Ω p(ρ)div u∞ − µ 0 Ω ∂i u∞ · ∂i (u − u∞ )

t

−(µ + λ) 0 Ω div u∞ div (u − u∞ ) a.e. in I, where

q = ρu, E (c) (ρ, q) = E (Ω,c) (ρ, q) = P (c) (ρ) =

1 γ γ−1 [ρ

1 2

Ω

|q−ρu∞ )|2 1{ρ>0} ρ (c)

+ (γ − 1)cγ − γcγ−1 ρ], E0

+

Ω

P (c) (ρ),

= E (Ω,c) (ρ0 , u0 ).

(7.12.37)

In what follows, we explain the main points of the proof of this result in the nontrivial case ρ∞ > 0. To start with, we take Ω a bounded domain of type (7.12.32). We set V = B2R′ (7.12.38) and consider the following penalized system ∂t (ρu) + ∂j (ρuuj ) − µ∆u − (µ + λ)∇div u + ∇ργ

+δ∇ρβ + ǫ∇ρ · ∇u + m1V \Ω (u − u∞ ) = ρf + g in V × I, m ∈ IN,

(7.12.39)

∂t ρ + div (ρu) − ǫ∆ρ = 0 in V × I,

(7.12.40)

u = 0 in ∂V × I, ∂n ρ = 0 in ∂V × I

(7.12.41)

where f , g are extended by 0 outside Ω, with the boundary conditions and the initial conditions ρ(0) = ρ0 ∈ C ∞ (V ), 0 < ρ ≤ ρ0 ≤ ρ < ∞, ρu(0) = q 0 ∈ (C ∞ (V ))3 .

(7.12.42)

The extra term m1V \Ω (u − u∞ ) does not prevent the Galerkin approximation in Section 7.7 from working. As in that section and in the same way as in Section 7.8, 41 The

existence of a finite energy weak solution is an open problem, even if Ω is smooth.


417

we obtain a couple (ρm , um ) which belongs on V to the class (i) of Proposition 7.31, satisfies system (7.12.39)–(7.12.42) and obeys the energy inequalities

d (V ) 2 2 dt Eδ (ρm , q m ) + µ V |∇(um − u∞ )| + (µ + λ) Ω |div (u − u∞ )|

2 2 +ǫδβ V ρβ−2 m |∇ρm | + m V \Ω |um − u∞ |

≤ Ω (ρm f + g) · (um − u∞ ) + ǫ V ∇ρm · ∇(um − u∞ ) · u∞

+ V ρm ui (u∞ − um ) · ∂i u∞ − V pδ (ρm )div u∞

−µ V ∂i u∞ · ∂i (um − u∞ ) − (µ + λ) V div u∞ div (um − u∞ ) in D′ (I), (7.12.43) and

t

t

(V ) Eδ (ρ(t), q(t)) + µ 0 Ω |∇(um − u∞ )|2 + (µ + λ) 0 Ω |div (um − u∞ )|2

t

t

2 2 +ǫδβ 0 V ρβ−2 m |∇ρm | + m 0 V \Ω |um − u∞ |

t

(V ) ≤ Eδ0 + 0 Ω (ρm f + g) · (um − u∞ )

t

t

+ǫ 0 V ∇ρm · ∇(um − u∞ ) · u∞ + 0 V ρui (u∞ − um ) · ∂i u∞

t

t

− 0 V pδ (ρm )div u∞ − µ 0 V ∂i u∞ · ∂i (um − u∞ )

t

−(µ + λ) 0 V div u∞ div (um − u∞ ) a.e. in I, (7.12.44) where

2 (V ) ∞| 1{ρ>0} + Pδ (ρ)], Eδ (ρ, q) = V [ 21 |q−ρu ρ (7.12.45) (V ) (V ) 1 1 Pδ (ρ) = γ−1 ργ + β−1 ρβ , pδ (ρ) = ργ + δρβ , Eδ0 = Eδ (ρ0 , u0 ). (V,c)

Inequalities (7.12.43), (7.12.44) hold also with Eδ (V ) in place of Eδ0 , where42 (V,c)

c > 0, Eδ (c)

Pδ (ρ) =

(ρ, q) =

1 γ γ−1 [ρ

2 ∞| 1{ρ>0} [ 1 |q−ρu ρ V 2

(V,c)

(V,c)

and Eδ0

(c)

+ Pδ (ρ)],

+ (γ − 1)cγ − γcγ−1 ρ] +

+(β − 1)cβ − βcβ−1 ρ], Eδ0

(V )

in place of Eδ

1 β β−1 [ρ (V,c)

= Eδ

(7.12.46)

(ρ0 , u0 ).

Formula (7.12.43) is obtained by subtracting from the “usual” energy inequality

1

d 2 2 2 dt V [ 2 ρm |um | + Pδ (ρm )] + µ V |∇um | + (µ + λ) Ω |div um |

2 +ǫδβ V ρβ−2 m |∇ρm | + m V \Ω (um − u∞ ) · um ≤ Ω (ρm f + g) · um

42 E (V ) (ρ, q) is equal to 1 δ 2 V (c) u∞ |2 + V Pδ (ρ) a.e. in I.

ρ|u − u∞ |2 +

V

(V,c)

Pδ (ρ) a.e. in I and Eδ

(ρ, q) to

1 2

V

ρ|u −

418


(which corresponds to system (7.12.39)–(7.12.41)), the identity

d dt V ρm um · u∞ + µ V ∂i um · ∂i u∞ + (µ + λ) V div um div u∞

− V ρm uim ujm ∂i uj∞ − V pδ (ρm )div u∞ + ǫ V ∇ρm · ∇um · u∞

+m V \Ω (um − u∞ ) · u∞ = Ω (ρm f + g) · um

(which is obtained from testing of (7.12.39) by u∞ ). Similar approach leads to (7.12.44). From inequalities (7.12.43) and (7.12.44) it is not difficult to derive estimates (7.4.33)–(7.4.38) on V , (7.4.40)–(7.4.42) independent of m, and a bound for

m V \Ω |um − u∞ |2 independent of m, as well. With this information, we can easily pass to the limit m → ∞ both in system (7.12.39)–(7.12.42) and in the energy inequalities. We obtain a couple (ρǫ , uǫ ) which possesses the regularity properties of Proposition 7.31 on V × I (except for ρǫ ∈ Lβ+1 (V × I) which is replaced only by ρǫ ∈ Lβ+1 (I, Lβ+1 loc (Ω))), and which satisfies ∂t (ρǫ uǫ ) + ∂j (ρǫ uǫ ujǫ ) − µ∆uǫ − (µ + λ)∇div uǫ

+∇ργǫ + δ∇ρβǫ + ǫ∇ρǫ · ∇uǫ = ρǫ f + g in Ω × I,

(7.12.47)

∂t ρǫ + div (ρǫ uǫ ) − ǫ∆ρǫ = 0 in V × I,

(7.12.48)

uǫ = 0 in ∂V × I, ∂n ρǫ = 0 in ∂V × I,

(7.12.49)

ρǫ (0) = ρ0 , (ρǫ uǫ )(0) = q 0 ,

(7.12.50)

and moreover uǫ = u∞ in (V \ Ω) × I.

(7.12.51) (c) Eδ and Eδ ,

In addition, energy inequalities (7.12.43)–(7.12.44), both with where we replace (ρm , um ) by (ρǫ , uǫ ) and neglect the positive term with m V \Ω |um − u∞ |2 , hold. These inequalities yield bounds (vi) of Proposition 7.31 on V × I with the exception of estimate (7.4.39) which holds only on K × I, where K ∈ K, cf. (7.12.10), namely ρǫ 0,β+1,K×I ≤ L(E 0 , f , g, δ, T, K, u∞ ),

(7.12.52)

where E 0 = supδ∈(0,1) Eδ0 . Proofs of these facts can be copied, with minor changes, from Section 7.8.3 (for all inequalities except for (7.12.52)) and from Section 7.8.4 (for (7.12.52)). In order to get (7.12.52), we use in momentum equation (7.12.47) the test function43 Ω is at least a Lipschitz domain, we can take ϕ = ψBΩ (ρǫ η − Ω [ρǫ η]) with η ∈ D(Ω), 0 ≤ η ≤ 1, η = 1 in K. The presence of η in the test function is, however, inevitable. Indeed, Ω (ρuη) in order to estimate one of the terms in ∂t ϕ by using Lemma 3.17, the normal trace γn is required to be zero. This is true with η = 1 only provided u∞ = 0. See Sections 7.8.4, 7.9.5 for more details and Sections 7.11.4 and 7.12.4 for a similar phenomenon. 43 If


ϕ = ψBΩ′ (ρǫ η −

Ω′

[ρǫ η]),

419

(7.12.53)

extended by 0 outside Ω′ , where BΩ′ is again the Bogovskii operator, ψ ∈ D(I), and Ω′ is a bounded Lipschitz domain η ∈ D(Ω′ ), 0 ≤ η ≤ 1, η = 1 in K, K ∈ K such that K ⊂ Ω′ ⊂ Ω. Now, we set (ρǫ , uǫ ) = (0, 0) outside V and let ǫ → 0+ . One repeats the argument of Section 7.9. After the obvious standard reasoning, we arrive at the point proving the strong convergence of ρǫ , where we need the effective viscous flux. Using Proposition 7.36 we arrive at the identity for the effective viscous flux in the form [ργ + δρβ − (2µ + λ)div u]ρ = ργ+1 + ρβ+1 − (2µ + λ)ρdiv u

(7.12.54)

only a.e. in Ω × I. The missing information is provided by (7.12.51) which yields ρdiv u = ρdiv u = 0 in (V \ Ω) × I. This allows us to finish the argument of Section 7.9.3 and to get ρǫ → ρ strongly in L1 (V ×I) and consequently in Lp (V ×I), 1 ≤ p < β as well as in Lp (I, Lploc (Ω)), 1 ≤ p < β + 1. As a result of these considerations, and similarly as in Proposition 7.27, we obtain a couple (ρδ , uδ ) with the following properties: (i) β 0 p 0 ρδ ∈ Lβ+1 (I, Lβ+1 loc (Ω)), ρδ ∈ C (I, Lweak (V )) ∩ C (I, L (V )),

1 ≤ p < β, ρδ (0) = ρ0 a.e. in V , ρδ ≥ 0 a.e. in V × I, ρδ = 0 in (IR3 \ V ) × I, uδ ∈ L2 (I, (W 1,2 (IR3 ))3 ),

uδ = u∞ in (V \ Ω) × I, uδ = u∞ = 0 in (IR3 \ V ) × I, q δ := ρδ uδ ∈ L2 (I, (L

6β 6+β

(7.12.55)

2β β+1

(IR3 ))3 ) ∩ C 0 (I, (Lweak (Ω))3 ),

(ρδ uδ )(0) = q 0 a.e. in Ω, 6β

3β

ρδ |uδ |2 ∈ L∞ (I, L1 (IR3 )) ∩ L2 (I, L 4β+3 (IR3 )) ∩ L1 (I, L β+3 (IR3 )). (ii) ∂t (ρδ uδ ) + ∂j (ρδ uδ ujδ ) − µ∆uδ

−(µ + λ)∇div uδ + ∇ργδ + δ∇ρβδ = ρf + g in D′ (Ω × I).

(7.12.56)

(iii) ∂t ρδ + div (ρδ uδ ) = 0 in D′ (IR3 × I).

(7.12.57) β

λ1 +1 (V )) (iv) For any b from (6.2.9)–(6.2.11), the function b(ρδ ) belongs to C 0 (I,Lweak β 0 p ∩C (I, L (V )), 1 0 defined by (6.2.22), bk (ρδ ) belongs to C 0 (I, Lp (V )), 1 ≤ p < ∞ and ∂t bk (ρδ ) + div (bk (ρδ )uδ ) (7.12.59) +{ρδ (bk )′+ (ρδ ) − b(ρδ )}div uδ = 0 in D′ (IR3 × I). (v)

t

t

(ρδ , q δ ) + µ 0 Ω |∇(uδ − u∞ )|2 + (µ + λ) 0 Ω |div (uδ − u∞ )|2

t

t

≤ Eδ0 + 0 Ω (ρf + g) · (uδ − u∞ ) + 0 Ω ρuiδ (u∞ − uδ ) · ∂i u∞ (7.12.60)

t

t

− 0 Ω pδ (ρδ )div u∞ − µ 0 Ω ∂i u∞ · ∂i (uδ − u∞ )

t

−(µ + λ) 0 Ω div u∞ div (uδ − u∞ ) a.e. in I.

(V )

Eδ

(V )

(vi) Inequality (7.12.60) holds also if we replace Eδ

(V,c)

by Eδ

(V )

(V,c)

and Eδ0 by Eδ0

.

Due to lack of the strong convergence of ρǫ in Lp (V × I) with some p > β, the energy inequality in the differential form, in general, cannot be deduced. The reasons for this difficulty are explained in Section 7.12.4. In the sequel, we show how to transform identities (7.12.57), (7.12.58) and inequality (7.12.60) to forms (7.12.21), (7.12.23) and (7.12.26), respectively. By virtue of (7.12.34)–(7.12.35), 3

u∞ = a∞ in B12

(R+T |a∞ |)

.

Therefore, regularizing (7.12.57) with respect to space variables, we, in particular, get ∂t Sα (ρδ ) + a∞ · Sα (ρδ ) = 0 a.e. in Uα × I, (7.12.61) where 0 < α < Uα =

T 2

and

t∈(0,T −2α)

S(t), S(t) =

r∈(1, 23 ) {x;

x ∈ rΓ − (T − t)a∞ }.

(7.12.62)

If B is a ball in S(0) and B(t) = B + ta∞ , by virtue of (7.12.61), we have

d dt B(t) Sα (ρδ ) = 0, t ∈ (0, T − 2α).

Once this formula is integrated, and after letting α → 0+ , one gets B(t) ρδ (t) =

ρ , t ∈ I. Therefore, if x is a Lebesgue point of ρ0 in S(0), the point x + ta∞ B 0 is a Lebesgue point of ρδ (t) in S(t), or, in the other words, ρδ (x + ta∞ , t) = ρ0 (x), x ∈ S(0). If we choose ρ0 in such a way that


ρ0 = ρ∞ in U = U0

421

(7.12.63)

(which is evidently possible, see later), then ρδ = ρ∞ in S(T ) × I.

(7.12.64)

Using in (7.12.57) a test function φ ∈ D(I, D(BR ∪ S(T ) ∪ Γ) ∩ D(IR3 × I)), after applying the Stokes formula, we obtain (iii′ )

[ρ ∂ φ + ρδ uδ · ∇φ] = I Γ ρ∞ u∞ · nφ (7.12.65) I BR δ t

which implies (7.12.21) with ρ replaced by ρδ . In an analogous way, formula (7.12.58) implies (iv′ )

[b(ρδ )∂t φ + b(ρδ )uδ · ∇φ] I BR (7.12.66)

= I BR {ρδ b′ (ρδ ) − b(ρδ )}div u φ + I Γ b(ρ∞ )u∞ · nφ which yields (7.12.23) with ρ replaced by ρδ . Finally, we set

φn (x) = w(ndist (x, ∂(V \ BR )), x ∈ V \ BR , n CV,− = {x ∈ V \ BR ;

n CB,+ = {x ∈ V \ BR ;

1 n

1 n

≤ dist (x, ∂V ) ≤ ≤ dist (x, ∂BR ) ≤

2 n },

2 n },

n n n n n CB,+,Γ = CB,+ ∩ S(T ), CB,+,Γ c = CB,+ \ CB,+,Γ ,

where w ∈ C ∞ (IR), 0 ≤ w ≤ 1, w′ ≥ 0, w(s) =

0 if s ≤ 1 1 if s ≥ 2

(7.12.67)

n and take n large enough so that CV,− ∩ supp u∞ = ∅ and (1 + n2 )Γ ⊂ S(T ). Since n n n u∞ = 0 in CV,− , u∞ · ∇φn ≥ 0 in CB,+,Γ c and ρδ = ρ∞ , uδ = a∞ in CB,+,Γ , due to Exercise 6.13 and by using the Stokes formula, we get

d dt V \BR Pδ (ρδ )φn = V \BR Pδ (ρδ )u∞ · ∇φn

Pδ (ρδ )u∞ · ∇φn = C n Pδ (ρδ )u∞ · ∇φn + C n B,+,Γ V,−

+ C n c Pδ (ρδ )u∞ · ∇φn B,+,Γ

≥ (1+ 2 )Γ Pδ (ρδ )u∞ · nφn . n

Integrating this inequality from s to t, s, t ∈ I, s < t, we get

t

P (ρ (t))φn − V \BR Pδ (ρδ (s))φn ≥ s (1+ 2 )Γ Pδ (ρ∞ )u∞ · nφn V \BR δ δ n

422


for a.a. s, t ∈ I, cf. Exercise 7.4. Letting n → ∞ in the last inequality and applying the lower semicontinuity of the map s → V \BR Pδ (ρ(s)), cf. Exercise 7.20, we get

V \BR

Pδ (ρ(t)) −

V \BR

Pδ (ρ0 ) ≥

t

0

Γ

Pδ (ρ∞ )u∞ · n for a.a. t ∈ I.

Using the last inequality in (7.12.60), we obtain that (v′ ) inequality (7.12.26) holds with ρ replaced by ρδ , (Ω)

p by pδ , P by Pδ , E by Eδ

(7.12.68)

(Ω)

and E0 by Eδ0

(c)

(see (7.12.45)). Effectuating the same procedure with Pδ (c) taking into account the fact Pδ (c) = 0, we get ′ (vi )

in place of Pδ , and

inequality (7.12.36) holds with ρ replaced by ρδ , (ρ∞ )

p by pδ , P (ρ∞ ) by Pδ

(Ω,ρ∞ )

, E (ρ∞ ) by Eδ

(ρ∞ )

and E0

(Ω,ρ∞ )

by Eδ0

(7.12.69)

(see (7.12.46)). The energy inequality (7.12.68) yields the bounds (vii) of Proposition 7.27 with the exception of bound (7.4.17). The latter estimate holds true only in local form, namely ρδ 0,γ+θ,K×I ≤ L(E 0 , f , g, K, T, a∞ ), 1 β+θ ρδ 0,β+θ,K×I ≤ L(E 0 , f , g, K, T, a∞ ) δ

/

, θ=

2 (7.12.70) γ − 1, K ∈ K. 3

For the proof of this inequality see Section 7.9.5 whose argument has to be modified by considering test function (7.12.11) with n replaced by δ. (The reasons for this choice were discussed in the footnote related to (7.12.53).) Now, it is easy to repeat, with small changes, the reasoning of Section 7.10 to let δ → 0+ and thus complete the proof of Theorem 7.77. The argument goes as follows. First, we extend (ρδ (t)|BR , uδ (t)|BR ) by (ρ∞ , u∞ ) outside BR and, from now on, denote the new so-redefined couple again by (ρδ , uδ ). In accordance with Remark 7.75, letting δ → 0+ in (7.12.65), (7.12.66), we get ∂t ρ + div (ρu) = 0 in D′ ([IR3 \ (∂BR \ Γ)] × I).

(7.12.71)

and ∂t b(ρ) + div (b(ρ)u) (ρb′ (ρ) − b(ρ))div u = 0 in D′ ([IR3 \ (∂BR \ Γ)] × I),

(7.12.72)

where ρ, u and overlined quantities denote weak limits of corresponding sequences and b satisfies (6.2.9), (6.2.10) with λ1 + 1 < γ2 .


423

We still have the identity for the effective viscous flux, namely ργ Tk (ρ) − (2µ + λ)Tk (ρ)div u = ργ Tk (ρ) − (2µ + λ)Tk (ρ)div u a.e. in Ω × I (where Tk is defined in (7.10.19)), see Section 7.10.2, and therefore, the quantity sup lim sup Tk (ρδ ) − Tk (ρ)0,γ+1,QT k>0 δ→0+

is bounded, see Section 7.10.3. Thus, from (7.12.71), by the same reasoning as that in Section 7.10.4, the renormalized continuity equation ∂t b(ρ) + div (b(ρ)u) +(ρb′ (ρ) − b(ρ))div u = 0 in D′ ([IR3 \ (∂BR \ Γ)] × I)

(7.12.73)

can be deduced. In (7.12.73), b satisfies (6.2.9), (6.2.10) with 1 + λ1 ≤ γ2 . The key point of the proof in Section 7.10 is equation (7.10.43). It was obtained from the renormalized continuity equations for Lk (ρ) and Lk (ρ) (for the definition of Lk see (7.10.36)) by using the test functions φ ∈ D(IR3 ), φ = 1 in Ω. These test functions are not admissible in (7.12.73). To modify the argument, we have to use the test functions φm (x) = ηw(mdist (x, ∂BR \ Γ), m ∈ IN, x ∈ IR3 , where w is defined in (7.12.67) and η ∈ D(IR3 ), η = 1 in BR . Taking into account the values of weak limits outside BR , in place of (7.10.43), we shall have

[Lk (ρ(T )) − Lk (ρ(T ))]φm

= BR ×I [Lk (ρ) − Lk (ρ)]u · ∇φm + Ω×I [Tk (ρ)div u − Tk (ρ)div u]φm .

Ω

We write the first term at the right-hand side as

BR ×I

[Lk (ρ) − Lk (ρ)](u − u∞ ) · ∇φm +

QT

[Lk (ρ) − Lk (ρ)]u∞ · ∇φm .

Since u∞ · ∇φm ≤ 0 in BR for all m large enough, and since | BR [Lk (ρ) −

∞ Lk (ρ)](u − u∞ ) · ∇φm | = | BR [l(ρ) − l(ρ)](u − u∞ ) · ∇φm | ≤ c(k) distu−u (x,∂BR ) 0,2 1

|supp∇φm | 2 , cf. (7.10.37), we obtain the inequality

Ω×I

[Lk (ρ(T )) − Lk (ρ(T ))] =

Ω×I

[Tk (ρ)div u − Tk (ρ)div u]

as m → ∞ (see Section 1.3.5.9). This is the same formula as (7.10.43). The rest of the proof is the same as in Section 7.10.5.

424


Up to now, we considered smooth initial conditions (ρ0 , q 0 ) belonging to (7.12.42) and (7.12.63). In this paragraph we describe how to deal with the general initial conditions (7.12.30). First, we solve the penalized system (7.12.38)– (7.12.41) as well as systems (7.12.47)–(7.12.51) and (7.12.56)–(7.12.57) with initial conditions (ρ0δ , q 0δ ) satisfying (7.12.42) and (7.12.63), where (ρ0δ , q 0δ ) is defined as follows. We set ρ0δ = S δ (ρ0 1(BR )(δ,−) + ρ∞ 1U( δ ,+) ) + δ, 4

(7.12.74)

2

where ρ0 is extended by 0 outside Ω and A(δ,−) = {x ∈ A; dist (x, ∂A) > δ}, A(δ,+) = {x ∈ A; dist (x, A) < δ}. Next, we put  √ ρ   q 0 √ρ0δ0 in {x ∈ Ω; ρ0 > 0} ˜ 0δ = 0 (7.12.75) q in {x ∈ Ω; ρ0 = 0}   ρ u in V \ Ω. 0δ ∞

By density, there exists hδ ∈ (C ∞ (V ))3 such that ˜

√qρ0δ0δ − hδ 0,2,V ≤ δ. Finally we set q 0δ =

√

ρ0δ hδ ∈ (C ∞ (V ))3 .

(7.12.76) (7.12.77)

With this choice we easily complete the proof in the spirit of Section 7.10.7. 7.12.6 Unbounded domains and non-zero velocity at infinity In this section we generalize Theorem 7.15 to the case of prescribed nonzero velocity at infinity and to the case of an arbitrary exterior domain (no regularity assumption). To start with we shall suppose Ω = IR3 \ S, where S ⊂ B 21 is a bounded domain, ρ∞ > 0, a∞ ∈ IR3

(7.12.78) (7.12.79)

We want to solve in Ω the problem (7.1.1)–(7.1.7) completed with conditions at infinity lim ρ(x) = ρ∞ , lim u(x) = a∞ . (7.12.80) |x|→∞, x∈Ω

|x|→∞, x∈Ω

To start with, we take the vector field U ∞ constructed in (7.12.34) which corresponds to R = 1. Then we set u∞ =

U ∞ in B1+|a∞ |T a∞ in B 1+|a∞ |T .

(7.12.81)

Obviously, u∞ ∈ C ∞ (IR3 ) and

u∞ = 0 in B 34 , div u∞ = 0 in B 1 , supp∇u∞ ⊂ B1+|a∞ |T .

