Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1911
C.I.M.E. means Centro Internazionale Matematico Estivo, that is, International Mathematical Summer Center. Conceived in the early fifties, it was born in 1954 and made welcome by the world mathematical community where it remains in good health and spirit. Many mathematicians from all over the world have been involved in a way or another in C.I.M.E.’s activities during the past years. So they already know what the C.I.M.E. is all about. For the benefit of future potential users and cooperators the main purposes and the functioning of the Centre may be summarized as follows: every year, during the summer, Sessions (three or four as a rule) on different themes from pure and applied mathematics are offered by application to mathematicians from all countries. Each session is generally based on three or four main courses (24−30 hours over a period of 6-8 working days) held from specialists of international renown, plus a certain number of seminars. A C.I.M.E. Session, therefore, is neither a Symposium, nor just a School, but maybe a blend of both. The aim is that of bringing to the attention of younger researchers the origins, later developments, and perspectives of some branch of live mathematics. The topics of the courses are generally of international resonance and the participation of the courses cover the expertise of different countries and continents. Such combination, gave an excellent opportunity to young participants to be acquainted with the most advance research in the topics of the courses and the possibility of an interchange with the world famous specialists. The full immersion atmosphere of the courses and the daily exchange among participants are a first building brick in the edifice of international collaboration in mathematical research. C.I.M.E. Director Pietro ZECCA Dipartimento di Energetica “S. Stecco” Università di Firenze Via S. Marta, 3 50139 Florence Italy e-mail:
[email protected] C.I.M.E. Secretary Elvira MASCOLO Dipartimento di Matematica Università di Firenze viale G.B. Morgagni 67/A 50134 Florence Italy e-mail:
[email protected] For more information see CIME’s homepage: http://www.cime.unifi.it CIME’s activity is supported by: – Istituto Nationale di Alta Mathematica “F. Severi” – Ministero dell’Istruzione, dell’Università e delle Ricerca – Ministero degli Affari Esteri, Direzione Generale per la Promozione e la Cooperazione, Ufficio V This CIME course was partially supported by: HyKE a Research Training Network (RTN) financed by the European Union in the 5th Framework Programme “Improving the Human Potential” (1HP). Project Reference: Contract Number: HPRN-CT-2002-00282
Alberto Bressan · Denis Serre Mark Williams · Kevin Zumbrun
Hyperbolic Systems of Balance Laws Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14–21, 2003 Editor: Pierangelo Marcati
ABC
Editor and Authors Alberto Bressan
Kevin Zumbrun
Department of Mathematics McAllister Building, Room 201 Penn State University University Park PA 16802, USA e-mail:
[email protected] Department of Mathematics Rawles Hall 225 Indiana University Bloomington IN 47405, USA e-mail:
[email protected] Denis Serre
Pierangelo Marcati
Unité de mathématiques pures et appliquées Ecole Normale Supérieure de Lyon (UMR 5669 CNRS) 46, allée d’Italie 69364 Lyon Cedex 07, France e-mail:
[email protected] Department of Pure and Applied Mathematics University of L’Aquila Via Vetoio – Loc. Coppito 67100 L’Aquila, Italy e-mail:
[email protected] URL: http://univaq.it/ ∼ marcati
Mark Williams Department of Mathematics CB 3250 University of North Carolina Chapel Hill NC 27599, USA e-mail:
[email protected] Library of Congress Control Number: 2007925049 Mathematics Subject Classification (2000): 35L60, 35L65, 35L67, 35B35, 76L05, 35L50, 65M06
ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-72186-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-72186-4 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-72187-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 ° The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the authors and SPi using a Springer LATEX package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper
SPIN: 12055890
41/SPi
543210
Preface
This volume includes the lecture notes delivered at the CIME Course “Hyperbolic Systems of Balance Laws” held July 14-21, 2003 in Cetraro (Cosenza, Italy). The present volume includes lectures notes by A. Bressan, D. Serre, K. Zumbrun and M. Williams and an appendix by A. Bressan on the center manifold theorem. These are among the “hot topics” in this field and can be of great interest, not only to professional mathematicians, but also for physicists and engineers. The concept of hyperbolic systems of balance laws was introduced by the works of natural philosophers of the eighteenth century, predominantly L. Euler (1755), and has over the past one hundred and fifty years become the natural framework for the study of gas dynamics and, more broadly, of continuum physics. During this period of time great personalities like Stokes, Challis, Riemann, Rankine, Hugoniot, Lord Rayleigh and later Prandtl, Hadamard, H. Lewy, G.I. Taylor and many others wrote several fundamental papers, thus laying the groundwork for the further development of the mathematical theory. However the first part of the past century did not see much activity on the part of mathematicians in this field and it was only during the Second World War, in connection with the Manhattan Project, that associated research received a great impetus. Many important scientists like J. Von Neumann, R. Courant, K.O. Friedrichs, H. Bethe and Ya. Zeldowich became interested in this field and proposed many new key concepts, the influence of which remains very great to the present day. Immediately after the Second World War there was a considerable development in mathematical theory, with key results being obtained by a new generation of great mathematicians like S.K. Godunov, P. Lax, F. John, C. Morawetz and O. Oleinik, who led the field until the mid 1960s, when J. Glimm published an outstanding paper which marked the most important breakthrough in the history of this field. Glimm was able to prove the global existence of general systems in one space dimension, with small BV data. This result introduced a new approach to nonlinear wave interaction, but the
VI
Preface
proof was not fully deterministic. Tai-Ping Liu was later able to remove the probabilistic part of the proof, thus making it completely deterministic. The relation between hyperbolic balance laws and continuum physics is not covered in any of the lectures in the present volume, but was the core topic of a series of lectures delivered in the Cetraro School by C. Dafermos entitled “Conservation Laws on Continuum Mechanics.” In his wonderful monograph, published by Springer-Verlag in the Grundlehren der Mathematischen Wissenschaften, vol. 325, Dafermos provides an extremely thorough account of the most relevant aspects of the theory of hyperbolic conservation laws and systematically develops their ties to classical mechanics. The notes by Alberto Bressan in this volume are intended to provide a self-contained presentation of recent results on hyperbolic conservation laws, based on the vanishing viscosity approach. Glimm’s aforementioned theory was based on the construction of partially smooth approximating solutions with a locally self-similar structure. In order to get a uniform bound in BV norm, interaction potential was a crucial tool, an idea Glimm borrowed from physics. This potential, though a nonlinear functional, displays quadratic behaviour and decreases with time, provided the initial data have a small total variation. In the 1990s Bressan and Tai-Ping Liu, together with various collaborators, completed this theory by proving the continuous dependence on initial data. The relations with the theory of compressible fluids have raised, since the very beginning of the theory, the question whether the inviscid solutions are in practice the same as the solutions with low viscosity. Although this fact had been established for various specific situations, it was only very recently that Bianchini and Bressan discovered a way to prove that, if the total variation of initial data remains sufficiently small, then the solutions of a viscous system of conservation laws converge to the solutions of the inviscid system, as long as the viscosity tends to zero. This approach allows the stability results obtained using the previous theories to be generalized. The results are based on various technical steps, which in the present lecture notes Bressan describes in great detail, making a remarkable effort to make this difficult subject also accessible to non-specialists and young doctoral students. The notes of D. Serre cover the existence and stability of discrete shock profiles, another very exciting topic which, since the 1940s, has greatly interested applied mathematicians, including Von Neumann, Godounov and Lax, who were motivated by the need for efficient numerical codes to approximate the solutions of compressible fluid systems, including situations where shocks are present. It was immediately clear to them that a number of challenging and difficult mathematical problems needed to be solved. Partial differential equations are often approximated by finite difference schemes. The consistency and stability of a given scheme are usually studied through a linearization along elementary solutions such as constants, travelling waves, and shocks.
Preface
VII
The study of the existence and stability of travelling waves faces significant difficulties; for example, existence may fail in rather natural situations because of small divisors problems. Ties to many branches of mathematics, ranging from dynamical systems to arithmetic number theory, prove to be relevant in this field. Serre’s notes greatly emphasize this interdisciplinary aspect. His lecture notes not only provide a very useful and comprehensive introduction to this specific topic, but moreover propose a class of truly interesting and challenging problems in modern spectral theory. The analysis of the vanishing viscosity limit is far from being fully understood in the multidimensional setting. It is, in any case, important to understand the presence of stable and unstable modes along boundaries and shock profiles, where the most relevant linear and nonlinear phenomena take place. As such, the stability of viscous shock waves was the main focus of the lectures delivered by M. Williams and K. Zumbrun. This topic started, for the inviscid case, with the pioneering papers of Kreiss, Osher, Rauch and Majda. Later Metivier brought into the field a number of far-reaching ideas from microlocal analysis, in particular the paradifferential calculus introduced by Bony. The stability condition is expressed in terms of the so-called Kreiss-Lopatinskiˇı determinant. The viscous case can benefit from many of these ideas, but new tools are also needed. Linearizing the system about a given profile (made stationary by Galilean invariance), and taking the Laplace transform in time and the Fourier transform in the hyperplane orthogonal to the direction of propagation, allows the formulation of an eigenvalue equation for a differential operator with variable coefficients. A necessary condition for the viscous profile to be stable is that these eigenvalue equations do not have (nontrivial) solutions. The Evans function technique provides a means to quantify this criterion. But some rather subtle issues, in particular regarding regular dependence on parameters, call for a cautious approach. This was understood in a celebrated paper by J. Alexander, Gardner and C. Jones. Necessary stability conditions are expressed in terms of: (1) the transversality of the connection in the travelling wave ODE, and (2) the Kreiss-Lopatinskiˇı condition, which is known to ensure weak inviscid stability. The argument relies on the low frequency behaviour of the Evans function. Unlike the Kreiss-Lopatinskiˇı determinant ∆, encoding the linearized stability of the inviscid shock, the Evans function is not explicitly computable. But Zumbrun and Serre’s result shows that the Evans function is tangent to ∆ in the low frequency limit. Kevin Zumbrun’s lectures focused on the planar stability for viscous shock waves in systems with real viscosity. His course provided an extensive overview of the technical tools and central concepts involved. He took great care to make such a difficult matter comparatively simple and approachable to an audience of young mathematicians. M. Williams’ course focused on the short time existence of curved multidimensional viscous shocks and the related small viscosity limit. It provided an accessible account of the main ideas and methods, trying to avoid most of the
VIII
Preface
technical difficulties connected with the use of paradifferential calculus. His final lecture introduced the analysis of long time stability for planar viscous shocks. A fairly complete list of references to books and research articles is included at the bottom of all of the lecture notes. The course was attended by several young mathematicians from various European countries, who worked very hard during the whole period of the Summer School. However they were also able to enjoy the beautiful hosting facility provided by CIME, in the paradise-like sea resort of Cetraro, under the Calabrian sun. This course was organized with the collaboration and financial support of the European Network on Hyperbolic and Kinetic Equations (HyKE). I would like to express my gratitude to the CIME Foundation, to the CIME Director Prof. Pietro Zecca and to the CIME Board Secretary Prof. Elvira Mascolo, for their invaluable help and support, and for the tremendous efforts they have invested to return the CIME Courses to their traditional greatness. Pierangelo Marcati
Contents
BV Solutions to Hyperbolic Systems by Vanishing Viscosity Alberto Bressan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Review of Hyperbolic Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . 2.1 Centered Rarefaction Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Shocks and Admissibility Conditions. . . . . . . . . . . . . . . . . . . . . . 2.3 Solution of the Riemann Problem. . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Glimm and Front Tracking Approximations. . . . . . . . . . . . . . . . 2.5 A Semigroup of Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Uniqueness and Characterization of Entropy Weak Solutions . 3 The Vanishing Viscosity Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Parabolic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Decomposition by Traveling Wave Profiles . . . . . . . . . . . . . . . . . . . . . . 6 Interaction of Viscous Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Stability of Viscous Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Vanishing Viscosity Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete Shock Profiles: Existence and Stability Denis Serre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Existence Theory Rational Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Steady Lax Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More Complex Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Rational Values of η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit Profiles for the Godunov Scheme . . . . . . . . . . . . . . . . . DSPs for Strong Steady Shocks Under the Lax–Wendroff Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scalar Shocks Under Monotone Schemes . . . . . . . . . . . . . . . . . . 1.2 Under-Compressive Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Example from Reaction-Diffusion . . . . . . . . . . . . . . . . . . . . .
