Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors: Advances in Partial Differential Equations

Operator Theory: Advances and Applications Vol. 184 Editor: I. Gohberg Editorial Office: School of Mathematical Sciences...

Author: Yuming Qin

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Operator Theory: Advances and Applications Vol. 184 Editor: I. Gohberg Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel

Editorial Board: D. Alpay (Beer Sheva, Israel) J. Arazy (Haifa, Israel) A. Atzmon (Tel Aviv, Israel) J.A. Ball (Blacksburg, VA, USA) H. Bart (Rotterdam, The Netherlands) A. Ben-Artzi (Tel Aviv, Israel) H. Bercovici (Bloomington, IN, USA) A. Böttcher (Chemnitz, Germany) K. Clancey (Athens, GA, USA) R. Curto (Iowa, IA, USA) K. R. Davidson (Waterloo, ON, Canada) M. Demuth (Clausthal-Zellerfeld, Germany) A. Dijksma (Groningen, The Netherlands) R. G. Douglas (College Station, TX, USA) R. Duduchava (Tbilisi, Georgia) A. Ferreira dos Santos (Lisboa, Portugal) A.E. Frazho (West Lafayette, IN, USA) P.A. Fuhrmann (Beer Sheva, Israel) B. Gramsch (Mainz, Germany) H.G. Kaper (Argonne, IL, USA) S.T. Kuroda (Tokyo, Japan) L.E. Lerer (Haifa, Israel) B. Mityagin (Columbus, OH, USA)

V. Olshevski (Storrs, CT, USA) M. Putinar (Santa Barbara, CA, USA) A.C.M. Ran (Amsterdam, The Netherlands) L. Rodman (Williamsburg, VA, USA) J. Rovnyak (Charlottesville, VA, USA) B.-W. Schulze (Potsdam, Germany) F. Speck (Lisboa, Portugal) I.M. Spitkovsky (Williamsburg, VA, USA) S. Treil (Providence, RI, USA) C. Tretter (Bern, Switzerland) H. Upmeier (Marburg, Germany) N. Vasilevski (Mexico, D.F., Mexico) S. Verduyn Lunel (Leiden, The Netherlands) D. Voiculescu (Berkeley, CA, USA) D. Xia (Nashville, TN, USA) D. Yafaev (Rennes, France) Honorary and Advisory Editorial Board: L.A. Coburn (Buffalo, NY, USA) H. Dym (Rehovot, Israel) C. Foias (College Station, TX, USA) J.W. Helton (San Diego, CA, USA) T. Kailath (Stanford, CA, USA) M.A. Kaashoek (Amsterdam, The Netherlands) P. Lancaster (Calgary, AB, Canada) H. Langer (Vienna, Austria) P.D. Lax (New York, NY, USA) D. Sarason (Berkeley, CA, USA) B. Silbermann (Chemnitz, Germany) H. Widom (Santa Cruz, CA, USA)

Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze Universität Potsdam, Germany Sergio Albeverio Universität Bonn, Germany Michael Demuth Technische Universität Clausthal, Germany

Jerome A. Goldstein The University of Memphis, TN, USA Nobuyuki Tose Keio University, Yokohama, Japan

Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors Yuming Qin

Birkhäuser Basel · Boston · Berlin

A P D E

Advances in Partial Differential Equations

Author: Yuming Qin Donghua University College of Science 201620 Shanghai People’s Republic of China e-mail: [email protected]

2000 Mathematical Subject Classification: 35Bxx, 35Lxx, 35Qxx, 35-99, 74, 76Nxx, 76-99

Library of Congress Control Number: 2008927171

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-7643-8813-3 Birkhäuser Verlag AG, Basel - Boston - Berlin 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2008 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-7643-8813-3

e-ISBN 978-3-7643-8814-0

987654321

www.birkhauser.ch

To my Parents Zhenrong Qin and Xilan Xia

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Preliminary 1.1 Sobolev Spaces and Their Basic Properties . . . . . . . . . . . . . . 1.1.1 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Weak Derivatives and Sobolev Spaces . . . . . . . . . . . . 1.1.3 Sobolev Inequalities, Embedding Theorems and the Trace Theorem . . . . . . . . . . . . . . . . . . . . 1.1.4 Interpolation Inequalities . . . . . . . . . . . . . . . . . . . 1.1.5 The Poincaré´ Inequality . . . . . . . . . . . . . . . . . . . 1.2 Some Inequalities in Analysis . . . . . . . . . . . . . . . . . . . . 1.2.1 The Classical Bellman-Gronwall Inequality . . . . . . . . . 1.2.2 The Generalized Bellman-Gronwall Inequalities . . . . . . 1.2.3 The Uniform Bellman-Gronwall Inequality . . . . . . . . . 1.2.4 The Nakao Inequalities . . . . . . . . . . . . . . . . . . . . 1.3 Some Differential Inequalities for Nonexistence of Global Solutions 1.4 Other Useful Inequalities . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Young Inequalities . . . . . . . . . . . . . . . . . . . . 1.4.2 The Hö¨ lder Inequality . . . . . . . . . . . . . . . . . . . . 1.4.3 The Minkowski Inequalities . . . . . . . . . . . . . . . . . 1.4.4 The Jensen Inequality . . . . . . . . . . . . . . . . . . . . 1.5 C0 -Semigroups of Linear Operators . . . . . . . . . . . . . . . . . 1.5.1 C0 -Semigroups of Linear Operators . . . . . . . . . . . . . 1.6 Global Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Compact Semigroups (Semiflows) for Autonomous Systems 1.6.2 Weakly Compact Semigroups (Semiflows) for Autonomous Systems . . . . . . . . . . . . . . . . . . . . 1.7 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . .

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9 17 17 18 18 19 20 23 25 26 27 28 29 30 31 31 37 39

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2 A One-dimensional Nonlinear Viscous and Heat-conductive Real Gas 2.1 Fixed and Thermally Insulated Boundary Conditions . . . . . . . . 2.1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Uniform A Priori Estimates . . . . . . . . . . . . . . . . . 2.2 Clamped and Constant Temperature Boundary Conditions . . . . . 2.3 Exponential Stability in H 1 and H 2 . . . . . . . . . . . . . . . . . 2.3.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Exponential Stability in H 1 . . . . . . . . . . . . . . . . . 2.3.3 Exponential Stability in H 2 . . . . . . . . . . . . . . . . . 2.4 Exponential Stability in H 4 . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Global Existence in H 4 . . . . . . . . . . . . . . . . . . . . 2.4.2 A Nonlinear C0 -Semigroup S(t) on H 4 . . . . . . . . . . . 2.4.3 Exponential Stability in H 4 . . . . . . . . . . . . . . . . . 2.5 Attractors in H 1 and H 2 . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 An Absorbing Set in H 1 . . . . . . . . . . . . . . . . . . . 2.5.2 An Absorbing Set in H 2 . . . . . . . . . . . . . . . . . . . 2.6 Universal Attractor in H 4 . . . . . . . . . . . . . . . . . . . . . . . 2.7 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . .

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46 46 49 71 78 78 80 89 97 100 111 119 123 126 132 135 138

3 A One-dimensional Polytropic Viscous and Heat-conductive Gas 3.1 Initial Boundary Value Problems . . . . . . . . . . . . . . . . . 3.1.1 Global Existence and Asymptotic Behavior of Solutions 3.1.2 Exponential Stability . . . . . . . . . . . . . . . . . . . 3.1.3 Universal Attractors . . . . . . . . . . . . . . . . . . . 3.2 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Global Existence in H 2(R) . . . . . . . . . . . . . . . . 3.2.2 Large-Time Behavior of Solutions . . . . . . . . . . . . 3.3 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . .

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143 143 153 154 154 154 159 164

4 A Polytropic Ideal Gas in Bounded Annular Domains in Rn 4.1 Global Existence and Asymptotic Behavior in H 1 and H 2 4.1.1 Uniform A Priori Estimates in H 1 . . . . . . . . . 4.1.2 Uniform a priori estimates in H 2 . . . . . . . . . . 4.1.3 Results in Eulerian Coordinates . . . . . . . . . . 4.2 Exponential Stability in H 4 . . . . . . . . . . . . . . . . . 4.2.1 Main Results . . . . . . . . . . . . . . . . . . . . 4.2.2 Global Existence in H 4 . . . . . . . . . . . . . . . 4.2.3 A Nonlinear C0 -Semigroup S(t) on H 4 . . . . . . 4.2.4 Exponential Stability in H 4 . . . . . . . . . . . . 4.3 Universal Attractors . . . . . . . . . . . . . . . . . . . . . 4.3.1 Nonlinear Semigroups on H 2 . . . . . . . . . . .

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167 175 187 199 200 200 202 211 222 227 230

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4.3.2

4.4

(1)

Existence of an Absorbing Set in Hδ

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4.3.3

Existence of an Absorbing Set in

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4.3.4

Results of the Eulerian Coordinates . . . . . . . . . . . . . . . . 241

4.3.5

Attractor in H 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Hδ(2)

Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 243

5 A Polytropic Viscous Gas with Cylinder Symmetry in R3 5.1

Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

5.2

Global Existence and Exponential Stability in H 1 . . . . . . . . . . . . . 249

5.3


5.4


5.5

5.4.1

Global Existence of Solutions in H 4 . . . . . . . . . . . . . . . . 268

5.4.2

Exponential Stability in H+4 . . . . . . . . . . . . . . . . . . . . 285


6 One-dimensional Nonlinear Thermoviscoelasticity 6.1

Global Existence and Asymptotic Behavior of Solutions . . . . . . . . . 293

6.2

Uniform A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 297

6.3

Exponential Stability and Maximal Attractors . . . . . . . . . . . . . . . 325

6.4 6.5

Exponential Stability in H 1 and H 2 . . . . . . . . . . . . . . . . . . . . 331 Exponential Stability in H 4 . . . . . . . . . . . . . . . . . . . . . . . . . 332

6.6

Universal Attractors in H i (i = 1, 2, 4) . . . . . . . . . . . . . . . . . . 332

6.7

6.6.1

Existence of An Absorbing Set in Hδ1 . . . . . . . . . . . . . . . 332

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6.6.3



7 A Nonlinear One-dimensional Thermoelastic System with a Thermal Memory 7.1

Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

7.2

Global Existence and Exponential Stability . . . . . . . . . . . . . . . . 342

7.3


8 One-dimensional Thermoelastic Equations of Hyperbolic Type 8.1

Global Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

8.2

Global Existence and Exponential Stability . . . . . . . . . . . . . . . . 365

8.3


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9 Blow-up for the Cauchy Problem in Nonlinear One-dimensional Thermoelasticity 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Main Results – Case I . . . . . . . . . . . . . . . . . . . . . . 9.3 Main Results – Case II . . . . . . . . . . . . . . . . . . . . . 9.4 Bibliographic Comments . . . . . . . . . . . . . . . . . . . .

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381 382 399 408

10 Large-Time Behavior of Energy in Multi-Dimensional Elasticity 10.1 Polynomial Decay of Energy . . . . . . . . . . . . . . . . . . 10.1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Proof of Theorem 10.1.3 . . . . . . . . . . . . . . . . 10.2 Exponential Decay of Energy . . . . . . . . . . . . . . . . . . 10.2.1 Main Results . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Proof of Theorem 10.2.3 . . . . . . . . . . . . . . . . 10.3 Bibliographic Comments . . . . . . . . . . . . . . . . . . . .