(7.12.82)

Now, we are in position to define weak solutions, renormalized weak solutions, bounded energy and finite energy weak solutions in this situation.


425

Definition 7.78 We take Definition 7.3 with the following changes: (i) In (7.1.19), u − u∞ ∈ L2 (I, (D01,2 (Ω))3 ) and ρ|u − u∞ |2 ∈ L∞ (I, L1 (Ω)). (ii) The continuity equation and renormalized continuity equations (7.1.23), (7.1.24) are satisfied in D′ (IR3 × I) provided (ρ, u) is extended by (ρ∞ , u∞ ) = (ρ∞ , 0) outside Ω. (iii) In (7.1.25), the energy inequality in differential form is not (7.1.13) but

|∇(u − u∞ )|2 + (µ + λ) Ω |div (u − u∞ )|2

≤ Ω (ρf + g) · (u − u∞ ) + Ω ρui (u∞ − u) · ∂i u∞

− Ω p(ρ)div u∞ − µ Ω ∂i u∞ · ∂i (u − u∞ )

−(µ + λ) Ω div u∞ div (u − u∞ ) in D′ (I),

d dt E(ρ, q)

where

+µ

Ω

E(ρ, q) = (iv) In (7.1.26), one sets E0 =

2 ∞| 1{ρ>0} [ 1 |q−ρu ρ Ω 2

2 [ 1 |q0 −ρρ00u∞ | 1{ρ0 >0} Ω 2

+ P (ρ∞ ) (ρ)].

(7.12.83)

(7.12.84)

+ P (ρ∞ ) (ρ0 )]

and one supposes that it is finite. The energy inequality in integral form is not (7.1.14) but

t

|∇(u − u∞ )|2 + (µ + λ) 0 Ω |div (u − u∞ )|2

t

t

≤ E0 + 0 Ω (ρf + g) · (u − u∞ ) + 0 Ω ρui (u∞ − u) · ∂i u∞

t

t

− 0 Ω p(ρ)div u∞ − µ 0 Ω ∂i u∞ · ∂i (u − u∞ )

t

−(µ + λ) 0 Ω div u∞ div (u − u∞ ) a.e. in I.

E(ρ, q) + µ

t

0

Ω

(7.12.85)

The following theorem holds true.

Theorem 7.79 Let Ω be an exterior domain (no regularity assumption). Suppose that f , g satisfy (7.1.40), p(ρ) satisfies (7.1.4), µ, λ satisfy (7.1.5), ρ∞ > 0, a∞ ∈ IR3 and 2γ

ρ0 ≥ 0 a.e. in Ω, P (ρ0 ) = P (ρ∞ ) (ρ0 ) ∈ L1 (Ω), |q 0 −ρ0 a∞ |2 1{ρ0 >0} ρ0

∈ L1 (Ω). (7.12.86) (i) If Ω is a Lipschitz domain, then there exists a bounded energy renormalized weak solution (ρ, u) of problem (7.1.1), (7.1.2), (7.1.6), (7.1.7), (7.12.80) which is such that44 γ+1 (Ω), q 0 1{ρ0 =0} = 0 a.e. in Ω, q 0 ∈ Lloc

44 This

line means that u − u∞ ∈ L2 (I, (D01,2 (Ω))3 ).

426

WEAK SOLUTIONS FOR NONSTEADY NAVIER–STOKES EQUATIONS s(γ)

ρ ∈ Lloc (IR3 × I)), s(γ) =

5γ−3 3 ,

P (ρ) = P (ρ∞ ) (ρ) ∈ L∞ (I, L1 (IR3 )),

ρ ∈ C 0 (I, Lγweak (Ω′ )) ∩ C 0 (I, Lp (Ω′ )),

1 ≤ p < γ, Ω′ a bounded subdomain of Ω,

ρ ≥ 0 a.e. in Ω × I, ρ = ρ∞ in (IR3 \ Ω) × I,

(7.12.87)

u − u∞ ∈ L2 (I, (D1,2 (IR3 ))3 ), u = u∞ = 0 in (IR3 \ Ω) × I, 2γ

γ+1 (Ω′ ))3 ), ρu ∈ C 0 (I, (Lweak

6γ

4γ+3 ρ|u − u∞ |2 ∈ L∞ (I, L1 (IR3 )) ∩ L2 (I, Lloc (IR3 )).

The renormalized continuity equation holds with any b satisfying (6.2.9), (6.2.10), (6.2.11), where β = s(γ). (ii) If Ω is an exterior domain (no regularity assumption), then statement (i) res(γ) mains valid with the following exceptions: (a) ρ does not belong to Ls(γ) (I, Lloc s(γ) (IR3 )) but only to Ls(γ) (I, Lloc (Ω)), (b) u − u∞ does not belong to L2 (I, (D01,2 3 (Ω)) ) but only u − u∞ ∈ L2 (I, (D1,2 (IR3 ))3 ) and u = u∞ in S, (c) the renormalized continuity equation holds with any b satisfying (6.2.9), (6.2.10), (6.2.11), where β = γ. We shall outline the proof for an arbitrary exterior domain Ω (no regularity assumption) and for a∞ = 0. As in Section 7.11, we shall suppose, without loss of generality that ρ∞ ∈ (0, 1). We take a sequence {(ρR , uR )}R>1+|a∞ |T of renormalized bounded energy weak solutions to problem (7.1.1)–(7.1.4), (7.1.7), (7.12.13)–(7.12.15) on ΩR = BR \ S. The existence of such a sequence is guaranteed by Theorem 7.77. We shall start with energy inequality (7.12.36) written with ΩR and (ρR , uR ). Using properties (7.12.82) of u∞ , H¨ older’s, Sobolev’s and Young’s inequalities at the right-hand side of (7.12.36), we finally obtain

(ρ

)

(ρ ) ER ∞ (ρR , q R ) + 2c I ΩR |∇(uR − u∞ )|2

≤ c′ I EΩR (ρR , uR ) + L(E0 , f , g, T, u∞ ),

where ER ∞ = E (ΩR ,ρ∞ ) , E0 = E (Ω,ρ∞ ) (ρ0 , q 0 )(cf. (7.12.37)), with some c, c′ > 0 independent of R. Using Gronwall’s lemma, we get the following bounds: ρR |uR − u∞ |2 L∞ (I,L1 (IR3 )) ≤ L(E0 , f , g, T, u∞ ), ∇uR L2 (IR3 ×I) ≤ L(E0 , f , g, T, u∞ ), 1{|ρ−ρ∞ |1 R→∞

Consequently, the renormalized continuity equation ∂t Lk (ρ) + div (Lk (ρ)u) + Tk (ρ)div u = 0 in D′ (IR3 × I), k > 1 holds, cf. Section 7.11.6.3. We are now in a position to prove the strong convergence of the sequence ρR . We shall follow the lines of Section 7.11.7. Several arrangements, however, will be necessary. Starting from now, to simplify, without loss of generality, we may suppose that (7.12.92) a∞ = (a∞ , 0, 0), a∞ > 0. Instead of test functions (7.11.43), we take Φm = ηm (x1 )ζm (x′ ), x′ = (x2 , x3 ), m ∈ IN,

(7.12.93)

where ′

ζm = ζ( mxα ), ζ ∈ D(IR2 ), 0 ≤ ζ ≤ 1, ζ = ηm (s) =

s η( m ),

η ∈ D(IR), 0 ≤ η ≤ 1 η =

1 in B1 0 in IR2 \ B 2 , 1 in (−1, 1) 0 in IR \ (−2, 2)

(7.12.94)

with α ∈ (0, 1) to be specified later. We observe that |supp ∇Φm | ≤ cm1+2α , ∇Φm 0,p,IR3 ≤ cm |∂1 Φm | ≤

c m,

′

|∇ Φm | ≤

c mα ,

1+2α−αp p

, 1 ≤ p < ∞,

where c > 0.

(7.12.95)


429

In our situation, equation (7.11.47) takes the form

[L (ρ(T )) − Lk (ρ(T ))]Φm + QT [Tk (ρ)div u − Tk (ρ)div u]Φm Ω k

= QT [Lk (ρ) − Lk (ρ)](u − u∞ ) · ∇Φm + QT [Lk (ρ) − Lk (ρ)]u∞ · ∇Φm . (7.12.96) The first two terms in (7.12.96) are treated in exactly the same way as is done in Section 7.11.7. For the remaining terms, we have by H¨ older’s inequality

| QT [Lk (ρ) − Lk (ρ)](u − u∞ ) · ∇Φm |

T ≤ (∇Φm 0,∞ + ∇Φm 0,6 ) 0 Lk (ρ) − Lk (ρ) 32 u − u∞ L62 (Ω) ,

L2 (Ω)

[Lk (ρ) − Lk (ρ)]u∞ · ∇Φm |

T 1 ≤ ∂1 Φm 0,∞ (1 + |supp ∇Φm | 2 )u∞ 0,∞ 0 Lk (ρ) − Lk (ρ)L12 (Ω) . |

QT

By virtue of (7.12.95), (7.12.89) and Exercise 7.65, they tend to zero as m → ∞ provided α ∈ ( 14 , 12 ). We are back in the situation of Section 7.11.7. Therefore, (with K ∈ K if Ω is Lipschitz). Lemma 7.68 holds true with any compact K ∈ K γ γ Consequently, ρ = ρ which means that the momentum equation is satisfied. Finally, we can show energy inequality (7.12.85) by copying (with small changes) the reasoning of Section 7.11.8 thus completing the proof of Theorem 7.79. 7.12.7

Domains with nonsmooth boundaries

In this section we shall sketch some existence results for domains with nonsmooth boundaries. Some of them need more precise formulation and their detailed proof is left to the reader as an exercise. In order to avoid useless complexity, we suppose that f , g are smooth (of compact support if Ω is unbounded) and ρ0 , q 0 are smooth (and satisfy in a convenient way compatibility conditions with data at infinity if Ω is unbounded). (a) If a domain Ω can be approximated by smooth bounded or exterior domains in the sense of Definition 7.71, then in Ω there exists a bounded energy weak solution in the sense of Definition 7.3 (where we take s = 5γ−3 3 ) of problem (7.1.1)–(7.1.7) (and (7.1.8) with ρ∞ = 0, u∞ = 0, if Ω is unbounded). This statement is a consequence of Theorem 7.72. (b) If Ω is a bounded domain (no regularity assumption), then in Ω there exists a bounded energy weak solution in the sense of Definition 7.3 (where we take s = 5γ−3 3 ) of problem (7.1.1)–(7.1.7). The velocity u, however, may not belong to L2 (I, (W01,2 (Ω))3 ) but it satisfies only u ∈ L2 (I, (W 1,2 (IR3 ))3 ), u = 0 in IR3 \ Ω. This statement can be obtained by the penalization method (7.12.38)–(7.12.42) (where we take ρ∞ = 0, u∞ = 0) following Section 7.12.5. (c) If Ω is an exterior domain (no regularity assumption), then in Ω there exists a bounded energy weak solution in the sense of Definition 7.3 (where we take s = 5γ−3 3 ) of problem (7.1.1)–(7.1.8). The velocity u, however, may not belong

430


to L2 (I, (D01,2 (Ω))3 ) but it satisfies u ∈ L2 (I, (D1,2 (IR3 ))3 ), u = 0 in IR3 \ Ω. This statement can be obtained by applying the method of invading domains as in Section 7.11 to result (b). (d) If Ω is an exterior domain (no regularity assumption), then in Ω there exists a bounded energy weak solution in the sense of Definition 7.78 (where we take s = 5γ−3 3 ) of problem (7.1.1)–(7.1.7), (7.1.9), (7.12.79). The velocity u − u∞ , however, may not belong to L2 (I, (D01,2 (Ω))3 ) but we know only that u − u∞ ∈ L2 (I, (D1,2 (IR3 ))3 ), u = u∞ in IR3 \ Ω. For this statement, see Theorem 7.79.

8 GLOBAL BEHAVIOR OF WEAK SOLUTIONS In this chapter the asymptotic behavior of weak solutions to problem (7.1.1), (7.1.2), (7.1.3), (7.1.5), (7.1.6), (7.1.7), as time tends to infinity will be investigated. The solution will be supposed to have the regularity as in Theorem 7.7. The corresponding theorem on asymptotic behavior (Theorem 8.1 below) will be formulated for a general pressure law p = p(ρ) satisfying a certain growth condition. The hypotheses are formulated so that we have generic coincidence of hypotheses assumed in known existence theorems and those assumed here. This allows us to have both the existence of a solution and its asymptotic behavior, unconditionally. For a discussion of the existence for general pressure laws (i.e. not necessarily γ-law p(ρ) = Cργ ) see Sections 7.12.1.2 and 7.12.3. 8.1 Formulation of the problem We consider a slightly modified problem (7.1.1)–(7.1.7), namely ∂t (ρu) + div (ρu ⊗ u) + ∇p(ρ) − µ∆u − (µ + λ)∇(div u) = ρf , ∂t ρ + div (ρu) = 0, x ∈ Ω, t > 0, u(x, t) = 0, x ∈ ∂Ω (ρu)(x, 0) = q 0 (x), ρ(x, 0) = ρ0 (x), x ∈ Ω

(8.1.1) (8.1.2) (8.1.3) (8.1.4)

with µ > 0, µ +

2 λ ≥ 0, p = p(·) : [0, ∞) → [0, ∞), N f = ∇Φ, Φ = Φ(x), x ∈ Ω,

(8.1.5) (8.1.6)

and a bounded domain Ω ⊂ IRN , N = 2, 3. By Theorem 7.7 we know the conditions under which there exists a finite and bounded energy weak solution (see Definition 7.3) at least for N = 3. (The discussion for N = 2 is performed in Section 7.12.1.2.) We are going to show that this solution has an asymptotically compact trajectory. More precisely, we will prove that for any sequence tn → ∞ and some q, r ≥ 1 there is a subsequence {sn } and a function ρ∞ ∈ Lq (Ω) such that lim ρ(sn ) − ρ∞ Lq (Ω) = 0,

(8.1.7)

lim (ρu)(sn )Lr (Ω) = 0.

(8.1.8)

n→∞

n→∞

Moreover, the function ρ∞ is an equilibrium density with the same total mass as ρ0 , i.e. 431

432

GLOBAL BEHAVIOR OF WEAK SOLUTIONS

∇p(ρ∞ ) = ρ∞ ∇Φ Ω

ρ∞ dx =

Ω

ρ0 dx, ρ∞ ≥ 0.

(8.1.9) (8.1.10)

In other words, we can say that the solution of the evolutionary problem sequentially stabilizes to an equilibrium state, i.e., to the state (ρ, ρu) = (ρ∞ , 0). The results and the method presented here were published in (Novotn´ y and Straˇskraba, 2001) (periodic boundary conditions) and (Novotn´ y and Straˇskraba, 2000) (Dirichlet and other boundary conditions). 8.2

Basic assumptions

We make the following basic assumptions: Ω ⊂ IRN (N = 2, 3) is a bounded domain, ∂Ω ∈ C 2,α (α ∈ (0, 1]); (8.2.1) f = ∇Φ, where Φ ∈ W 2,∞ (Ω); (8.2.2) 1 ′ p(·) ∈ C ([0, ∞)), p(0) = 0, p (r) > 0 for r > 0, p(∞) = ∞ and there exist constants γ > 1 and C such that 0 r p(s) − p(1) γ ds (≥ 0), r ≤ C(1 + P (r)) for r ≥ 0, where P (r) := r s2 1 γ > 1 for N = 2, γ > 3/2 for N = 3; (8.2.3) γ ρ0 ∈ L (Ω), ρ0 ≥ 0 a.e. in Ω, (8.2.4) 2 2γ | |q q 0 ∈ (L γ+1 (Ω))N , q0 1{ρ0 =0} = 0 a.e. in Ω, 0 1{ρ0 >0} ∈ L1 (Ω). ρ0 For a discussion of the relation between q 0 and ρ0 u0 see (7.1.7). 8.3

Sequential stabilization of the weak solution

Now we can formulate the main result of this section. Theorem 8.1 Let assumptions (8.2.1)–(8.2.4) be satisfied. Then for the finite and bounded energy weak solution (ρ, u) of (8.1.1)–(8.1.6), the couple (ρ, q) sequentially stabilizes to (ρ∞ , 0) with ρ∞ , ∇p(ρ∞ ) ∈ Lγ (Ω), satisfying (8.1.9), (8.1.10), in the following sense: Given tn → ∞ there is a subsequence {sn } ⊂ {tn } and ρ∞ satisfying (8.1.9), (8.1.10), such that (8.1.7) holds with q ∈ [1, γ). If, in addition, γ > 5/3, then 2γ ]. (8.1.7) holds with q ∈ [1, γ] and in addition, (8.1.8) holds with r ∈ [1, γ+1 If ρ∞ is determined by (8.1.9), (8.1.10) uniquely, then ρ(t) → ρ∞ , (ρu)(t) → 0

as t → ∞,

(8.3.1)

(i.e. without restrictions to subsequences) in the respective spaces Lq (Ω) and (Lr (Ω))N .

AUXILIARY FUNCTIONS

433

Proof First, observe that since (ρ, u) is a finite and bounded energy weak solution, we have u ∈ L2 ((0, ∞), (W01,2 (Ω))N ).

ρ, P (ρ) ∈ L∞ ((0, ∞), L1 (Ω)),

(8.3.2)

This is due to the fact that the term Ω ρ∇Φ · u dx in energy inequality (7.1.13)

d ρΦ dx (in the sense of distributions) when using the can be written as − dt Ω equation of continuity. In particular, due to assumption (8.2.3) and inclusions (8.3.2) we have (8.3.3) ρ ∈ L∞ ((0, ∞), Lγ (Ω)), limt→∞

t+a t−a

∇u(s)20,2 ds = 0 for any fixed a > 0.

(8.3.4)

By the Poincaré inequality (1.3.61), u0,6 ≤ C∇u0,2 , so that it follows from (8.3.4) that limt→∞

t+a t−a

u(s)20,r ds = 0 for r ∈ [1, 6] and a > 0.

(8.3.5)

For technical reasons let us prolong the state equation function p(·) to the negative part of the real axis by p(r) = −p(−r),

r < 0 (r ∈ IR).

(8.3.6)

Let us note that since the solution is renormalized in the sense of Definition 7.3, the total mass of the fluid in Ω is conserved, i.e.

Ω

ρ(t) dx =

Ω

ρ0 (x) dx =: M0 , t > 0.

(8.3.7)

We are going to perform operations on the solution which require more regularity than the solution has. That is why we regularize ρ and other quantities in time by the usual mollifier (Sε z)(t) :=

∞

−∞

ωε (t − s)z(s) ds :=

where ω0 ∈ C0∞ (−1, 1),

∞

−∞

1 ε

∞

−∞

ω0

t−s ε

z(s) ds, z ∈ L1 (IR), (8.3.8)

ω0 (s) ds = 1. Prolong ρ by zero onto Ω × IR and put

ρε (x, t) = Sε (ρ(x, ·))(t), x ∈ Ω, t > 0, ε > 0. 8.4

(8.3.9)

Auxiliary functions

In what follows, we construct four auxiliary functions wε , mε , ρε and ψ ε to show that ρ(t) is close to a function ρ(t) as t → ∞, where {ρ(t)}t>0 is relatively γ compact in Lr (Ω), r ∈ [1, NN−γ ) if γ < N, r ∈ [1, ∞) if γ = N, r = ∞ if γ > N. We start with the construction of wε .

434


Lemma 8.2 Let ρε be given by (8.3.9), f := ∇Φ. Then for any ε ∈ (0, 1], s ∈ IR there is a unique generalized solution wε ∈ W 1,γ (Ω) of the weak Neumann problem

γ ∇wε (s) · ∇η dx = Ω ρε (s)f · ∇η dx for all η ∈ W 1, γ−1 (Ω), Ω (8.4.1)

w (x, s) dx = 0, Ω ε and

wε (s)1,γ ≤ C < ∞, s > 0, ε ∈ (0, 1],

1 ∂t wε (s)0,r ≤ C Sε (ρu(s)0,r ) + ω0 (s/ε) , ε where the constant C > 0 depends only on the data.

(8.4.2) (8.4.3)

Proof of Lemma 8.2. The existence and uniqueness of wε (s) follows from Lemma 4.27 and the regularity of Φ and ρ. The estimate (8.4.2) follows from (4.5.40), the regularity property of the operator Sε (see Lemma 1.34), and the uniform estimate of ρ(s), as follows: wε (s)1,γ ≤ Cρε (s)f 0,γ ≤ C sup ρε (τ )0,γ f 0,∞ τ >0

≤ C sup ρ(τ )0,γ ≤ C < ∞,

s > 0, ε > 0,

τ >0

where C is a generic constant. Note that by (8.4.2), sup{wε (s)0,γ ; s ≥ 0, ε ∈ (0, 1]} ≤ C < ∞, γ if γ < N, γ := q > 1 sufficiently large if γ = N, and γ = ∞ if where γ := NN−γ γ > N (cf. (1.3.64) and (1.3.62)). We prove now estimate (8.4.3). To this purpose, ′ ′ let r ∈ (0, ∞), η ∈ Lr (Ω), and ζ ∈ W 2,r (Ω) be a solution of the problem

∆ζ = η − Ω η dx, ∂n ζ|∂Ω = 0, Ω ζ dx = 0.

Such a solution exists by virtue of Lemma 4.27. Then by the differentiability of wε and (8.4.1) we have

(∂ ρ f − ∂t ∇wε )∇ζ dx = 0. Ω t ε Consequently by the equation of continuity (8.1.2),

∂ w η dx = − Ω ∂t ρε f · ∇ζ dx Ω t ε ≤ ∂t ρε −1,r f · ∇ζ1,r′

≤ Cf 1,∞ ∂t ρε −1,r η0,r′

≤ C Sε (ρu(s)0,r ) + 1ε ω0 (s/ε) η0,r′ .

2 In the following lemma an auxiliary function θ is introduced which will be helpful later.

EXISTENCE AND ESTIMATES OF AUXILIARY FUNCTIONS

435

Lemma 8.3 There exists a positive constant c0 and a bounded increasing continuously differentiable function θ on IR with limr→∞ rθ′ (r) = 0, such that

2 p(r1 ) − p(r2 ) θ(r1 ) − θ(r2 ) ≥ c0 θ(r1 ) − θ(r2 )

for all

r1 , r2 ∈ IR.

Proof of Lemma 8.3. Let r0 > 0, θ(r) = p(r) for 0 ≤ r ≤ r0 and θ be concave, bounded with θ′ (r) < p′ (r) in (r0 , ∞), limr→∞ rθ′ (r) = 0. Extend θ to (−∞, 0) by θ(r) = −θ(−r) for r < 0. Clearly θ has the desired property. 2 8.5

Existence and estimates of auxiliary functions

Next, we introduce the auxiliary functions mε (s) and ρε (x, t) with the help of the following lemma. Lemma 8.4 Let wε be the function from Lemma 8.2 and θ the function from Lemma 8.3. Define the function G = Gε (s, m) by 0 Gε (s, m) := (θ ◦ p−1 )(wε (s) + m) dx, s ∈ IR, m ∈ IR, ε ∈ (0, 1]. (8.5.1) Ω

Then there is a uniquely determined family of functions m = mε (s), s ∈ IR such that

Gε (s, mε (s)) = Ω Sε θ(ρ(x, ·))(s) dx, s ∈ IR, ε ∈ (0, 1], (8.5.2) {mε }ε∈(0,1] is bounded in L∞ (IR) " dmε # is bounded in L1 (IR) ⊕ L2 (IR). ds ε∈(0,1]

(8.5.3)

(Here B = B1 ⊕ B2 for Banach spaces B1 , B2 denotes the Banach space B = {z := z1 + z2 ; zj ∈ Bj , j = 1, 2, zB := inf{z1 B1 + z2 B2 ; zj ∈ Bj , z1 + z2 = z}.) Proof of Lemma 8.4. Clearly, for each s > 0, ε > 0, we have limm→±∞ Gε (s, m) ε = ±|Ω| supr≥0 θ(r) and ∂G ∂m (s, m) > 0 for m ∈ IR. Since for almost all (x, s), ρ(x, s) is finite we have

∞

S θ(ρ(x, ·))(s) dx = Ω −∞ ωε (s − τ )θ(ρ(x, τ )) dτ dx Ω ε

∞ < θ(∞) Ω −∞ ϕε (s − τ ) dτ dx = θ(∞)|Ω|.