1 1 6 7 8 11 12 15 17 19 25 34 48 67 70 76 79 81 86 87 91 92 92 95 97 98 99
X
Contents
Homoclinic and Chaotic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Exponentially Small Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2 Existence Theory the Irrational Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.1 Obstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 The Small Divisors Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 The Function Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Counter-Examples to (2.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 The Lax–Friedrichs Scheme with an Almost Linear Flux . . . . 111 The Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.2 The Approach by Liu and Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3 Semi-Discrete vs Discrete Traveling Waves . . . . . . . . . . . . . . . . . . . . . . 117 3.1 Semi-Discrete Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.2 A Strategy Towards Fully Discrete Traveling Waves . . . . . . . . 118 3.3 Sketch of Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 120 The Richness of Discrete Dynamics . . . . . . . . . . . . . . . . . . . . . . . 123 4 Stability Analysis: The Evans Function . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.1 Spectral Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.2 The Essential Spectrum of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.3 Construction of the Evans Function . . . . . . . . . . . . . . . . . . . . . . 131 The Gap Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 The Geometric Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5 Stability Analysis: Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.1 Calculations with Lax Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 The Homotopy from ζ = 1 to ∞ . . . . . . . . . . . . . . . . . . . . . . . . . 139 The Large Wave-Length Analysis . . . . . . . . . . . . . . . . . . . . . . . . 140 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2 Calculations with Under-Compressive Shocks . . . . . . . . . . . . . . 144 5.3 Results for the Godunov Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 145 The Case of Perfect Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.4 The Role of the Functional Y in the Nonlinear Stability . . . . . 151 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Stability of Multidimensional Viscous Shocks Mark Williams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 1 Lecture One: The Small Viscosity Limit: Introduction, Approximate Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 1.1 Approximate Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 2 Lecture Two: Full Linearization, Reduction to ODEs, Conjugation to a Limiting Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 2.1 Full Versus Partial Linearization . . . . . . . . . . . . . . . . . . . . . . . . . 167 2.2 The Extra Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 169 2.3 Corner Compatible Initial Data and Reduction to a Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Contents
2.4 2.5 2.6 2.7
3
4
5
6 7 8 9 10
11
XI
Principal Parts, Exponential Weights . . . . . . . . . . . . . . . . . . . . . 171 Some Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Semiclassical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Frozen Coefficients; ODEs Depending on Frequencies as Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 2.8 Three Frequency Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 2.9 First-Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 2.10 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 2.11 Conjugation to HP Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Lecture Three: Evans Functions, Lopatinski Determinants, Removing the Translational Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . 178 3.1 Evans Functions, Instabilities, the Zumbrun-Serre Result . . . . 179 3.2 The Evans Function as a Lopatinski Determinant . . . . . . . . . . 182 3.3 Doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 3.4 Slow Modes and Fast Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3.5 Removing the Translational Degeneracy . . . . . . . . . . . . . . . . . . . 184 Lecture Four: Block Structure, Symmetrizers, Estimates . . . . . . . . . . 187 4.1 The MF Regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.2 The SF Regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4.3 The Sign Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.4 Glancing Blocks and Glancing Modes . . . . . . . . . . . . . . . . . . . . . 193 4.5 Auxiliary Hypothesis for Lecture 5 . . . . . . . . . . . . . . . . . . . . . . . 195 4.6 The SF Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 4.7 The HF Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 4.8 Summary of Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Lecture Five: Long Time Stability via Degenerate Symmetrizers . . . 200 5.1 Nonlinear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.2 L1 − L2 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.3 Proof of Proposition 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.4 The Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.5 Decomposition of UH± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.6 Interior Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.7 L∞ Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.8 Nonlinear Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Appendix A: The Uniform Stability Determinant . . . . . . . . . . . . . . . . 212 Appendix B: Continuity of Decaying Eigenspaces . . . . . . . . . . . . . . . . 213 Appendix C: Limits as z → ±∞ of Slow Modes at Zero Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Appendix D: Evans ⇒ Transversality + Uniform Stability . . . . . . . . 216 Appendix E: Proofs in Lecture 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 10.1 Construction of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 10.2 Propositions 3.4 and 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Appendix F: The HF Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 11.1 Block Stucture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.2 Symmetrizer and Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
XII
Contents
12 Appendix G: Transition to PDE Estimates . . . . . . . . . . . . . . . . . . . . . . 225 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Planar Stability Criteria for Viscous Shock Waves of Systems with Real Viscosity Kevin Zumbrun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 1 Introduction: Structure of Physical Equations . . . . . . . . . . . . . . . . . . . 230 2 Description of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3 Analytical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 4 Reduction to Low Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 5 Low Frequency Analysis/Completion of Proofs . . . . . . . . . . . . . . . . . . 284 6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 6.1 Appendix A: Semigroup Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 6.2 Appendix B: Proof of Proposition 1.21 . . . . . . . . . . . . . . . . . . . . 315 6.3 Appendix C: Proof of Proposition 5.15 . . . . . . . . . . . . . . . . . . . . 317 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 Tutorial on the Center Manifold Theorem . . . . . . . . . . . . . . . . . . . . . 327 A.1 Review of Linear O.D.E’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 A.2 Statement of the Center Manifold Theorem . . . . . . . . . . . . . . . . . . . . . 329 A.3 Proof of the Center Manifold Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 331 A.3.1 Reduction to the Case of a Compact Perturbation. . . . . . . . . . 331 A.3.2 Characterization of the Global Center Manifold. . . . . . . . . . . . 332 A.3.3 Construction of the Center Manifold. . . . . . . . . . . . . . . . . . . . . . 333 A.3.4 Proof of the Invariance Property (ii). . . . . . . . . . . . . . . . . . . . . . 335 A.3.5 Proof of (iv). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 A.3.6 Proof of the Tangency Property (iii). . . . . . . . . . . . . . . . . . . . . . 335 A.3.7 Proof of the Asymptotic Approximation Property (v). . . . . . . 336 A.3.8 Smoothness of the Center Manifold. . . . . . . . . . . . . . . . . . . . . . . 337 A.4 The Contraction Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
BV Solutions to Hyperbolic Systems by Vanishing Viscosity Alberto Bressan Department of Mathematics, Penn State University University Park, Pa. 16802 U.S.A.
[email protected] 1 Introduction The aim of these notes is to provide a self-contained presentation of recent results on hyperbolic systems of conservation laws, based on the vanishing viscosity approach. A system of conservation laws in one space dimension takes the form ut + f (u)x = 0 .
(1.1)
Here u = (u1 , . . . , un ) is the vector of conserved quantities while the components of f = (f1 , . . . , fn ) are called the fluxes. Integrating (1.1) over a fixed interval [a, b] we find d dt
b b u(t, x) dx = a ut (t, x) dx = − a f u(t, x) x dx = f u(t, a) − f u(t, b) = [inflow at a] − [outflow at b] .
b a
Each component of the vector u thus represents a quantity which is neither created nor destroyed: its total amount inside any given interval [a, b] can change only because of the flow across boundary points. Systems of the form (1.1) are commonly used to express the fundamental balance laws of continuum physics, when small viscosity or dissipation effects are neglected. For a comprehensive discussion of conservation laws and their derivation from basic principles of physics we refer to the book of Dafermos [D1]. Smooth solutions of (1.1) satisfy the equivalent quasilinear system ut + A(u)ux = 0 ,
(1.2)
. where A(u) = Df (u) is the Jacobian matrix of first order partial derivatives of f . We notice, however, that if u has a jump at some point x0 , then the left hand side of (1.2) contains a product of the discontinuous function x → A u(x)
2
Alberto Bressan
with the distributional derivative ux , which in this case contains a Dirac mass at the point x0 . In general, such a product is not well defined. The quasilinear system (1.2) is thus meaningful only within a class of continuous functions. On the other hand, working with the equation in divergence form (1.1) allows us to consider discontinuous solutions as well, interpreted in distributional sense. We say that a locally integrable function u = u(t, x) is a weak solution of (1.1) if t → u(t, ·) is continuous as a map with values in L1loc and moreover (1.3) uφt + f (u)φx dx dt = 0 for every differentiable function with compact support φ ∈ Cc1 . . The above system is called strictly hyperbolic if each matrix A(u) = Df (u) has n real, distinct eigenvalues λ1 (u) < · · · < λn (u). One can then find dual bases of left and right eigenvectors of A(u), denoted by l1 (u), . . . , ln (u) and r1 (u), . . . , rn (u), normalized according to 1 if i = j, |ri | = 1 , li · rj = (1.4) 0 if i = j.
To appreciate the effect of the non-linearity, consider first the case of a linear system with constant coefficients ut + Aux = 0.
(1.5)
Call λ1 < · · · < λn the eigenvalues of the matrix A, and let li , ri be the corresponding left and right eigenvectors as in (1.4). The general solution of (1.5) can be written as a superposition of independent linear waves: . u(t, x) = φi (x − λi t)ri , φi (y) = li · u(0, y) . i
Notice that here the solution is completely decoupled along the eigenspaces of A, and each component travels with constant speed, given by the corresponding eigenvalue of A. In the nonlinear case (1.2) where the matrix A depends on the state u, new features will appear in the solutions. (i) Since the eigenvalues λi now depend on u, the shape of the various components in the solution will vary in time (fig. 1). Rarefaction waves will decay, and compression waves will become steeper, possibly leading to shock formation in finite time. (ii) Since the eigenvectors ri also depend on u, nontrivial interactions between different waves will occur (fig. 2). The strength of the interacting waves may change, and new waves of different families can be created, as a result of the interaction.
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
3
u(t)
u(0)
x Fig. 1.
t
nonlinear
linear
x Fig. 2.
The strong nonlinearity of the equations and the lack of regularity of solutions, also due to the absence of second order terms that could provide a smoothing effect, account for most of the difficulties encountered in a rigorous mathematical analysis of the system (1.1). It is well known that the main techniques of abstract functional analysis do not apply in this context. Solutions cannot be represented as fixed points of continuous transformations, or in variational form, as critical points of suitable functionals. Dealing with vector valued functions, comparison principles based on upper or lower solutions cannot be used. Moreover, the theory of accretive operators and contractive nonlinear semigroups works well in the scalar case [C], but does not apply to systems. For the above reasons, the theory of hyperbolic conservation laws has largely developed by ad hoc methods, along two main lines. 1. The BV setting, considered by Glimm [G]. Solutions are here constructed within a space of functions with bounded variation, controlling the BV norm by a wave interaction functional. 2. The L∞ setting, considered by Tartar and DiPerna [DP2], based on weak convergence and a compensated compactness argument.
4
Alberto Bressan
Both approaches yield results on the global existence of weak solutions. However, the method of compensated compactness appears to be suitable only for 2×2 systems. Moreover, it is only in the BV setting that the well-posedness of the Cauchy problem could recently be proved, as well as the stability and convergence of vanishing viscosity approximations. Throughout the following we thus restrict ourselves to the study of BV solutions, referring to [DP2] or [Se] for the alternative approach based on compensated compactness. Since the pioneering work of Glimm, the basic building block toward the construction and the analysis of more general solutions has been provided by the Riemann problem, i.e. the initial value problem with piecewise constant data u− if x < 0, (1.6) u(0, x) = u ¯(x) = u+ if x > 0. This was first introduced by B.Riemann (1860) in the context of isentropic gas dynamics. A century later, P.Lax [Lx] and T.P.Liu [L1] solved the Riemann problem for more general n × n systems. The new approach of Bianchini [Bi] now applies to all strictly hyperbolic systems, not necessarily in conservation form. Solutions are always found in the self-similar form u(t, x) = U (x/t). The central position taken by the Riemann problem is related to a symmetry of the equations (1.1). If u = u(t, x) is a solution of (1.1), then for any θ > 0 the function . uθ (t, x) = u(θt, θx) provides another solution. The solutions which are invariant under these rescalings of the independent variables are precisely those which correspond to some Riemann data (1.6). For a general Cauchy problem, both the Glimm scheme [G] and the method of front tracking [D1], [DP1], [B1], [BaJ], [HR] yield approximate solutions of a general Cauchy problem by piecing together a large number of Riemann solutions. For initial data with small total variation, this approach is successful because one can provide a uniform a priori bound on the amount of new waves produced by nonlinear interactions, and hence on the total variation of the solution. It is safe to say that, in the context of weak solutions with small total variation, nearly all results on the existence, uniqueness, continuous dependence and qualitative behavior have relied on a careful analysis of the Riemann problem. In [BiB] a substantially different perspective has emerged from the study of vanishing viscosity approximations. Solutions of (1.1) are here obtained as limits for ε → 0 of solutions to the parabolic problems uεt + A(uε )uεx = εuεxx
(1.7)
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
5
. with A(u) = Df (u). This approach is very natural and has been considered since the 1950’s. However, complete results had been obtained only in the scalar case [O], [K]. For general n × n systems, the main difficulty lies in establishing the compactness of the approximating sequence. We observe that uε (t, x) solves (1.7) if and only if uε (t, x) = u(t/ε, x/ε) for some function u which satisfies (1.8) ut + A(u)ux = uxx . In the analysis of vanishing viscosity approximations, the key step is to derive a priori estimates on the total variation and on the stability of solutions of (1.8). For this parabolic system, the rescaling (t, x) → (θt, θx) no longer determines a symmetry. Hence the Riemann data no longer hold a privileged position. The role of basic building block is now taken by the viscous traveling profiles, i.e. solutions of the form u(t, x) = U (x − λt). Of course, the function U must then satisfy the second order O.D.E. U ′′ = A(U ) − λ U ′ .
In this new approach, the profile u(·) of a viscous solution is viewed locally as a superposition of viscous traveling waves. More precisely, let a smooth function u : R → Rn be given. At each point x, looking at the second order jet (u, ux , uxx ) we seek traveling profiles U1 , . . . , Un such that (1.9) Ui′′ = A(Ui ) − σi Ui′ for some speed σi close to the characteristic speed λi , and moreover Ui (x) = u(x)
i = 1, . . . , n , Ui′′ (x) = uxx (x) .