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409 411 413 421 421 426 433

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

Preface This book is designed to present some recent results on some nonlinear parabolic-hyperbolic coupled systems arising from physics, mechanics and material science such as the compressible Navier-Stokes equations, thermo(visco)elastic systems and elastic systems. Some of the content of this book is based on research carried out by the author and his collaborators in recent years. Most of it has been previously published only in original papers, and some of the material has never been published until now. Therefore, the author hopes that the book will benefit both the interested beginner in the field and the expert. All the models under consideration in Chapters 2–10 are built on nonlinear evolution equations that are parabolic-hyperbolic coupled systems of partial differential equations with time t as one of the independent variables. This type of partial differential equations arises not only in many fields of mathematics, but also in other branches of science such as physics, mechanics and materials science, etc. For example, some models studied in this book, such as the compressible Navier-Stokes equations (a 1D heat conductive viscous real gas and a polytropic ideal gas) from fluid mechanics, and thermo(visco)elastic systems from materials science, are typical examples of nonlinear evolutionary equations. It is well known that the properties of solutions to nonlinear parabolic-hyperbolic coupled systems are very different from those of parabolic or hyperbolic equations. Since the 1970s, more and more mathematicians have begun to focus their interests on the study of local well-posedness, global well-posedness and blow-up of solutions in a finite time. Local well-posedness means that, for any given initial datum, a solution exists locally in time, and if it exists locally in time, it is unique and stable in some sense in the considered class. Generally speaking, we have two powerful tools to derive the local existence of solutions to a wide of class of nonlinear evolutionary equations, i.e., the contraction mapping theorem and the Leray-Schauder fixed point theorem. Once a local solution in some sense has been established, we may talk about the global well-posedness of solutions, i.e., the global-in-time existence, uniqueness and stability of global solutions. Since the 1960s, many methods of studying global well-posedness have been developed, among which are two powerful tools to derive the global existence of solutions; one is continuation of local solutions, the other is the global iteration method. In the 1980s, more interest was focused on the global existence of “small solutions”. However, knowledge about the global existence of a “small solution” is usually far from being enough for physical and mechanical problems. Thus we have to look for global

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solutions with arbitrary (not necessarily small) initial data. It turns out that an important step is to derive uniform a priori estimates on the solutions by using the special constitutive relations of the equations under consideration. Once global existence and uniqueness have been established, then the main interest should be focused on topics related to the asymptotic behavior of solutions, multiplicity of equilibria, convergence to an equilibrium, dynamical systems such as absorbing sets, the maximal compact attractor, etc. The study of asymptotic behavior of solutions can be divided into two categories. The first category comprises investigations of asymptotic behavior of the global solution for any given initial datum. The second category comprises investigations of asymptotic behavior of all solutions when the initial data vary in any bounded set. There are essential differences between these two categories. The first category deals with only one orbit starting from the datum in the phase space, while the second category deals with a family of orbits starting from any bounded set in the phase space. For the basic theories of infinite-dimensional dynamical systems, we refer readers to the works by Babin [16], Babin and Vishik [17, 18], Ball [22, 23], Bernard and Wang [38], Chepyzhov, Gatti, Grasselli, Miranville and Pata [56], Chepyzhov and Vishik [57], Constantin and Foias [63], Constantin, Foias and Temam [64], Dlotko [84], Eden and Kalantarov [90], Edfendiev, Zelik and Miranville n [92], Feireisl [97, 98, 100], Feireisl and Petzeltova [101, 102], Ghidaglia [117, 118], Ghidaglia and Temam [119], Goubet [125], Goubet and Moise [126], Hale [135], Hale and Perissinotto [136], Haraux [138], Hoff and Ziane [150, 151], Ladyzhenskaya [207], Liu and Zheng [240], Lu, Wu and Zhong [242], Ma, Wang and Zhong [246], Miranville [265, 266], Miranville and Wang [267], Moise and Rosa [269], Moise, Rosa and Wang [270], Pata and Zelik [307], Robinson [362], Rosa [363], Sell [369], Sell and You [370, 371], Temam [407], Vishik and Chepyzhov [413, 414], Wang [421], Wang, Zhong and Zhou [422], Wu and Zhong [429], Zhao and Zhou [445], Zheng [450], Zheng and Qin [451, 452], Zhong, Yang and Sun [457], and references therein. There are 10 chapters in this book. Chapter 1 is a preliminary chapter in which we collect some basic results from nonlinear functional analysis, basic properties of Sobolev spaces, some differential and integral inequalities in analysis, the basic theory of semigroups of linear operators and the basic theory for global attractors. Some results in this chapter will be used in the subsequent chapters, other results, though not used in the subsequent chapters, will be very beneficial to the readers for further study. The first topic studied in this book is compressible Navier-Stokes equations which describe the fluid motion of conservation of mass, momentum and energy. Chapters 2–5 are devoted to the study of this challenging topic. Chapter 2 will concern the global existence, asymptotic behavior of solutions and the existence of universal attractors for the compressible Navier-Stokes equations of a nonlinear 1D viscous and heat-conductive real gas. In Chapter 3, we shall establish the global existence, asymptotic behavior of solutions to initial boundary value problems and the Cauchy problem of the compressible NavierStokes equations of a 1D polytropic viscous and heat-conductive gas. In Chapter 4, we shall investigate the global existence, asymptotic behavior of solutions and the existence of maximal attractors for the compressible Navier-Stokes equations of a polytropic vis-