So Ω Sε θ(ρ(x, ·))(s) dx lies in the range of Gε (s, ·) and for any fixed s ≥ 0 and ε > 0 equation (8.5.1) has a unique solution mε (s). We show that |mε (s)| ≤ C < ∞, If we had

ε > 0, s > 0.

(8.5.4)

436


|Ω|θ(−∞) + δ <

Ω

Sε θ(ρ(·))(s) dx < |Ω|θ(∞) − δ for ε ∈ (0, 1), s > 0 (8.5.5)

with some positive δ, then (8.5.4) would clearly follow. Denoting Ω(s) := {x ∈ Ω; ρ(x, s) ≤ M } we have

S θ(ρ) dx = Ω(s) Sε θ(ρ) dx + Ω\Ω(s) Sε θ(ρ) dx Ω ε

≤ θ(M )|Ω(s)| + θ(∞) |Ω| − |Ω(s)| .

We show that there exists M > 0 such that inf s∈IR |Ω(s)| > 0. If this is not the case then there are sn such that |Ω(sn )| → 0 and ρ(x, sn ) ≥ n for any x ∈ Ω \ Ω(sn ). Hence n|Ω \ Ω(sn )| ≤

Ω\Ω(sn )

ρ(x, sn ) dx ≤ |Ω|

γ−1 γ

ρ(·, sn )0,γ

≤ C sups ρ(., s)0,γ < ∞ which is contradictory when n → ∞. So we have proved the right-hand part of the inequality (8.5.5). Since the left-hand part is trivial, due to positivity of ρ, (8.5.4) readily follows. Let us prove the second part of (8.5.3). Since ρ satisfies the renormalized q equation of continuity, given q ∈ (1, ∞) and ζ ∈ W 1, q−1 (Ω) we have

∞ θ(ρ(τ )) ζ dτ dx S ∂ θ(ρ)(s)ζ dx = ε12 Ω 0 ϕ′0 s−τ ε Ω ε t

∞ θ(ρ(τ ))u(τ ) · ∇ζ dx dτ + 1ε ϕ0 (s/ε) Ω θ(ρ0 )ζ dx = 1ε 0 Ω ϕ0 s−τ ε

′

∞

ρθ (ρ) − θ(ρ) div u(τ ) ζ dx dτ − 1ε 0 Ω ϕ0 s−τ ε

= Ω Sε (θ(ρ)u(·))(s) · ∇ζ dx − Ω Sε (ρθ′ (ρ) − θ(ρ))div u (·)(s)ζ dx

+ 1ε ϕ0 (s/ε) Ω ρ0 ζ dx, s > 0, ε small.

So, in the sense of distributions we have

∂t Sε θ(ρ) = −div Sε θ(ρ)u − Sε ρθ′ (ρ) − θ(ρ) div u + 1ε ϕ0 (s/ε)ρ0 ; (8.5.6) ∞ (notice that div Sε (θ(ρ)u) ∈ Cloc (0, ∞; Lr (Ω)) for some r > 1). Further, we have ∂Gε ∂m (s, m)

=

We claim that c0 := inf ε,s

Ω

Ω

θ ◦ p−1

′

wε (s) + m dx.

(θ ◦ p−1 )′ (wε (s) + mε (s)) dx > 0.

(8.5.7)

(8.5.8)

Indeed, if this was not true, then wn → w weakly in W 1,γ (Ω), mn → m ∈ IR such that, by Fatou’s lemma,

0 < Ω (θ ◦ p−1 )′ (w + m) dx ≤ lim inf n→∞ Ω (θ ◦ p−1 )′ (wn + mn ) dx = 0

COMPARISON DENSITY AND A TEST FUNCTION

437

which is a contradiction. Consequently, by the implicit function theorem (see Theorem 1.3), there exists the derivative m′ε (s) and we have

′

−1

∂ Sε θ(ρ) θ ◦ p−1 wε (s) + mε (s) dx Ω t

′

− Ω θ ◦ p−1 wε (x, s) + mε (s) ∂t wε (x, s) dx

m′ε (s) =

Ω

(8.5.9)

which with the help of (8.5.6) and (8.4.3) yields

supr∈R (θ ◦ p−1 )′ (r)∂t wε (s)0,1 + Ω ∂t Sε θ(ρ) dx |m′ε (s)| ≤ c−1 0

≤ C ∂t wε (s)0,1 + ∇u(s)0,2 + 1ε ϕ0 (s/ε)

(8.5.10) ≤ C Sε (ρu0,1 ) + ∇u0,2 + 1ε ϕ0 (s/ε)

≤ C(ρ0,6/5 u0,6 + ∇u0,2 + 1ε ϕ0 (s/ε)

≤ C ∇u0,2 + 1ε ϕ0 (s/ε) , s > 0, ε ∈ (0, 1].

By integration over s, the second part of (8.5.3) immediately follows. 8.6

2

Comparison density and a test function

Now, having defined the functions wε and mε we can put ρε (x, s) := p−1 (wε (x, s) + mε (s)),

x ∈ Ω, s > 0, ε ∈ (0, 1].

(8.6.1)

Finally, we define a test function ψ ε = ψ ε (x, s) as a solution of the problem div ψ ε (s) = Sε θ(ρ)(s) − θ(ρε (s)) in Ω ψ ε (x, s) = 0, x ∈ ∂Ω, s > 0, ε ∈ (0, 1].

(8.6.2) (8.6.3)

To resolve (8.6.2), (8.6.3) we use Lemma 3.17 according to which we know that there is a bounded operator B : Lr (Ω) → (W 1,r (Ω))3 , 1 < r < ∞, such that

ψ ε (s) = B Sε θ(ρ)(s) − θ(ρε (s)) (8.6.4)

satisfies (8.6.1) a.e. in Ω for each s > 0 and ε ∈ (0, 1] and in addition, for any q ∈ (1, ∞), we have ψ ε (s)1,q ≤ CSε θ(ρ)(s) − θ(ρε (s))0,q ≤ C sup θ(r) < ∞

(8.6.5)

r>0

with a generic constant C independent of s and ε. 8.7

Passing to the limit with the regularization parameter

The following lemma allows us to pass to the limit with respect to the regularization parameter ε in the uniform estimates we are going to derive.

438


Lemma 8.5 Let the basic assumptions (8.2.1)–(8.2.4) be satisfied. Then there are functions w = w(x, s) and m = m(s) such that for any r ∈ [1, ∞) we have (8.7.1) wε → w strongly in Lrloc ([0, ∞); W 1,γ (Ω)) and a.e. in Ω × (0, ∞) + ∞ mεn → m strongly in Lloc ([0, ∞)) and a.e. in (0, ∞) for some εn → 0 . (8.7.2) Proof Relation (8.7.1) follows from Lemma 8.2, the Aubin–Lions lemma 1.71 and uniqueness for the generalized Neumann problem. Relation (8.7.2) follows from Lemma 8.4 and the Aubin–Lions lemma. 2 Remark 8.6 Notice that by Lemma 8.2 we can write

wε (s) = Aρε (s), where A ∈ L Lγ (Ω), W 1,γ (Ω) , s > 0, ε ∈ (0, 1].

(8.7.3)

Since ρ ∈ L∞ (0, ∞; Lγ (Ω)), by the continuity of mollifiers (see Lemma 1.34) we have (8.7.4) lim+ ρε = ρ in any Lrloc ([0, ∞); Lγ (Ω)), r ∈ [1, ∞). ε→0

Since (8.7.1) holds, passing to the limit as ε → 0+ in (8.7.3) we find w(s) = Aρ(s),

s > 0.

(8.7.5)

Putting ρ(x, s) := p−1 (w(x, s) + m(s)),

x ∈ Ω, s > 0,

(8.7.6)

we find from (8.6.1), (8.7.1) and (8.7.2) p(ρεn ) → p(ρ) strongly in Lqloc ([0, ∞); W 1,γ (Ω)), r ∈ [1, ∞). We call ρ the comparison density. 8.8

(8.7.7) 2

Comparison density is close to the density as t → ∞.

Now put

Q(t) :=

t

t−1 Ω

p(ρ(s)) − p(ρ(s)) (θ(ρ(s)) − θ(ρ(s))) dx ds,

t ≥ 1.

(8.8.1)

By monotonicity of p(·) and θ we have Q(t) ≥ 0. Our intention now is to prove the following global property of Q. Proposition 8.7 Let ρ be a function defined in (8.7.6), where w and m are given by (8.7.1), (8.7.2). Then the function Q(t) defined by (8.8.1) satisfies lim Q(t) = 0.

t→∞

(8.8.2)

COMPARISON DENSITY IS CLOSE TO THE DENSITY AS T → ∞.

439

Proof Let a > 1, ϕ ∈ C0∞ (−a, a), ϕ ≥ 0, ϕ(σ) = 1 for σ ∈ (−1, 0). Put Qεa (t) := Then clearly

t+a t−a

ϕ(s − t)

Ω

p(ρ(s)) − p(ρε (s)) (Sε θ(ρ(s)) − θ(ρε (s))) dx ds. (8.8.3)

p(ρ(s)) − p(ρε (s)) θ(ρ(s)) − θ(ρε (s)) dx ds

t+a + t−a ϕ(s − t) Ω p(ρ(s)) − p(ρε (s)) Sε θ(ρ(s)) − θ(ρ(s)) dx ds, (8.8.4) where the last term on the right-hand side of (8.8.4) tends to zero as ε → 0+ . By Lemma 8.5

t limn→∞ t−1 Ω p(ρ(s)) − p(ρεn (s)) θ(ρ(s)) − θ(ρεn (s)) dx ds = Q(t), t > 1 (8.8.5) for some εn ↓ 0. Now we wish to estimate Qεa (t). Denote Qεa (t) =

t+a t−a

ϕ(s − t)

Ω

Va (t) := {(x, s); x ∈ Ω, t − a < s < t + a}.

(8.8.6)

By the definition of the weak solution, and, with regard to the definition and smoothness of ψ ε , we can write

ϕ(s − t)p(ρ(s))(Sε θ(ρ)(s) − θ(ρε (s))) dx ds Va (t)

= Va (t) ϕ(s − t)p(ρ(s)) div ψ ε (s) dx ds

= Va (t) ϕ(s − t) −ρu(s)∂t ψ ε (s) − ρu · (u · ∇)ψ ε (s) + µ∇u(s) · ∇ψ ε (s)

+(µ + λ) div u(s)(Sε θ(ρ)(s) − θ(ρε (s)) − ρ(s)f · ψ ε (s) dx ds

− Va (t) ϕ′ (s − t)ρu(s) · ψ ε (s) dx ds. (8.8.7) Now, take the Helmholtz decomposition of ψ ε (s), that is ψ ε (s) = ∇zε (s) + v ε (s),

div v ε (s) = 0 in Ω,

v ε (s) · n = 0 in ∂Ω.

The existence of the Helmholz decomposition of an Lq (Ω)-function (1 < q < ∞) follows from Lemma 4.27 and Lemma 5.7 Since

ψ ε (s) = B Sε θ(ρ) − θ(ρε ) (s) and θ is bounded, by Lemma 3.17,

ψ ε (s)1,q ≤ C

∀q ∈ [1, ∞).

By the usual construction of the decomposition and by (8.8.8) we have

z dx = 0, zε ∈ W 2,q (Ω), ∂n zε |∂Ω = 0, v ε ∈ W 1,q (Ω) Ω ε

(8.8.8)

440


(q ∈ [1, ∞) arbitrary). Take into account the generalized formulation (8.4.1), namely

γ (8.8.9) ∇wε (s) − ρε (s)f · ∇η dx = 0 for η ∈ W 1, γ−1 (Ω). Ω

Then we find

ϕ(s − t)p(ρε (s))(Sε θ(ρ)(s) − θ(ρε (s)) dx ds Va (t)

= Va (t) ϕ(s − t)wε (s) div ψ ε (s) dx ds

= − Va (t) ϕ(s − t)∇wε (s) · ψ ε (s) dx ds

= − Va (t) ϕ(s − t)∇wε · ∇zε (s) + v ε (s) dx ds

= − Va (t) ϕ(s − t)∇wε · ∇zε (s) dx ds = − Va (t) ϕ(s − t)ρε (s)f · ∇zε (s) dx ds. (8.8.10) Subtracting (8.8.7) and (8.8.10) we obtain

ϕ(s − t) p(ρ(s)) − p(ρε (s)) Sε θ(ρ(s)) − θ(ρε (s)) dx ds Va (t)

= µ Va (t) ϕ(s − t)∇u(s) : ∇ψ ε (s) dx ds

+(µ + λ) Va (t) ϕ(s − t) div u(s)(Sε θ(ρ(s)) − θ(ρε (s))) dx ds

− Va (t) ϕ(s − t)ρu(s) · (u(s) · ∇)ψ ε (s) dx ds

− Va (t) ϕ′ (s − t)ρu(s) · ψ ε (s) dx ds − Va (t) ϕ(s − t)ρu(s) · ∂t ψ ε (s) dx ds

+ Va (t) ϕ(s − t)(ρε (s) − ρ(s))f · ∇zε (s) dx ds

7 − Va (t) ϕ(s − t)ρ(s)f · v ε (s) dx ds =: j=1 Ijε (t). (8.8.11) The estimate the integrals Ijε (t) in (8.8.11) one by one. Starting with I1ε (t) and taking into account (8.8.8), (8.4.2) we find

t+a |I1ε (t)| ≤ C|ϕ|∞ t−a ∇u(s)0,2 ψ ε (s)1,2 ds (8.8.12)

t+a ≤ C t−a ∇u(s)0,2 ds ≤ Cσa (t), where we denote

σa (t) := Quite analogously we get To estimate

I3ε (t),

t+a t−a

|I2ε (t)| ≤ Cσa (t).

12

.

(8.8.13) (8.8.14)

notice that having γ > 1 for N = 2,

we have

∇u(s)20,2 ds

γ>

3 2

for N = 3,

(8.8.15)


t+a

|I3ε (t)| = t−a ϕ(s − t) Ω ρuj uk ∂k ψεj (s) dx ds

t+a ≤ Cϕ∞ t−a ρ(s)0,γ u(s)20,2q ψ ε (s)1,r ds

t+a ≤ C t−a ∇u(s)20,2 ds ≤ Cσa (t)2

441

(8.8.16)

with r arbitrarily large and q = rγ(rγ − r − γ)−1 (with obvious modifications when rγ − r − γ ≤ 0) which requires γ > 1 for N = 2 and γ ≥ 3/2 for N = 3. Continuing with I4ε (t) we notice that if γ ≥ 6/5 in the case N = 3, then

t+a γ ψ (s) |I4ε (t)| ≤ ϕ′ 0,∞ t−a ρ(s)0,γ u(s)0, γ−1 0,∞ ds ε

t+a ≤ C t−a ∇u(s)0,2 ρ(s)0,γ ds ≤ Cσa (t).

(8.8.17)

Next we estimate I5ε (t). First, by (8.6.4) and the properties of the solution operator B it is clear that the function ∂t ψ ε exists and is sufficiently regular so that the integral I5ε (t) is well defined. In addition, ∂t ψ ε ∈ W 1,r (Ω) for any r > 1. Denote (Dh z)(t) := h1 (z(t + h − z(t)) for any Banach space valued function z = z(t). The linearity of (8.4.1), estimate (8.4.2) and the regularity of ρε provide that (D1/n ρε )(s) is a Cauchy sequence in Lγ (Ω) for each fixed s, and consequently D1/n wε is a Cauchy sequence in W 1,γ (Ω). Thus ∂t wε (t) exists and belongs to W 1,γ (Ω) for each ε > 0, t > 0. Further, the regularity of wε (s) implies continuous differentiability of the function Gε (s, m), given by (8.5.1), with respect to s and since, as we have already proved, ∂Gε /∂m is positive, the function mε (s) given as the unique solution of (8.5.1) is continuously differentiable. It follows that the function θ(ρε ) with ρε given by (8.6.1) has derivative with respect to s 1,γ (Ω)). Then, in the same way as we proved the existence and in L∞ loc ([0, ∞); W regularity of ∂wε /∂t we can show that ∂t ψ ε (s) exists in the sense of W 1,r (Ω) with r > 1 arbitrary and for each s and ε > 0 it is an element of the same space. To estimate I5ε (t) we need to estimate ∂t ψ ε 0,q for suitable q ≥ 1. So let us turn to the estimate of ∂t ψ ε . By (8.6.4) and (8.5.4) we have

∂t ψ ε = B div z + B h − ∂t θ(ρε ) , (8.8.18)

z = −Sε θ(ρ)u , h = Sε θ(ρ) − ρθ′ (ρ) div u + 1ε ω0 (s/ε)ρ0 . ∞

6

(8.8.19)

Firstly, it can be easily checked that z belongs to {ζ ∈ C (0, ∞; L (Ω)); div ζ ∈ C ∞ (0, ∞; L2 (Ω)), ζ · n|∂Ω = 0}. Secondly, if y = div η, where η ∈ Lr (Ω) with q ≤ r < ∞ and η · ν|∂Ω = 0, then, again by Lemma 3.17, By1,r ≤ const η0,r . By differentiation, (8.5.8) and (8.4.3) we get

∂t θ(ρε (s))0,N q/(N +q) ≤ C ∂t wε (s)0,N q/(N +q) + |m′ε (s)| (8.8.20)

1 ≤ C Sε (ρu)(s)0,N q/(N +q) + ∇u(s)0,2 + ω0 (s/ε) . ε Putting together (8.8.18)–(8.8.20) we finally obtain

442


∂t ψ ε 0,q ≤ C Sε ρu(s)0,

1 + ∇u(s)0,2 + ω0 (s/ε) , s > 0, ε ∈ (0, 1]. ε (8.8.21) With (8.8.21) in hand we estimate I5ε (t) :

t+a |I5ε (t)| ≤ ϕ0,∞ sups≥0 ρ(s)0,γ t−a u(s)0,r ∂t ψ ε (s)0,q ds (8.8.22)

t+a ≤ C t−a ∇u(s)0,2 ∂t ψ ε (s)0,q ds, Nq N +q

γ where 1r + 1q < γ−1 γ and, for N = 2 we can take r ∈ ( γ−1 , ∞) arbitrarily large so γ γ can be chosen arbitrarily close to γ−1 , and for N = 3, assuming that q > γ−1 6γ γ > 3/2 we fix r = 6, q = 5γ−6 . Putting together (8.8.11), (8.8.12), (8.8.14), (8.8.16), (8.8.17), (8.8.22) and (8.8.21) we arrive at

ϕ(s − t) p(ρ(s)) − p(ρε (s)) Sε θ(ρ(s)) − θ(ρε (s)) dx Va (t)

t+a ≤ C σa (t) + t−a Sε (ρu(s)0, N q ) + ∇u(s)0,2 + 1ε ϕ0 (s/ε) ∇u(s)0,2 ds N +q

+|I6ε (t)|

+

|I7ε (t)|

(8.8.23) with σa (t) given by (8.8.13). Since we assume t > a and ϕ0 has finite support we have limε→0+ 1ε ϕ0 (s/ε) = 0 for s ≥ t − a. (8.8.24)

Moreover, 1ε ϕ0 (s/ε) ≤ C < ∞ for ε > 0, s ≥ t − a. By continuity of the mollifier Sε we have Sε (ρu(·)0,

Nq N +q

) → ρu(·)0,

Nq N +q

as ε → 0+ in L1 ((t − a, t + a), IR+ ). (8.8.25)

q Indeed, by our choice of q, NN+q < 2, we have 1/2

ρu0,N q/(N +q) ≤ ρ0,

Nq 2N +(2−N )q

√ ρu0,2 ,

Nq 11 5 where 2N +(2−N )q is less than 5 γ − 1 if N = 2 and less than 3 γ − 1 if N = 3, so that regularity guaranteed by the existence theorem suffices to pass to the limit in (8.8.25). As

t+a |I6ε (t)| ≤ Cϕ0,∞ f 0,∞ sups,ε ψ ε (s)0,∞ t−a ρε (s) − ρ(s)0,1 ds,

we have limε→0+ I6ε (t) = 0. In addition, there exist εn ↓ 0 such that (8.8.5) holds true. So putting in (8.8.23) ε := εn and n → ∞ we finally obtain

0 ≤ Va (t) p(ρ(s)) − p(ρ(s)) θ(ρ(s))(s) − θ(ρ(s)) dx

1/2

s+a √ Nq ≤ C σa (t) + σa (t) sups>0 ρu(s)0,2 sups≥a s−a ρ(τ )0, dτ 2N +(2−N )q

+ lim supn→∞ |I7εn (t)|.

(8.8.26)


443

It remains to estimate I7ε (t) = where we know that sup v ε (s)1,q < ∞, ε,s

ϕ(s − t)ρ(s)f · v ε (s) dx ds,

(8.8.27)

div v ε (s) = 0 and v ε (s) · n = 0 on ∂Ω.

(8.8.28)

Va (t)

Let η > 0 and κ ∈ C0∞ (Ω) be such that |supp (1 − κ)| ≤ η. Then I7ε (s) can be decomposed as

t+a

I7ε (t) = t−a ϕ(s − t) Ω ρ(s)f · v ε (s)κ dx ds (8.8.29)

t+a

+ t−a ϕ(s − t) Ω ρ(s)f · v ε (s)(1 − κ) dx ds =: J1 + J2 ,

and clearly

|J2 | ≤ 2aϕ0,∞ f 0,∞ sup v ε (s) |supp (1 − κ)| ≤ Cη.

(8.8.30)

ε,s

Since (ρ, u) is a weak solution of (8.1.1)–(8.1.4) we can write J1 in the form

t+a

J1 = t−a ϕ(s − t) Ω µ∇u∇(κv ε ) + (µ + λ) div u div (κv ε ) − p(ρ) div (κv ε )

t+a

−ρu · ((u · ∇)(κv ε )) − κρu · ∂t v ε dxds − t−a ϕ′ (s − t) Ω (ρu · v ε )(s)dxds

t+a = t−a ϕ(s − t) Ω µκ∇u · ∇v ε − κρu · ((u · ∇)v ε ) − κρu · ∂t v ε dx ds

t+a

t+a

− t−a ϕ′ (s − t) Ω κ(ρu · v ε )(s) dx ds + t−a ϕ(s − t) Ω µ(∇κ · ∇u) · v ε

+(µ + λ)(∇κ · v ε ) div u − ρ(u · ∇κ)(u · v ε ) − p(ρ)(∇κ · v ε ) dx ds 8 =: k=1 J k .

(8.8.31) The integrals J 1 , . . . , J 4 can be estimated quite analogously to the integrals I1ε (t), . . . , I4ε (t) in (8.8.11) and we leave the details to the reader. To estimate the remaining terms J 5 to J 8 we need the following lemma.