Ui′ (x) = ux (x) ,
i
(1.10) (1.11)
i
It turns out that this decomposition is unique provided that the traveling profiles are chosen within suitable center manifolds. We let r˜i be the unit vector parallel to Ui′ , so that Ui′ = vi r˜i for some scalar vi . One can show . that r˜i remains close to the eigenvector ri (u) of the Jacobian matrix A(u) = Df (u), but r˜i = ri (u) in general. The first equation in (1.11) now yields the decomposition ux = vi r˜i . (1.12) i
If u = u(t, x) is a solution of (1.8), we can think of vi as the density of iwaves in u. The remarkable fact is that these components satisfy a system of evolution equations ˜ i vi )x − vi,xx = φi vi,t + (λ
i = 1, . . . , n ,
(1.13)
6
Alberto Bressan
where the source terms φi on the right hand side are INTEGRABLE over the whole domain {x ∈ R , t > 0}. Indeed, we can think of the sources φi as new waves produced by interactions between viscous waves. Their total strength is controlled by means of viscous interaction functionals, somewhat similar to the one introduced by Glimm in [G] to study the hyperbolic case. Since the left hand side of (1.13) is in conservation form and the vectors r˜i have unit length, for an arbitrarily large time t we obtain the bound t φi (s, x) dxds . (1.14) vi (t) 1 ≤ vi (t0 ) + ux (t) L1 ≤ L i
i
t0
This argument yields global BV bounds and stability estimates for viscous solutions. In turn, letting ε → 0 in (1.7), a standard compactness argument yields the convergence of uε to a weak solution u of (1.1). The plan of these notes is as follows. In Section 2 we briefly review the basic theory of hyperbolic systems of conservation laws: shock and rarefaction waves, entropies, the Liu admissibility conditions and the Riemann problem. For initial data with small total variation, we also recall the main results concerning the existence, uniqueness and stability of solutions to the general Cauchy problem. Section 3 contains the statement of the main new results on vanishing viscosity limits and an outline of the proof. In Section 4 we derive some preliminary estimates which can be obtained by standard parabolic techniques, representing a viscous solution in terms of convolutions with a heat kernel. Section 5 discusses in detail the local decomposition of a solution as superposition of viscous traveling waves. The evolution equations (1.13) for the components vi and the strength of the source terms φi are then studied in Section 6. This will provide the crucial estimate on the total variation of viscous solutions, uniformly in time. In Section 7 we briefly examine the stability and some other properties of viscous solutions. The existence, uniqueness and stability of vanishing viscosity limits are then discussed in Section 8.
2 Review of Hyperbolic Conservation Laws In most of this section, we shall consider a strictly hyperbolic system of conservation laws satisfying the additional hypothesis (H)For each i = 1, . . . , n, the i-th field is either genuinely nonlinear, so that Dλi (u)·ri (u) > 0 for all u, or linearly degenerate, with Dλi (u)·ri (u) = 0 for all u. We observe that the i-th characteristic field is genuinely nonlinear iff the eigenvalue λi is strictly increasing along each integral curve of the corresponding field of eigenvectors ri . It is linearly degenerate when λi is constant along each such curve.
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
7
2.1 Centered Rarefaction Waves. We begin by studying a special type of solutions, in the form of centered rarefaction waves. Fix a state u0 ∈ Rn and an index i ∈ {1, . . . , n}. Let ri (u) be a field of i-eigenvectors of the Jacobian matrix A(u) = Df (u). The integral curve of the vector field ri through the point u0 is called the i-rarefaction curve through u0 . It is obtained by solving the initial value problem for the O.D.E. in state space: du = ri (u), ds We shall denote this curve as
u(0) = u0 .
(2.1)
s → Ri (s)(u0 ). Clearly, the parametrization depends on the choice of the eigenvectors ri . In particular, if we impose the normalization ri (u) ≡ 1, then the rarefaction curve Ri will be parametrized by arc-length. Let the i-th field be genuinely nonlinear. Given two states u− , u+ , assume that u+ lies on the positive i-rarefaction curve through u− , i.e. u+ = Ri (σ)(u− ) for some σ > 0. For each s ∈ [0, σ], define λi (s) = λi Ri (s)(u− ) .
By genuine nonlinearity, the map s → λi (s) is strictly increasing. Hence,
for every λ ∈ λi (u− ), λi (u+ ) , there is a unique value s ∈ [0, σ] such that λ = λi (s). We claim that, for t > 0, the function ⎧ − ⎪ if x/t < λi (u− ), ⎨u
− u(t, x) = Ri (s)(u ) if x/t = λi (s) ∈ λi (u− ), λi (u+ ) , (2.2) ⎪ ⎩ + u if x/t > λi (u+ ),
is a continuous solution of the Riemann problem (1.1), (1.6). Indeed, by construction it follows ¯ L1 = 0. lim u(t, ·) − u t→0+
Moreover, the equation (1.2) is trivially satisfied in the sectors where x/t < λi (u− ) or x/t > λi (u+ ), since here ut = ux = 0. Next, consider what happens in the intermediate sector λi (u− ) < x/t < λi (u+ ) . Since u is constant along each ray through the origin {x/t = c}, we have x ut (t, x) + ux (t, x) = 0. t Our construction at (2.2) now yields x/t = λi u(t, x) . Moreover, ux is parallel to ri (u), hence it is an eigenvector of the Jacobian matrix A(u) with eigenvalue λi (u). This implies ut + A(u)ux = 0 , proving our claim. Notice that the assumption σ > 0 is essential for the validity of this construction. In the opposite case σ < 0, the definition (2.2) would yield a triple-valued function
in the region where x/t ∈ λi (u+ ), λi (u− ) .
8
Alberto Bressan
2.2 Shocks and Admissibility Conditions. The simplest type of discontinuous solution of (1.1) is given by a shock: u+ if x > λt, U (t, x) = (2.3) − if x < λt, u for some left and right states u− , u+ ∈ Rn and a speed λ ∈ R. Using the divergence theorem, one checks that the identity (1.2) is satisfied for all test functions φ if and only if the following Rankine-Hugoniot conditions hold: λ (u+ − u− ) = f (u+ ) − f (u− ) .
(2.4)
As usual, we denote by A(u) = Df (u) the n × n Jacobian matrix of f at u. For any u, v, consider the averaged matrix 1 . A θu + (1 − θ)v dθ A(u, v) = 0
and call λi (u, v), i = 1, . . . , n its eigenvalues. We can then write (2.4) in the equivalent form 1 + − λ (u −u ) = Df θu+ +(1−θ)u− ·(u+ −u− ) dθ = A(u+ , u− )·(u+ −u− ). 0
(2.5) In other words, the Rankine-Hugoniot conditions hold if and only if the jump u+ − u− is an eigenvector of the averaged matrix A(u+ , u− ) and the speed λ coincides with the corresponding eigenvalue. The representation (2.5) is very useful for studying the set of pairs (u− , u+ ) that can be connected by a shock. Given a left state u− = u0 , for each i = 1, . . . , n one can prove that there exists a one-parameter curve of right states u+ = Si (s)(u0 ) which satisfy the Rankine-Hugoniot conditions (2.4) for a suitable speed λ = λi (s). This curve can also be parametrized by arc-length: s → Si (s)(u0 ). It has a second order tangency with the corresponding rarefaction curve Ri at the point u0 (see fig. 3). To study the behavior of a general weak solution at a point of discontinuity, we first recall some definitions. A function u = u(t, x) has an approximate jump discontinuity at the point (τ, ξ) if there exists vectors u+ = u− and a speed λ such that r r 1 u(τ + t, ξ + x) − U (t, x) dx dt = 0 , (2.6) lim r→0+ r 2 −r −r
with U as in (2.3). We say that u is approximately continuous at the point (τ, ξ) if the above limit hold with u+ = u− (and λ arbitrary).
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
9
Si Ri
u
ri (u 0) 0
Fig. 3.
If u is now a solution of the system of conservation laws (1.1) having an approximate jump at a point (τ, ξ), one can prove that the states u− , u+ and the speed λ again satisfy the Rankine-Hugoniot conditions (see Theorem 4.1 in [B3]). In order to achieve uniqueness of solutions to initial value problems, one needs to supplement the conservation equations (1.1) with additional admissibility conditions, to be satisfied at points of jump. We recall here two basic approaches. A continuously differentiable function η : Rn → R is called an entropy for the system of conservation laws (1.1), with entropy flux q : Rn → R, if it satisfies the identity Dη(u) · Df (u) = Dq(u).
(2.7)
An immediate consequence of (2.7) is that, if u = u(t, x) is a C 1 solution of (1.1), then (2.8) η(u)t + q(u)x = 0. For a smooth solution u, not only the quantities u1 , . . . , un are conserved, but the additional conservation law (2.8) holds as well. However one should be aware that, for a discontinuous solution u, the quantity η(u) may not be conserved. A standard admissibility condition for weak solutions can now be formulated as follows. Let η be a convex entropy for the system (1.1), with entropy flux q. A weak solution u is entropy-admissible if η(u)t + q(u)x ≤ 0 in distribution sense, i.e.
η(u)ϕt + q(u)ϕx dx dt ≥ 0
(2.9)
(2.10)
10
Alberto Bressan
for every function ϕ ≥ 0, continuously differentiable with compact support. In analogy with (2.4), if u is an entropy admissible solution, at every point of approximate jump one can show that (2.9) implies
λ η(u+ ) − η(u− ) ≥ q u+ ) − q(u− ). (2.11) The above admissibility condition can be useful only if some nontrivial convex entropy exists. This is not always the case. Indeed, according to (2.7), to construct an entropy we must solve a system of n equations for the two scalar functions η, q. This system is overdetermined whenever n ≥ 3. In continuum physics, however, one finds various systems of conservation laws endowed with significant entropies. An alternative admissibility condition, due to Liu [L2], is particularly useful because it can be applied to any system, even in the absence of nontrivial entropies. To understand its meaning, consider first a scalar conservation law. In this case, any two states u− , u+ satisfy the equations (2.4), provided that we choose the speed f (u+ ) − f (u− ) λ= . (2.12) u+ − u− To test the stability of the shock (2.3) w.r.t. a small perturbation, fix any intermediate state u∗ ∈ [u− , u+ ]. We can slightly perturb the initial data by splitting the initial jump in two smaller jumps (fig. 4), say ⎧ − ⎪ if x < 0, ⎨u u(0, x) = u∗ if 0 < x < ε, ⎪ ⎩ + u if x > ε.
u+
+ λ
+ P
P*
f
f(u)
u* λ− − P
u− 0ε Fig. 4.
x
u−
u*
u+
Fig. 5
u
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
11
Clearly, the original shock can be stable only if the speed λ− of the jump behind is faster than the speed λ+ of the jump ahead, i.e. f (u+ ) − f (u∗ ) . f (u∗ ) − f (u− ) λ− = ≥ = λ+ . ∗ − u −u u+ − u∗
(2.13)
In fig. 5, the first speed corresponds to the slope of the segment P − P ∗ while the second speed is the slope of P ∗ P + . On the other hand, by (2.12) the speed λ of the original shock is the slope of the segment P − P + . Therefore the inequality (2.13) holds if and only if f (u∗ ) − f (u− ) f (u+ ) − f (u− ) ≥ . u∗ − u− u+ − u−
(2.14)
The condition (2.13) cannot be extended to the vector valued case, because in general one cannot find intermediate states u∗ such that both couples (u− , u∗ ) and (u∗ , u+ ) are connected by a shock. However, the condition (2.14) has a straightforward generalization to n × n systems. Denote by s → Si (s)(u− ) the i-shock curve through the point u− , and let λi (s) be the speed of the shock connecting u− with Si (s)(u− ). Now consider any right state u+ = Si (σ)(u− ) on this shock curve. Following [L2], we say that the shock (2.3) satisfies the Liu admissibility condition if λi (s) ≥ λi (σ)
for all s ∈ [0, σ]
(2.15)
(or for all s ∈ [σ, 0], in case where σ < 0). In other words, if u∗ = Si (s)(u− ) is any intermediate state along the shock curve through u− , the speed of the shock (u− , u∗ ) should be larger or equal to the speed of the original shock (u− , u+ ). 2.3 Solution of the Riemann Problem. Let the assumptions (H) hold, so that each characteristic field is either genuinely nonlinear or linearly degenerate. Relying on the previous analysis, the solution of the general Riemann problem u− if x < 0, ut + f (u)x = 0 , u(0, x) = (2.16) + u if x > 0. can be obtained by finding intermediate states ω0 = u− , ω1 , . . . , ωn = u+ such that each pair of adjacent states ωi−1 , ωi can be connected by an i-shock, or by a centered rarefaction i-wave. By the implicit function theorem, this can always be done provided that the two states u− , u+ are sufficiently close.
12
Alberto Bressan
Following [Lx], the complete solution is now obtained by piecing together the solutions of the n Riemann problems ωi−1 if x < 0, u(0, x) = ut + f (u)x = 0, (2.17) ωi if x > 0, on different sectors of the t-x plane. A typical situation is illustrated in fig. 6. If the assumption (H) is removed, one can still solve the Riemann problem in terms of n + 1 constant states ω0 , . . . , ωn . However, in this case each couple of states ωi−1 , ωi can be connected by a large number of adjacent shocks and rarefaction waves (fig. 7). The analysis of the Riemann problem thus becomes far more complicated [L1]. shock
rarefaction
t
shock
ω1
t
ω2
u −= ω0
ω1
u+= ω 3
u−= ω0
u+= ω3
x
Fig. 6.
ω2
x
Fig. 7
2.4 Glimm and Front Tracking Approximations. Approximate solutions for the general Cauchy problem can be constructed by patching together solutions of several Riemann problems. In the Glimm scheme (fig. 8), one works with a fixed grid in the t-x plane, with mesh sizes ∆t, ∆x. At time t = 0 the initial data is approximated by a piecewise constant function, with jumps at grid points. Solving the corresponding Riemann problems, a solution is constructed up to a time ∆t sufficiently small so that . waves emerging from different nodes do not interact. At time t1 = ∆t, we replace the solution u(∆t, ·) by a piecewise constant function having jumps exactly at grid points. Solving the new Riemann problems at every one of these points, one can prolong the solution to the next time interval [∆t, 2∆t]. . At time t2 = 2∆t, the solution is again approximated by a piecewise constant functions with jumps exactly at grid points, etc. . . A key ingredient of the . Glimm scheme is the restarting procedure. At each time tj = j ∆t, a natural way to approximate a BV function with a piecewise constant one is by taking . its average value on each subinterval Jk = [xk−1 , xk ]. However, this procedure may generate an arbitrarily large amount of oscillations. Instead, the Glimm scheme is based on random sampling: a point yk is selected at random inside
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
13
each interval Jk and the old value u(tj −, yk ) at this particular point is taken as the new value of u(tj , x) for all x ∈ Jk . An excellent introduction to the Glimm scheme can be found in the book by J. Smoller [Sm].