Preface

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cous and heat-conductive gas in bounded annular domains in Rn (n = 2, 3). Chapter 5 will be concerned with the global existence and asymptotic behavior of solutions to a polytropic viscous and heat-conductive gas with cylinder symmetry in R3 . For the compressible Navier-Stokes equations, we consult the works by Duan, Yang and Zhu [87], Ducomet and Zlotnik [88], Feireisl and Petzeltova [103], Feireisl, Novotny and Petzeltova [104], Frid and Shelukhin [106], Fujita-Yashima and Benabidallah [110, 111], Fujita-Yashima, Padula and Novotny [112], Galdi [115], Hoff [142–146], Hoff and Serre [147], Hoff and Smoller [148], Hoff and Zarnowski [149], Hsiao and Luo [158], Huang, Matsumura and Xin [160], Itaya [161], Jiang [164–167, 169–171], Jiang and Zhang [174–177], Jiang and Zlotnik [178], Kanel [182], Kawashima [188, 189], Kawashima, Nishibata and Zhu [190], Kawashima and Nishida [191], Kawohl [192], Kazhikhov [193–195], LeFloch and Shelukhin [219], Lions [235], Matsumura [252], Matsumura and Nishida [253–257], Nagasawa [283–287], Novotny and Stra˘s˘ kraba [301, 302], Okada and Kawashima [303], Padula [305], Qin [323, 325, 326], Qin and Hu [329], Qin, Huang and Ma [330], Qin and Jiang [331], Qin and Kong [332], Qin, Ma, Cavalcanti and Andrade [335], Qin, Ma and Huang [336], Qin, Muñ˜ oz Rivera [337, 339], Qin and Song [343], Qin and Wen [344], Qin, Wu and Liu [345], Qin and Zhao [346], Valli and Zajaczkowski [412], and the references therein. The second topic studied in this book is a 1D thermoviscoelastic system which describes the motion of conservation of mass, momentum and energy in the thermoviscoelastic media. Chapter 6 will be devoted to the study of global existence, asymptotic behavior and the existence of universal attractors for a 1D thermoviscoelastic model in materials science. The third topic considered in this book is that of some viscoelastic models. In Chapter 10, we shall obtain the large-time behavior of energy of multi-dimensional nonhomogeneous anisotropic elastic system. For the related (thermo)(visco)elastic models, we refer to Andrews [12], Andrews and Ball [13], Chen and Hoffmann [54], Coleman and Gurtin [62], Dafermos [69, 75, 76], Dafermos and Nohel [79, 80], Fabrizio and Lazzari [95], Giorgi and Naso [121], Greenberg and MacCamy [129], Guo and Zhu [132], Kim [197], Lagnese [209], Liu and ´ and Sprekels [293], Niezgodka, ´ Zheng and Sprekels [294], Zheng [239, 240], Niezgodka Qin, Ma and Huang [336], Racke and Zheng [355], Renardy, Hrusa and Nohel [361], Shen and Zheng [373], Shen, Zheng and Zhu [376], Shibata [377], Sprekels and Zheng [390, 391], Sprekels, Zheng and Zhu [392], Watson [424], Zheng [447, 448, 450], Zheng and Shen [453, 454], Zhu [460], and the references therein. The fourth topic under consideration is an investigation of a classical 1D thermoelastic model. Such a model describes the elastic and the thermal behavior of elastic, heat conductive media, in particular the reciprocal actions between elastic stresses and temperature differences. The classical thermoelastic system is such a thermoelastic model that the elastic part is the usual second-order one in the space variable and the heat flux obeys Fourier’s law, which means that the heat flux is proportional to the temperature gradient. In Chapter 7, we shall establish the global existence and exponential stability of solutions to a 1D classical thermoelastic system of equations with a thermal memory. In

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Chapter 9, we shall study the blowup phenomena of solutions to the Cauchy problem of a 1D non-autonomous classical thermoelastic system. There is much literature on classical thermoelastic model; we refer the readers to Burns, Liu and Zheng [46], Dafermos [67], Dafermos and Hsiao [78], Hale and Perissinotto [136], Hansen [137], Hoffmann and Zochowski [153], Hrusa and Messaoudi [155], Hrusa and Tarabek [156], Jiang, Muñ˜ oz Rivera and Racke [172], Jiang and Racke [173], Kim [198], Kirane and Kouachi and Tatar [199], Kirane and Tatar [200], Lebeau and Zuazua [216], Liu and Zheng [238, 240], Messaoudi [260], Muñ˜ oz Rivera [274, 275], Munoz ˜ Rivera and Barreto [277], Muñoz ˜ Rivera and Oliveira [278], Muñoz ˜ Rivera and Qin [279], Qin [315], Qin and Muñ˜ oz Rivera [341], Racke [348], Racke and Zheng [355], Slemord [378], Zheng [450], and the references therein. Recently, Green and Naghdi [127, 128] re-examined the classical thermoelastic models and introduced the so-called models of thermoelasticity of types II and III for which the heat fluxes are different from Fourier’s law. Chapter 8 will concern the global existence and exponential stability of solutions to the 1D thermoelastic equations of hyperbolic type, which is in fact a 1D thermoelastic system of type II with a thermal memory. We consult the works by Messaoudi [261], Racke [350, 351], Racke and Wang [354] for thermoelastic models with second sound, which means that the heat flux is given by Cattaneo’s law (i.e., the heat flux q satisfies τ qt + q + κ∇θ = 0 with τ > 0, κ > 0 constants), instead of Fourier’s law of the classical thermoelastic models in which τ = 0. For the thermoelastic models of type II, we refer to the works by Green and Naghdi [127, 128], Gurtin and Pipkin [133], and Qin and Muñ˜ oz Rivera [340], and the references therein. For the thermoelastic models of type III, we refer to the works by Green and Naghdi [127, 128], Quintanilla and Racke [347], Reissig and Wang [360], and Zhang and Zuazua [444], and the references therein. I sincerely hope that readers will learn the main ideas and essence of the basic theories and methods in deriving global well-posedness, asymptotic behavior and existence of global (universal) attractors for the models under consideration in this book. Also I hope that readers will be stimulated by some ideas from this book and undertake further study and research after having read the related references. I appreciate my former Ph.D. advisor, Professor Songmu Zheng from Fudan University for his constant encouragement, useful advice and great support and help. Special thanks go to Professor Bert-Wolfgang Schulze for his interest in my research and for acting as the initiator for publication of this book. I would like also to acknowledge the NNSF of China for its support. Currently, this book project is being supported by the National Jie Chu Qing Nian Grant (No. 10225102), Grant (No. 10571024) of the NNSF of China, by a grant from the Institute of Mathematical Sciences, The Chinese University of Hong Kong, and by Grant (No. 0412000100) of Prominent Youth from Henan Province of China. Also I hope to take this opportunity to thank my teachers Professors Daqian Li (Ta-tsien Li) (one of my former advisors for the Master Degree), Jiaxing Hong, Weixi Shen, Tiehu Qin, Shuxing Chen, Yongji Tan, Jin Cheng from Fudan University. I appreciate the help from Professors Boling Guo, Ling Hsiao, Zhouping Xin, Tong Yang, Yi Zhou, Hua Chen,