Lemma 8.8 Let Ω be of class C 2 . Then for any sufficiently small η > 0 there exists a domain Ωη ⊂ Ω such that Ωη ⊂ Ω, |Ω \ Ωη | ≤ Cη and if x ∈ ∂Ω, then there is a unique y = y(x) ∈ ∂Ωη such that n(x) = n(y(x)) and |x − y(x)| = η for all x ∈ ∂Ω. In addition, there is a function κ ∈ W 1,∞ (Ω) such that κ(x) = 1 for x ∈ Ωη , κ(x) = 0 for x ∈ Ω \ Ωη/2 , |∇κ(x)| ≤ Cη −1 for x ∈ Ωη/2 \ Ωη and dκ dτ |∂Ωη = 0, where τ is any unit vector tangential to the boundary. Proof Assume that Ω is of class C 2 , i.e., there are open sets Dr ⊂ IRN −1 , functions ar∈ C 2 (Dr , IR) and coordinate systems (x′r , xrN ), r = 1, . . . , m, such m that ∂Ω = r=1 {x ∈ IRN : xrN = ar (x′r )}, where x′r = (xr1 , . . . , xr,N −1 ). Further, we assume that there exist βr > 0, r = 1, . . . , m, such that {x ∈ IRN ; x′r ∈

444


Dr , ar (x′r ) < xrN < ar (x′r ) + βr } ⊂ Ω. In the sequel we omit the indeces when working on a local part of the boundary. For x ∈ ∂Ω the unit normal vector n(x) = (1 + |∇a(x′ )|2 )−1/2 (−∇a(x′ ), 1)T points out of Ω. Let η be small enough (i.e. so small that xN − δ < ar (x′ − δn(x)′ ) + β still holds for all δ ∈ (0, η); by continuity, η can be chosen uniformly with respect to x′ ∈ Dr , r = 1, . . . , m). Then Sη := {y = x − δn(x); 0 < δ ≤ η, x ∈ ∂Ω} ⊂ Ω. Define Ωη := Ω \ Sη . Then ∂Ωη = {y ∈ IRN ; y = x − ηn(x), x ∈ ∂Ω}. This can be proved by a rather lengthy but routine argument and so we do not present it in detail. Denote by nη (y) the unit exterior normal vector at y to ∂Ωη . Show that nη (y) = n(x), where y = x − ηn(x). Indeed, solving the system y ′ = x′ − ηn(x)′ , we get x′ = φ(y ′ ) and by differentiation

∂φl ∂φk ∂ ∂a = δkj + η (1 + |∇a|2 )−1/2 . ∂yj ∂xl ∂xk ∂yj

Further, the local equation for ∂Ωη near y is

−1/2 . yN = a(φ(y ′ )) − η 1 + |∇a(φ(y ′ ))|2

So by an elementary calculation after evident cancellations we obtain

∂yN ∂a ∂a ′ = (x ), φ(y ′ ) = ∂yj ∂xj ∂xj

and the assertion is proved. Now, let k be a smooth function such that k(ξ) = 0 for ξ ≤ η/2 and k(ξ) = 1 for ξ ≥ η. Define κ(x) = 1 on Ωη and for x ∈ Ω \ Ωη , κ(x) := k(|x − y(x)|), where (y(x), µ(x)) is a solution of the system Fj (y, µ) := yj − xj − µνj (y) = 0,

j = 1, . . . , N,

FN +1 (y, µ) := yN − a(y ′ ) = 0

near the point (y 0 , µ0 ) := (x, 0). By the implicit function theorem (Theorem 1.3) the solution exists since ' ( ∂F IN −n(x)T = (1 + |∇a(x)|2 )1/2 det det . n(x) 0 ∂x (x,0)

and the rows in the matrix can be shown to be linearly independent. Finally, we prove that ∂τ κ∂Ω = 0, where τ is any vector orthogonal to η nη (x) with x ∈ ∂Ωη . Since ∂y xk − yk ∂xk ∂yk ∂κ k = k ′ (|x − y(x)|) − − νj , = k ′ (|x − y(x)|) νk ∂xj |x − y(x)| ∂xj ∂xj ∂xj

by orthogonality of n(y(x)) and τ = τ η (x) we have ∂τ κ =

∂yk ∂κ τj = k ′ (|x − y(x)|) νk τ j . ∂xj ∂xj


445

Differentiating the relations yk = xk − η

∂a (1 + |∇a|2 )−1/2 , k = 1, . . . , N − 1, yN = xN + η(1 + |∇a|2 )−1/2 , ∂xk

by an elementary calculation we obtain ∂yk νk τj = 0. ∂xj

Finally, |∇κ(x)| ≤ C 1+sup{|∇x y(ξ)|; ξ ∈ Ωη/2 \Ωη } supξ∈(η/2,η) |k ′ (ξ)| for x ∈ Ωη/2 \Ωη and it is clear that k can be chosen in such a way that supξ∈(η/2,η) |k ′ (ξ)| ≤ Cη −1 . Since Ω is C 2 , by the preceding formulas we have |∇κ(x)| ≤ Cη −1 for all x ∈ Ω. 2 All integrals J 5 to J 8 are of the type

t+a t−a

(8.8.32)

a(x, s)∇κ(x) · v ε (x, s) dx ds.

Given x ∈ Ωη/2 \ Ωη , issue from x a ray which is a normal to ∂Ωη at x1 and to ∂Ω at x2 . Then |x − x2 | ≤ η. Further, since v(x2 ) · n(x2 ) = 0, n(x2 ) = n(x1 ) and ∇κ(x1 ) ⊥ n(x2 ), by Lemma 8.6, we have ∇κ(x) · v(x2 ) = 0. Indeed, we might construct Ωα with α = |x − x2 | and use the same argument as in Lemma 8.6 for Ωη to show that ∇κ(x) · τ (x) = 0 for any vector τ tangential to ∂Ωα at x. Consequently, by the imbedding theorem (see (1.3.62)), we find |∇κ(x) · v ε (x, s)| = |∇κ(x) · (v ε (x, s) − v ε (x2 , s))| N C ≤ v ε (s)1,q |x − x2 |1− q ≤ Cη −N/q , η

x ∈ Ωη \ Ωη/2

with q > 1 arbitrary but fixed. So the integral (8.8.32) is estimated as follows t+a

a(x, s)∇κ(x) · v ε (x, s) dx ds ≤ Cη −N/q t−a

Va (t)

|a(x, s)| dx ds,

and we get

max {|J 5 |, |J 6 |} ≤ Cη −N/q |J 7 | ≤ Cη −N/q

t+a

t−a

Ω

t+a

t−a

1

Ωη/2 \Ωη

N

|∇u| dx ds ≤ Cη 2 − q σa (t), (8.8.33)

ρ|u|2 dx ds 1

N

≤ Cη −N/q sups ρ(s)0,γ 1Lα (Ωη/2 \Ωη ) σa (t)2 ≤ Cη α − q σa (t)2 ,

(8.8.34)

6γ γ where α > γ−1 for N = 2 and α = 5γ−6 (γ > 6/5) for N = 3. Taking q sufficiently large we see that J k , k = 5, 6, 7 tend to zero as η → 0+ . To estimate

446


the last integral J 8 in (8.8.31) we again use the space-time integrability of ρ guaranteed by the existence theorem. We find

t+a

|J 8 | ≤ Cη −N/q t−a Ωη/2 \Ωη ργ dx ds (8.8.35)

γ/β β−γ β−γ N |Ωη/2 \ Ωη | γ ≤ Cη γ − q ≤ Cη −N/q Va (t) ρβ dx ds

whenever

Va (t)

ρβ dx ds ≤ C < ∞.

(8.8.36)

Since Theorem 7.7 guarantees the existence of β > γ such that estimate (8.8.36) holds true, we conclude that due to the estimates for J 1 to J 4 , (8.8.29)–(8.8.35) we have |I7ε (t)| ≤ Cσa (t) + ω(η), where limη→0+ ω(η) = 0 and in particular, lim supn→∞ |I7εn (t)| ≤ Cσa (t). By (8.8.26), (8.8.4), (8.8.1), regularity of ρε (s) and Lemma 8.5 combined with (8.5.8), (8.4.3) we have 0 ≤ Q(t) ≤ lim Qεan (t) ≤ Cσa (t) n→∞

and (8.8.2) follows from (8.3.4), (8.8.13). The proof of Proposition 8.7 is complete. 2 8.9

Convergence of the density

In this section we finally show that under the basic assumptions of Section 8.2, for any sequence {tn } there is a subsequence {sn } ⊂ {tn } such that there exists an equilibrium state ρ∞ ≥ 0 satisfying lim ρ(sn ) − ρ∞ 0,r = 0

(8.9.1)

n→∞

with r ∈ [1, γ] arbitrary. Hence, if the equilibrium state is unique, then it follows that (8.9.2) lim ρ(t) − ρ∞ 0,r = 0. t→∞

We start with the following trivial observation. 1,1 Lemma 8.9 Let q ∈ Wloc (a, ∞) (a ∈ IR) be such that q(s) ≥ 0 for s ≥ a and

t ′ limt→∞ t−1 q(s) + |q (s)| ds = 0. Then

lim q(t) = 0.

(8.9.3)

t→∞

Proof Since

t q(t) = q(s) + s q ′ (τ ) dτ

t for a + 1 ≤ s < t < ∞, by integration t−1 ds we get q(t) ≤

t

t−1

q(s) ds +

t t−1 s

t

and (8.9.3) follows immediately.

|q ′ (τ )| dτ ds ≤

t t−1

q(s) + |q ′ (s)| ds,

2

CONVERGENCE OF THE DENSITY

447

Put q(t) := θ(ρ(t)) − θ(ρ(t))20,2 ,

t ≥ 1,

(8.9.4)

where ρ(t) is the function from Remark 8.1.6. Then by Lemma 8.3 we have

q(t) ≤ C Ω p(ρ(t)) − p(ρ(t)) θ(ρ(t)) − θ(ρ(t)) dx for t ≥ 0. (8.9.5)

So it follows from Proposition 8.7 that

t limt→∞ t−1 q(s) ds = 0.

(8.9.6)

Now we are going to prove that

limt→∞

t

t−1

|q ′ (s)| ds = 0.

(8.9.7)

This will be a consequence of the following lemma. Lemma 8.10 Let ρ(t) = limn→∞ p−1 (wεn + mεn ) (cf. Remark 8.6). Then we have

∞ d θ(ρ(t)) − θ(ρ(t))20,2 2 dt ≤ C ∞ ∇u(t)20,2 dt < ∞. (8.9.8) dt 1 1

Proof It suffices to prove that ∞ ′

2 η (s) Ω θ(ρ(s)) − θ(ρ(s)) dx ds ≤ CηL2 (0,∞) ∇uL2 ((0,∞),L2 (Ω)) , 1 (8.9.9) for any η ∈ C0∞ (1, ∞). First, by the renormalized equation of continuity we have ∞ ′

η (s) Ω θ(ρ(s))2 dx ds 1 ∞

= 1 η(s) Ω 2ρ(s)θ(ρ(s))θ′ (ρ(s)) − θ(ρ(s))2 div u(s) dx ds (8.9.10)

∞ ≤ C 1 |η(s)| ∇u(s)0,2 ds ≤ CηL2 (0,∞) ∇uL2 ((0,∞),L2 (Ω)) .

Secondly, we know that θ(ρε ) = (θ ◦ p−1 )(wε + mε ), wεn → w in Lr (Ω) with γ r < NN−γ and weakly in W 1,γ (Ω), mεn → m (n → ∞). Hence w ∈ W 1,γ (Ω) and consequently (θ ◦ p−1 )(w + m) = θ(ρ) ∈ W 1,γ (Ω) as well. Now, again by the renormalized equation of continuity, taking φ := η(s)θ(ρ(x, s)) as a test function we get ∞ ∞ ′

η (s) Ω θ(ρ(s))θ(ρ(s)) dx ds = 1 η(s) Ω θ(ρ(s))(u · ∇)θ(ρ(s)) dx ds 1

∞

+ 1 η(s) Ω θ(ρ(s))∂t θ(ρ(s)) dx ds

∞

+ 1 η(s) Ω ρ(s)θ′ (ρ(s)) − θ(ρ(s)) div u(s) θ(ρ(s)) dx ds

∞ ≤ C 1 |η(s)| u(s)0,6 ∇θ(ρ(s))0,6/5 + ∂t θ(ρ(s))0,1 + ∇u(s)0,1 ds

≤ CηL2 ((0,∞)) ∇uL2 ((0,∞),L2 (Ω)) + ∂t θ(ρ)L2 ((0,∞),L2 (Ω)) , (8.9.11)

448


(for the estimate of the last term see the analogous estimate below). Finally, we have to estimate

∞ ′

∞

η (s) Ω θ(ρ(s))2 dx ds = limn→∞ 1 η ′ (s) Ω θ(ρεn (s))2 dx ds. (8.9.12) 1

It is clear that ∞ ′

∞

η (s) Ω θ(ρε (s))2 dx ds = 2 1 η(s) Ω θ(ρε (s))∂t θ(ρε (s)) dx ds 1 (8.9.13)

∞ ≤ C 1 |η(s)| ∂t θ(ρε (s))0,1 ds.

By (8.5.8), (8.4.3) we have (δ > 0 small enough)

∂t θ(ρε (s))0,1 ≤ C ρu(s)0,1+δ + ∇u(s)0,2 + 1ε ϕ0 (s/ε)

≤ C ∇u(s)0,2 + 1ε ϕ0 (s/ε)

(8.9.14)

having thus still the only restriction γ > 3/2 for N = 3. Thus by (8.9.12), (8.9.13), (8.9.14) we find ∞ ′

∞

η (s) Ω θ(ρ(s))2 dx ds = limn→∞ 1 η ′ (s) Ω θ(ρεn (s))2 dx ds 1

∞ ≤ C 1 η(s)∇u(s)0,2 ds ≤ CηL2 (0,∞) ∇uL2 ((0,∞),L2 (Ω)) .

This completes the proof of Lemma 8.9.

2

Now, by (8.9.4), (8.9.5), (8.9.6), (8.9.8) and Lemma 8.9 we get the following

t

t result. (Observe that t−1 |q ′ (s)| ds ≤ ( t−1 |q ′ (s)|2 ds)1/2 → 0 as t → ∞ in virtue of (8.9.8).) Lemma 8.11 Let γ > 1 for N = 2 and γ > 3/2 for N = 3. Then for any r ∈ [1, ∞), (8.9.15) lim θ(ρ(t)) − θ(ρ(t))0,r = 0, t→∞

where θ is a function defined in Lemma 8.3. Proof The convergence of q(t) given by (8.9.4) to zero follows from Proposition 8.7, Lemma 8.9 and Lemma 8.10. Since θ is bounded, (8.9.15) follows immediately. 2 Now we are in a position to complete the proof of Theorem 8.1. In Remark 8.6 we have defined the operator A ∈ L(Lγ (Ω), W 1,γ (Ω)), and by (8.7.5) we have w(t) = Aρ(t) for t ≥ 0. Let tn ↑ ∞ be arbitrary. Then we can select {sn } ⊂ {tn } such that ρ(sn ) → ρ∞ weakly in Lγ (Ω), w(sn ) → γ , w∞ = A(ρ∞ ) ∈ W 1,γ (Ω) weakly in W 1,γ (Ω), strongly in Lq (Ω) with q < NN−γ and almost everywhere in Ω and also such that m(sn ) → m∞ (since m(·) is bounded) and by Lemma 8.11, θ(ρ(sn )) − θ(ρ(sn )) → 0 a.e. in Ω. Hence given r ≥ 1, θ(ρ(sn )) → θ ◦ p−1 (w∞ + m∞ ), θ(ρ(sn )) → θ ◦ p−1 (w∞ + m∞ ) a.e in Ω and in Lr (Ω), and ρ(sn ) → p−1 (w∞ + m∞ ) a.e. in Ω and by boundedness of


449

ρ(sn )0,γ also in Lq (Ω), 1 ≤ q < γ. Since ρ(sn ) → ρ∞ weakly in Lγ (Ω) we find ρ∞ = p−1 (A(ρ∞ ) + m∞ ), or p(ρ∞ ) = A(ρ∞ ) + m∞ .

(8.9.16)

Since w(sn ) → w∞ weakly in W 1,γ (Ω) and w(tn ) satisfy (8.8.9), we obtain

(∇w∞ − ρ∞ f ) · ∇Θ dx = 0 for all Θ ∈ C ∞ (Ω), Ω

or, since w∞ = p(ρ∞ ) − m∞ ,

∇p(ρ∞ ) − ρ∞ f · ∇Θ dx = 0 for all Θ ∈ C ∞ (Ω). Ω

(8.9.17)

Now, given z in (C0∞ (Ω))N arbitrary, we have z = ∇Θ + v, where Θ is a solution of the problem

Θ dx = 0, ∆Θ = div z in Ω, ∂n Θ = 0 on ∂Ω, (8.9.18) Ω

v is smooth enough, and div v = 0 in Ω and v · n = 0 on ∂Ω. Let us prove ∇p(ρ∞ ) = ρ∞ f . Since (8.9.17) is valid, it is only left to prove that

∇p(ρ∞ ) − ρ∞ f · v dx = 0. Ω As we have

it remains to show

Ω

∇p(ρ∞ ) · v dx = −

Ω

Ω

p(ρ∞ )div v dx = 0,

ρ∞ f · v dx = 0.

(8.9.19)

(8.9.20)

(8.9.21) (8.9.22)

It is clear that the proof will be complete when we show that

limn→∞ Ω ρ(sn )f · v dx = 0, (8.9.23)

since then 0 = limn→∞ Ω ρ(sn )f · v dx = Ω ρ∞ f · v dx by the weak

convergence of ρ(sn ) to ρ∞ in Lγ (Ω). To prove (8.9.23) it suffices to show that Ω ρ(·)f ·v dx ∈ W 1,2 (0, ∞), or equivalently, ∞ ∞

η(s) Ω ρ(s)f · v dx ds + 0 η ′ (s) Ω ρ(s)f · v dx ds 0 (8.9.24) ≤ CηW 1,2 (0,∞) for any η ∈ C0∞ (0, ∞). The following estimate is almost a repetition of the estimate of the integral I7ε (t) given by (8.8.29) with the function κ defined in Lemma 8.8. By this technique we obtain the estimate

450


∞

η(s) Ω ρ(s)f · v dx ds ≤ µ∇v0,2 ∇uL2 ((0,∞),L2 (Ω)) ηL2 (0,∞) 0

∞ 1/2 √ + 0 |η(s)| ∇v0,∞ ρ(s)0,γ ρ(s) u(s)0,2 u(s)0,6 ds

∞ +v0,∞ 0 |η ′ (s)| ρu(s)0,1 ds ≤ CηW 1,2 (0,∞)

(if γ > 3/2).

Further, from the weak equation of continuity ∞ ′

∞

η (s) Ω ρ(s)f · v dx ds = 0 η(s) Ω ρ(s)(u(s) · ∇)(f · v) dx 0 ≤ CηL2 (0,∞) if γ ≥ 6/5.

So, (8.9.23) is proved and this yields (8.9.22). Thus (8.9.20) is established. It remains to show that under the additional assumptions given in the the

orem, ρ(sn ) → ρ∞ in Lγ (Ω) and limt→∞ Ω ρ|u|2 dx = 0. To this end consider the functions ρn = ρn (τ ) = ρ(sn + τ ), un = un (τ ) = u(sn + τ ), τ ∈ [0, 1]. By the equation of continuity (8.1.2), ∂τ ρn

L1 (0,1;W

−1,

6γ γ+1

(Ω))

≤ Cρn un

6γ

L1 ((0,1),L γ+1 (Ω))

≤ Cρn L2 ((0,1),Lγ (Ω)) un L2 (0,1;L6 (Ω))

≤ C < ∞. 6γ

Put X = B = Lγ (Ω) and Y = W −1, γ+1 (Ω) in Theorem 1.71. Then Lγ (Ω) ֒→֒→ 6γ 3γ 6γ < 3−γ , i.e. γ > 5/3. Moreover, we W −1, γ+1 (Ω) if γ ≥ 3 or γ < 3 and γ+1 ∞ have ρn bounded in L ((0, 1), B) and ∂τ ρn bounded in L1 ((0, 1), Y ). It follows that {ρn } is compact in Lp ((0, 1), B) for any 1 ≤ p < ∞. Hence we can assume ρn → ρ in Lp ((0, 1), B) for some ρ ∈ Lp ((0, 1), Lγ (Ω)) and 1 ≤ p < ∞. Clearly, ρn , un satisfy the energy inequality

sn +τ

1 µ|∇un |2 + (µ + λ)|∇div un | dx ρ |u |2 + P (ρn ) − ρn Φ (τ )dx + sn−1 +τ Ω 2 n n

un−1 |2 + P (ρ ≤ Ω 12 ρn−1ρ|n−1 n−1 ) − ρn−1 Φ dx. (8.9.25) By monotonicity

1 (8.9.26) ρ |u |2 + P (ρn ) − ρn Φ (τ ) dx ↓ E ∞ (n → ∞) Ω 2 n n with some E ∞ ∈ IR. Integrate (8.9.26)

1

0

1 ρ |u | dx dτ Ω 2 n n

1 0

dτ. By the Poincaré inequality we find

≤ Cρn L∞ ((0,1);Lγ (Ω)) ∇un 2L2 (Ω×(0,1)) → 0 (n → ∞). (8.9.27)


By the equation of continuity (8.1.2) we have

1 ρ ∂ ϕ + ρn un · ∇ϕ dx dτ = 0 for all ϕ ∈ D(Ω × (0, 1)). 0 Ω n τ

451

(8.9.28)

Using compactness of ρn in Lp ((0, 1); Lγ (Ω)) and (8.9.27) we can pass to the limit in (8.9.28) and obtain

1

ρ∂τ ϕ dx dτ = 0 for all ϕ ∈ D(Ω × (0, 1)) 0 Ω

which yields ρ ∈ Lγ (Ω) independent of τ. We have proved that sn can be chosen so that ρ( sn ) → ρ∞ in Lr (Ω), 1 ≤ r < γ with ρ∞ satisfying (8.1.9), (8.1.10). Since θ(ρn ) satisfy the renormalized equation of continuity

∂τ θ(ρn ) + div θ(ρn )un + ρn θ′ (ρn ) − θ(ρn ) div un = 0 in D′ (Ω × (0, 1)), we have

θ(ρn ) bounded in L∞ (Ω × (0, 1)), ∂τ θ(ρn ) bounded in L2 (0, 1; W −1,2 (Ω)). Setting B = L2 (Ω), X = W −1,2 (Ω) in Theorem 1.70, we obtain that θ(ρn ) is compact in C([0, 1]; W −1,2 (Ω)). It follows that θ( ρ) ∈ C([0, 1]; W −1,2 (Ω)). Since q ρn (0) = ρ(sn ) and ρ(sn ) → ρ∞ in L (Ω), 1 ≤ q < γ, we get θ( ρ) = θ(ρ∞ ), or ρ = ρ∞ . We have proved that ρn → ρ∞ in Lp ((0, 1), Lγ (Ω)), 1 ≤ p < ∞.

(8.9.29)

Besides, (8.9.26) and (8.9.27) imply

1 P (ρn ) − ρn Φ dx dτ → E ∞ . 0 Ω

γ−2 κ ργ + κ κ−1 ρ + κ and so by (8.9.29) Assume that p(ρ) = κργ . Then P (ρ) = γ−1 ρn Lp ((0,1),Lγ Ω)) → ρ∞ Lp ((0,1),Lγ (Ω)) which implies

1

0

It follows that

Ω

P (ρ(sn )) dx dτ → E∞ =

1 0

Ω

1

0

Ω

P (ρ∞ ) dx dτ.

P (ρ∞ ) − ρ∞ Φ .

Then we have

E ∞ = Ω P (ρ∞ ) − ρ∞ Φ dx ≤ lim inf n→∞ Ω P (ρ(sn )) − ρ(sn )Φ dx

≤ lim supt→∞ Ω P (ρ(t)) − ρ(t)Φ dx

≤ ess limt→∞ Ω 12 ρ|u|2 + P (ρ) − ρΦ (t) dx = E ∞ .

(8.9.30)

Consequently,

limn→∞ Ω P (ρ(sn )) − ρ(sn )Φ dx = E ∞ and limn→∞ Ω ρ|u|2 (sn ) dx = 0. (8.9.31)

452


Since ρ(sn ) → ρ∞ in L1 (Ω), by (8.9.30) we have also

P (ρ(sn )) dx → Ω P (ρ∞ ) dx Ω

(8.9.32)

which implies

ρ(sn )0,γ → ρ∞ 0,γ . By Theorem 1.4.5.22 we conclude that ρ(sn ) → ρ∞ in Lγ (Ω). If ρ(t) → ρ∞ in Lq (Ω), q ∈ [0, γ), then we can consider sn := t in (8.9.30) and obtain (8.9.31) with sn := t. Then (8.3.1) immediately follows. In the general case p = p(ρ) we use condition (8.2.3) and the comparison between the space Lγ (Ω) and Orlicz space generated by the function P (see (Kufner et al., 1977), Theorem 3.17.1). We arrive at the same conclusion. The details are left to the reader as an excersise in the theory of Orlicz spaces (note that P is a Young function satisfying the ∆2 -condition and P (r) ∼ rγ at r ∼ ∞). Finally, ρ(t)u(t)0, 2γ ≤ ρ(t)0,γ ρ(t)u(t)0,2 → 0 as t → ∞. γ+1

This completes the proof of Theorem 8.1.