2∆ t
*
∆t
*
0
x
*
*
y k * k−1
2∆x
x
* k
4∆ x
Fig. 8.
An alternative technique for constructing approximate solutions is by wave-front tracking (fig. 9). This method was introduced by Dafermos [D1] in the scalar case and later developed by various authors [DP1], [B1], [BaJ]. It now provides an efficient tool in the study of general n × n systems of conservation laws, both for theoretical and numerical purposes [B3], [HR]. The initial data is here approximated with a piecewise constant function, and each Riemann problem is solved approximately, within the class of piecewise constant functions. In particular, if the exact solution contains a centered rarefaction, this must be approximated by a rarefaction fan, containing several small jumps. At the first time t1 where two fronts interact, the new Riemann problem is again approximately solved by a piecewise constant function. The solution is then prolonged up to the second interaction time t2 , where the new Riemann problem is solved, etc. . . Comparing this method with the previous one of Glimm, we see that in the Glimm scheme one specifies a priori the nodal points where the the Riemann problems are to be solved. On the other hand, in a solution constructed by wave-front tracking the locations of the jumps and of the interaction points depend on the solution itself. No restarting procedure is needed and the map t → u(t, ·) is thus continuous with values in L1loc . In the end, both algorithms produce a sequence of approximate solutions, whose convergence relies on a compactness argument based on uniform bounds on the total variation. We sketch the main idea involved in these a priori BV bounds. Consider a piecewise constant function u : R → Rn , say with jumps at points x1 < x2 < · · · < xN . Call σα the amplitude of the jump at xα . The total strength of waves is then defined as
14
Alberto Bressan
t t4 t3 xβ xα t2 t1
σ
σ’
0
x Fig. 9.
. V (u) = |σα | .
(2.18)
α
Clearly, this is an equivalent way to measure the total variation. Along a solu- tion u = u(t, x) constructed by front tracking, the quantity V (t) = V u(t, ·) may well increase at interaction times. To provide global a priori bounds, following [G] one introduces a wave interaction potential, defined as |σα σβ |, (2.19) Q(u) = (α,β)∈A
where the summation runs over the set A of all couples of approaching waves. Roughly speaking, we say that two wave-fronts located at xα < xβ are approaching if the one at xα has a faster speed than the one at xβ (hence the two fronts are expected to collide at a future time). Now consider a time τ where two incoming wave-fronts interact, say with strengths σ, σ ′ (for example, take τ = t1 in fig. 9). The difference between the outgoing waves emerging from the interaction and the two incoming waves σ, σ ′ is of magnitude O(1) · |σσ ′ |. On the other hand, after time τ the two incoming waves are no longer approaching. This accounts for the decrease of the functional Q in (2.19) by the amount |σσ ′ |. Observing that the new waves generated by the interaction could approach all other fronts, the change in the functionals V, Q across the interaction time τ is estimated as ∆V (τ ) = O(1) · |σσ ′ | ,
∆Q(τ ) = −|σσ ′ | + O(1) · |σσ ′ | V (τ −).
If the initial data has small total variation, for a suitable constant C0 the quantity . Υ (t) = V u(t, ·) + C0 Q u(t, ·)
is monotone decreasing in time. This argument provides a uniform BV bound on all approximate solutions. Using Helly’s compactness theorem, one obtains
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
15
the strong convergence in L1loc of a subsequence of approximate solutions, and hence the existence of a weak solution. Theorem 1. Consider the Cauchy problem ut + f (u)x = 0 ,
u(0, x) = u ¯(x) ,
(2.20)
for a system of conservation laws with a smooth flux f defined in a neighborhood of the origin. Assume that the system is strictly hyperbolic and satisfies the conditions (H). Then, for a sufficiently small δ > 0 the following holds. For every initial condition u(0, x) = u ¯(x) with Tot.Var.{¯ u} < δ ,
¯ uL∞ < δ ,
(2.21)
the Cauchy problem has a weak solution, defined for all times t ≥ 0, obtained as limit of front tracking approximations. 2.5 A Semigroup of Solutions. For solutions obtained as limits of Glimm or front tracking approximations, the uniqueness and the continuous dependence on the initial data were first established in [BC] for systems of two equations and then in [BCP] for n × n systems. Relying on some major new ideas introduced by Liu and Yang in [LY1], [LY2] a much simpler proof could then be worked out in [BLY]. Here the key step is the construction of a Lyapunov functional Φ(u, v), equivalent to the L1 distance 1 u − vL1 ≤ Φ(u, v) ≤ Cu − vL1 , C and almost decreasing in time along each pair of ε-approximate solutions constructed by front tracking d Φ u(t), v(t) ≤ O(1) · ε . dt
A comprehensive treatment of the above existence-uniqueness theory for BV solutions can be found in [B3], [HR]. The main result is as follows. Theorem 2. With the same assumptions as Theorem 1, the weak solution of the Cauchy problem (2.20) obtained as limit of Glimm or front tracking approximations is unique and satisfies the Liu entropy conditions. Adopting a semigroup notation, this solution can be written as u(t, ·) = St u ¯. The semigroup S : D × [0, ∞[ → D can be defined on a closed domain D ⊂ L1 containing all functions with sufficiently small total variation, It is uniformly Lipschitz continuous w.r.t. both time and the initial data:
16
Alberto Bressan
St u ¯ − Ss v¯ L1 ≤ L ¯ u − v¯L1 + L′ |t − s| .
(2.22)
We refer to the flow generated by a system of conservation laws as a Riemann semigroup, because it is entirely determined by specifying how Riemann problems are solved. As proved in [B2], if two semigroups S, S ′ yield the same solutions to all Riemann problems, then they coincide, up to the choice of their domains.
w(T)
S
w(t+h) w(t)
w(t+h)
T−t−h
S w(t) T−t
S w(t) h
S w(0) T
w(0) Fig. 10.
From (2.22) one can deduce the error bound (fig. 10) T w(t + h) − Sh w(t) 1 L w(T ) − ST w(0) 1 ≤ L · dt , (2.23) lim inf L h→0+ h 0
valid for every Lipschitz continuous map w : [0, T ] → D taking values inside the domain of the semigroup. We can think of t → w(t) as an approximate solution of (1.1), while t → St w(0) is the exact solution having the same initial data. The distance w(T ) − ST w(0) L1 is clearly bounded by the length of . the curve γ : [0, T ] → L1 , with γ(t) = ST −t w(t). According to (2.23), this length is estimated by the integral of an instantaneous error rate, amplified by the Lipschitz constant L of the semigroup. Using (2.23), one can estimate the distance between a front tracking approximation and the corresponding exact solution. For approximate solutions constructed by the Glimm scheme, a direct application of this same formula is not possible because of the additional errors introduced by the restarting . procedure at each time tk = k ∆t. However, relying on a careful analysis of
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
17
Liu [L3], one can construct a front tracking approximate solution having the same initial and terminal values as the Glimm solution. By this technique, in [BM] the authors proved the estimate Glimm u (τ, ·) − uexact (τ, ·) L1 √ =0 (2.24) lim ∆x→0 ∆x · | ln ∆x| for every τ > 0. In other words, letting the mesh sizes ∆x, ∆t → 0 while keeping their ratio ∆x/∆t constant, the L1 norm of the error in √ the Glimm approximate solution tends to zero at a rate slightly slower than ∆x. 2.6 Uniqueness and Characterization of Entropy Weak Solutions The stability result stated in Theorem 1 refers to a special class of weak solutions: those obtained as limits of Glimm or front tracking approximations. It is desirable to have a uniqueness theorem valid for general weak solutions, without reference to any particular constructive procedure. Results in this direction were proved in, [BLF], [BG], [BLw]. They are all based on the error formula (2.23). In the proofs, one considers a weak solution u = u(t, x) of the Cauchy problem ut + f (u)x = 0 ,
u(0, x) = u ¯(x) .
(2.25)
Assuming that u satisfies suitable entropy and regularity conditions, one shows that u(t + h) − Sh u(t) 1 L =0 (2.26) lim inf h→0+ h
at almost every time t. By (2.23), u thus coincides with the semigroup tra¯ obtained as limit of front tracking approximations. jectory t → St u(0) = St u Of course, this implies uniqueness. We state here the main result in [BG]. A function u = u(t, x) is said to have tame oscillation if there exist constants C ′ , β > 0 such that the following holds. For every τ > 0 and a < b, the oscillation of u over the triangle . ∆τa,b = (s, y) ; s ≥ t , a + β(s − τ ) < y < b − β(s − τ ) satisfies
Osc.{u; ∆τa,b } ≤ C ′ · Tot.Var. u(τ, ·); ]a, b[ .
(2.27)
We recall that the oscillation of a u on a set ∆ is defined as . u(s, y) − u(s′ , y ′ ) . Osc.{u; ∆} = sup (s,y), (s′ ,y ′ )∈∆
Theorem 3. In the same setting as Theorem 1, let u = u(t, x) be a weak solution of the Cauchy problem (2.20) satisfying the tame oscillation condition
18
Alberto Bressan
(2.27) and the Liu admissibility condition (2.15) at every point of approximate jump. Then u coincides with the corresponding semigroup trajectory, i.e. u(t, ·) = St u ¯
(2.28)
for all t ≥ 0. In particular, the solution that satisfies the above conditions is unique. An additional characterization of these admissible weak solutions, based on local integral estimates, was given in [B2]. To state this result, we need to introduce some notations. Given a function u = u(t, x) and a point (τ, ξ), we ♯ denote by U(u;τ,ξ) the solution of the Riemann problem with initial data u− = lim u(τ, x), x→ξ−
u+ = lim u(τ, x). x→ξ+
(2.29)
♭ In addition, we define U(u;τ,ξ) as the solution of the linear hyperbolic Cauchy problem with constant coefficients
wt + A♭ wx = 0, w(0, x) = u(τ, x). (2.30) . Here A♭ = A u(τ, ξ) . Observe that (2.30) is obtained from the quasilinear system (1.2) by “freezing” the coefficients of the matrix A(u) at the point (τ, ξ) and choosing u(τ ) as initial data. A notion of “good solution” can now be introduced, by locally comparing a function u with the self-similar solution of a Riemann problem and with the solution of a linear hyperbolic system with constant coefficients. More precisely, we say that a function u = u(t, x) is a viscosity solution of the system (1.1) if t → u(t, ·) is continuous as a map with values into L1loc , and moreover the following integral estimates hold. (i) At every point (τ, ξ), for every β ′ > 0 one has 1 h→0+ h lim
β′ h
−β ′ h
♯ u(τ + h, ξ + x) − U(u;τ,ξ) (h, x) dx = 0.
(2.31)
(ii) There exist constants C, β > 0 such that, for every τ ≥ 0 and a < ξ < b, one has 1 b−βh ♭ lim sup (h, x) dx u(τ + h, x) − U(u;τ,ξ) h→0+ h a+βh (2.32) 2 ≤ C · Tot.Var. u(τ ); ]a, b[ .
The above definition was introduced in [B2], with the expectation that the “viscosity solutions” in the above integral sense should precisely coincide with the limits of vanishing viscosity approximations (1.7). This conjecture was proved only many years later in [BiB]. The main result in [B2] shows
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
19
that the integral estimates (2.31)–(2.32) completely characterize the semigroup trajectories. Theorem 4. Let S : D × [0, ∞[×D be the semigroup generated by the system of conservation laws (1.1), whose trajectories are limits of front tracking approximations. A function u : [0, T ] → D is a viscosity solution of (1.1) if and only if u(t) = St u(0) for all t ∈ [0, T ].
3 The Vanishing Viscosity Approach In view of the previous uniqueness and stability results, it is natural to expect that the entropy weak solutions of the hyperbolic system (1.1) should coincide with the unique limits of solutions to the parabolic system uεt + f (uε )x = ε uεxx
(3.1)
letting the viscosity coefficient ε → 0. For smooth solutions, this convergence is easy to show. However, one should keep in mind that a weak solution of the hyperbolic system (1.1) in general is only a function with bounded variation, possibly with a countable number of discontinuities. In this case, as the smooth functions uε approach the discontinuous solution u, near points of jump their gradients uεx must tend to infinity (fig. 11), while their second derivatives uεxx become even more singular. Therefore, establishing the convergence uε → u is a highly nontrivial matter. In earlier literature, results in this direction relied on three different approaches: 1 - Comparison principles for parabolic equations. For a scalar conservation law, the existence, uniqueness and global stability of vanishing viscosity solutions was first established by Oleinik [O] in one space dimension. The famous paper by Kruzhkov [K] covers the more general class of L∞ solutions and is also valid in several space dimensions. 2 - Singular perturbations. This technique was developed by Goodman and Xin [GX], and covers the case where the limit solution u is piecewise smooth, with a finite number of non-interacting, entropy admissible shocks. See also [Yu] and [Ro], for further results in this direction. 3 - Compensated compactness. With this approach, introduced by Tartar and DiPerna [DP2], one first considers a weakly convergent subsequence uε ⇀ u. For a class of 2 × 2 systems, one can show that this weak limit u actually provides a distributional solution to the nonlinear system (1.1). The proof relies on a compensated compactness argument, based on the representation of the weak limit in terms of Young measures, which must reduce to a Dirac mass due to the presence of a large family of entropies.