Preface

xv

Jingxue Yin, Song Jiang, Ping Zhang, Changxing Miao, Zheng-an Yao, Junning Zhao, Weike Wang, Huijiang Zhao, Changjiang Zhu, Zhong Tan, Jinghua Wang, Guowang Chen, Mingxin Wang, Sining Zheng, Chengkui Zhong, Xiaoping Yang, Huicheng Yin, Daoyuan Fang, Dexing Kong, Ting Wei, Y Yachun Li, Shu Wang, Xiangao Liu, Yaguang Wang, Yongqian Zhang, Wenyi Chen, Yaping Wu, Quansen Jiu, Hailiang Li, Xi-nan Ma, Feimin Huang, Xiaozhou Yang, Ganshan Yang, Lixin Tian, Yong Zhou, Hao Wu, Zhenhua Guo, Yeping Li, Xiongfeng Yang, Feng Xie, Jing Wang, Chunjing Xie and Ting Zhang for their constant help. Also I would like to thank Professors Herbert Amann, Michel Chipot from Switzerland, Professors Guiqiang Chen, Irena Lasiecka, Chun Liu, Hailiang Liu, Tao Luo and Dening Li from the USA, Professor Hugo Beirao da Veiga, Maurizio Grasselli, Cecilia Cavaterra from Italy, Professors Jaime E. Muñ˜ oz Rivera, Abimael F. Dourado Loula, Alexandre L. Madureira, Fré´ déric ´ G. Christian Valentin, Tofu Ma, M.M. Cavalcanti, D. Andrade from Brazil, Professors Tzon Tzer Lü¨ , Jyh-Hao Lee, Chun-Kong Law, Ngai-Ching Wong, John Men-Kai Hong and Kin-Ming Hui from Chinese Taiwan, Professors Reinhard Racke, Michael Reissig, Jü¨ rgen Sprekels, Pavel Krejci and Peicheng Zhu from Germany, and Professors Alain Miranville, Yuejun Peng, Bopeng Rao from France for their constant and great help. Last but not least, I want to take this opportunity to express my deepest thanks to my parents, Zhenrong Qin and Xilan Xia, and to my elder brother Yuxing Qin and sisters Yujuan Qin and Yuzhou Qin for their constant concern, encouragement and great help in all aspects of my life. My deepest gratitude goes to my wife, Yu Yin and my son, Jia Qin, for their constant advice and support in my career. Professor Yuming Qin Department of Applied Mathematics College of Science Donghua University Shanghai 201620, China E-mails: [email protected] yuming [email protected] and Visiting Professor Yuming Qin The Institute of Mathematical Sciences The Chinese University of Hong Kong Shatin, N.T., Hong Kong, China E-mail: [email protected]

Chapter 1

Preliminary In this chapter, we recall some basic results, most of which will be used in subsequent chapters. The reader can easily find detailed proofs in the related literature, see, e.g., Adams [1], Friedman [107, 108], Gagliardo [113, 114], Maz’ja [258], Nirenberg [297– 300], Tanabe [403] and Yosida [438].

1.1 Sobolev Spaces and Their Basic Properties In this section, we use the notation D = (D1 , . . . , Dn ) = (∂/∂ x 1 , . . . , ∂/∂ x n ) and D α = D1α1 · · · Dnαn , |α| = α1 + · · · + αn for a vector α = (α1 , . . . , αn ) with integral components αi ≥ 0. We often write D m to denote mth-order derivatives, i.e., D m is one of D α with |α| = m. Let be a nonempty open subset of Rn and m be a non-negative integer. Then C m () denotes the set of all functions whose derivatives of up to m are all continuous in , and C0m () the totality of functions belonging to C m () and with compact support in . We denote by B m () the set of all functions which are bounded and continuous in together with their derivatives of order up to m. For 0 < h < 1, we denote by B m+h () the set of all functions belonging to B m () whose mth-order derivatives are all uniformly Hö¨ lder continuous in with exponent h. Similarly, the sets B m () and B m+h () are defined replacing by . We put |u|m,∞, = max sup |D α u(x)|, |α|=m x∈

(1.1.1)

um,∞, = max |u| j,∞, .

(1.1.2)

u0,∞, = |u|0,∞, = sup |u(x)|.

(1.1.3)

j =0,...,m

In particular,

x∈

2

Chapter 1. Preliminary

If a = m + h with an integer m and 0 < h < 1, we set |u|a,∞, = max

sup

|α|=m x,y∈,x = y

|D α u(x) − D α u(y)| , |x − y|h

ua,∞, = max{um,∞, , |u|a,∞, }. B m (),

¯ B m (),

B a (),

(1.1.4) (1.1.5)

¯ B a ()

It is to verify that are Banach spaces with norm (1.1.2) or (1.1.5). If = Rn , we write | · |m,∞ , · m,∞ , | · |a,∞ , · a,∞ instead of | · |m,∞,Rn , · m,∞,Rn , | · |a,∞,Rn , · a,∞,Rn , respectively.

1.1.1 Distributions In this subsection we present a very brief review of some elementary concepts and techniques of the Schwartz theory of distributions. The notion of weak or distributional derivative will be of special importance. Definition 1.1.1. Let ⊂ Rn be an open set. The space D() is the set of all φ in C0∞ () endowed with a topology so that a sequence {φi } converges to an element φ in D() if and only if (1) there exists a compact set K ⊂ such that supp φi ⊂ K for every i , and (2) limi→∞ D α φi = D α φ uniformly on K for each multi-index α. The definition above does not attempt to actually define the topology on D() but merely states a consequence of the rigorous definition which requires the concept of generalized sequences or nets, a topic that we do not wish to pursue in this brief treatment. For our purpose, it will suffice to only consider ordinary sequences. It turns out that D() is a topological vector space with a locally convex topology but is not a normable space. The dual space, D (), of D() is called the space of (Schwartz) distributions and is given the weak*-topology. Thus, Ti ∈ D () converges to T if and only if Ti (φ) → T (φ) for every φ ∈ D(). We now consider some important examples of distributions. Example 1.1.1 Let μ be a Radon measure on and define the corresponding distribution by T (φ) = φ(x)dμ for all φ ∈ D(). Clearly T is a linear functional on D() and |T (φ)| ≤ |μ|(supp φ)φ L ∞ , from which it is easily seen that T is continuous, and thus a distribution. In this way, we will make an identification of Radon measures and the associated distributions. Here |μ|(S) denotes the measure of the set S. p

Example 1.1.2 Let f ∈ L loc (), p ≥ 1, and consider the corresponding signed measure μ defined for all Borel sets E ⊂ Rn by μ(E) = f (x)d x E

1.1. Sobolev Spaces and Their Basic Properties

3

and pass to the associated distribution f (φ) =

Rn

φ(x) f (x)d x.