2

Remark 8.12 The problem of convergence (8.3.1) with f ≡ 0 and large data was first studied in 1D in (Straˇskraba and Valli, 1988). The present method has been succesfully applied in 1D also for problems with other boundary conditions (e.g. free boundary) and/or density dependent viscosity (Straˇskraba, 1997), (Straˇskraba, 1998), (Penel and Straˇskraba, 1999), (Penel and Straˇskraba, 2003). Let us also note that the rate of convergence in (8.3.1) is an open question unless the data are small or N = 1. On the other hand, in the 1D case results on the rate of convergence are fairly complete as is shown in (Straˇskraba and Zlotnik, 2002), (Straˇskraba and Zlotnik, 2003a), (Straˇskraba and Zlotnik, 2003b), (Straˇskraba and Zlotnik, 2001). The spherically symmetric case were studied in (Matuˇs˚ u et al., 1997). 8.10

Uniqueness of equilibrium

In view of the second assertion of Theorem 8.1 the uniqueness of the equilibrium density given by equations (8.1.9), (8.1.10) is important. Notice that a consequence of Theorem 8.1 is that problem (8.1.9), (8.1.10) has at least one solution. As for uniqueness, the first observation was made in (Beir˜ ao da Veiga, 1987), where a necessary and sufficient condition for the existence of a unique strictly positive solution to (8.1.9), (8.1.10) is found. Moreover, if the condition (on Φ and p(·)) is satisfied, then the solution is unique. Next, it was shown in (Lovicar and Straˇskraba, 1991) that in the presence of vacuum zones the uniqueness and nonuniqueness may hold accordingly and supporting examples are given. Since then this situation has been analyzed more precisely and we are going to give here the present state of affairs in this respect.

UNIQUENESS OF EQUILIBRIUM

Define Q(r) := Q(r) :=

r 1

dp(s) s ,

0

dp(s) s ,

r

r ≥ 0,

if

r ≥ 0,

if

1 0

dp(s) s

0

dp(s) s

1

453

= ∞, < ∞.

Theorem 8.13 ((Feireisl and Petzeltov´ a, 1998)) Let Ω ⊂ IRN be an arbitrary open set. Suppose p ∈ C([0, ∞)) ∩ C 1 (0, ∞) (8.10.1)

and

p′ (r) > 0 for any r > 0.

(8.10.2)

Let Φ be a locally Lipschitz continuous function on Ω. 1 If 0 dp(s) = ∞, assume, in addition, that Ω is connected. s

1 dp(s) If 0 s < ∞, assume, in addition, that the upper level sets {x ∈ Ω; g(x) > k} are connected in Ω for any constant k ∈ IR. Then, given m > 0, there is at most one function ρ∞ , ρ∞ ∈ L∞ loc (Ω),

′

ρ∞ ≥ 0

satisfying (8.1.9) in D (Ω) and such that

ρ (x) dx = m. Ω ∞

(8.10.3) (8.10.4)

Moreover, if such a function exists, it is given by the formula ρ∞ (x) = Q−1 ([g(x) − kΩ ]+ )

(8.10.5)

+

for a certain constant kΩ . (Here [z] := max{z, 0}.)

Proof Let ρ∞ be a function satisfying (8.1.9), (8.10.3) and (8.10.4). Since Φ is 1,∞ (Ω). Consequently, by (8.1.9), locally Lipschitz continuous, we have Φ ∈ Wloc ∇p(ρ∞ ) ∈ L∞ loc (Ω) and so due to Sobolev imbedding in Section 1.3.5.8, p(ρ∞ ) is locally Lipschitz continuous in Ω. Since by assumption (8.10.2), p is increasing, ρ∞ is continuous in Ω and the set {x ∈ Ω; ρ∞ (x) > 0} is open. Let O ⊂ Ω be its maximal connected component and consider a ball B such that B ⊂ O. It follows that there are constants ρ, ρ such that 0 < ρ ≤ ρ∞ ≤ ρ < ∞ for all x ∈ B.

Hence, by local Lipschitz continuity of p(ρ∞ ) in Ω and (8.10.2), ρ∞ is Lipschitz continuous on B. Therefore the equation (8.1.9) restricted to B takes the form ∇Q(ρ∞ ) = ∇Φ.

This implies (see (Ziemer, 1989), Corollary 2.1.9) that there is a constant kB such that Q(ρ∞ ) = Φ − kB on B.

Moreover, as ρ∞ and Φ are continuous, the constants kB are independent of B and so there exists a constant kO such that Q(ρ∞ ) = Φ − kO Let us distinguih two cases:

on O.

(8.10.6)

454

(i)


1 0

dp(s) s

= ∞. We show that in this case necessarily ρ∞ > 0 in Ω and Q(ρ∞ ) = Φ − kΩ

on Ω,

where kΩ is uniquely determined by condition (8.10.4). Assume on the contrary that there is x0 ∈ Ω such that ρ∞ (x0 ) = 0. Since (8.10.4) holds, x0 can be chosen so that there is a connected domain of positivity O of ρ∞ such that x0 ∈ Ω ∩ ∂O. But then there exists a sequence xn → x0 , xn ∈ O such that, by (8.10.6), Φ(xn ) − kO = Q(ρ∞ (xn )), where the left-hand side has a finite limit while the right-hand side tends to −∞. This is a contradiction. Thus we have ρ∞ > 0 in Ω. Since Ω is connected we have Q(ρ∞ ) = Φ − kΩ in Ω which is certainly equivalent to (8.10.5) in this case.

1 < ∞. We show that (ii) Let 0 dp(s) s O = {x ∈ Ω; Φ(x) > kO }

(8.10.7)

for any nonempty positivity component of ρ∞ . Thus it follows from (8.10.6) that O ⊂ {x ∈ Ω; Φ(x) > kO }. On the other hand, Q(ρ∞ (x)) = Φ(x) − kO = 0 for any x ∈ ∂O ∩ Ω, and in particular, ∂O ∩ {x ∈ Ω; Φ(x) > kO } = ∅. Consequently, O is both an open and closed subset of the connected set {x ∈ Ω; Φ(x) > kO } which yields (8.10.7). The set {ρ∞ > 0} is a union of maximal connected components Cj , j ∈ S (S is finite or infinite). So we have {ρ∞ > 0} = j∈S {Φ > kj }.

If inf j∈S kj < inf x∈Ω g, then {ρ∞ > 0} = Ω and Ω is connected. If, on the other hand, inf j∈S kj ≥ inf x∈Ω g, then there are lj ↓ inf i∈S ki such that Cl1 ⊂ Cl2 ⊂ · · · ⊂ {ρ∞ > 0} and j∈S Clj = {ρ∞ > 0} and so {ρ∞ > 0} is connected. We leave the proof of the fact that j∈S Clj is connected

UNIQUENESS OF EQUILIBRIUM

455

when Clj is an “increasing” sequence of connected sets to the reader as an excercise. (Hint: Use the definition in Section 1.4.4.11.) This yields Q(ρ∞ (x)) = Φ(x) − kΩ

in {ρ∞ > 0},

or equivalently

ρ∞ (x) = Q−1 [Φ(x) − kΩ ]+

on Ω

for a certain constant kΩ . To conclude the uniqueness of ρ∞ , it suffices to observe that

Q−1 [Φ(x) − k1 ]+ dx = Ω Q−1 [Φ(x) − k2 ]+ dx for k1 = k2 Ω provided that at least one of the integrals is positive and finite.

2

It should be analyzed whether we need an additional condition (cf. (8.10.3)) in Theorem 8.1 to get uniqueness of ρ∞ according to Theorem 8.13, to obtain complete convergence ρ(t) → ρ∞ as t → ∞ (in Lr (Ω) with r ∈ [0, γ) arbitrary). This amounts to analyzing conditions under which ρ∞ ∈ L∞ loc (Ω). The following theorem sheds some light on this question. Theorem 8.14 Let the assumptions (8.10.1), (8.10.2) be satisfied and let ρ∞ satisfy (8.1.9), (8.10.4). Assume, in addition, that one of the following conditions is satisfied: (i) There are constants C, r0 > 0 and γ > 1 +

r rγ ≤ C 1 + r 1

(ii)

p(s)−p(1) s2

N −1 N

ds

such that (8.10.8)

for r ≥ r0 ;

ρ∞ ∈ Lqloc (Ω) for some q > 1;

(iii) p(r) ≥ c1 rγ − c2 with c1 > 0, c2 ∈ IR and γ >

N −1 N .

Then ρ∞ ∈ L∞ loc (Ω).

Proof First, if we only use condition (8.10.4), we deduce from (8.1.9) that ∇p(ρ∞ ) ∈ L1 (Ω). We prove that p(ρ∞ ) ∈ L1loc (Ω). If Ωc ⊂ Ω is a compact subset of Ω, then there is a domain ΩL with a Lipschitz continuous boundary such that Ωc ⊂ ΩL ⊂ Ω. This fact

follows from the Morse–Sard theorem (see Theorem 1.4). Let h ∈ Lq (ΩL ), ΩL h dx = 0. Then by Lemma 3.17 there is a function ϕ ∈ (D(Ω))3 such that

div ϕ = h, x ∈ ΩL , ϕ dx = 0, ϕ| = 0, ϕW 1,q(ΩL ) ≤ ChLq (ΩL ) (1 < q < ∞) ∂Ω L Ω

with a constant C = C(q, Ω) independent of ϕ and h. By (8.1.9),

456

Hence


ΩL

ΩL

p(ρ∞ )h dx =

ΩL

p(ρ∞ )div ϕ dx =

ΩL

ρ∞ ϕ · f dx.

p(ρ∞ )h dx ≤ ρ∞ 0,1 f 0,∞ ϕL∞ (ΩL ) ≤ ChLq (ΩL ) ,

where q ∈ (N, ∞) is arbitrary. It follows that p(ρ∞ ) ∈ Lsloc (Ω) for any s ∈ 1,1 (Ω). This yields p(ρ∞ ) ∈ L∞ [1, NN−1 ). In particular, p(ρ∞ ) ∈ Wloc loc (Ω) and ∞ ′ consequently ρ∞ ∈ Lloc (Ω) (p > 0, p(∞) = ∞!), if N = 1. If N > 1, then we use first e.g. assumption (8.10.8) which yields (notice that by p′ > 0 we have p(s) ≤ p(r) for s ≤ r)

r γ−1 − c2 for r ≥ r0 p(r) 1 − 1r ≥ 1 p(s) s2 ds ≥ c1 r with some positive constants c1 , c2 , r0 . Choosing for example " 1 # r0 > max 2, cc21 γ−1

we find p(r) ≥ c0 rγ−1 with some c0 > 0 for r ≥ r0 . As a consequence we find N (γ−1) N −1 ρ∞ ∈ Lq (ΩL ) for any q ∈ [1, NN(γ−1) −1 ), assuming N −1 > 1, i.e. γ > 1 + N . q1 Then ρ∞ ∈ Lloc (Ω) with some q1 > 1. We clearly arrive at the same conclusion if we assume (ii) or (iii) instead of (i). Since p(ρ∞ ) ∈ L1 (ΩL ), by the Poincaré inequality (1.3.61),

p(ρ∞ ) − |Ω1L | ΩL p(ρ∞ ) dx ∈ Lq2 (ΩL ) with q2 =

N qj N −qj , j = a qj > NN+a

N q1 N −q1

≤ ∞ if q1 ≤ N, and belongs to L∞ (ΩL ), if q1 > N. Define qj+1 =

1, 2, . . . . The condition qj+1 > a (a > 0) leads to the requirement

a and by induction to qj+1−k > NN+ka . Choosing k = j, a = N we N obtain q1 > j+1 . Putting j = N − 1 we find that after N steps we have qN > N,

1,qN ∞ (Ω) ֒→ L∞ i.e. p(ρ∞ ) ∈ Wloc loc (Ω), which yields ρ∞ ∈ Lloc (Ω). The above procedure for raising the regularity of p(ρ∞ ) and ρ∞ is called in the literature the bootstrap argument and is useful in all cases in which part of an equation has a regularizing effect (here the operator ∇), and the rest raises the regularity of the right-hand side (here ρ∞ f ), once plugged in after the preceeding regularization step. As we have seen, a certain “triggering” regularity is necessary 1 (here ρ∞ ∈ Lqloc (Ω)). Next, if we assume that condition (ii) is satisfied, then the above bootstrap argument can be performed with q1 = γ. Finally, if (iii) holds, then, similarly as above, we get ρ∞ ∈ Lq1 (Ω) with 2 q1 = NγN −1 > 1 and again use the bootstrap argument.

8.11

Global behavior of weak solutions in time in bounded domains – arbitrary forces In Section 8 we studied the global stabilization in time of a finite energy weak solution of problem (8.1.1)–(8.1.6). The stabilization of the evolutionary solution

BOUNDED ABSORBING SETS

457

to an equilibrium state was a consequence of the dissipativity of equation (8.1.1) and the potential form of f given by (8.1.6). The dissipativity is caused by viscous terms which control the change of mechanical energy to the internal energy (heat). This dissipative mechanism works of course also in the case of a general external force f = f (x, t) and the question arises of how to describe its eventual stabilizing effect. In this section we present several recent results in this direction. 8.12

Bounded absorbing sets

The following result can be found in (Feireisl and Petzeltov´ a, 2001b). Theorem 8.15 Let Ω ⊂ IRN , N = 2, 3 be a bounded domain with a Lipschitz continuous boundary and I ⊂ IR an interval such that inf{x ∈ I} > −∞. Consider system (8.1.1)–(8.1.3) with p(ρ) = κργ , γ > 1 if N = 2, γ > 5/3 for N = 3,

(8.12.1)

and f ∈ (L∞ (Ω × (0, ∞)))N . Denote, for a finite energy weak solution of the problem, E(ρ, ρu)(t) :=

1 |ρu|2 (x, t) ρ(x,t)>0 2 ρ

+

κ γ γ−1 ρ (x, t)

dx.

Then there exists a constant E∞ , depending solely on the norm f L∞ (Ω×(0,∞)) of the driving force f and the total mass m = Ω ρ(x, t) dx(= const) with the following property: Given E0 , there exists a time T = T (E0 ) such that E(t) := E(ρ, u)(t) ≤ E∞ for all t ∈ I, t > T + inf{x ∈ I}

(8.12.2)

for any finite energy weak solution of problem (8.1.1)–(8.1.4) satisfying

ρ dx = m. lim supt→inf{I}+ E(ρ, ρu)(t) ≤ E0 , Ω The proof of Theorem 8.15 is based on the global a priori estimate

t+1

supt≥0 t ρ(x, s)γ+θ dx ds < ∞ Ω

(8.12.3)

with θ = min{ 41 , 2γ−3 3γ }. The technique of deriving such an estimate is the same as in Section 7.9.5. Thorough investigation of the energy inequality (7.1.37) yields the estimate 4γ−3 3(γ+θ−1) E(t+ ) + sup ρ(t)0,γ . (8.12.4) sup E(t+ ) ≤ C 1 + sup t∈[T,T +1]

t∈[T,T +1]

t∈[T,T +1]

4γ−3 < γ, it follows from (8.12.4) that there exists a constant L such Since 3(γ+θ−1) that if

458

then


E((T + 1)− ) > E(T + ) − 1 for a certain T > 0, sup t∈(T,T +1)

E(t+ ) ≤ L.

This implies that there exists T = T (E0 ) and t0 < T such that E(t+ 0 ) ≤ L. By induction it can then be proved that

E (t0 + n)+ ≤ L, n = 1, 2, . . . . The rest follows from an easy consequence of the energy inequality, namely,

√2mK(t2 −t1 ) + , 0 < t1 < t2 . E(t− 2 ) ≤ 1 + E(t1 ) e √

The choice E∞ := (1 + L)e 2mK − 1 completes the proof of (8.12.2). The intermediate details can be found in (Feireisl and Petzeltov´ a, 2001b). 2 Remark 8.16 Relation (8.12.2) may be interpreted as follows: After a finite time, any finite energy weak solution is absorbed in the set " (r, v); r ≥ 0, r ∈ L∞ ((0, ∞); Lγ (Ω)), v ∈ L2 ((0, ∞), (W01,2 (Ω))N , (8.12.5)

#

κ r|v|2 ∈ L∞ ((0, ∞); L1 (Ω)), Ω 12 r|v|2 + γ−1 rγ dx ≤ E∞ with a suitable constant E∞ .

8.13

Asymptotically closed trajectories

In order to give a true picture of the asymptotic behavior of trajectories generated by finite energy weak solutions, the possibility of multiple solutions must be taken ∞ N into account. To this end consider a set F ⊂ L∞ loc (IR, (L (Ω)) ) of right-hand sides f and introduce the evolution operator U (t0 , t), related to problem (8.1.1)– (8.1.3): " (8.13.1) U (E0 , F)(t0 , t) := (ρ, ρu)(t); (ρ, u) is a finite energy # weak solution of (8.1.1)–(8.1.3) . Consider the so-called short trajectory (a notion introduced in (M´ alek and Neˇcas, 1996)) defined by " # (ρ(t + τ ), (ρu)(t + τ )) τ ∈[0,1] ; ρ, u is a finite energy U s (E0 , F)(t0 , t) :=

weak solution of problem (8.1.1)–(8.1.3) on an open interval

I, (t0 , t0 + 1) ⊂ I, with f ∈ F and such that lim supt→t0 E(t) ≤ E0 .

(8.13.2)

The following result is a consequence of Theorem 7.7 and Theorem 8.15 (see (Feireisl and Petzeltová, 2001a), Theorem 1.1 and (Feireisl, 2000), Proposition 10).

GLOBAL ATTRACTOR OF SHORT TRAJECTORIES

459

Theorem 8.17 Let Ω ⊂ IRN , N = 2, 3 be a bounded domain with a Lipschitzcontinuous boundary and p as in (8.12.1). Let F be bounded in (L∞ (Ω × IR))N . Consider a sequence

ρn , (ρn un ) ∈ U s (E0 , F)(a, tn ) (a ∈ IR) for certain tn → ∞. Then there is a subsequence (not relabeled) such that

ρn → ρ in Lγ (Ω × (0, 1)) and in C([0, 1], Lα (Ω)) for 1 ≤ α < γ, 2γ

γ+1 (Ω))N ) for 1 ≤ p < ρn un → ρu in, (Lp (Ω × (0, 1)))N ∩ C([0, 1], (Lweak

2γ , γ+1

and E(ρn , ρn un ) → E(ρ, ρu)

in L1 (0, 1),

where ρ, u is a finite energy weak solution of problem (8.1.1)–(8.1.3) defined on the whole real line I = IR such that E ∈ L∞ (IR) and with f ∈ F + , where " F + := f ; f = lim hn (· + τn ) weak-∗ in (L∞ (Ω × IR))N τn →∞ # for certain hn ∈ F and τn → ∞ . (8.13.3) 8.14

Global attractor of short trajectories

Theorem 8.17 has an immediate consequence for construction of a set of short trajectories to which any finite energy weak solution is asymptotically attracted. Define

(8.14.1) ρ(τ ), (ρu)(τ ) τ ∈[0,1] ; ρ, u is a finite energy As (F) := weak solution of problem (8.1.1)–(8.1.3)

on the interval I = IR with f ∈ F + and E(ρ, ρu) ∈ L∞ (IR) .

The following property of the set As (F) can be shown (see (Feireisl, 2003c)): Theorem 8.18 Let Ω be a bounded domain with a Lipschitz continuous boundary and p as in (8.12.1). Then the set As (F) defined by (8.14.1) is compact in Lγ (Ω×(0, 1))×(Lp (Ω× (0, 1)))N and ρ − ρLγ (Ω×(0,1)) + ρu − ρ uLp (Ω×(0,1)) sup inf s (ρ,ρu)∈U (E0 ,F )(t0 ,t) (ρ,ρ u)∈A (F )

−→ 0

as

t −→ ∞

for any 1 ≤ p
k} are connected for any k ∈ IR. Then given R > 0, ε > 0 there exists δ = δ(R, ε) > 0 such that lim sup ρ(t) − ρ∞ 0,γ + ρu(t)0,1 < ε t→∞

for any finite energy weak solution ρ, u of problem (8.1.1)–(8.1.3) provided that lim sup g(t)L∞ (Ω×(t,∞)) < R, t→∞

lim sup dR [g(t + s)|s∈[0,1] , 0] < δ. t→∞

Here ρ∞ is the unique solution of equilibrium problem (8.1.9), (8.1.10) (cf. Theorem 8.13). 8.16

Attractors

For a general dynamical system a set A is called a global attractor if it is compact, attracting all trajectories and minimal in the sense that if a set A1 is compact and attracting all trajectories, then A ⊂ A1 . Despite possible nonuniqueness of finite energy weak solutions with fixed initial data the notion of the global attractor may make sense and has reasonable properties which we briefly describe here. Let

ρ(0), (ρu)(0) ; ρ, u is a finite energy weak solution of problem A(F) = (8.1.1)–(8.1.3) on I = IR with f ∈ F + and E ∈ L∞ (IR) . The following assertion is proved in (Feireisl, 2003c), Theorem 4.1.

Theorem 8.20 Let Ω ⊂ IRN , N = 2, 3 be a bounded domain with a Lipschitz continuous boundary and p as in (8.12.1). Let F be a bounded subset of (L∞ (Ω× IR)N .

TIME-PERIODIC SOLUTIONS

461

2γ

γ+1 Then A(F) is compact in Lα (Ω) × (Lweak (Ω))N and 0 sup inf ρ − ρ0,α + (ρu − ρ u) · ϕ dx → 0 (t → ∞)

(ρ,ρu)∈U (E0 ,F )(t0 ,t) (ρ,ρ u)∈A(F )

Ω

2γ

for any 1 ≤ α < γ and any ϕ ∈ (L γ−1 (Ω))N .

Notice that A is a weak attractor with respect to the momentum ρu. A stronger conclusion is proved in (Feireisl, 2000), Theorem 17 under the assumption of additional smoothness of A. Theorem 8.21 Let the assumptions of Theorem 8.20 be satisfied. Assume that, in addition, the energy E = E(ρ, ρu) is sequentially continuous on A(F), namely, for any sequence " # (ρn , ρn un ) ⊂ A(F) such that ρn → ρ in L1 (Ω), ρn un → ρu weak in (L1 (Ω))N , we require

E(ρn , ρn un ) → E(ρ, ρu).

Then sup

inf

(ρ,ρu)∈U (E0 ,F )(t0 ,t) (ρ,ρ u)∈A(F )

ρ − ρ0,γ + ρu − ρ u0,1

→ 0 (t → ∞.)

8.17 Time-periodic solutions A natural question of the existence of time-periodic solutions arises when the external force is time-periodic, i.e. f (x, t + ω) = f (x, t) for a.e. x ∈ Ω, t ∈ IR,

(8.17.1)

with some constant (period) ω > 0. The corresponding periodic solution should satisfy ρ(t + ω) = ρ(t), (ρu)(t + ω) = (ρu)(t) for all t ∈ IR, and

Ω

ρ dx = m

(8.17.2) (8.17.3)

with a given fixed mass m > 0. In (Feireisl et al., 1999), Theorem 1.1, the existence of a finite energy weak solution of (8.1.1), (8.1.2) satisfying (8.17.2) has been proved for the case of so-called no-stick boundary conditions (1.2.101). Combining the method of (Feireisl et al., 1999) with the existence theory given in Sections 7.1–7.11 the following result can be proved. Theorem 8.22 Let Ω ⊂ IRN , N = 2, 3 be a bounded domain with boundary of class C 2+a with some a > 0. Consider problem (8.1.1)–(8.1.3) with p as in (8.12.1). Then given f ∈ (L∞ (Ω × IR))N satisfying (8.17.1) and m > 0, there exists a finite energy weak solution (ρ, ρu) of (8.1.1)–(8.1.3) on Ω × IR which satisfies (8.17.2) and (8.17.3).

462


Theorem 8.22 in combination with Theorem 8.18 can be used to an alternative proof of the existence of a solution to problem (4.1.1)–(4.1.5). Indeed, if we take F = F + = {f } and ωn := k −n , where k > 1 is an integer, then 1 ≥ ωn ↓ 0 and by Theorem 8.22 problem (8.1.1)–(8.1.3)

has time-periodic solutions (ρn , ρn un ) with respective periods ωn , such that Ω ρn dx = m. Theorem 8.18 implies that the restriction ρn , ρn un to the time-interval [0, 1] belongs to As . Since the set As 2γ is compact in Lγ (Ω × (0, 1)) × Lp (Ω × (0, 1)), 1 ≤ p < γ+1 , the sequence ρn , ρn un has an accumulation point which is a global finite energy weak solution (ρ, ρu) of (8.1.1)–(8.1.3). Moreover, this solution is periodic with each period ωn . So, for example, given t, s ∈ IR arbitrary, we have α ρ(s) = limn→∞ ρ s + t−s ωn ωn = ρ(t) in L (Ω), (1 ≤ α < γ) (cf. Theorem 8.17). Consequently, (ρ, ρu) is independent of t and it is a solution of the stationary problem (4.1.1)–(4.1.5). 8.18

Uniqueness of equilibrium revisited

In this chapter the convergence to an equilibrium state has been established (see Theorem 8.1) and it appeared that uniqueness of equilibrium plays an important role since Theorem 8.1 claims only sequential convergence to equilibrium unless it is a priori unique. In the latter case 2γ

ρ(t) → ρ∞ in Lγ (Ω) and ρ(t)u(t) → 0 in L γ+1 (Ω) as t → ∞.