20
Alberto Bressan
u
ε
u
x Fig. 11.
Since the hyperbolic Cauchy problem is known to be well posed within a space of functions with small total variation, it seems natural to develop a theory of vanishing viscosity approximations within the same space BV. This was indeed accomplished in [BiB], in the more general framework of nonlinear hyperbolic systems not necessarily in conservation form. The only assumptions needed here are the strict hyperbolicity of the system and the small total variation of the initial data. Theorem 5. Consider the Cauchy problem for the hyperbolic system with viscosity uε (0, x) = u ¯(x) . (3.2) uεt + A(uε )uεx = ε uεxx Assume that the matrices A(u) are strictly hyperbolic, smoothly depending on u in a neighborhood of the origin. Then there exist constants C, L, L′ and δ > 0 such that the following holds. If Tot.Var.{¯ u} < δ ,
¯ uL∞ < δ,
(3.3)
then for each ε > 0 the Cauchy problem (3.2)ε has a unique solution uε , defined for all t ≥ 0. Adopting a semigroup notation, this will be written as . ¯. In addition, one has t → uε (t, ·) = Stε u BV bounds : Tot.Var. Stε u ¯ ≤ C Tot.Var.{¯ u} . (3.4) ε St u ¯ − v¯ L1 , L1 stability: ¯ − Stε v¯ L1 ≤ L u (3.5) √ ε √ St u (3.6) ¯ − Ssε u ¯ L1 ≤ L′ |t − s| + εt − εs .
Convergence: As ε → 0+, the solutions uε converge to the trajectories of a semigroup S such that St u ¯ − Ss v¯ L1 ≤ L ¯ u − v¯L1 + L′ |t − s| . (3.7)
These vanishing viscosity limits can be regarded as the unique vanishing viscosity solutions of the hyperbolic Cauchy problem ut + A(u)ux = 0,
u(0, x) = u ¯(x) .
(3.8)
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
21
In the conservative case A(u) = Df (u), every vanishing viscosity solution is a weak solution of ut + f (u)x = 0,
u(0, x) = u ¯(x) ,
(3.9)
satisfying the Liu admissibility conditions. Assuming, in addition, that each field is genuinely nonlinear or linearly degenerate, the vanishing viscosity solutions coincide with the unique limits of Glimm and front tracking approximations. In the genuinely nonlinear case, an estimate on the rate of convergence of these viscous approximations was provided in [BY]: Theorem 6. For the strictly hyperbolic system of conservation laws (3.9), assume that every characteristic field is genuinely nonlinear. At any time t > 0, the difference between the corresponding solutions of (3.2) and (3.9) can be estimated as ε √ u (t, ·) − u(t, ·) 1 = O(1) · (1 + t) ε| ln ε| Tot.Var.{¯ u} . L The following remarks relate Theorem 5 with previous literature.
(i) Concerning the global existence of weak solutions to the system of conservation laws (3.9), Theorem 5 contains the famous result of Glimm [G]. It also slightly extends the existence theorem of Liu [L4], which does not require the assumption (H). Its main novelty lies in the case where the system (3.8) is non-conservative. However, one should be aware that for non-conservative systems the limit solution might change if we choose a viscosity matrix different from the identity, say ut + A(u)ux = ε B(u)ux x .
The analysis of viscous solutions, with a more general viscosity matrix B(u), is still a wide open problem.
(ii) Concerning the uniform stability of entropy weak solutions, the results previously available for n × n hyperbolic systems [BC1], [BCP], [BLY] always required the assumption (H). For 2 × 2 systems, this condition was somewhat relaxed in [AM]. On the other hand, Theorem 5 established this uniform stability for completely general n × n strictly hyperbolic systems, without any reference to the assumption (H). (iii) For the viscous system (1.10), previous results in [L5], [SX], [SZ], [Yu] have established the stability of special types of solutions, such as traveling viscous shocks or viscous rarefactions, w.r.t. suitably small perturbations. Taking ε = 1, our present theorem yields at once the uniform Lipschitz stability of all viscous solutions with sufficiently small total variation, w.r.t. the
22
Alberto Bressan
L1 distance. On the other hand, our approach has not yet been used to study the stability and convergence of viscous solutions with large total variation. For a result in this direction using singular perturbations, see [R]. We give below a general outline of the proof of Theorem 5. The main steps will then be discussed in more detail in the following sections. 1. As a preliminary, we observe that uε is a solution of (3.2) if and only if . the rescaled function u(t, x) = uε (εt, εx) is a solution of the parabolic system with unit viscosity (3.10) ut + A(u)ux = uxx , with initial data u(0, x) = u ¯(εx). Clearly, the stretching of the space variable has no effect on the total variation. Notice however that the values of uε on a fixed time interval [0, T ] correspond to the values of u on the much longer time interval [0, T /ε]. To obtain the desired BV bounds for the viscous solutions uε , we can confine all our analysis to solutions of (3.10), but we need estimates uniformly valid for all times t ≥ 0 , depending only on the total variation of the initial data u ¯. 2. Consider a solution of (3.10), with initial data u(0, x) = u ¯(x)
(3.11)
having small total variation, say Tot.Var.{¯ u} ≤ δ0 . Using standard parabolic estimates, one can then prove that the Cauchy problem (3.10)–(3.11) has a unique solution, defined on an initial time interval [0, tˆ] with tˆ ≈ δ0−2 . For t ∈ [0, tˆ] the total variation remains small: Tot.Var. u(t) = ux (t) L1 = O(1) · δ0 , while all higher derivatives decay quickly. In particular δ0 uxx (t) 1 = O(1) · √ , L t
uxxx (t) 1 = O(1) · δ0 . L t
As long as the total variation remains small, one can prolong the solution also for larger times t > tˆ. In this case, uniform bounds on higher derivatives remain valid. 3. To establish uniform bounds on the total variation, valid for arbitrarily large times, we decompose the gradient along a basis of unit vectors r˜i = r˜i (u, ux , uxx ), say ux = vi r˜i (3.12) i
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
23
We then derive an evolution equation for these gradient components, of the form ˜ i vi )x − vi,xx = φi i = 1, . . . , n , (3.13) vi,t + (λ 4. A careful examination of the source terms φi in (3.13) shows that they arise mainly because of interactions among different viscous waves. Their total strength can thus be estimated in terms of suitable interaction potentials. This will allow us to prove the implication T φi (t, x) dx dt ≤ δ0 ux (t) 1 ≤ δ0 for all t ∈ [tˆ, T ], i tˆ L =⇒
T φi (t, x) dx dt = O(1) · δ02 ˆ t
Since the left hand side of (3.13) is in conservation t ˆ vi (t) + vi (t) L1 ≤ ux (t) L1 ≤ tˆ
i
i
i = 1, . . . , n .
(3.14) form, one has φi (s, x) dxds .
(3.15)
By the implication (3.14), for δ0 sufficiently small we obtain uniform bounds on the total strength of the source terms φi , and hence on the BV norm of u(t, ·).
5. Similar techniques can also be applied to a solution z = z(t, x) of the variational equation
zt + DA(u) · z ux + A(u)zx = zxx , (3.16)
which describes the evolution of a first order perturbation to a solution u of (3.11). Assuming that the total variation of u remains small, one proves an estimate of the form z(t, ·) 1 ≤ L z(0, ·) 1 for all t ≥ 0 , (3.17) L
L
valid for every solution u of (3.10) having small total variation and every L1 solution of the corresponding system (3.16).
6. Relying on (3.17), a standard homotopy argument yields the Lipschitz continuity of the flow of (3.10) w.r.t. the initial data, uniformly in time. Indeed, let any two solutions u, v of (3.10) be given (fig. 12). We can connect them by a smooth path of solutions uθ , whose initial data satisfy . uθ (0, x) = θu(0, x) + (1 − θ)v(0, x) , θ ∈ [0, 1] . The distance u(t, ·) − v(t, ·) L1 at any later time t > 0 is clearly bounded by the length of the path θ → uθ (t). In turn, this can be computed by integrating . the norm of a tangent vector. Calling z θ = duθ /dθ, each vector z θ is a solution
24
Alberto Bressan
of the corresponding equation (3.19), with u replaced by uθ . Using (3.18) we thus obtain u(t, ·) − v(t, ·) 1 ≤ 1 d uθ (t) 1 dθ = 1 z θ (t) 1 dθ dθ 0 0 L L L (3.18) 1 θ ≤ L 0 z (0) L1 dθ = L u(0, ·) − v(0, ·) L1 .
v(t)
zθ (t) u θ(t)
v(0) θ
zθ(0)
u (0) u(0)
u(t) Fig. 12.
7. By the simple rescaling of coordinates t → εt, x → εx, all of the above estimates remain valid for solutions uε of the system (3.2)ε . In particular this yields the bounds (3.4) and (3.5). 8. By a compactness argument, these BV bounds imply the existence of a strong limit uεm → u in L1loc , at least for some subsequence εm → 0. In the conservative case where A = Df , it is now easy to show that this limit u provides a weak solution to the Cauchy problem (3.9). 9. At this stage, it only remains to prove that the limit is unique, i.e. it does not depend on the choice of the sequence εm → 0. For a system in conservative form, and with the standard assumption (H) that each field is either genuinely nonlinear or linearly degenerate, we can apply the uniqueness theorem in [BG] and conclude that the limit of vanishing viscosity approximations coincides with the limit of Glimm and of front tracking approximations. 10. To handle the general non-conservative case, some additional work is required. We first consider Riemann initial data and show that in this special case the vanishing viscosity solution is unique and can be accurately described. Then we prove that any weak solution obtained as limit vanishing viscosity approximations is also a “viscosity solution” in the sense that it satisfies the local integral estimates (2.31)–(2.32), where U ♯ is now the unique non-conservative
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
25
solution of a Riemann problem. By an argument introduced in [B2], a Lipschitz semigroup is completely determined as soon as one specifies its local behavior for piecewise constant initial data. Characterizing its trajectories as “viscosity solutions” we thus obtain the uniqueness of the semigroup of vanishing viscosity limits.
4 Parabolic Estimates Following a standard approach, one can write the parabolic system (3.11) in the form ut − uxx = −A(u)ux (4.1)
regarding the hyperbolic term A(u)ux as a first order perturbation of the heat equation. The general solution of (4.1) can then be represented as t u(t) = G(t) ∗ u(0) − G(t − s) ∗ A(u(s) ux (s) ds 0
in terms of convolutions with the Gauss kernel
2 1 G(t, x) = √ e−x /4t . 2 πt
(4.2)
From the above integral formula, one can derive local existence, uniqueness and regularity estimates for solutions of (4.1). Since we shall be dealing with a solution u = u(t, x) having small total variation, a more effective representation is the following. Consider the state . u∗ = lim u(t, x) , (4.3) x→−∞
. which is independent of time. We then define the matrix A∗ = A(u∗ ) and let ∗ ∗ ∗ λi , ri , li be the corresponding eigenvalues and right and left eigenvectors, normalized as in (1.4). It will be convenient to use “•” to denote a directional . derivative, so that z • A(u) = DA(u) · z indicates the derivative of the matrix valued function u → A(u) in the direction of the vector z. The systems (3.10) and (3.16) can now be written respectively as ut + A∗ ux − uxx = A∗ − A(u) ux , (4.4) ∗ ∗ zt + A zx − zxx = A − A(u) zx − z • A(u) ux . (4.5)
Observe that, if u is a solution of (4.4), then z = ux is a particular solution of the variational equation (4.5). Therefore, as soon as one proves an a priori bound on zx or zxx , the same estimate will be valid also for the corresponding derivatives uxx , uxxx . In both of the equations (4.4), (4.5), we regard the right hand side as a perturbation of the linear parabolic system with constant coefficients
26
Alberto Bressan
wt + A∗ wx − wxx = 0 .
(4.6)
We denote by G∗ the Green kernel for (4.6), so that any solution admits the integral representation w(t, x) = G∗ (t, x − y) w(0, y) dy . The matrix valued function G∗ can be explicitly computed. If w solves (4.6), . then the i-th component wi = li∗ · w satisfies the scalar equation wi,t + λ∗i wi,x − wi,xx = 0 . Therefore wi (t) = G∗i (t) ∗ wi (0), where
1 (x − λ∗i t)2 G∗i (t, x) = √ exp − . 4t 2 πt
It is now clear that this Green kernel G∗ = G∗ (t, x) satisfies the bounds ∗ G (t) 1 ≤ κ , L
∗ κ Gx (t) 1 ≤ √ , L t
for some constant κ and all t > 0.
∗ Gxx (t) 1 ≤ κ , L t
(4.7)
In the following, we consider an initial data u(0, ·) having small total variation, but possibly discontinuous. We shall prove the local existence of solutions and some estimates on the decay of higher order derivatives. To get a feeling on this rate of decay, let us first take a look at the most elementary case. Example 4.1. The solution to the Cauchy problem for the heat equation 0 if x < 0, ut − uxx = 0 , u(0, x) = δ0 if x > 0, is computed explicitly as u(t, x) = δ0
0
∞
G(t, x − y) dy .