In the sequel we shall often identify locally integrable functions with their corresponding distributions without explicitly indicating the identification. We note the following two facts about distributions that will be of importance. (1) A distribution T on an open set is said to be positive if T (φ) ≥ 0 whenever φ ≥ 0, φ ∈ D(). A fundamental result in distribution theory states that a positive distribution is a measure. Of course, not all distributions are measures. For example, the distribution defined on R1 by T (φ) =

φ (x)d x

is not a measure since it is not continuous on D() when endowed with the topology of uniform convergence on compact sets. (2) Another important fact is that distributions are determined by their local behavior. By this we mean that if two distributions T and S on have the property that for every x ∈ there is a neighborhood U such that T (φ) = S(φ) for all φ ∈ D() supported by U , then T = S. For example, this implies that if {α } is a family of open sets such that ∪α α = and T is a distribution on such that T is a measure on α , then T is a measure on . This also implies that if a distribution T vanishes on each open set of some family F, it then vanishes on the union of all elements of F . The support of a distribution T is thus defined as the complement of the largest open set on which T vanishes. We now introduce the convolution of a distribution with a test function φ ∈ D(). ˜ For this purpose, we introduce the notation φ(x) = φ(−x) and τx φ(y) = φ(y − x). The convolution of a distribution T defined on Rn with φ ∈ D() is a function of class C ∞ given by ˜ T ∗ φ(x) = T (ττx φ). An important observation is ˜ = T (φ). ˜ T ∗ φ(0) = T (ττ0 φ) If the distribution T is given by a locally integrable function f , then we have (T ∗ φ)(x) = f (x − y)φ(y)d y which is the usual definition for the convolution of two functions. It is easy to verify that T ∗ (φ ∗ ψ) = (T ∗ φ) ∗ ψ whenever φ, ψ ∈ D().

4


Let T be a distribution on an open set . The partial derivative of T is defined as Di T (φ) = −T (Di φ) for φ ∈ D(). Since Di φ ∈ D(), it is clear that Di T is again a distribution. Since the test function φ is smooth, the mixed partial derivatives are independent of the order of differentiation: Di D j φ = D j D i φ and therefore the equation holds for distributions: Di D j T = D j Di T. Consequently, for any multi-index α the corresponding derivative of T is given by the equation D α T (φ) = (−1)|α| T (D α φ). Finally, we note that a distribution on can be multiplied by smooth functions. Thus, if T ∈ D () and f ∈ C ∞ (), then the product f T is a distribution defined by ( f T )(φ) = T ( f φ),

φ ∈ D().

1.1.2 Weak Derivatives and Sobolev Spaces Let u ∈ L 1loc (). For a given multi-index α, a function v ∈ L 1loc () is called the α t h weak derivative of u if

φvd x = (−1)|α|

u D α φd x

for all φ ∈ C0∞ (). v is also referred to as the generalized derivative of u and we write v = D α u. Clearly, D α u is uniquely determined up to sets of Lebesgue measure zero. We say that the α t h weak derivative of u is a measure if there exists a regular Borel (signed) measure μ on such that φudμ = (−1)|α| u D α φdμ

for all φ ∈ C0∞ (). In most applications, |α| = 1 and then we speak of u whose partial derivatives are measures. Now we introduce the definition of Sobolev spaces. Definition 1.1.2. Let be a bounded or unbounded domain of Rn with smooth boundary . For 1 ≤ p ≤ +∞ and m a non-negative integer, W m, p () is defined to be the space of functions u in L p () whose distribution derivatives of order up to m are also in L p (). That is, W m, p () = L p () ∩ u : D α u ∈ L p (), |α| ≤ m .


5

The space W m, p (), called a Sobolev space, is equipped with a norm 1/ p um, p, = |D α u| p d x , if 1 ≤ p < ∞, or

(1.1.6)

um, p, = max esssup x∈ |D α u(x)|, if p = ∞

(1.1.7)

|α|≤m

|α|≤m

which is clearly equivalent to

D α u p, .

(1.1.8)

|α|≤m

If = Rn , we simply write um, p = um, p,Rn ,

u0, p = u p . m, p

It is an easy matter to verify that W m, p () is a Banach space. The space W0 defined as the closure of C0∞ () relative to the norm (1.1.8). Clearly, we have

() is

W 0, p () = L p () with norm · 0, p, ≡ · p, . In case p = 2, W m,2 (), usually written as H m (), is a Hilbert space with the scalar product (u, v)m = (D α u, D α v) L 2 () |α|≤m

with ( f, f g) L 2 () =

f gd ¯ x ; here g¯ denotes the conjugate function of g. The homogeneous Sobolev space W˙ m, p () (1 ≤ p < ∞) is defined as

W˙ m, p () = u ∈ C0∞ () : D α u p, < ∞, |α| ≤ m

(1.1.9)

which is the completion space in terms of the norm p 1/ p uW˙ m, p () = D α u p, .

(1.1.10)

Moreover, when p = ∞, the completion space of the set of smooth functions u ∈ C ∞ () : D α u p, < ∞, |α| ≤ m

(1.1.11)

|α|=m

in terms of the norm |α|≤m supx∈ |∂ α u| is the space Cbm (). Generally, Cbm () = W m,∞ (). Thus W m,∞ () is not the completion space of the set (1.1.11) of smooth functions, but it is still a Banach space. We now define the space W −m, p () with a negative integer −m as follows.