(8.18.1)

In the general case we obtain the so-called ω-limit set of problem (8.1.1)–(8.1.6) defined as the set of all limits limt→∞ ρ(tn ) = ω[ρ], where tn → ∞. It may be proved (see e.g. (Feireisl, 2000), Proposition 12) that the ω-limit set is a bounded in Lγ (Ω), compact in Lα (Ω) for any α ∈ [1, γ) and connected (in the topology of L1 (Ω)) subset of the set of all solutions to the equilibrium problem κ∇ργ = ρ∇Φ in (D′ (Ω))3 ,

ρ ≥ 0, Ω ρ dx = m,

κ γ ρ − ρ∇Φ dx = E∞ , Ω γ−1

(8.18.2) (8.18.3) (8.18.4)

where f = ∇Φ and E∞ = ess limt→∞ E(ρ, ρu)(t). Let us note that E∞ is the limit total energy of system (8.1.1)–(8.1.4) and it is a constant for any fixed data. It is clear that for full convergence (8.18.1) it suffices that system (8.18.2) has only isolated solutions. In fact a more general result due to (Feireisl and Petzeltov´ a, 1998) and (Erban, 2001) holds true: Theorem 8.23 Let Ω be a domain and let Φ ∈ W 1,∞ (Ω) be such that −∞ < inf Ω Φ ≤ supΩ Φ < ∞. Suppose that m ∈ (0, ∞), γ ∈ (1, ∞), and denote M = {k ∈ (−∞, sup Φ); level set {x ∈ Ω; Φ(x) > k} is not connected }. Ω

UNIQUENESS OF EQUILIBRIUM REVISITED

463

If we set K=

inf M if M = ∅ m = supΩ Φ if M = ∅,

0 1 γ−1 γ−1 (Φ − K)+ , γ Ω

then: (i) If m ≥ m, then there exists at most one nonnegative function ρ ∈ L∞ (Ω) which satisfies (8.18.2), (8.18.3). (ii) If 0 < m < m < ∞, then there exists a continuum of nonnegative functions ρξ ∈ L∞ (Ω) satisfying (8.18.2), (8.18.3).

Due to statement (i) of the above theorem, the following uniqueness result is valid. Corollary 8.24 Let all assumptions of Theorem 8.23 be satisfied. If M = ∅ then m = 0. Moreover, given m ∈ (0, ∞), there exists km ∈ (−∞, supΩ Φ) such that ρ(x) = [

1 γ−1 (Φ(x) − km )+ ] γ−1 γ

is a unique solution of problem (8.18.2), (8.18.3) in L∞ (Ω). This statement holds true due to the continuity and strict monotonicity of func

1 + γ−1 which maps (−∞, supΩ Φ) on (0, ∞). tion k → Ω [ γ−1 γ (Φ(x) − k) ]

Remark 8.25 Another possibility of how to reduce the number of equilibria to one is to fix up the energy of the solution to (8.18.2), (8.18.3) by condition (8.18.4). The following assertion holds true: There exists a critical mass mc such that, (i) if m ∈ [mc , ∞), then problem (8.18.2)–(8.18.4) has at most two solutions; (ii) if m ∈ (0, mc ), then there exists an energy E∞ ∈ (0, ∞) such that system (8.18.2)–(8.18.4) has a continuum of solutions with this energy.

The critical mass mc is explicitly determined by the function Φ and the constants κ, γ. For details see (Feireisl and Petzeltová, 1999) and (Erban, 2001), Theorems 1–3.

9 STRONG SOLUTIONS OF NONSTEADY COMPRESSIBLE NAVIER–STOKES EQUATIONS In this chapter we prove a version of the global existence theorem for compressible Navier–Stokes equations assuming the initial data sufficiently small in suitable norms. We choose the equations of viscous barotropic flow (see Section 1.2.18) in three space dimensions as a relatively simple illustrative example rather than equations including temperature as is the case in the pioneering papers (Matsumura and Nishida, 1979) and (Matsumura and Nishida, 1980); cf. also (Valli, 1983). For many other versions we refer the reader to references given in comments at the end of this chapter (see Section 9.10). 9.1

Problem formulation

In this section we are interested in more regular solutions of the same problem as in Section 7.1, namely, ∂t (ρu) + div (ρu ⊗ u) − µ∆u − (µ + λ)div u + ∇p(ρ) = 0, ∂t ρ + div (ρu) = 0, x ∈ Ω, t > 0, u(x, t) = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u0 (x), ρ(x, 0) = ρ0 (x) ≥ ρ0 , x ∈ Ω.

(9.1.1) (9.1.2) (9.1.3) (9.1.4)

Here u is the velocity, ρ the density, p(·) the pressure – a given function of ρ, µ and λ are the first and second viscosity coefficient, respectively, u0 and ρ0 are given initial data and Ω is a bounded domain in IR3 . Precise assumptions on the given data will be given below in the main theorem of this section (Theorem 9.1). Notice that we impose the initial condition for the velocity since we anticipate ρ(x, t) ≥ ρ > 0 due to the greater regularity of (u, ρ) considered below. In what follows we shall assume p(·) ∈ C 1 ((0, ∞)),

p′ (r) > 0 for r > 0.

(9.1.5)

To establish the existence theorem we need several spaces. Given 1 ≤ p, q ≤ ∞ and T > 0 we denote I = (0, T ), QT = Ω × I, W p,q (0, T ) := W 1,p (I, (Lq (Ω))N ) ∩ Lp (I, (W 2,q (Ω))N ) ∩ (W01,q (Ω))N ) (9.1.6) with the norm uW p,q (0,T ) := uW 1,p (I,Lq (Ω)) + uLp (I,W 2,q (Ω)) , 464

SIMILARITY TRANSFORMATION

and

465

1 1 1− 1 ,q V0p,q := (W 2(1− p ),q (Ω))N ∩ (W0 p (Ω))N × W 1− p ,q (Ω)

with the norm

(u0 , ρ0 )V0p,q := u0

W

2(1− 1 ),q p (Ω)

+ ρ0 W 1,q (Ω) .

The Sobolev–Slobodetskii spaces of fractional order used here are defined in Section 1.3.5.10. Theorem 9.1 Assume that (9.1.5) holds. Let N = 3, p ∈ [2, ∞), q ∈ (3, ∞), a > 0, ∂Ω ∈ C 2+a , 1 1 (u0 , ρ0 ) ∈ V0p,q := (W 2(1− p ),q (Ω))N ∩ (W01,q (Ω))N × W 1− p ,q (Ω). (9.1.7)

Then there exists δ0 > 0 such that if 0 0

0 u ,ρ − 1 |Ω| Ω ρ dx V p,q ≤ δ0 ,

(9.1.8)

0

then problem (9.1.1)–(9.1.4) has a unique solution

(u, ρ) ∈ W p,q (0, T ) × W 1,p (I, Lq (Ω)) ∩ Lp (I, W 1,q (Ω))

for each T > 0. (9.1.9)

Proof The proof will be divided into several steps and will be performed in Sections 9.2–9.9. First, we will transform problem (9.1.1)–(9.1.4) with the help of a similarity transformation to an analogous problem with a small parameter in the pressure term. Then we use a result on maximal parabolic regularity to rewrite the problem as a “fixed point like” equation. To this end some imbeddings and a detailed study of the transport (continuity) equation will be necessary. As the next step an appropriate fixed point theorem will be applied to get a solution local in time. After a uniqueness argument we construct a suitable maximal solution on a time interval (0, Tmax ). Finally, a global estimate in time will ensure that the solution remains uniformly bounded in time which by a standard 2 argument leads to the conclusion Tmax = ∞. 9.2 Similarity transformation Define new variables 1 s := ν α t, y := ν β x, v(y, s) := u(x, t), r(y, s) := ρ(x, t), ν

where ν > 0 is fixed. (9.2.10)

Then ∂/∂t = ν α ∂/∂s, ∂/∂xj = ν β ∂/∂yj , and (9.1.1)–(9.1.4) read

ν α+1 ∂s (rv) + ν β+2 div y (rv ⊗ v) − ν 2β+1 µ∆y v + (µ + λ)∇y div y v

+ν β ∇y p(r) = 0, (9.2.11) y ∈ Ων := {y ∈ IRN ; ν −β y ∈ Ω}, s > 0,

ν α ∂s r + ν β+1 div y (rv) = 0, v(y, s) = 0, y ∈ ∂Ων , s > 0, 1 v(y, 0) = v 0 (y) := u0 (ν −β y), ν

r(y, 0) = r0 (y) := ρ0 (ν −β y),

y ∈ Ων .

466

STRONG SOLUTIONS OF NONSTEADY EQUATIONS

Here the subscript y on the differential operators means that it operates with respect to the variable y. To homogenize the maximum of the terms in (9.2.11) choose α, β so that we have α + 1 = β + 2 = 2β + 1,

α = β + 1.

Then α = 2,

β=1

(9.2.12)

and system (9.2.11) reads ∂s (rv) + div y (rv ⊗ v) − µ∆y v − (µ + λ)∇y div y v + ν −2 ∇y p(r) = 0, ∂s r + div y (rv) = 0, y ∈ Ων , s > 0, (9.2.13) v(y, s) = 0, y ∈ ∂Ων , s > 0, 1 v(y, 0) = v 0 (y) = u0 (ν −1 y), r(y, 0) = r0 (y) = ρ0 (ν −1 y), y ∈ Ων . ν Note that the positive constant ν in (9.2.13) can be chosen arbitrarily. We define the equilibrium state for (9.1.1), (9.1.2) as a function ρe independent of s and satisfying (9.1.1), (9.1.2) with u ≡ 0. This definition trivially yields ρe = const.

(9.2.14)

If a global evolutionary solution (u, ρ) of (9.1.1)–(9.1.4) is sufficiently regular (for example of regularity (9.1.9)), then by integration of (9.1.2) over Ω we obtain

ρ dx = Ω ρ0 dx for all t > 0. (9.2.15) Ω

So

ρe :=

1 |Ω|

Ω

ρ0 dx

(9.2.16)

is an appropriate equilibrium state for problem (9.1.1)–(9.1.4). As a consequence we obtain corresponding equilibrium state for the transformed problem (9.2.13):

0 1 re = re (y) = ρe (ν −1 y) = ρe = |Ω| ρ dx (9.2.17) Ω

Notice that re is independent of ν. Note also that condition (9.1.8) from Theorem 9.1 includes the assumption that the initial density is close to the equilibrium density. 9.3 Maximal parabolic regularity Define the following unbounded linear operator in Lq (Ω) by the relation Aw := −µ∆y w−(µ+λ)∇y div y w,

w ∈ D(A) := W 2,q (Ω)∩W01,q (Ω). (9.3.18)

Further, we define another operator on functions of the space and time variable by (9.3.19) w ∈ W p,q (0, T ). Lw := ρe dw dt + Aw, The following theorem can be deduced from (Amann, 1995), Remark 4.10.9 and (Lunardi, 1995), Theorem 3.2.3.

RESOLUTION OF CONTINUITY EQUATION

467

Theorem 9.2 Given 1 < p < ∞, w0 ∈ V0p,q and f ∈ Lp (I, Lq (Ω)N ), the Cauchy problem dw dt

+ Aw = f (t),

has a unique solution w ∈ W wW p,q (0,T )

t ∈ (0, T ),

w(0) = w0 ,

(9.3.20)

p,q

(0, T ), and ≤ C f Lp (I,Lq (Ω)) + w0 V0p,q ,

(9.3.21)

where C is independent of w0 ∈ V0p,q , f ∈ Lp (I, Lq (Ω)N ). In addition, there exists a positive constant c0 independent of f such that wW p,q (0,T ) ≥ c0 sup w(t)V0p,q , t∈(0,T )

f ∈ (Lp (I, Lq (Ω))N ).

In fact Theorem 9.2 claims that the operator L considered as an operator from V0p,q × W p,q (0, T )

into Lp (I, (Lq (Ω))N )

has an inverse which is continuous from Lp (I, (Lq (Ω))N ) into V0p,q × W p,q (0, T ), i.e.

L−1 ∈ L Lp (I, (Lq (Ω))N ), V0p,q × W p,q (0, T ) . (9.3.22) This fact will be useful in further transformation of the problem (9.2.13).

9.4 Resolution of the continuity equation with a given velocity Our intention in transforming the problem (9.2.13) is to resolve the equation (9.2.13)2 with respect to r and insert the result into (9.2.13)1 . To this end we formulate the following lemma. Lemma 9.3 Let N = 3, 2 ≤ p < ∞, 3 < q < ∞, v ∈ W p,q (0, T ), r0 ∈ W 1,q (Ω). Then there is a unique strictly positive function r =: S(v) ∈ W 1,p (I, Lq (Ω)) ∩ L∞ (I, W 1,q (Ω))

(9.4.23)

which is away from zero and such that y ∈ Ω, s ∈ (0, T ), y ∈ Ω.

∂s r + div y (rv) = 0, 0

r(y, 0) = r (y),

(9.4.24)

Proof of Lemma 9.3. Write in brief Ω instead of Ων . First, assume that v ∈ C 1 (I, C ∞ (Ω)), v|∂Ω = 0, r0 ∈ C ∞ (Ω). Then by Theorem 2.16 applied successively, we get a solution ∞ r ∈ j=1 C 1 (I, H j−1 (Ω)) ∩ C(I, H j (Ω)) . (9.4.25) Hence by the Sobolev imbedding theorems, ∞ r ∈ k=1 C 1 (I, C k (Ω)) = C 1 (I, C ∞ (Ω)). p,q

(9.4.26) n

Let now v ∈ W (0, T ). Then, by a density argument there are v in the space C 1 (I, C ∞ (Ω)) with v n |∂Ω = 0, v n → v in W p,q (0, T ) and rn0 ∈ C ∞ (Ω), rn0 → r0

468


in W 1,q (Ω). Notice that due to imbedding theorems 1.72, 1.73, for any Lipschitz domain Ω, v n → v in C(Ω × (0, T )). According to the previous step, there are rn satisfying ∂s rn + divy (rn v n ) = 0, rn (y, 0) = rn0 (y), rn ∈ C 1 (I, C ∞ (Ω)).

(9.4.27)

Multiplying the first equation in (9.4.27) by |rn |β−2 rn (β ≥ 2) and integrating over Ω, after two integrations by parts we obtain

1−β 1 d n β |rn |β div y v n dy β ds Ω |r (s)| dy = β Ω (9.4.28)

n n β ≤ β−1 β supy∈Ω |div y v (y, s)| Ω |r | dy. Since v n → v in Lp (I, W 2,q (Ω)), we have div y v n bounded in Lp (I, W 2,q (Ω)) ֒→ Lp (I, C 1 (Ω)) (q > 3). Then by Gronwall’s inequality (see Lemma 1.2), and the continuous imbedding W 2,q (Ω) into C 1 (Ω),

s |rn (s)|β dy ≤ exp (β − 1) 0 supy∈Ω,n=1,2,... |div y v n (y, σ)| dσ Ω |r0 (y)|β dy Ω

T ≤ exp C(β − 1) 0 v n (σ)2,q dσ r0 β0,β

p−1 ≤ exp C(β − 1)T p v n Lp (I,W 2,q (Ω)) r0 β0,β . (9.4.29) So rn is bounded in L∞ (I, Lβ (Ω)) and rn (s)Lβ (Ω) ≤ C(T, v n Lp (I,W 2,q (Ω)) )r0 0,β for all s ∈ (0, T ), β ∈ [2, ∞). (9.4.30) Letting β → ∞ in (9.4.30) we obtain the uniform estimate (see Section 1.1.14.1, vii) ), rn (s)0,∞ ≤ C(T, vLp (I,W 2,q (Ω)) )rn0 0,∞ , s ∈ (0, T ). Consequently

rn L∞ (QT ) ≤ C(T, vLp (I,W 2,q (Ω)) )rn0 0,∞ ≤ C1 (T ) < ∞

(9.4.31)

and we may assume that v n were chosen so that rn → r

weak-∗ in L∞ (QT ).

(9.4.32)

Let ϕ ∈ C0∞ (Ω × [0, T )). Then

0 = QT ∂s rn + div y (rn v n ) ϕ dy ds = − QT rn ∂s ϕ + rn (v n · ∇y )ϕ dy ds

− Ω rn0 ϕ(y, 0) dy → − QT r∂s ϕ + r(v · ∇y )ϕ dy ds − Ω r0 ϕ(y, 0) dy.

The last relation yields

∂s r + div y (rv) = 0 in the sense of distributions on QT .

(9.4.33)

FURTHER TRANSCRIPTION OF THE PROBLEM

469

Applying the gradient operator ∇y to (9.4.27) we obtain ∂s ∇y rn + (v n · ∇y )(∇y rn ) + ∇y v n · ∇y rn +rn ∇y (div y v n ) + ∇y rn div y v n = 0.

(9.4.34)

Multiplying (9.4.34) by |∇y rn |q−2 ∇y rn and integrating over Ω, we get 1 d n q q ds ∇y r 0,q

1

· ∇y )(|∇y rn |q ) + rn |∇y rn |q−2 ∇y (div y v n ) · ∇y rn

+|∇y rn |q div y v n + |∇y rn |q−2 ∇y rn · (∇y v n · ∇y rn ) dy = 0. (9.4.35) Bearing in mind the identity +

Ω

we find

(v Ω q

n

(v n · ∇y ) |∇y rn |q + |∇y rn |q div y v n dy = 0,

d n ds ∇y r 0,q

≤ C∇y rn 0,q supn∈N ∇y v n (s)0,∞

+rn (s)0,∞ ∇y div y v n 0,q .

(9.4.36)

Since v n are bounded in Lp (I, W 2,q (Ω)N ) and rn = S(v n ) satisfy (9.4.32), again with the help of Gronwall’s inequality we conclude

∇y rn (s)0,q ≤ C(T ) ∇y r0 0,q + 1 .

(9.4.37)

So rn can be chosen so that

∇y rn → ∇y r weak-∗ in L∞ (I, (Lq (Ω))3 ). By (9.4.31) we have r ∈ L∞ (QT ). Further, v∈Lp (I,(W 2,q (Ω))3 ) and by (9.4.37), ∇y r ∈ L∞ (I, (Lq (Ω))3 ). It follows from (9.4.27)1 that ∂s r ∈ Lp (I, Lq (Ω)) and, (see Section 1.4.5.8), ∂s rLp (I,Lq (Ω)) ≤ lim sup ∂s rn Lp (I,Lq (Ω)) n→∞

≤ rdiv y vLp (I,Lq (Ω)) + (v · ∇y )rLp (I,Lq (Ω)) ≤ vLp (I,W 2,q (Ω)) rL∞ (I,Lq (Ω)) + vLp (I,W 1,q (Ω)) ∇y rL∞ (I,Lq (Ω)) . The proof of Lemma 9.3 is complete. 9.5

2

Further transcription of the problem

Now we use Lemma 9.3 and Theorem 9.2 for the final transcription of problem (9.2.13). First of all to a given v smooth enough, we prescribe r = S(v) the

470


existence of which is guaranteed by Lemma 9.3. Then we write the first equation in (9.2.13) in the form Lv = F (v), (9.5.38) where L is the operator given by (9.3.19). Since L is invertible we can finally write problem (9.2.13) in the form of one equation v = G(v) := L−1 F (v 0 , v) (9.5.39)

= L−1 v 0 , ∂s (re − S(v))v − div y (S(v)v ⊗ v) + ν −2 ∇y p(re ) − p(S(v)) ,

provided that v is such that the argument of L−1 belongs to the space V0p,q × Lp (I, (Lq (Ω))3 ). 9.6

Fixed point argument and the existence of a local solution

Define for given v 0 ∈ W p,q (0, T ) BR (v 0 ) := {v ∈ W p,q (0, T ); v − v 0 W p,q (0,T ) ≤ R}.

(9.6.40)

We intend to solve equation (9.5.39) in BR (0) for sufficiently small R > 0. To this end let us first show that there are δ0 , T > 0 and R > 0 such that G(BR (0)) ⊂ BR (0).

(9.6.41)

To prove (9.6.41), let T > 0, R > 0 and v ∈ BR (0). First of all notice that since r = S(v) satisfies (9.4.24), the operator G can be written in the form G(v) = L−1 v 0 , (re − S(v))∂s v − S(v)(v · ∇y )v − ν −2 ∇y p(S(v)) . (9.6.42)

To show (9.6.41) it suffices to prove that the terms in the second argument of the operator L−1 are small in the norm of Lp (I, (Lq (Ω))3 ). Starting with the first term, put r := S(v) − re . Clearly, r satisfies the equations ∂s r + div y (rv) = −re div y v,

r(y, 0) = r0 (y) − re ,

(9.6.43)

y ∈ Ω, s ∈ (0, T ).

By the same procedure as used in the proof of Lemma 9.3 we can obtain an analogous estimate with the difference that now on the right-hand side will be an additional term produced by the right-hand side of (9.6.43)1 , i.e. we get

β−1 1 d β β β ds Ω |r(s)| dy ≤ β supy∈Ω |div y v(y, s)| Ω |r| dy +|re | div y v(s)0,β rβ−1 0,β

which yields d ds r(s)0,β

e ≤ C β−1 β v(s)2,q r(s)0,β + C|r | v(s)2,q .

LOCAL EXISTENCE

471

Using Gronwall’s lemma we further obtain

s r(s)0,β ≤ exp C β−1 v(σ)2,q dσ r0 − re 0,β β 0

s

s +C|re | 0 v(σ)2,q exp C β−1 v(τ 2,q dτ dσ β σ

p−1 p−1 p R ≤ exp C β−1 δ0 + C|re |T p R . β T

(9.6.44)

Letting β → ∞ and taking δ0 sufficiently small we obtain r − re L∞ (QT ) ≤ C(T )R.

The inclusion v ∈ BR (0) and (9.6.45) yields e

r − S(v) ∂s v p ≤ CR2 . L (I,Lq (Ω))

(9.6.45)

(9.6.46)

Further, by Sobolev imbedding,

|vDv|q dy ≤ vq0,q Dvq0,∞ ≤ Cvq0,q vq2,q (D ∈ {∂j }3j=1 ) Ω

and so with the help of the imbedding W 1,p (I, Lq (Ω)) ֒→ L∞ (I, Lq (Ω)), we get

T

0

Ω

|vDv|q dy

p/q

ds ≤ C

T 0

vp0,q vp2,q ds

≤ CvpL∞ (I,Lq (Ω)) vpLp (I,W 2,q (Ω)) ≤

Cv2p W p,q (0,T )

(9.6.47)

≤ CR2p .

By (9.4.31) and (9.6.47) we find S(v)(v · ∇y )vLp (I,Lq (Ω)) ≤ CR2 .

(9.6.48)

Finally, with the help of (9.4.31) and (9.4.37) we estimate ∇y p(S(v)) as follows: ∇y p(S(v))Lp (I,Lq (Ω)) ′

−1

≤ C(T ) sup{p (η); C(T )

≤ η ≤ C(T )} ∇y r0 0,q + 1 .