Observing that the Gauss kernel (4.2) satisfies G(t, x) = t−1/2 G(1, t−1/2 x), for every k ≥ 1 one obtains the estimates k k−1 ∂ ∂ −k/2 . ∂xk−1 G(t) ∞ ≤ ∂xk G(t) 1 = O(1) · t L L
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
27
In the present example we have ux (t, x) = δ0 G(t, x). Therefore δ0 ux (t) ∞ ≤ uxx (t) 1 = O(1) · √ , L L t
uxx (t) ∞ ≤ uxxx (t) 1 = O(1) · δ0 , L L t δ uxxx (t) ∞ = O(1) · √0 . L t t
(4.8) (4.9) (4.10)
In the remainder of this chapter, our analysis will show that the same decay rates hold for solutions of the perturbed equation (4.4), restricted to some initial interval [0, tˆ]. More precisely, let δ0 measure the order of magnitude of the total variation in the initial data, so that Tot.Var. ux (0, ·) = O(1) · δ0 . (4.11)
Then: • •
There exists an initial interval [0, tˆ], with tˆ = O(1) · δ0−2 on which the solution of (4.4) is well defined. Its derivatives decay according to the estimates (4.8)–(4.10). As long as the total variation remains small, say ux (t) 1 ≤ δ0 , (4.12) L
the solution can be prolonged in time. In this case, for t > tˆ the derivatives satisfy the bounds ux (t) ∞ , uxx (t) 1 = O(1) · δ02 , L L uxx (t) ∞ , uxxx (t) 1 = O(1) · δ03 , L L uxxx (t) ∞ = O(1) · δ04 . L
higher (4.13) (4.14) (4.15)
The situation is summarized in fig. 13. All the estimates drawn here as solid lines will be obtained by standard parabolic-type arguments. If any breakdown occurs in the solution, the first estimate that fails must be the one in (4.12), concerning the total variation. Toward a proof of global existence of solutions, the most difficult step is to show that (4.12) remains valid for all times t ∈ [tˆ, ∞[ (the broken line in fig. 13). This will require hyperbolic-type estimates, based on the local decomposition of the gradient ux as a sum of traveling waves, and on a careful analysis of all interaction terms. In the remainder of this section we work out a proof of local existence and a priori estimates for solutions of (4.4) and (4.5). Proposition 4.2 (Local existence). For δ0 > 0 suitably small, the following holds. Consider initial data
28
Alberto Bressan
δ0
ux u xx
1
L
u xxx ^t ~ δ− 2 ~
0
1
L
1
L
t
0
Fig. 13.
u(0, x) = u ¯(x) ,
z(0, x) = z¯(x) .
(4.16)
such that
δ0 , z¯ ∈ L1 . (4.17) 4κ Then the equations (4.4), (4.5) have unique solutions u = u(t, x), z = z(t, x) defined on the time interval t ∈ [0, 1]. Moreover Tot.Var.{¯ u} ≤
ux (t) 1 ≤ δ0 , L 2
z(t) 1 ≤ δ0 L 2
for all
t ∈ [0, 1] .
(4.18)
Proof. The couple (u, ux ), consisting of the solution of (4.2) together with its derivative, will be obtained as the unique fixed point of a contractive transformation. For notational convenience, we assume here . u∗ = lim u ¯(x) = 0. x→−∞
Of course this is not restrictive, since it can always be achieved by a translation of coordinates. Consider the space E of all mappings (u, z) : [0, 1] → L∞ × L1 ,
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
with norm
29
. (u, z) = sup max u(t) L∞ , z(t) L1 . E t
On E we define the transformation T (u, z) = (ˆ u, zˆ) by setting . ¯+ u ˆ(t) = G∗ (t) ∗ u . zˆ(t) = G∗x (t) ∗ u ¯+
t
0
t
0
G∗ (t − s) ∗ A∗ − A(u(s)) z(s) ds ,
G∗x (t − s) ∗ A∗ − A(u(s)) z(s) ds .
Of course, the above definition implies zˆ = u ˆx . The assumption Tot.Var.{¯ u} ≤ δ0 /4κ together with the bounds (4.7) yields
Moreover, if
then
∗ δ0 G (t) ∗ u ¯ L∞ ≤ G∗x (t) ∗ u ¯ L1 ≤ . 4
u(s) ∞ ≤ δ0 , L 2
z(s) 1 ≤ δ0 L 2
(4.19)
for all s ∈ [0, 1] ,
t
∗ ∗ G (t − s) ∗ A − A(u(s)) z(s) ds ∞ 0 L t κ DA u(s) L∞ z(s) L1 ds ≤ 0
≤ tκκA
δ02 4
and similarly t
∗ ∗ Gx (t − s) ∗ A − A(u(s)) z(s) ds 1 0 L t κ √ DA u(s) L∞ z(s) L1 ds ≤ t−s 0 √ δ2 ≤ 2 t κκA 0 . 4 Combining the above estimates with (4.19), we see that the transformation T maps the domain . D = (u, z) : [0, 1] → L∞ ×L1 ; u(t) L∞ , z(t) L1 ≤ δ0 /2 for all t ∈ [0, 1] into itself. To prove that T is a strict contraction, we compute the difference T (u, z) − T (u′ , z ′ ). The norm of the first component is estimated as
30
Alberto Bressan
t ∗ ′ u A∗ − A(u(s)) z(s) − z ′ (s) ˆ−u ˆ L∞ = G (t − s) ∗ 0
+ A(u′ (s)) − A(u(s)) z ′ (s) ds L∞ t ∗ G (t − s) ∞ DA u(s) ∞ z(s) − z ′ (s) 1 ≤ L L L 0 ′ ′ +DA u(s) − u (s) L∞ z (s) L1 ds √ ≤ 2 tκκA δ0 (u − u′ , z − z ′ ) . E
An entirely similar computation yields √ zˆ(t) − zˆ′ (t) 1 ≤ 2 tκκA δ0 (u − u′ , z − z ′ ) . L E
Therefore, for δ0 small enough, the map T is a strict contraction. By the contraction mapping theorem, a unique fixed point exists, inside the domain D. Clearly, this provides the solution of (4.4) with the prescribed initial data. Having constructed a solution u of (4.4), we now prove the existence of a solution z of the linearized variational system (4.5), with initial data z¯ ∈ L1 . Consider the space E ′ of all mappings z : [0, 1] → W 1,1 with norm √ . t zx (t) L1 . zE ′ = sup max z(t) L1 , t
′
On E we define the transformation T (z) = zˆ by setting t
. ∗ zˆ(t) = G (t)∗¯ G∗ (t−s)∗ A∗ −A(u(s)) zx (s)− z(s)•A(u(s)) ux (s) ds . z+ 0
The bounds (4.7) now yield ∗ G (t) ∗ z¯ 1 ≤ κ¯ z L1 , L
∗ κ Gx (t) ∗ z¯ 1 ≤ √ ¯ z L1 . L t
Moreover, using the identity t 1 1 1 ds = dσ = π < 4 s(t − s) σ(1 − σ) 0 0
(4.20)
(4.21)
one obtains the bound t
∗ ∗ Gx (t − s) ∗ A − A(u(s)) zx (s) − z(s) • A(u(s)) ux (s) ds 1 0 L t κ √ DA u(s) L∞ zx (s) L1 + DA zx (s) L1 ux (s) L1 ≤ t−s 0 ≤ 4κ κA δ0 zE ′ .
Assuming that δ0 is small enough, it is now clear that the linear transformation T ′ is a strict contraction in the space E ′ . Hence it has a unique fixed point, which provides the desired solution of (4.5). •
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
31
Having established the local existence of a solution, we now study the decay of its higher order derivatives. Proposition 4.3. Let u, z be solutions of the systems (4.4) and (4.5), satisfying the bounds ux (t) 1 ≤ δ0 , z(t) 1 ≤ δ0 , (4.22) L L
for some constant δ0 < 1 and all t ∈ [0, tˆ], where 2
1 . . ˆ , κA = sup DA + D2 A t= 400κ κA δ0 u
(4.23)
and κ is the constant in (4.7). Then for t ∈ [0, tˆ] the following estimates hold. uxx (t) 1 , zx (t) 1 ≤ 2κδ √0, L L t
2 uxxx (t) 1 , zxx (t) 1 ≤ 5κ δ0 , L L t 3 δ uxxx (t) ∞ , zxx (t) ∞ ≤ 16κ √ 0. L L t t
(4.24) (4.25) (4.26)
Proof. We begin with (4.24). The function zx can be represented in terms of convolutions with the Green kernel G∗ , as t ∗ G∗x (t − s) ∗ A∗ − A(u) zx (s) − z • A(u) ux (s) ds . zx (t) = Gx (t) ∗ z(0) + 0
Using (4.7) and (4.22) we obtain t ∗ ∗ Gx (t − s) ∗ A − A(u) zx (s) − z • A(u) ux (s) ds 1 0 L t ∗ Gx (t − s) L1 · ux (s) L1 DA L∞ zx (s) L1 ≤ 0 + z(s) L∞ DA L∞ ux (s) L1 ds t 1 zx (s) 1 ds . √ ≤ 2δ0 κ DA L∞ · L t−s 0
(4.27)
Consider first the case of smooth initial data. We shall argue by contradiction. If (4.24) fails for some t ≤ tˆ, by continuity will be a first time τ < tˆ at which (4.24) is satisfied as an equality. In this case, recalling the identity (4.21) we compute τ 1 2δ0 κ zx (τ ) 1 ≤ √κ δ0 + 2κδ0 DA ∞ · √ √ ds L L τ s τ − s 0 2κδ κδ0 0 2 2 < √ + 16κ κA δ0 DA L∞ ≤ √ , τ τ
32
Alberto Bressan
reaching a contradiction. Hence, (4.24) must be satisfied as a strict inequality for all t ∈ [0, tˆ]. A similar technique is used to establish (4.25). Indeed, we can write zxx (t) = G∗x (t/2) ∗ zx (t/2) t G∗x (t − s) ∗ z • A(u) ux (s) + A(u) − A∗ zx (s) ds . − t/2
x
(4.28) We prove (4.16) first in the case zxx = uxxx , then in the general case. If (4.25) is satisfied as an equality at a first time τ < tˆ, using (4.7), (4.24) and recalling the definitions of the constants tˆ, κA in (4.23), we compute τ κ 2κδ0 zx • A(u)ux (s) 1 zxx (τ ) 1 ≤ κ · √ · + L L τ − s τ /2 τ/2 τ /2 + z • ux • A(u) ux (s) L1 + z • A(u)uxx (s) L1 ∗ ds + ux • A(u)zx (s) L1 + A(u) − A zxx (s) L1 τ 2 2κ δ0 κ √ ≤ + · δ0 DAL∞ zxx (s) L1 τ /2 τ −s τ /2 2 +δ0 D2 AL∞ uxx (s) L1 + δ0 DAL∞ uxxx (s) L1 +δ0 DAL∞ zxx (s) L1 + δ0 DAL∞ zxx (s) L1 ds τ 1 4κ2 δ0 √ +κδ0 4κ2 δ02 D2 AL∞ +20κ2 δ0 DAL∞ ≤ ds τ s τ −s τ /2 4κ2 δ0 4 5κ2 δ0 + 20κ3 κA δ02 · . < < τ τ τ /2 Again, we conclude that at time τ the bound (4.16) is still satisfied as a strict inequality, thus reaching a contradiction.
Finally, assume that τ < tˆ is the first time where the bound (4.26) is satisfied as an equality. Using (4.7) and the previous bounds (4.24)–(4.25), from the representation (4.28) we now obtain τ 2 κ zxx (τ ) ∞ ≤ κ · 5κ δ0 + √ · zx • A(u)ux (s) L∞ L τ /2 τ − s τ /2 τ /2 + z • (ux • A(u))ux (s) L∞ + z • A(u)uxx (s) L∞ + ux • A(u)zx (s) L∞ + (A(u) − A∗ )zxx (s) L∞ ds √ τ 10 2κ3 δ0 4 3 2 1 4 2 √ √ leq + 8κ δ0 D A+46κ δ0 DA ds 3/2 τ − s τ τ s τ /2 3 3 16κ δ0 15κ δ0 4 √ . < ≤ √ + 46κ4 κA δ02 · τ /2 τ τ τ τ Therefore, the bound (4.26) must be satisfied as a strict inequality for all t ≤ tˆ. •
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
33
Corollary 4.4. In the same setting as Proposition 4.2, assume that the bounds (4.12) hold on a larger interval [0, T ]. Then for all t ∈ [tˆ, T ] there holds uxx (t) 1 , ux (t) ∞ , zx (t) 1 = O(1) · δ02 , (4.29) L L L uxxx (t) 1 , uxx (t) ∞ , zxx (t) 1 = O(1) · δ03 , (4.30) L L L 4 uxxx (t) ∞ , zxx (t) ∞ = O(1) · δ0 . (4.31) L L Proof. This follows by applying Proposition 4.3 on the interval [t − tˆ, t].
•
The next result provides a first estimate on the time interval where the solution u of (4.4) exists and remain small. Proposition 4.5. For δ0 > 0 sufficiently small, the following holds. Consider initial data u ¯, z¯ as in (4.16), satisfying the bounds δ0 Tot.Var. u ¯ ≤ , 4κ
¯ z L1 ≤
δ0 . 4κ
(4.32)
Then the systems (4.4), (4.5) admit unique solutions u = u(t, x), z = z(t, x) defined on the whole interval [0, tˆ], with tˆ defined at (4.23). Moreover, one has z(t)
ux (t) 1 ≤ δ0 , L 2
L1
≤
δ0 2
for all t ∈ ]0, tˆ]. (4.33)
Proof. Because of the local existence result proved in Proposition 4.2, it suffices to check that the bounds (4.32) remain valid for all t ≤ tˆ. The solution of (4.5) can be written in the form z(t) = G∗ (t) ∗ z¯ +
0
t
G∗ (t − s) ∗
A∗ − A(u) zx (s) − z • A(u) ux (s) ds .