6


Definition 1.1.3. Let 1 < p < ∞. We define W −m, p () = u : u = D α gα (x), gα (x) ∈ L p (), |α|≤m

u−m, p, < ∞, 1/ p + 1/q = 1

where u−m, p, =

sup

f m,q, =1 |α|≤m

(−1)|α| gα D α f d x < ∞.

We also introduce the space BV () of integrable functions whose partial derivatives are (signed measures) with finite variation; thus, BV () = L 1 () ∩ {u : D α u is a measure, |D α u|() < ∞, |α| = 1}. A norm on BV () is defined by u B V () = u1, +

|D α u|().

|α|=1

Observe that if u ∈ W m, p ()∩ BV (), then u is determined only up to a set of Lebesgue measure zero. We agree to call these functions u continuous, bounded, etc. if there is a function u¯ such that u¯ = u a.e. and u¯ has these properties. We shall see that elements in W m, p () have representatives that permit us to regard them as generalizations of absolutely continuous functions on R1 . Let φ be a non-negative, real-valued function in C0∞ () with the property that φ(x)d x = 1 Rn

where supp φ ⊂ B(0, 1), i.e., the support of φ is in the closed unit ball. A typical example of such a function is given by C exp[−1/(1 − |x|2 )] if |x| < 1, φ(x) = 0 if |x| ≥ 1 where the constant C is chosen so that Rn φ(x)d x = 1. For ε > 0, the function φε (x) ≡ ε−n φ(x/ε) belongs to C0∞ () and supp φε = B(0, ε). The function φε is called a regularizer (or mollifier) and the convolution u ε (x) = φε ∗ u(x) ≡ φε (x − y)u(y)d y (1.1.12) Rn

defined for functions u for which the right-hand side of (1.1.12) has meaning, is called the regularization (mollification) of u. Regularization has several important and useful properties that are summarized in the following theorem.


7

Theorem 1.1.1. (1) If u ∈ L 1loc (Rn ), then for any ε > 0, u ε ∈ C ∞ (Rn ) and D α (φε ∗ u) = (D α φε ) ∗ u for each multi-index α. (2) u ε (x) → u(x) whenever x is a Lebesgue point for u. In case u is continuous, then u ε converges uniformly to u on compact subsets of Rn . (3) If u ∈ L p (Rn ), 1 ≤ p < ∞, then u ε ∈ L p (Rn ), u ε p ≤ u p , and limε→0 u ε − u p = 0. Note that if u ∈ L 1 (), then u ε (x) ≡ φε ∗ u(x) is defined provided x ∈ and ε < dist(x, ∂). It is easy to verify that the above theorem still holds for this case with obvious modification. For example, if u ∈ C() and ⊂⊂ , then u ε converges uniformly to u on as ε → 0. Moreover, (3) of Theorem 1.1.1 implies the modification does not increase the norm. This is intuitively clear since the norm must take into account the extremities of the function and modification, which is an averaging operation, does not increase the extremities. Now let us give an important result concerning the convergence of regularizers of Sobolev functions. Theorem 1.1.2. Suppose that u ∈ W m, p (), p ≥ 1. Then the regularizers, u ε , of u, have the property that lim u ε − um, p; = 0 ε→0

whenever

⊂⊂ . In case = Rn , then limε→0 u ε − um, p = 0.

Since the definition of a Sobolev function requires that its distributional derivatives belong to L p , it is natural to inquire whether the function possesses any classical differentiability properties. To this end, we shall state that its partial derivatives of u¯ exist almost everywhere. That is, there is a function u¯ such that u¯ = u a.e. and the partial derivatives of u¯ exist almost everywhere. Theorem 1.1.3. Suppose that u ∈ L p (). Then u ∈ W 1, p (), p ≥ 1, if and only if u has a representative u¯ that is absolutely continuous on almost all line segments in parallel to the coordinate axes and whose (classical) partial derivatives belong to L p (). This theorem can also be stated in the following way. Corollary 1.1.1. If u ∈ L p (), then u ∈ W 1, p () if and only if u has a representative u¯ such that u¯ ∈ W 1, p ( ) for almost all line segments in parallel to the coordinate axes and |D u| ¯ ∈ L p (). For an equivalent statement, an application of Fubini’s Theorem allows us to replace almost all line segments by almost all k-dimensional planes k in that are parallel to the coordinate k-planes. However, if u were not an element of W 1, p (), but merely an

8


element of L 1 (), Fubini’s Theorem would imply that the convergence occurs only H 1a.e. on almost all lines. Thus, the assumption u ∈ W 1, p () implies that the regularizers converge on a relatively large set of points. This is an interesting fact of Sobolev functions. Recall that if u ∈ L p (Rn ), then u(x + h) − u(x) p → 0 as h → 0. The next similar result provides a very useful characterization of W 1, p (Rn ). Theorem 1.1.4. Let 1 < p < ∞. Then u ∈ W 1, p (Rn ) if and only if u ∈ L p (Rn ) and