(9.6.49)

Putting together (9.6.46), (9.6.48), (9.6.49), using the continuity of L−1 from V0p,q × Lp (I, (Lq (Ω))3 ) into W p,q (0, T ), and assuming ν sufficiently large and δ0 , R sufficiently small we get G(v)W p,q (0,T ) ≤ C(δ0 + R2 + ν −2 ) ≤ R. We are going to use the Tikhonov fixed point theorem (see Section 1.4.11.6). Put in the theorem B := W p,q (0, T ), F := G, K := BR (0). To verify the assumptions of the theorem we need only to show that G is weakly continuous

472


from W p,q (0, T ) into itself. To this purpose, let v n → v weakly in W p,q (0, T ) and rn := S(v n ). Then by Lemma 9.3, {rn } is bounded in L∞ (I, W 1,q (Ω)). Further ∂s rn = −(v n · ∇y )rn − rn div v n

(9.6.50)

and we have ∇y rn bounded in L∞ (I, (Lq (Ω))3 ); (v n , rn ) bounded in (L∞ (QT ))4 and div v n bounded in Lp (I, W 1,q (Ω)). So the result is that ∂s rn is bounded in Lp (I, Lq (Ω)). In summary, rn is bounded in W 1,p (I, Lq (Ω)) ∩ L∞ (I, W 1,q (Ω)), and consequently weakly-∗ compact in that space. Passing to the limit we obtain rnk → r weakly(-∗) in W 1,p (I, Lq (Ω)) ∩ L∞ (I, W 1,q (Ω)), and strongly in C(QT ), and at least the same holds for v n . So it easily follows that ∂s r + div (rv) = 0. Since r is uniquely determined, we have rn → r in the respective spaces. So we have p′ (S(v n ))∇y S(v n ) → p′ (S(v))∇y S(v) weakly in Lp (I, (Lq (Ω))3 ) which implies L−1 0, ∇y p(S(v n )) → L−1 0, ∇y p(S(v)) weakly in W p,q (0, T ),

since the strong continuity of L−1 from Lp (I, (Lq (Ω))3 ) into W p,q (0, T ) implies also weak continuity in these spaces. Further, since ∂s v n → ∂s v weakly in Lp (I, (Lq (Ω))3 ) and rn → r in C(QT ), we have (re −rn )∂s v n → (re −r)∂s v weakly in Lp (I, (Lq (Ω))3 ) and consequently weakly in W p,q (0, T ). L−1 0, (re − rn )∂s v n → L−1 0, (re − r)∂s v Finally, since ∇y v n → ∇y v weakly in

W 1,p (I, (W −1,q (Ω))3 ) ∩ Lp (I, (W 1,q (Ω))3 ) ֒→ C(I, (Lq (Ω))3 ),

(the last imbedding being a consequence of Theorems 1.72, 1.73), and v n → v strongly in L∞ (I, (Lq (Ω))3 ), we have also rn (v n · ∇y )v n → r(v · ∇y )v weakly in Lp (I, (Lq (Ω))3 ) v n · (v n · ∇y )rn → v · (v · ∇y )r weakly in Lp (I, (Lq (Ω))3 ). So in conclusion G(v n ) → G(v) weakly in W p,q (0, T ). By Theorem 1.4.11.6 there exists a fixed point v = G(v) ∈ BR (0) ⊂ W p,q (0, T )

(9.6.51)

of the mapping G. More precisely, we have proved that for any T > 0 there are δ0 and R > 0 such that (9.6.51) holds true. The above fixed point v produces a solution (u, ρ) of problem (9.1.1)–(9.1.4) local in time through the relations (9.2.10) with ν sufficiently large (see above).

UNIQUENESS

9.7

473

Uniqueness

Now we shall prove uniqueness of the solution found in the previous section. So assume we have v 1 , v 2 satisfying (9.6.51) for some T > 0, r := S(v 1 )−S(v 2 ), v := v 1 − v 2 . Then we have ∂s r + (v 1 · ∇y )r + (v 1 − v 2 ) · ∇y S(v 2 ) + rdiv y v 1 + S(v 2 )div y (v 1 − v 2 ) = 0, r(0) = 0. (9.7.52) Multiplying (9.7.52) by r, and integrating over Ω with the help of Green’s theorem we get

1 d 1 2 2 2 1 2 ds r0,2 − 2 Ω |r| div y v dy + Ω (v · ∇y )S(v ) · r dy

+ Ω |r|2 div y v 1 dy + Ω rS(v 2 )div y v dy = 0. This yields

d 2 ds r0,2

≤ div y v 1 0,∞ r20,2 + ε∇v20,2 + C(ε)∇y S(v 2 )r20,6/5 +ε∇y v20,2 + C(ε)S(v 2 )20,∞ r20,2

(9.7.53)

≤ η(ε, s)r20,2 + 2ε∇y v20,2 ,

where η(ε, s) = div y v 1 (s)0,∞ + C(ε) ∇y S(v 2 (s))20,3 + S(v 2 (s))0,∞ . Now v j satisfy S(v j )∂s v j − µ∆y v j − (µ + λ)∇y div y v j = −S(v j )(v j · ∇y )v j − ∇y p(S(v j )) v j (0) = v 0 , j = 1, 2. Subtracting these equations we obtain

S(v 1 )∂s v 1 − S(v 2 )∂s v 2 − µ∆y (v 1 − v 2 ) − (µ + λ)∇y div y (v 1 − v 2 ) 1

1

1

2

2

2

1

(9.7.54) 2

= −S(v )(v · ∇y )v + S(v )(v · ∇y )v − ∇y p(S(v )) + ∇y p(S(v )).

System (9.7.54) can be written in the form (9.7.55) S(v 1 )∂s v −µ∆y v − (µ + λ)∇y div y v

2 2 1 1 1 2 = −r∂s v − S(v )(v · ∇y )v − S(v ) − S(v ) (v · ∇)v 1 −S(v 2 )(v 2 · ∇y )v − ∇y p(S(v 1 )) + ∇y p(S(v 2 )).

Multiply (9.7.55) by v and integrate over Ω to obtain

1 d 1 2 2 2 dy 2 ds Ω S(v )|v| dy + Ω µ|∇y v| + (µ + λ)|div y v|

= Ω S(v 1 )(v 1 · ∇y )v · v − r∂s v 2 · v − S(v 1 )(v · ∇y )v 1 · v

−r(v 2 · ∇y )v 1 · v − S(v 2 )(v 2 · ∇y )v · v + p(S(v 1 )) − p(S(v 2 )) div y v dy

474


≤ ε∇y v20,2 + C(ε)S(v 1 )20,∞ v 1 20,∞ v20,2

+ε∇y v20,2 + C(ε)∂s v 2 20,3 r20,2

+S(v 1 )0,∞ ∇y v 1 0,∞ v20,2 + v 2 0,∞ ∇y v 1 0,∞ r20,2

+v20,2 + ε∇y v20,2 + C(ε)S(v 2 )20,∞ v 2 20,∞ v20,2

with

(9.7.56)

+ε∇y v20,2 + C(ε) sup{p′ (η); C(T )−1 ≤ η ≤ C(T )}r20,2

≤ 4ε∇y v20,2 + ω(ε, s) r20,2 + v20,2 ω(ε, s) = C(ε) S(v 1 (s))20,∞ v 1 (s)20,∞ + ∂s v 2 (s)20,3

+S(v 1 (s))0,∞ ∇y v 1 (s)0,∞ + v 2 (s)0,∞ ∇y v 1 (s)0,∞

+S(v 2 (s))20,∞ v 2 (s)20,∞ + 1 .

Notice that here we used the assumption q > 3. Summing up (9.7.56) and (9.7.53) µ and taking ε = 12 we obtain

d 1 2 2 dy + µ Ω |∇v|2 dy ds Ω S(v )|v| + |r|

≤ 2 η(ε, s) + ω(ε, s) v(s)20,2 + r(s)20,2 (9.7.57)

≤ ζ(µ, T, s) Ω S(v 1 )|v(s)|2 + |r(s)|2 dy. It is a routine matter to establish the integrability with respect to s of the function ζ on the interval (0, T ). This is a consequence of the regularity of v 1 , v 2 ∈ W p,q (0, T ) and the estimates which then follow for S(v j ), j = 1, 2. The condition p ≥ 2 has also to be used. Using the Gronwall lemma in (9.7.57) we obtain

S(v 1 (s))|v(s)|2 + |r(s)|2 dy = 0 for s ∈ (0, T ) Ω

and consequently

1

2

1

v ≡ 0, 2

r≡0

or equivalently v ≡ v , S(v ) ≡ S(v ). The proof of uniqueness is finished. 9.8

Global a priori estimate

In this part of the proof of Theorem 9.1 we derive a global estimate which will allow us to keep the solution in a small ball globally in time. To start with, for a given R and δ0 let T = T (R) be the maximal T such that there is a solution of the equation v = G(v) in BR (0). Further, put σ := ∇y ln r,

(9.8.58)

multiply (9.2.13)1 by σ|σ|q−2 and integrate over Ω. Taking into account that from (9.2.13)2

GLOBAL A PRIORI ESTIMATE

−∇y (div y v) = ∂s σ + ∇y (v · σ),

475

(9.8.59)

in the sense of distributions, we obtain (µ+λ) d q q ds σ(s)0,q

+ ν −2

rp′ (r)|σ|q dy

(9.8.60) = µ Ω ∆y v · σ|σ|q−2 dy − Ω r∂s v · σ|σ|q−2 dy

− Ω r(v · ∇y )v · σ|σ|q−2 dy − (µ + λ) Ω ∇y (v · σ) · σ|σ|q−2 dy. Ω

In fact, to obtain (9.8.60) rigorously, we must proceed as follows: Denote σ ε := Sε σ, where Sε is the usual mollifier in IR3 and σ extended by zero outside of Ω. If we extend also v by zero onto IR3 then by Lemma 7.49, (9.8.59) holds in the sense of distributions on IR3 ×(0, T ). Applying Sε to (9.8.59) we get ∂s σ ε + ∇(v · σ ε ) = Rε − Sε ∇div v, where Rε = ∇(v · σ ε ) − Sε ∇(v · σ)

= v · ∇σ ε − Sε (v · ∇σ) + σ ε ∇v − Sε (σ · ∇v) =: Rε1 + Rε2 .

Since by Lemma 9.3, σ ∈ L∞ (I, (Lq (Ω))3 ) and v ∈ Lp (I, (W 1,∞ (Ω))3 ) we obtain by Lemma 6.7, Rε1 → 0 in Lp (I, L∞ (Ω)). Further, (σ ε − σ)∇vL1 (I,Lq (Ω)) ≤ σ − σ ε

p

L p−1 (I,Lq (Ω))

∇vLp (I,L∞ (Ω)) → 0

and Sε (σ·∇v) → σ·∇v in L1 (I, Lq (Ω)) since σ·∇v is in Lp (I, Lq (Ω)) (p ≥ 2). So we have rε2 → 0 in Lp (I, Lq (Ω)). Similarly Sε ∇div v → ∇div v in L1 (I, Lq (Ω)). Now we multiply the perturbed equation by |σ ε |q−2 σ ε and integrate over Ω to obtain

q 1 d q−2 σ ε dx − Ω Rε |σ ε |q−2 σ ε dx q ds σ ε 0,q + Ω ∇(σ ε · v)|σ ε |

+ Ω (Sε − I)(∇div v)|σ ε |q−2 σ ε dx = − Ω ∇div v|σ ε |q−2 σ ε dx.

Then we proceed as follows below with σ ε instead of σ and with the perturbation term

Rε + (Sε − I)∇div v |σ ε |q−2 σ ε dx Ω

up to the moment when the identity is integrated over s, and pass to the limit as ε → 0+ . By the preceding considerations the perturbation term disappears in the limit and it is a routine matter to pass to the limit in the remaining terms. We leave these details to the reader and continue our considerations only with

476


σ despite the fact that ∂s σ is only a distribution. So let us return to (9.8.60). Since q > 3, we have r − re 0,∞ ≤ k∇y r0,q and consequently re − k∇y r0,q ≤ r ≤ re + k∇y r0,q ,

(9.8.61)

and since σ = r−1 ∇y r, also ∇y r0,q ≤ r0,∞ σ0,q ≤ (re + k∇y r0,q )σ0,q . If σ0,q < k −1 , then it follows that ∇y r0,q ≤ re σ0,q (1 − kσ0,q )−1 . From (9.8.61) and (9.8.62) we obtain α(σ0,q ) := re 1 − kσ0,q (1 − kσ0,q )−1 ≤ r ≤ re 1 + kσ0,q (1 − kσ0,q )−1

(9.8.62)

(9.8.63)

=: β(σ0,q ).

Also, given z ∈ C 1 ((0, ∞)), we have

1 z(r) = z(re ) + 0 z ′ (ηr + (1 − η)re ) dη (r − re ) for r close to re and consequently

z(r) − z(re )0,∞ ≤ β(σ0,q ) sup{z ′ (ζ); α(σ0,q ) ≤ ζ ≤ β(σ0,q )}σ0,q .

(9.8.64)

Now, estimate the terms on the right-hand side of (9.8.60) one by one:

∆y v · σ|σ|q−2 dy ≤ ∆y v0,q σq−1 (9.8.65) 0,q Ω

r∂s v · σ|σ|q−2 dy ≤ β(σ0,q )∂s v0,q σq−1 0,q Ω rv · ∇y v · σ|σ|q−2 dy ≤ β(σ0,q )v · ∇y v0,q σq−1 0,q

≤ β(v0,q )v0,q v2,q σq−1 0,q .

(9.8.66)

(9.8.67)

Finally, we have (using the Einstein summation convention)

∇y (v · σ) · σ|σ|q−2 dy = Ω ∂j vk σk σj |σ|q−2 + I, Ω where

3

q−2

2 dy vk ∂j ∂k (ln r)∂j ln r |σ|q−2 dy = 12 Ω vk ∂k j=1 σj |σ|

q−1

dy = 1q Ω vk ∂k |σ|q dy = − 1q Ω |σ|q divy v dy. = Ω vk ∂k |σ| |σ|

I :=

Ω

GLOBAL A PRIORI ESTIMATE

So we have

∇y (v · σ) · σ|σ|q−2 dy ≤ C∇y v0,∞ σq ≤ Cv2,q σq . 0,q 0,q Ω

477

(9.8.68)

Using (9.8.65)–(9.8.68) in (9.8.60) we obtain q µ+λ d q ds σ(s)0,q

+ ν −2

Ω

rp′ (r)|σ|q dy

q−1 ∆y v0,q + β(σ0,q )∂s v0,q ≤ σ0,q

+β(σ0,q )v0,q v2,q + Cv2,q σ0,q

q−1 ∂s v0,q + v2,q v0,q + σ0,q , ≤ γ(σ0,q )σ0,q

(9.8.69)

where γ has similar properties as β, in particular, we choose it nondecreasing. Putting z(r) := rp′ (r) in (9.8.64) we obtain rp′ (r) = re p′ (re ) + p1 (r) with p1 (r)0,∞ ≤ γ(σ0,q )σ0,q , if γ is appropriately corrected. So we can write (9.8.69) in the form q µ+λ d q ds σ(s)0,q

+ ν −2 re p′ (re )σq0,q

q−1 ∂s v0,q + v2,q v0,q + σ0,q + σ20,q . ≤ γ(σ0,q )σ0,q

(9.8.70)

p−q Multiplying (9.8.70) by σ(s)0,q we obtain (again with the correction of γ), p µ+λ d p ds σ(s)0,q

+ ν −2 re p′ (re )σ(s)p0,q

p−1 ≤ γ(R)σ0,q ∂s v0,q + v2,q v0,q + σ0,q + σ20,q .

Integrate (9.8.71) p µ+λ p σ(s)0,q

s 0

dτ and estimate:

+ ν −2 re p′ (re )

s 0

σ(τ )p0,q 1 −

γ(R) r e p′ (r e ) σ(τ )0,q dτ

s

p−1 p

1/p ∂s vp0,q dτ

s

1/p + vL∞ ((0,s);Lq (Ω)) + σL∞ ((0,s),Lq (Ω)) 0 vp2,q dτ ∇y r 0

p−1 0 + γ(R)R s σp dτ p ≤ µ+λ 0,q p r 0 0,q

× 1 + σL∞ ((0,s),Lq (Ω)) + vL∞ ((0,s),Lq (Ω)) ,

≤

p µ+λ p σ(0)0,q

(9.8.71)

+ γ(R)

s 0

σp0,q dτ

0

where s ∈ [s, T (R)) is arbitrary. But given t ∈ (0, T (R)), we can write

(9.8.72)

478


t d v(t)p0,q = v(0)p0,q + 0 dτ v(τ )p0,q dτ

s p−1 = v 0 p0,q + p 0 v(τ )0,q ∂s v(τ )0,q dτ

1/p s

p−1 s 3−q 3−q ≤ δ0p ν q + p 0 vp0,q dτ p ∂s v(τ )p0,q dτ ≤ δ0p ν q + pRp . 0 Consequently

vL∞ ((0,s);Lq (Ω)) ≤ (δ0 + pRp )1/p , Inserting (9.8.73) into (9.8.72), we obtain

s σ(s)p0,q + 0 σ(τ )p0,q 1 − ≤ Cδ0p + CR

s 0

σp0,q dτ

p−1 p

s ∈ (0, T (R)).

γ(R) r e p′ (r e ) σ(τ )0,q dτ

1 + σL∞ ((0,s);Lq (Ω))

with some constant C which, by the Young inequality, yields

s σ(s)p0,q + 0 σ(τ )p0,q p1 − κ(R)σ(τ )0,q dτ

p ≤ Cδ0p + CRp 1 + σL∞ ((0,s);Lq (Ω)) ,

0 ≤ s ≤ s < T (R) κ(R) := reγ(R) p′ (r e )

(9.8.73)

(9.8.74)

with some other constant C. Assume that

−1

−1 −1 . 1 + pκ(R) R < C −1/p pκ(R) 3−q

By (9.2.10), (9.2.12) and (9.1.8), σ(0)0,q < δ0 ν q supΩ ρ0 with δ0 small enough, q > 3 and ν large. It may be proved from (9.8.59) that the function s → σ(s)0,q is a continuous function. Indeed, writing (9.8.59) in the form ∂s σ + ∇y (v · σ) = g := −∇y div y v ∈ Lp ((0, T (R)), Lq (Ω)),

(9.8.75)

we multiply (9.8.75) by |σ|q−2 σ, integrate over Ω and treat the integral with v similarly as we have treated the integral I above. We get (using the Einstein summation convention) q

q 1 d |σ| g · σ − ∂j vk σj σk |σ|q−2 − 1q div y v|σ|q dy q ds σ0,q = Ω q−1 ≤ g0,q σ0,q + ∇y v0,∞ σq0,q + 1q div y v0,∞ σq0,q q−1 ≤ Cv2,q σ0,q (1 + σ0,q ).

Dividing (9.8.76) by σq−1 0,q we obtain d σ0,q ≤ Cv2,q (1 + σ0,q ). ds By integration of (9.8.77) we find

(9.8.76) (9.8.77)

GLOBAL EXISTENCE

479

s

1 + σ(s)0,q ≤ 1 + σ(0)0,q exp 0 v(τ )2,q dτ 3−2q q

≤ (1 + Cν

δ0 ) exp(RT

p−1 p

(9.8.78)

).

Hence σ ∈ L∞ (0, T (R); Lq (Ω)). Then, from (9.8.77), it may be easily seen that d p ds σ0,q ∈ L (0, T (R)). So by the Sobolev imbedding σ0,q ∈ C(0, T (R)). Having the continuity of s → σ(s)0,q we can assume that σ(s)0,q <

−1 in some maximal interval (0, smax ) ⊂ (0, T (R)). If smax < T (R), then pκ(R)

−1 and by (9.8.74), σ(smax )0,q = pκ(R)

−1

−1

−1 < pκ(R) pκ(R) = σ(smax )0,q ≤ C 1/p R 1 + pκ(R)

which is a contradiction. Hence smax = T (R) and σ(s)0,q ≤ KR,

s ∈ [0, T (R))

(9.8.79)

with a constant K. So by (9.8.74) max{σL∞ ((0,s);Lq (Ω)) , σLp ((0,s);Lq (Ω)) } ≤ CR

for all s ∈ (0, T (R))

with a constant C > 0. 9.9

Global existence

Define for a given R > 0 small enough in the above sense Tmax := sup{T > 0; there exists v = G(v) ∈ W p,q (0, T ) satisfying vW p,q (0,T ) < R}.

(9.9.80)

Let t ∈ (0, Tmax ). By (9.5.39), (9.6.42) and Theorem 9.2 we have vW p,q (0,T ) = G(v)W p,q (0,T ) ≤ C(p, q) v 0 V0p,q + (re − S(v))∂s vLp (I,Lq (Ω))

(9.9.81)

+S(v)(v · ∇y )vLp (I,Lq (Ω)) + ν −2 ∇y p(S(v))Lp (I,Lq (Ω)) .

From (9.8.63), (9.8.58) and (9.8.79) we find (re − S(v))∂s vLp (I,Lq (Ω)) e

(9.9.82) −1

≤ r σL∞ (I,Lq (Ω)) (1 − kσL∞ (I,Lq (Ω)) ) e

2

−1

≤ r KR (1 − kKR)

vW p,q (0,T )

with a constant K. Similarly as in (9.6.47) and with the help of (9.8.63) we get

S(v)(v · ∇y )vLp (I,Lq (Ω)) ≤ Cre 1 + kσ0,q (1 − kσ0,q )−1 v2W p,q (0,T )

≤ C 1 + kKR(1 − kKR)−1 R2 ≤ CR2 . (9.9.83)

480


Finally, analogously as in (9.6.48), we obtain ν −2 ∇y p(S(v))Lp (I,Lq (Ω)) ≤ ν −2 sup{ηp′ (η); α(KR) ≤ η ≤ β(KR)}. (9.9.84) Inserting (9.9.82)–(9.9.84) into (9.9.81) we find vW p,q (0,T ) ≤ C(p, q, R)(R2 + ν −2 R),

T ∈ (0, Tmax ),

(9.9.85)

where C is a nondecreasing locally bounded function of R. Choose R and ν so that C(p, q, R)(R2 + ν −2 R) < R and assume that Tmax < ∞. Let Tn ↑ Tmax be arbitrary. Then necessarily vW p,q (0,Tn ) ↑ R as n → ∞. Indeed, if this was not the case, then sup vW p,q (0,Tn ) < R n

and the function v, defined on (0, Tmax ), would be a solution to the equation v = G(v) on (0, Tmax ) with v ∈ W p,q (0, Tmax ). Taking v 0 := v(Tmax ) as a new initial condition, from Theorem 9.2 we get v(Tmax ) ∈ V0p,q . But then by Section 9.6 there exists T > Tmax such that the solution of v = G(v) exists on (Tmax , T ) as well. Connecting these two solutions we obtain a solution on the interval (0, T ) which is a contradiction to the maximality of Tmax . So it cannot be Tmax < ∞, but Tmax = ∞, i.e. v and r = S(v) are defined on Ω × (0, ∞). The proof of Theorem 9.1 is finished. 2 9.10

Bibliographical remarks

In Sections 9.1–9.9 solutions of the barotropic equations (9.1.1)–(9.1.4) have been studied. Related results can be found in (Matsumura and Yamagata, 2001). The difference is that a large potential force on the right-hand side of the momentum equation is allowed, Ω = IRN , N = 2, 3, and the solution is weaker in the sense that 1 N 3 ∞ N ∞ ρ ∈ L∞ loc ((0, ∞); L (IR )), u ∈ Lloc ((0, ∞); (H (IR )) ),

and equations (9.1.1)–(9.1.4) hold in D′ (IRN × (0, ∞)). The main assumptions are that ρ0 − ρe 0,∞ + ρ0 − ρe 0,2 + u1,2 is small enough,

where ρe is an equilibrium density corresponding to the given potential force and the exponent γ of the γ-law for the pressure is close to 1. Comparable results are given in (Salvi and Straˇskraba, 1993), (Mucha and Zaj¸aczkowski, 2002) and (Choe and Kim, 2003). The physically important case of possibly discontinuous initial data is discussed in (Hoff, 1995a), (Hoff, 1995b).

BIBLIOGRAPHICAL REMARKS

481

Regularity of a weak solution on a short interval of time is proved in (Desjardins, 1997) for the problem with periodic boundary conditions and state equations including the polytropic gas equation p(ρ) = κργ . As for the existence of global regular (strong) solutions in the general case, there is a number of results concerning the equations for viscous compressible and heat-conductive fluids. As an example we introduce the following result from (Matsumura and Padula, 1992). We consider equations (1.2.77)–(1.2.80) in the form ∂t ρ + div (ρu) = 0,

∂t ρ( 12 |u|2

∂t (ρu) + div (ρu ⊗ u) + T = ρf ,

+ e) + div ρu( 12 |u|2 + e − k∇θ + T · u = ρf · u,

(9.10.86)

T = pI − λ(div u)I, ρ(x, t) ≥ 0, θ ≥ 0, x ∈ Ω ⊂ IR3 , t > 0

and the initial and boundary data ρ|t=0 = ρ0 , u|t=0 = u0 , θ|t=0 = θ0 , u|∂Ω = 0, θ|∂Ω = θ.

(9.10.87)

(I) Existence of regular flow near the equilibrium state We make the following assumptions: (i) 3λ + 2µ ≥ 0, µ > 0, k > 0 constants; (ii) there exists a smooth entropy S(ρ, θ) such that de = θdS − pd ρ1 , e = e(ρ, θ), p = p(ρ, θ);

(iii) e = cv θ, where cv > 0 is the specific heat at constant volume; (iv) p = p(ρ, θ), p > 0, pρ > 0 for ρ > 0, θ > 0.