As before, we first establish the result for z = ux , then for a general solution z of (4.5). Assume that there exists a first time τ < tˆ where the bound in (4.17) is satisfied as an equality. Using (4.7) and (4.24) and recalling the definition of tˆ at (4.23), from the above identity we obtain τ z(τ ) 1 ≤ κ ¯ 1 + κ · DA ux (s) L1 zx (s) L1 z L L 0 +DA zx (s) L1 ux (s) L1 ds τ κδ0 2κδ02 √ DA ds ≤ + 4κ s 0 √ δ0 δ0 2 + 4κ κA δ0 τ < , ≤ 4 2 reaching a contradiction.
•
34
Alberto Bressan
To simplify the proof, we assumed here the same bounds on the functions ux and z. However, observing that z solves a linear homogeneous equation, similar estimates can be immediately derived without any restriction on the initial size of z¯ ∈ L1 . In particular, from Proposition 4.5 it follows Corollary 4.6. Let u = u(t, x), z = z(t, x) be solutions of (4.4), (4.5) respectively, with ux (0) L1 ≤ δ0 /4κ. Then u, z are well defined on the whole interval [0, tˆ] in (4.13), and satisfy ux (t) 1 ≤ 2κ ux (0) 1 , z(t) 1 ≤ 2κ z(0) 1 , for all t ≤ tˆ. L L L L (4.34)
5 Decomposition by Traveling Wave Profiles Given a smooth function u : R → Rn , at each point x we want to decompose the gradient ux as a sum of gradients of viscous traveling waves, say ux (x) = vi r˜i = Ui′ (x) . (5.1) i
i
We recall that, for the parabolic system ut + A(u)ux = uxx ,
(5.2)
a traveling wave profile is a solution of the special form u(t, x) = U (x − σt). For the application that we have in mind, we can assume that the function u has small total variation, and that all of its derivatives are uniformly small. However, in order to come up with a unique decomposition of the form (1.10)– (1.11), we still face two major obstacles. 1. The system (1.9)–(1.11) is highly underdetermined. Indeed, by (1.9)–(1.10) each traveling profile Ui provides a solution to the Cauchy problem Ui′′ = A(Ui ) − σi Ui′ , Ui (x) = u(x) . (5.3)
Since this is a second order equation, we can still choose the derivative U ′ (x) ∈ Rn arbitrarily, as well as the speed σi . The general solution of (5.3) thus depends on n + 1 scalar parameters. Summing over all n families of waves, we obtain n(n + 1) free parameters. On the other hand, the system Ui′ (x) = ux (x) , Ui′′ (x) = uxx (x) (5.4) i
i
consists of only n + n scalar equations. When n ≥ 2, these are not enough to uniquely determine the traveling profiles U1 , . . . , Un .
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
35
2. For particular values of ux , uxx , the equations (5.3)–(5.4) may not have any solution at all. This can already be seen for a scalar conservation law ut + λ(u)ux = uxx . In this case, the above equations reduce to U ′′ = λ(U ) − σ U ′ , U ′ (x) = ux (x) ,
U (x) = u(x) ,
U ′′ (x) = uxx (x) .
The solution is now unique. The only choice for the speed is uxx (x) . σ = λ u(x) − ux (x)
The smallness assumption on ux , uxx is not of much use here. If ux = 0 while uxx = 0, no solution is found. This typically occurs at points where u has a local maximum or minimum. To overcome the first obstacle, we shall consider not the family of all traveling profiles, but only certain subfamilies. More precisely, for each i = 1, . . . , n, our decomposition will involve only those traveling profiles Ui which lie on a suitable center manifold. The second difficulty will be removed by introducing a cutoff function. In a normal situation, our decomposition will be uniquely determined by (5.3)– (5.4). In the exceptional cases where the cutoff function is active, only the first equation in (5.4) will be satisfied, allowing for some error in the second equation. 5.1. Construction of a Center Manifold To carry out our program, we begin by selecting certain families of traveling waves, depending on the correct number of parameters to fit the data. Observing that the system (5.4) consists of n + n scalar equations, this number of parameters is easy to guess. To achieve a unique solution, through any given state u we need to construct n families of traveling wave profiles Ui , each depending on two scalar parameters. The construction will be achieved by an application of the center manifold theorem. Traveling waves for the parabolic system (5.2) correspond to (possibly unbounded) solutions of A(U ) − σ U ′ = U ′′ . (5.5)
We write (5.5) as a first order system on the space Rn × Rn × R:
36
Alberto Bressan
⎧ ⎨ u˙ = v , v˙ = A(u) − σ v , ⎩ σ˙ = 0 .
(5.6)
Let a state u∗ be given andfix an index i ∈ {1, . . . , n}. Linearizing (5.6) at . the equilibrium point P ∗ = u∗ , 0, λi (u∗ ) we obtain the linear system ⎧ ⎨ u˙ = v , v˙ = A(u∗ ) − λi (u∗ ) v , (5.7) ⎩ σ˙ = 0 .
Let {r1∗ , . . . , rn∗ } and {l1∗ , . . . , ln∗ } be dual bases of right and left eigenvectors of A(u∗ ) normalized as in (1.4). We call (V1 , . . . , Vn ) the coordinates of a vector v ∈ Rn w.r.t. this basis, so that . |ri∗ | = 1, v= Vj rj∗ , Vj = lj∗ · v . j
The center subspace N for (5.7) (see fig. 14) consists of all vectors (u, v, σ) ∈ Rn × Rn × R such that Vj = 0
for all j = i,
(5.8)
and therefore has dimension n + 2. By the center manifold theorem (see the Appendix), there exists a smooth manifold M ⊂ Rn+n+1 , tangent to N at the stationary point P ∗ , which is locally invariant under the flow of (5.6). This manifold has dimension n + 2 and can be locally defined by the n − 1 equations j = i. (5.9) Vj = ϕj (u, Vi , σ)
M
N P* Fig. 14.
We seek a set of coordinates on this center manifold, involving n + 2 free parameters. As a preliminary, observe that a generic point on the center subspace N can be described as P = (u, v, σ) = (u, vi ri∗ , σ) .
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
37
We can thus regard (u, vi , σ) ∈ Rn+2 as coordinates of the point P . Notice that vi = ±|v| is the signed strength of the vector v, while σ is the wave speed. Similarly, we will show that a generic point on the center manifold M can be written as P = (u, v, σ) = (u, vi r˜i , σ) , where again vi is the signed strength of v and r˜i is a unit vector parallel to v. In general, this unit vector is not constant as in the linear case, but depends on u, vi , σ. All the information about the center manifold is thus encoded in the map (u, vi , σ) → r˜i (u, vi , σ). We can assume that the n − 1 smooth scalar functions ϕj in (5.9) are defined on the domain . D = (u, Vi , σi ) ; |u − u∗ | < ǫ, |Vi | < ǫ, σi − λi (u∗ )| < ǫ ⊂ Rn+2 , (5.10) for some small ǫ > 0. The tangency condition implies 2 ϕj (u, Vi , σ) = O(1) · |u − u∗ |2 + |Vi |2 + σ − λi (u∗ ) .
(5.11)
By construction, every trajectory
. t → P (t) = u(t), v(t), σ(t)
. remains within a small neighborhood of the point P ∗ = of ∗(5.6), which u , 0, λi (u∗ ) for all t ∈ R, must lie entirely on the manifold M. In particular, M contains all viscous i-shock profiles joining a pair of states u− , u+ suffi∗ ∗ ciently close to∗ u . Moreover, all equilibrium points (u, 0, σ) with |u − u | < ǫ and σ − λi (u ) < ǫ must lie on M. Hence ϕj (u, 0, σ) = 0
for all j = i.
(5.12)
By (5.12) and the smoothness of the functions ϕj , we can “factor out” the component Vi and write ϕj (u, Vi , σ) = ψj (u, Vi , σ) · Vi , for suitable smooth functions ψj . From (5.11) it follows ψj → 0 as (u, Vi , σ) → u∗ , 0, λi (u∗ ) .
On the manifold M we thus have ⎞ ⎛ . v= ψj (u, Vi , σ) rj∗ ⎠ = Vi ri♯ (u, Vi , σ). Vk rk∗ = Vi · ⎝ri∗ + k
(5.13)
(5.14)
j =i
By (5.13), the function r♯ defined by the last equality in (5.14) satisfies as (u, Vi , σ) → u∗ , 0, λi (u∗ ) . (5.15) ri♯ (u, Vi , σ) → ri∗
38
Alberto Bressan
Notice that, for a traveling profile Ui of the i-th family, the derivative v = Ui′ should be “almost parallel” to the eigenvector ri∗ . This is indeed confirmed by (5.15). We can now define the new variable . (5.16) vi = vi (u, Vi , σ) = Vi · ri♯ (u, Vi , σ) . ∗ As (u, Vi , σ) ranges in a small neighborhood of u , 0, λi (u∗ ) , by (5.15) the . vector ri♯ remains close to the unit vector ri∗ = ri (u∗ ). In particular, its length |ri♯ | remains uniformly positive. Hence the transformation Vi ←→ vi is invertible and smooth. We can thus reparametrize the center manifold M in terms of the variables (u, vi , σ) ∈ Rn × R × R. Moreover, we define the unit vector ♯ . r r˜i (u, vi , σ) = i♯ . |ri |
(5.17)
Observe that r˜i is also a smooth function of its arguments. With the above definitions, in alternative to (5.9) we can express the manifold M by means of the equation (5.18) v = vi r˜i (u, vi , σ) . The above construction of a center manifold can be repeated for every i = 1, . . . , n. We thus obtain n center manifolds Mi ⊂ Rn+n+1 and vector functions r˜i = r˜i (u, vi , σi ) such that |˜ ri | ≡ 1,
(5.19)
Mi = (u, v, σi ) ; v = vi r˜i (u, vi , σi ) , (5.20) ∗ n ∗ as (u, vi , σi ) ∈ R × R × R ranges in a neighborhood of u , 0, λi (u ) .
Next, we derive some identities for future use. The partial derivatives of r˜i = r˜i (u, vi , σi ) w.r.t. its arguments will be written as . ∂ r˜i,u = r˜i , ∂u
. ∂ r˜i,v = r˜i , ∂vi
. ∂ r˜i,σ = r˜i . ∂σi
Of course r˜i,u is an n × n matrix, while r˜i,v , r˜i,σ are n-vectors. Higher order derivatives are denoted as r˜i,uσ , r˜i,σσ . . . We claim that r˜i (u, 0, σi ) = ri (u)
for all u, σi .
Indeed, consider again the equation for a viscous traveling i-wave: uxx = A(u) − σi ux .
(5.21)
(5.22)
For a solution contained in the center manifold, taking the derivative w.r.t. x of
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
ux = v = vi r˜i (u, vi , σi )
39
(5.23)
and using (5.22) one finds vi,x r˜i + vi r˜i,x = A(u) − σi vi r˜i .
(5.24)
Since |˜ ri | ≡ 1, the vector r˜i is perpendicular to its derivative r˜i,x . Taking the inner product of (5.24) with r˜i we thus obtain ˜ i − σi )vi , vi,x = (λ
(5.25)
˜ i (u, vi , σi ) as the inner ˜i = λ where we defined the “generalized eigenvalue” λ product # . " ˜i = r˜i , A(u)˜ ri . (5.26) λ
Writing out the derivative
r˜i,x = r˜i,u ux + r˜i,v vi,x + r˜i,σ σx and using (5.25), (5.23) and the identity σx = 0 clearly valid for a traveling wave, from (5.24) we finally obtain ˜ i − σi )vi r˜i + vi r˜i,u r˜i vi + r˜i,v (λ ˜ i − σi )vi = A(u) − σi vi r˜i . (λ (5.27)
This yields a fundamental identity satisfied by our “generalized eigenvectors” r˜i , namely ˜ i − σi ) . ˜ i r˜i = vi r˜i,u r˜i + r˜i,v (λ (5.28) A(u) − λ Notice how (5.28) replaces the familiar identity A(u) − λi ri = 0
(5.29)
satisfied by the usual eigenvectors and eigenvalues of the matrix A(u). The presence of some small terms on the right hand side of (5.28) is of great importance. Indeed, in the evolution equations (3.13), these terms achieve a crucial cancellation with other source terms that would otherwise not be integrable.
Comparing (5.28) with (5.29) we see that, as vi → 0, the unit vector ˜ i (u, vi , σi ) r˜i (u, vi , σi ) approaches an eigenvector of the matrix A(u), while λ approaches the corresponding eigenvalue. By continuity, this establishes our claim (5.21). By (5.21), when vi = 0 the vector r˜i does not depend on the speed σi , hence r˜i,σ (u, 0, σi ) = 0. In turn, by the smoothness of the vector field r˜i we also have r˜i (u, vi , σi ) − ri (u) = O(1) · vi , r˜i,uσ = O(1) · vi ,
r˜i,σ = O(1) · vi , r˜i,σσ = O(1) · vi .
(5.30)
40
Alberto Bressan
Observing that the vectors r˜i,v and r˜i,σ are both perpendicular to r˜i , from (5.28) and the definition (5.26) we deduce ˜ i (u, vi , σi ) − λi (u) = O(1) · vi , λ
˜ i,σ = O(1) · vi . λ
(5.31)
We conclude our analysis of trajectories on the center manifold by proving the identity ˜ i vi )x − vi,xx = 0, −σi vi,x + (λ (5.32) valid for a traveling wave profile. Toward this goal, we first differentiate (5.24) and obtain vi,xx r˜i + 2vi,x r˜i,x + vi r˜i,xx = A(u)vi r˜i x − σi vi,x r˜i − σi vi r˜i,x . (5.33) From the identities " # r˜i , r˜i,x = 0 ,
"
# " # r˜i , r˜i,xx = − r˜i,x , r˜i,x ,
taking the inner product of (5.24) with r˜i,x we find " # # " r˜i , r˜i,xx vi = − r˜i,x , A(u)˜ ri vi .