Theorem 9.4 Let f = −∇F , F ∈ H 4 (Ω), (ρ0 , u0 , θ0 ) ∈ H 3 (Ω)5 , u0 |∂Ω = 0, θ0 |∂Ω = θ, ∂t u(0)|∂Ω = 0, θt (0)|∂Ω = 0, where ∂t u(x, 0) and ∂t θ(x, 0) are computed from equations (9.10.86) in which we put ρ = ρ0 , θ = θ0 , u = u0 except for the time derivatives. In addition we assume that there is an equilibrium state (0, ρ, θ) (i.e. a time-independent solution of (9.10.86) with u = 0). 4

Then there exist constants ε, β and C0 = C0 (ρ, θ, F H (Ω) ) (where ρ := 1 |Ω| Ω ρ0 dx) such that if (ρ0 − ρ, u0 , θ0 − θ)H 3 (Ω)5 ≤ ε0 ,

then the initial boundary–value problem (9.10.86), (9.10.87) has a unique global solution (ρ, u, θ) in time satisfying

(ρ, u, θ) ∈ C 0 ∩ L∞ ([0, ∞); (H 3 (Ω))5 ), inf ρ(x, t) > 0, inf θ(x, t) > 0, x,t

x,t

and

sup{|ρ(x, t) − ρ| + |u(x, t)| + |θ(x, t) − θ|} ≤ C0 e−βt . x

482


A related result is proved in (Jiang, 1998), where the smallness of the data is measured in different and weaker norms. The result on global existence of the strong solution near equilibrium is proved in (Tanaka and Tani, 2003) for the same system with a free boundary condition. Further references are given in (Valli, 1992) and (Desjardins and Chi-Kun, 1999). Inflow and outflow is taken into account in (Valli and Zaj¸aczkowski, 1986). The Cauchy problem with possibly discontinuous initial data is considered in (Hoff, 1997). Critical spaces of Besov type for optimal global well-posedness of the Cauchy problem for system (9.10.86) in IRN are found in (Danchin, 2001). (II) Global existence of smooth solutions for large data Assume N = 2 and put u := (u, v). In (Vaigant and Kazhikhov, 1995) the following problem is considered: ∂t ρ + ∂x (ρu) + ∂y (ρv) = 0, ρ(∂t u + u∂x u + v∂y u) = ∂x (2µ∂x u) + ∂y (µ(∂y u + ∂x v)) + ∂x (λ(∂x u + ∂y u)) − ∂x p, ρ(∂t v + u∂x v + v∂y v)

= ∂x µ(∂y u + ∂x v) + ∂y (2µ∂y v) + ∂y λ(∂x u + ∂y v) − ∂y p,

(9.10.88)

x ∈ Ω := (0, 1) × (0, 1), t ∈ (0, T ),

µ = const = 1, λ = λ(ρ) = ρβ , β = const ≥ 3, p = p(ρ) = ργ , 0 ≤ γ < ∞, u(x, 0) = u0 (x), v(x, 0) = v0 (x), ρ(x, 0) = ρ0 (x) ≥ ρ = const > 0, u, v

periodic with period 1

or (u · n)|∂Ω = 0,

rot (u, v) ≡ (∂y u − ∂x v)|∂Ω = 0.

Theorem 9.5 (Existence of a generalized solution) If

ρ0 ∈ L∞ (Ω), u0 ∈ (W 1,2 (Ω))2 , T > 0, then problem (9.10.88) has at least one generalized solution in the sense that the momentum equation is satisfied a.e. in Q := Ω × (0, ∞) and the continuity equation in the sense of distributions, i.e. for any T > 0,

T

ρ(∂t ϕ + (u · ∇)ϕ)) dx dy dt + Ω (ρ0 ϕ)(x, 0) dx dy = 0 0 Ω for any ϕ ∈ C ∞ (Q), 1-periodic in (x, y) and such that ϕ(x, y, T ) ≡ 0. In addition we have ρ ∈ Lq (Q) ∀ q ∈ [1, ∞), u ∈ L2 (I, (W 1,2 (Ω))2 ), A := rot(u, v) = ∂y u − ∂x v ∈ L2 (I, W 1,r (Ω)) ∀ r ∈ [1, 2),

B := (2µ + λ(ρ))(ux + vy ) − p(ρ) ∈ Lq (Q) ∩ L2 (I, W 1,s (Ω)) ∀ q ∈ [1, ∞), s ∈ [1, 2).

BIBLIOGRAPHICAL REMARKS

483

Theorem 9.6 (Existence and uniqueness of strong solution) If ρ0 ∈ W 1,q (Ω), u0 ∈ W 2,q (Ω)2 with some q > 2, then there is a unique strong solution of the problem, i.e. all equations (9.10.88) are satisfied a.e. in Q and ρ ∈ L∞ (Q), ∂t ρ, ∇ρ ∈ (W 1,q (Q))2 , u ∈ (Lq (Q))2 ∩ Lq (I, (W 2,q (Ω))2 ). Theorem 9.7 (Regularity of the strong solution.) If ρ0 ∈ C 1+α (Ω), u0 ∈ (C 2+α (Ω))2 for some 0 < α < 1, then the strong solution (ρ, u) is classical: α

α

ρ ∈ C 1+α,1+ 2 (Q), u ∈ (C 2+α,1+ 2 (Q))2 . (III) Blow up results In (Vaigant, 1994) a counterexample with spherical symmetry is exhibited: For γ ∈ [1, NN−1 ) and all q > N, initial data (ρ0 , u0 ) and external forces f are found such that the unique local solution of (8.1.1)–(8.1.6) blows up in the L∞ -norm. Another result in this direction has been proved in (Xin, 1998): Theorem 9.8 For full Navier–Stokes system (9.10.86) without heat conduction (k = 0) and with p = Rρθ, e = cθ, p = AeS/c ργ , where A > 0 is a constant, γ > 1, S is the entropy, and c = R/(γ − 1), 1 m N there is no solution N in C ([0, ∞); H (IR )) if ρ0 has compact support and ρ0 ∈ m N H (IR ), m > 2 + 2.


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INDEX (bk )′+ , 162 1M , 2 A∗ , 69 BV (Ω), 53 BV −space, 53 BVloc (Ω), 53 Br , B r , 1 C(I, X∗−weak ), 57 C(I, Xweak ), 57 C(M ), C 0 (M ), 2 C([a, b], X), 54 C k (I, X), 57 C k (M ), 2 C k,µ (Ω), 6 CB (Ω), 52 Cbk (Ω), 2 Dα , 33 ′ D−1,q (Ω), 49 1,q D (Ω), 47 D01,q (Ω), 47 Dt , 16 E p (Ω), 164 E0p (Ω), 165 E q,p (Ω), 164 E0q,p (Ω), 165 H k (Ω), 41 H0k (Ω), 41 I (identity), 71 Lp ((a, b), X), 56 Lp (M ), 10 Lp (∂Ω), 10 Lpq (Ω), 395 Lploc (M ), 10 LΦ (Ω), 34 M (Ω), 52 Sε , 38 T VΩ , 53 W k,p ((a, b), X), 57 W k,p (Ω), 40, 42 W k,p (∂Ω), 46 WSk,p (∂Ω), 46

IR, IR+ , IRN , 1 a : b, 3 ab, 4 a · b, 3 a ⊗ b, 4 a × b, 4 Ω γ0 , 43 M , 10 ·, ·, 55 ∇, 3 ω-limit set, 462 ⊗, 4 Ω, 2 p, 45 Lp , 10 Lp (M ), 10 W k,p , 41 ∂Ω, 2 σ-algebra, 7 p, 45 | · |−1,q′ , 49 | · |1,q , 47 || · ||0,p,M , | · |0,p , 10 || · ||k,p,Ω , || · ||k,p , 41, 42, 46 p L weak (Ω), 37 {ρ > k}, 2 a∗ , 51 k-Riemann invariant, 119 p-system, 77 p′ , 45 p∗ , 45 p∗ , 45 D′ (Ω), 35 D(A), 68 D(Ω), 35 F (f ), 50 L(X), 69 L(X, Y ), 69 N (A), 68 R(A), 68 S ′ (IRN ), 50 S(IRN ), 50 W01,∞ (Ω), 213

W0k,p (Ω), 41 k,p Wloc (Ω), 42 X ֒→֒→ Y , 71 X ֒→ Y , 71 X ∗ , 55 ∆, 4 ∆2 condition, 34 N, 1

a.a., 10 a.e., 10 absolute temperature, 23 absolutely continuous function, 12 acceleration, 16

499

500 adiabatic exponent, 406 flow, 27 inviscid flow, 27 adjoint operator, 69 admissible BV-solution, 116 process, 23 weak solution, 115 almost all, 10 almost everywhere, 10 amplitude of oscillations, 401 approximations bounded domains, 200 exterior domains, 245 Galerkin, 353 artificial pressure, 200 attractor global, 459, 460 global of short trajectories, 459 weak, 461 ball, 62, 63 ball in IRN , 1 Banach –Alaoglu theorem, 68 fixed point theorem, 72 space, 64 barotropic process, 29 bijection, 2 blow up of solution, 149, 150, 483 Bochner integrable function, 55 integral, 55 space, 56 theorem, 55 body force, 20 Bogovskii operator, 169 Borel measure, 7 support of, 8 set, 7 boundary condition, 31 admissibly prolongable, 413 Dirichlet, 31 essential, 31 no-stick, 32, 461 slip, 32 cone property, 59 description of, 8 free, 32 Lipschitz continuous, 8 measure on, 8 of a set, 2, 61 smooth, 8

INDEX bounded energy weak solution, domain, 1 energy weak solution, energy weak solution, energy weak solution, linear operator, 69 set, 63 Brouwer theorem, 72 Burgers equation, 75

192 414 steady, 192 unsteady, 316

canonical mapping, 67 Cauchy –Riemann conditions, 101 –Schwartz inequality, 11 inequality, 11, 64 problem, 80 sequence, 64 characteristic, 80 field, 149 function, 2 method of ∼s, 80, 81 speed, 126 classical solution, 80 Clausius–Duhem inequality, 23, 25 closable linear operator, 69 closed subspace, 64 linear operator, 69 set, 61 closure, 63 of a set, 2, 61 sequentially strong, 63 strong, 63 commutator lemma Feireisl, 207 Friedrichs, 155, 304 compact nonlinear operator, 72 imbedding, 44, 71 operator, 70 set, 63 compactness argument, 114, 130 compensated, 154 unsteady, 369, 382 compactness of weak solutions, 194 compatibility, 32 complementary Riemann invariant, 85, 119 complete space, 64 complete system steady approximations, 218

INDEX completely continuous imbedding, 71 linear operator, 70 mapping, 72 nonlinear operator, 72 composed functions, 42 composition of functions, 2 compressible Euler equations, 28 cone, 179 cone property, 59 conjugate number, 45 connected set, 1, 63 conservation laws, 18, 75 of energy, 22 of mass, 19, 309, 320 of moment of momentum, 21 of momentum, 19 continuity equation, 19 dissipation, 212 prolongation, 306 prolongation to IRN , 158 relaxation, 202 relaxed with dissipation, 203 renormalized, 314 renormalized solutions, 159, 307 continuous imbedding, 44, 71 linear functional, 64 linear operator, 69 mapping, 61, 63 operator, 69 contractive mapping, 72 control volume, 19 convective derivative, 16 form of equations, 24 convex entropy, 111 entropy condition, 111 entropy solution, 111 entropy–entropy flux pair, 111 function, 2 functional, 184 set, 60 strictly, 2 convolution, 36 Courant–Friedrichs–Loewy condition, 146 curl, 3 cut-off function, 48 cylindrical exit, 179 dense set, 64 densely defined operator, 69 density, 23 comparison, 438

501 equilibrium, 431 oscillations, 405 derivative almost everywhere, 42 along trajectory, 17 convective, 16 distributional, 36 fractional, 45 generalized, 36 local, 16 material, 16, 17 of a distribution, 36 of abstract function, 56 strong, 55 total, 16 diagonally hyperbolic system, 79 diagonally strictly hyperbolic system, 79 diam (V), 11 diffeomorphism, 14 Dirac distribution, 36 Dirichlet boundary condition, 31 problem, 287 dissipation, 25 form of energy equation, 25 dist, 62 distance, 62 distributional derivative, 36 distributions, 35 div, 3, 4 div–curl lemma, 134, 206 divergence equation, 165 bounded domains, 166 domain with superconical exits, 181 exterior domains, 177 in cone, 180 in supercone, 180 invading domains, 175 nonhomogenous boundary conditions, 174 domain, 1 bounded, 1 conical exits, 179 exterior, 1 nonsteady, 393 invading growing, 398 nonsteady, 393 Lipschitz, 8 noncompact boundary, 275 of definition of function, 2 of operator, 68 subconical exits, 179 superconical exits, 179 unbounded, 1

502 dual imbedding, 71 of Lp , 10 of homogeneous Sobolev space, 49 of Sobolev space, 42 space, 65 duality, 65 duals to Sobolev spaces, 42 effective pressure, 196, 326 viscous flux, 196, 208 effective viscous flux, 285, 326, 338 invading domains, 255 vanishing artificial pressure, 241 vanishing dissipation, 225 vanishing relaxation, 233 Egoroff’s theorem, 15 eigenvalue, 80 energy equation, 22 internal, 22 kinetic, 22 potential, 22 total, 22 entropy, 111 –entropy flux pair, 111 -entropy flux pair, 126 condition, 110, 111 flux, 111 form of energy equation, 25 pair, 107 solution, 111 specific, 23 equation of continuity, 19 of energy, 22 of motion, 21 of state, 23 equilibrium state, 271, 452, 462, 466 ess sup, 10 essential boundary condition, 31 estimates invading domains, 247 of density, 195 of velocity, 195 vanishing artificial pressure, 236 vanishing dissipation, 222 Eulerian coordinates, 16 Eulerian description, 16 evolution operator, 458 existence bounded domain, 192 exterior domain, 193 noncompact boundaries, 264 extension operators, 47

INDEX exterior domains 2D, 274 nonzero velocity at infinity, 274 steady, 274 zero density at infinity, 275 Fatou’s lemma, 14 finite energy weak solution, 316, 413 measure, 133 fixed point, 71 flux, 75 effective viscous, 400, 419 of heat, 27 Fourier law, 26 multiplier, 51 transform, 50 transform, inverse, 50 free boundary, 32 friction shear forces, 21 Friedrichs inequality, 44 Friedrichs’ commutators lemma, 155, 304 front tracking approximations, 146 Fubini’s theorem, 13 function of bounded variation, 53 of locally bounded variation, 52 with values in Banach space, 53 functional convex, 184 lower semicontinuous, 184 fundamental sequence, 64 fundamental solution Laplace, 292 Oseen, 297 Stokes, 292 generalized derivative, 36 eigenvalues, 79 functions, 35 genuinely nonlinear eigenvector, 119 graph, 68 Green’s theorem, 9 Gronwall’s lemma, 5 H¨ older continuous function, 6 H¨ older space, 33 H¨ older’s inequality, 3, 11, 34 Hahn–Banach theorem, 65 Hardy-type imbedding, 45 heat conduction, 27, 32 heat flux, 22, 27 heat sources, 22

INDEX Helly’s compactness theorem, 53 Helmholtz decomposition, 283, 286, 439 Hilbert space, 64 homoentropic flow, 27 homogeneous Sobolev spaces, 47 homotopy, 72 hyperbolic system, 79 identity operator, 71 imbedding theorems, 44 impermeable wall, 31 inflow, 31 initial condition, 31, 80 data, 31 inlet, 31 inner force, 20 inner product, 64 integrals dependent on parameter, 13 integration by parts, 57 interior point, 61 internal energy, 22 interpolation of Lebesgue spaces, 46 interpolation of Sobolev spaces, 46 inverse Fourier transform, 50 inviscid fluid, 21 inviscid model, 31 isentropic flow, 27 isometric mapping, 67 isometrically isomorphic spaces, 11 isomorphism, 67 isothermal flow, 28 steady, 269 unsteady, 407 Jacobi matrix, 4 Jensen’s inequality, 34 Jensen’s integral inequality, 34 jump condition, 109 kernel, 68 kernel of Calder´ on–Zygmund type, 40 kinematic condition, 32 kinetic energy, 22 Kondrashov’s imbedding theorems, 44 Lagrange’s formula, 41 Lagrangian coordinates, 16 description, 16 mass coordinates, 76 Lam´ e system existence, 217 Laplace operator, 4 Laplacian, 4 Lax shock entropy condition, 111

503 Lax–Friedrichs approximations, 128 Lax–Milgram lemma, 70 Lebesgue dominated convergence theorem, 15 measure, 8 point, 12 spaces, 10 Leray–Schauder fixed point relaxed continuity equation, 215 vanishing dissipation, 221 Leray–Schauder theorem, 73 Levi–Civita tensor, 4 lifespan of a solution, 148 linear operator, 68 linear space, 60 linearly degenerate eigenvector, 119 Lipschitz domain, 8 Lipschitz-continuous boundary, 8 Lipschitz-continuous function, 6 local derivative, 16 locally integrable function, 36 locally uniform convergence, 35 lower semicontinuous functional, 184 lower semicontinuous mapping, 62 magnitude, 4 mapping canonical, 67 completely continuous, 72 continuous, 63 contractive, 72 duality, 65 isometric, 67 lower semicontinuous, 62 measurable, 7 sequentially lower semicontinuous, 62 weakly continuous, 72 material derivative, 16 Mazur’s theorem, 66 measurable mapping, 7 measurable sets, 7 measurable space, 7 measure, 7 measure on the boundary, 8 method of artificial viscosity, 112 method of characteristics, 80, 81 method of decomposition, 283 metric, 62, 63 metric linear space, 62 Milman–Pettis theorem, 67 Minkowski inequality, 3 Minti’s trick, 188 mixture of fluids, 278 mollifier, 38, 42, 303 moment of momentum, 21 monotone convergence lemma, 15

504

INDEX

monotone operator, 186 Morse–Sard theorem, 6 multiindex, 33

outer force, 20 outflow, 31 outlet, 31

negative part of function, 42 neighborhood, 61 Neumann problem, 286 eigenvalues, 212 regularity, 211 Newtonian fluid, 24 no-slip condition, 31 no-stick conditions, 32 nonexistence noncompact boundaries, 264 global strong solution, 483 nonlinear operator equations, 71 nonmonotone pressure steady, 270 unsteady, 412 nonuniqueness steady, 271 nonzero outflow-inflow steady, 270 unsteady, 407, 412 norm, 62 normal trace, 164 normed linear space, 62 null set of measure, 133 null space, 68

partition of unity, 7 perfect gas, 26 perfect viscous gas, 28 periodic boundary conditions global behavior, 432 in time, 461 steady, 270 unsteady, 407 physical entropy, 112 piecewise smooth function, 109 piecewise smooth weak solution, 109 piston problem, 84 plane flow, 30 Poincar´ e inequality, 44, 46 positive part of function, 42 potential energy, 22 power, 22 precompact set, 63 pressure, 21, 23 artificial, 330 gradient equations, 152

one-dimensional model of flow, 30 open set, 60, 63 operator adjoint, 69 bounded, 69 closable, 69 closed, 69 compact, 70 completely continuous, 70 continuous, 69 densely defined, 69 identity, 71 imbedding, 71 linear, 68 of extension, 47 regularizing, 38 selfadjoint, 70 symmetric, 70 Orlicz norm, 34 Orlicz space, 34, 395 orthonormal matrix, 6 oscillations density, 242 oscillations of density asymptotic behavior, 256 Oseen problem, 288

quasilinear system, 75 Radon measure, 8, 52 range of operator, 68 Rankine–Hugoniot condition, 109 rarefaction wave, 107 reflexive Banach space, 67 regular distribution, 36 regularity of weak solutions steady, 271 unsteady, 481 regularization, 38 regularizing operator, 38 relatively compact set, 63 Rellich’s theorem, 44 renormalized continuity equation invading domains, 255 oscillations, 243 vanishing dissipation, 226 vanishing relaxation, 234 transport inequality, 161 weak solution, 192 weak solution, steady, 192 weak solution, unsteady, 316, 413 representation formulas, 293 rheological equations, 21 Riemann problem, 118, 151 Riesz operator, 206

INDEX Riesz representation theorem, 68 Riesz theorem, 11 scalar product, 3, 64 Schauder theorem, 72 second law of thermodynamics, 28 selfadjoint operator, 70 separable space, 64 sequentially closed set, 61 weakly closed set, 66 shock tube problem, 151 shock wave, 107 simple function, 54 slip boundary conditions steady, 273 unsteady, 408 slip conditions, 32 small perturbations of large potential steady, 270 unsteady, 481 Sobolev –Slobodetskii spaces, 45 imbedding theorems, 44 imbeddings, 48 imbeddings of abstract spaces, 57 inequality, 49 spaces, 40 Sobolev’s inequality, 48 solution periodic, 461 space Orlicz, 395 space of functions of bounded variation, 53 space of functions of locally bounded variation, 53 specific entropy, 23 specific heat at constant volume, 26 spectral theory, 70 speed of propagation of discontinuity, 109 speed of sound, 30 spherically symmetric solution, 150 stabilization, 456 sequential, 432 star-shaped domain, 166 decomposition, 166 state variable, 23 stationary flow, 30 steady flow, 30 Stokes problem, 288 Stokes’ postulates, 24 stress tensor, 20 strictly convex function, 2 strictly hyperbolic system, 79 strong convergence, 63

505 density vanishing relaxation, 236 vanishing dissipation, 230 strong derivative, 55 strong solutions steady, 280 unsteady, 464 strongly measurable function, 54 subcone, 179 subspace, 64 substitution theorem, 14 supercone, 179 support of a Borel measure, 8 support of measure, 133 surface force, 20 integral, 9 symbols o, O, 7 symmetric hyperbolic system, 80, 89 operator, 70 system, 75 symmetrizable system, 77 system of conservation laws, 75 Tartar’s commutation relation, 134 tempered distributions, 50 tensor, 4 tensor product, 4 theorem of Aubin–Lions, 59 of Banach on fixed point, 72 of Banach–Alaoglu, 68 of Bochner, 55 of Brouwer, 72 of Egoroff, 15 of Fubini, 13 of Green, 9 of Hahn–Banach, 65 of Helly, 53 of Kondrashov, 44 of Lax–Milgram, 70 of Lebesgue, 15 of Leray–Schauder, 73 of Mazur, 66 of Milman–Pettis, 67 of Morse–Sard, 6 of Rellich, 44 of Riesz, 11 of Schauder, 72 of Tikhonov, 72 of Tonelli, 13 of Vitali, 15 on monotone convergence, 15 on substitution, 14

506 theorem (cont.) on traces, 43, 46 on transport, 18 Tikhonov theorem, 72 Tonelli’s theorem, 13 topological linear space, 60 topology, 60 total derivative, 16 energy, 22 variation, 53 trace theorem, 46 trajectory asymptotically closed, 458 asymptotically compact, 431 short, 458 transformation of Cartesian coordinates, 6 transonic nozzle flow, 151 transport equation, 404 steady, 284, 289 unsteady, 406 theorem, 18 travelling wave, 82 triangle inequality, 62 two–dimensional model of flow, 30 uniformly convex Banach space, 67 unit outer normal, 9 unit vector, 1 vanishing viscosity method, 153 vector fields with summable divergence, 163 vector product, 4 velocity, 16

INDEX viscous fluid, 21 Vitali’s theorem, 15 volume force, 20 weak -∗ convergence, 66 -∗ topology, 66 solution, 192 compactness steady, 268 unsteady, 405 convergence, 65 entropy inequality, 114 entropy–entropy flux pair, 127 solution, 108 bounded energy, steady, 192 bounded energy, unsteady, 316, 414 finite energy, 316, 413 noncompact boundary, 263 renormalized, steady, 192 renormalized, unsteady, 316, 413 unsteady, 316, 413 solution, steady, 192 topology, 65 weakly closed set, 66 Young function, 33 complementary, 33 inequality, 3 convolutions, 37 measure, 132 zero, 60 zero-pressure Euler equations, 152

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An Introduction to Galois Cohomology and its Applications (London Mathematical Society Lecture Note Series)

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Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models (Oxford Lecture Series in Mathematics and Its Applications, 3)

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