(5.34)
On the other hand, taking the inner product of (5.33) with r˜i we find # # " " vi,xx + r˜i , r˜i,xx vi = r˜i , (A(u)˜ ri vi )x − σi vi,x . By (5.34), this yields (5.32).
The significance of the identity (5.32) can be explained as follows. Consider a very special solution of the parabolic system (5.2), say u = u(t, x) = Ui (x − σi t) consisting of a traveling i-wave on the center manifold. The corresponding decomposition (5.1) will then contain one single term: ux (t, x) = vi (t, x)˜ ri (t, x) = Ui′ (x − σi t) , while vj ≡ 0 for all other components j = i. We seek the evolution equation satisfied by the scalar component vi . Since we are dealing with a traveling wave, we have vi,t + σi vi,x = 0 . Inserting this in (5.32) we thus conclude ˜ i vi )x − vi,xx = 0. vi,t + (λ
(5.35)
In other words, in connection with a traveling wave profile, the source terms φi in (3.13) vanish identically.
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
41
5.2. Wave Decomposition Let u : R → Rn be a smooth function with small total variation. At each point x, we seek a decomposition of the gradient ux in the form (5.1), where r˜i = r˜i (u, vi , σi ) are the vectors defining the center manifold at (5.20). To uniquely determine the r˜i , we must first define the wave strengths vi and speeds σi in terms of u, ux , uxx . Consider first the special case where u is precisely the profile of a viscous traveling wave of the j-th family (contained in the center manifold Mj ). In this case, our decomposition should clearly contain one single component: ux = vj r˜j (u, vj , σj ) .
(5.36)
It is easy to guess what vj , σj should be. Indeed, since by construction |˜ rj | = 1, the quantity vj = ±|ux | is the signed strength of the wave. Notice also that for a traveling wave the vectors ux and ut are always parallel, because ut = −σj ux where σj is the speed of the wave. We can thus write ut = uxx − A(u)ux = ωj r˜j (u, vj , σj )
(5.37)
for some scalar ωj . The speed of the wave is now obtained as σj = −ωj /vj . Motivated by the previous analysis, as a first attempt we define . ut = uxx − A(u)ux and try to find scalar quantities vi , ωi such that ux = i vi r˜i (u, vi , σi ), ut = i ωi r˜i (u, vi , σi ),
(5.38)
σi = −
ωi . vi
(5.39)
The trouble with (5.39) is that the vectors r˜i are defined only for speeds σi . close to the i-th characteristic speed λ∗i = λi (u∗ ). However, when ux ≈ 0 one has vi ≈ 0 and the ratio ωi /vi may become arbitrarily large. To overcome this problem, we introduce a cutoff function (fig. 15). Fix δ1 > 0 sufficiently small. Define a smooth odd function θ : R → [−2δ1 , 2δ1 ] such that s if |s| ≤ δ1 θ(s) = |θ′ | ≤ 1, |θ′′ | ≤ 4/δ1 . (5.40) 0 if |s| ≥ 3δ1 We now rewrite (5.39) in terms of the new variable wi , related to ωi by . ωi = wi − λ∗i vi . We require that σi coincides with −ωi /vi only when this ratio . is sufficiently close to λ∗i = λi (u∗ ). Our basic equations thus take the form
42
Alberto Bressan
η (s)
1
θ(s) −δ1 δ
s 1
Fig. 15.
where
ux = i vi r˜i (u, vi , σi ), ut = i (wi − λ∗i vi ) r˜i (u, vi , σi ), σi = λ∗i − θ
ut = uxx − A(u)ux ,
(5.41)
wi vi
.
(5.42)
Notice that σi is not well defined when vi = wi = 0. However, recalling (5.21), in this case we have r˜i = ri (u) regardless of σi . Hence the two equations in (5.41) are still meaningful. Remark 5.1. By construction, the vectors vi r˜i are the gradients of viscous traveling waves Ui such that Ui (x) = u(x), Ui′ (x) = vi r˜i , Ui′′ = A(u) − σi Ui′ . From the first equation in (5.41) it follows ux (x) = Ui′ (x) . i
If σi = λ∗i − wi /vi for all i = 1, . . . , n, i.e. if none of the cutoff functions is active, then (wi − λ∗i vi )˜ ri + A(u) vi r˜i uxx (x) = ut + A(u)ux = i i A(u) − σi vi r˜i = Ui′′ (x) . = i
i
In this case, both of the equalities in (5.4) hold. Notice however that the second equality in (5.4) may fail if |wi /vi | > δ1 for some i.
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
43
We now show that the system of equations (5.41)–(5.42) uniquely defines the components vi , wi . Moreover, we study the smoothness of these components, as functions of u, ux , uxx . Lemma 5.2. For |u − u∗ |, |ux | and |uxx | sufficiently small, the system of 2n equations (5.41) has a unique solution (v, w) = (v1 , . . . , vn , w1 , . . . , wn ). . The map (u, ux , uxx ) → (v, w) is smooth outside the n manifolds Zi = (v, w) ; vi = wi = 0 ; moreover it is C 1,1 , i.e. continuously differentiable with Lipschitz continuous derivatives on a whole neighborhood of the point (u∗ , 0, 0). Proof. Consider the mapping Λ : Rn × Rn × Rn → R2n defined by n
. Λ(u, v, w) = Λi (u, vi , wi ),
(5.43)
i=1
. Λi (u, vi , wi ) =
vi r˜i u, vi , λ∗i − θ(wi /vi ) . (wi − λ∗i vi ) r˜i u, vi , λ∗i − θ(wi /vi )
(5.44)
Given u close to u∗ and (ux , uxx ) in a neighborhood of (0, 0) ∈ Rn+n , we need to find vectors v, w such that Λ(u, v, w) = ux , uxx − A(u)ux . (5.45)
This will be achieved by applying the implicit function theorem. Observe that each map Λi is well defined and continuous also when vi = 0, because in this case (5.21) implies r˜i = ri (u). Computing the Jacobian matrix of partial derivatives w.r.t. (vi , wi ) we find % $ ∂Λi = ∂(vi , wi )
$
r˜i ,
0
+
−λ∗i r˜i , r˜i
vi r˜i,v + (wi /vi )θi′ r˜i,σ ,
−θi′ r˜i,σ
%
wi r˜i,v − λ∗i vi r˜i,v − λ∗i (wi /vi )θi′ r˜i,σ + (wi /vi )2 θi′ r˜i,σ , λ∗i θi′ r˜i,σ − (wi /vi )θi′ r˜i,σ . &i . = Bi + B (5.46)
Here and throughout the following, by θi , θi′ we denote the function θ and its derivative, evaluated at the point s = wi /vi . Notice that the matrix valued &i is well defined and continuous also when vi = 0. Indeed, either function B |wi /vi | > 2δ1 , in which case θi′ = 0, or else |wi | ≤ 2δ1 |vi |. In this second case, by (5.30) we have r˜i,σ = O(1) · vi . In all cases we have the estimate &i (u, vi , wi ) = O(1) · vi . B
(5.47)
According to (5.46), we can split the Jacobian matrix of partial derivatives of Λ as the sum of two matrices:
44
Alberto Bressan
∂Λ & v, w) . = B(u, v, w) + B(u, ∂(v, w)
(5.48)
& → 0 as (v, w) → For (v, w) small, B has a uniformly bounded inverse, while B 2n 0. Since Λ(u, 0, 0) = (0, 0) ∈ R , by the implicit function theorem we conclude that the map (v, w) → Λ(u, v, w) is C 1 and invertible in a neighborhood of the origin, for |u−u∗ | suitably small. We shall write Λ−1 for this inverse mapping. In other words, Λ−1 (u, ux , ut ) = (v, w)
if and only if
Λ(u, v, w) = (ux , ut ) .
Having proved the existence and uniqueness of the decomposition, it now remains to study its regularity. Since the cutoff function θ vanishes for |s| ≥ . 3δ1 , it is clear that each Λi is smooth outside the manifold Zi = (v, w) ; vi = wi = 0 having codimension 2. We shall not write out the second derivatives of the maps Λi explicitly. However, it is clear that ∂2Λ ∂2Λ ∂2Λ = = =0 ∂vi ∂vj ∂vi ∂wj ∂wi ∂wj
if i = j .
(5.49)
Moreover, recalling that θi ≡ 0 for |wi /vi | > 2δi and using the bounds (5.30), we have the estimates ∂2Λ ∂2Λ ∂2Λ , , = O(1) . 2 ∂vi ∂vi ∂wi ∂wi2
(5.50)
In other words, all second derivatives exist and are uniformly bounded outside the n manifolds Zi . Therefore, Λ is continuously differentiable with Lipschitz continuous first derivatives on a whole neighborhood of the point (u∗ , 0, 0). • Clearly the same holds for the inverse mapping Λ−1 . Remark 5.3. By possibly performing a linear transformation of variables, we can assume that the matrix A(u∗ ) is diagonal, hence its eigenvectors r1∗ , . . . rn∗ form an orthonormal basis: " ∗ ∗# (5.51) ri , rj = δij .
Observing that
r˜i (u, vi , σi ) − ri∗ = O(1) · |u − u∗ | + |vi | ,
from (5.21) and the above assumption we deduce " # r˜i (u, vi , σi ), r˜j (u, vj , σj ) = δij + O(1) · |u − u∗ | + |vi | + |vj | = δij + O(1) · δ0 , " # r˜i , A(u)˜ rj = O(1) · δ0 j = i .
(5.52)
(5.53) (5.54)
BV Solutions to Hyperbolic Systems by Vanishing Viscosity
45
Another useful consequence of (5.51)–(5.52) is the following. Choosing the bound on the total variation δ0 > 0 small enough, the vectors r˜i will remain close the an orthonormal basis. Hence the decomposition (5.1) will satisfy √ (5.55) |vi | ≤ 2 n|ux | . |ux | ≤ i
5.3. Bounds on Derivative Components Relying on the bounds on the derivatives of u stated in Corollary 4.4, in this last section we derive some analogous estimates, valid for the components vi , wi and their derivatives. Lemma 5.4. In the same setting as Proposition 4.2, assume that the bound ux (t) 1 ≤ δ0 L
holds on a larger time interval [0, T ]. Then for all t ∈ [tˆ, T ], the decomposition (5.41) is well defined. The components vi , wi satisfy the estimates vi (t) 1 , wi (t) 1 = O(1) · δ0 , (5.56) L L vi (t) ∞ , wi (t) ∞ , vi,x (t) 1 , wi,x (t) 1 = O(1) · δ02 , (5.57) L L L L vi,x (t) ∞ , wi,x (t) ∞ = O(1) · δ03 . (5.58) L L
Proof. By Lemma 5.2, in a neighborhood of the origin the map (v, w) → Λ(u, v, w) in (5.43) is well defined, locally invertible, and continuously differentiable with Lipschitz continuous derivatives. Hence, for δ0 > 0 suitably small, the L∞ bounds in (4.29) and (4.30) guarantee that the decomposition (5.41) is well defined. From the identity (5.45) we deduce vi , wi = O(1) · |ux | + |uxx | . By (4.12) and (4.29)–(4.30) this yields the L1 bounds in (5.56) and the L∞ bounds in (5.57). Differentiating (5.45) w.r.t. x we obtain ∂Λ ∂Λ ux + (vx , wx ) = uxx , uxxx − A(u)uxx − ux • A(u) ux . (5.59) ∂u ∂(v, w)
Using the estimate
∂Λ = O(1) · |v| + |w| , ∂u since the derivative ∂Λ/∂(v, w) has bounded inverse, from (5.59) we deduce (vx , wx ) = O(1) · |uxx | + |uxxx | + |ux |2 + |ux | |v| + |w| .
Together with the bounds (4.29)–(4.31), this yields the remaining L1 estimates in (5.57) and the L∞ estimates in (5.58). •
46
Alberto Bressan
Corollary 5.5. With the same assumptions as Lemma 5.2, one has the estimates r˜i,x (t) 1 , ˜ λi,x (t) L1 = O(1) · δ0 , (5.60) L 2 r˜i,x (t) ∞ , ˜ λi,x (t) L∞ = O(1) · δ0 . (5.61) L Proof. Recalling (5.30) we have r˜i,x = vj r˜i,u r˜j + vi,x r˜i,v + σi,x r˜i,σ j
= O(1) ·
= O(1) ·
j j
|vj | + O(1) · |vi,x | + O(1) · |σi,x | |vi | |vj | + O(1) · |vi,x | + |wi,x | .
The estimate for r˜i,x thus follows from the corresponding bounds in Lemma ˜ i,x = O(1) · |˜ 5.2. Clearly λ ri,x |, hence it satisfies the same bounds. • In the remaining part of this section we shall examine the relations between wi and vi,x . Due to the presence of a cutoff function, the results will depend on the ratio |wi /vi |. It is convenient to recall here the bounds ˜ i − λ∗ | = O(1) · |˜ |λ ri − r∗ | = O(1) · δ0 , |vi | , |wi | = O(1) · δ 2 , (5.62) i
i
0
which follow from (5.52) and (5.57). Moreover, one should keep in mind our choice of the constants 0 < δ0