u(x + h) − u(x) p | | dx |h|

1/ p

= |h|−1 u(x + h) − u(x) p

remains bounded for all h ∈ Rn . For a measurable function u : → R1 , let u + = max{u, 0}, u − = min{u, 0}. Corollary 1.1.2. Let u ∈ W 1, p (), p ≥ 1. Then u + , u − ∈ W 1, p () and Du + = Du if u > 0; Du + = 0 if u ≤ 0, Du − = 0 if u ≥ 0; Du − = Du if u < 0. Corollary 1.1.3. If is connected, u ∈ W 1, p (), p ≥ 1, and Du = 0 a.e. on , then u is a constant in . Remark 1.1.1. Corollary 1.1.2 states that elements of W 1, p () remain invariant under the operation of truncation. One of the interesting aspects of the theory is that this, in general, is no longer true for the space W m, p (). We refer the reader to a counterexample in Adams [1] (see also Ziemer [462]). Next we consider the problem of composition of a suitable function with u ∈ W 1, p (). Before doing so, let us recall the analogy in Real Variable Theory. In general, if f and g are both absolutely continuous functions, then the composition, f ◦ g, need not be absolutely continuous. Recall that a function, f , is absolutely continuous if and only if it is continuous, of bounded variation, and has the property that | f (E)| = 0 whenever |E| = 0 (|E| is the Lebesgue measure of E). Thus, the consideration that prevents f ◦ g from being absolutely continuous is that f ◦ g need not be of bounded variation. A result of Poussin [312] says that f ◦ g is absolutely continuous if and only if f ◦ g · g is integrable. An analogous result is valid in the context of Sobolev theory (see, e.g., Marcus and Mizel [247, 248]), but here we only state the case when the outer function is Lipschitz. Theorem 1.1.5. Let f : R1 → R1 be a Lipschitz function and u ∈ W 1, p (), p ≥ 1. If f ◦ u ∈ L p (), then f ◦ u ∈ W 1, p () and for almost all x ∈ , D( f ◦ u)(x) = f (u(x)) · Du(x).


9

1.1.3 Sobolev Inequalities, Embedding Theorems and the Trace Theorem In this subsection we shall review some Sobolev inequalities which are of fundamental importance in the investigation of the problems of partial differential equations. One of the important characteristics of these Sobolev inequalities is that they allow the L p -norm of a function to be estimated by the norm of its partial derivatives. First, we establish these 1, p inequalities for functions in the space W0 (). Theorem 1.1.6. (Sobolev’s Inequality) Let ⊆ Rn , n > 1, be an open domain. There is a constant C = C(n, p) such that ∗

1, p

(1) if n > p ≥ 1, and u ∈ W0 (), then u ∈ L p () and u p∗ , ≤

p(n − 1) √ Du p, 2(n − p) n

(1.1.13)

where p∗ = np/(n − p); 1, p

(2) if p > n and is bounded, and u ∈ W0 (), then u ∈ C() and sup |u| ≤ C||1/n−1/ p Du p, .

(1.1.14)

Particularly, if = Rn , then −1/ p

sup |u| ≤ Cωn Rn

where ωn =

2π n/2 n(n/2)

u1, p,Rn

(1.1.15)

is the measure of the n-dimensional unit ball, is the Euler

p−1 ( p−1)/ p gamma function and C = max{1, ( p−n ) }.

Remark 1.1.2. The Sobolev inequality (1.1.13) does not hold for p = n, p ∗ = ∞. Note that the first conclusion of Theorem 1.1.6, derived firstly by Sobolev [381] ∗ in 1938, states that the L p norm of u can be estimated by u1, p, or Du p, , the Sobolev norm of u. However, it is possible to bound a higher L p norm of u by exploiting higher-order derivatives of u as shown in the next theorem which generalizes Theorem 1.1.6 from m = 1, p > n to m ≥ 1 an integer. Theorem 1.1.7. Let ⊆ Rn be an open set. There is a constant C = C(n, m, p) such that m, p

(1) if mp < n, p ≥ 1, and u ∈ W0

(), then u ∈ L p∗ () and

u p∗ , ≤ Cum, p, where p∗ = np/(n − mp);

(1.1.16)

10

Chapter 1. Preliminary m, p

(2) if mp > n, and u ∈ W0

(), then u ∈ C() and

sup |u| ≤ C|K |1/ p

m−1 |α|=0

+(diam K )m

(diam K )|α|

1 D α u p,K α!

1 (m − n/ p)−1 D m u p,K (m − 1)!

(1.1.17)

where K = supp u, C = C(m, p, n) and diam K is the diameter of K . Remark 1.1.3. An important case to consider in Theorems 1.1.6–1.1.7 is = Rn . In m, p this situation, W m, p (Rn ) = W0 (Rn ) and therefore the results of Theorems 1.1.6–1.1.7 m, p n apply to W (R ). Note that for p > n, the results of Theorems 1.1.6–1.1.7 yield more than the fact that u is bounded. Indeed, u is Hö¨ lder continuous, which we shall state as follows. ¯ where α = 1 − n/ p. Theorem 1.1.8. If u ∈ W0 (), p > n, then u ∈ C 0,α () 1, p

In connection with (1.1.13) we would like to make some comments. When is an unbounded domain (in particular, exterior to a compact region) the investigation of the asymptotic properties of a solution u to a system of partial differential equations is strictly related to the Lebesgue space L p () to which u belongs and, roughly speaking, the behavior of u at large distances will be better known when the exponent p is lower. Indeed, the inherent information, derived from the Navier-Stokes equations in such domains in that u (a generic component of the velocity field) has first derivatives Di u summable with exponents pi which, however, may vary with x i , i = 1, 2, . . . , n. Thus we may wonder if (1.1.13) can be replaced by another inequality which takes into account this different behavior in different directions and leads to an exponent q of summability for u strictly less than the exponent p given in (1.1.13). This question finds its answer within the context of anisotropic Sobolev spaces (see, e.g., Nikol’skii [296]). Here we shall restrict ourselves to quoting without proof, an inequality due to Troisi [411] representing the natural generalization of (1.1.13) to the anisotropic case (see also e.g., Galdi [115]). Theorem 1.1.9. (Troisi’s Inequality) Let 1 ≤ pi < ∞, i = 1, . . . , n. Then for all u ∈ C0∞ (Rn ), the following inequality holds: us ≤ C

n

1/n

Di u pi ,

i=1

n i=1

pi−1 > 1, s = n i=1

n pi−1 − 1

.

(1.1.18)

If pi = p for all i = 1, 2, . . . , n, (1.1.18) reduces to (1.1.13). If for some i (= 1 say), p1 < p ≡ p2 = · · · = pn , then s 0 such that for any x ∈ A, there holds that I x B ≤ Cx A . If A is embedded into B, then we simply denote it by A →

B. A is said to be compactly embedded into B if and only if (1) A is embedded into B; (2) the identity mapping I : A → B is a compact operator.

→

B. If A is compactly embedded into B, then we simply denote it by A → Now we want to draw some consequences from Theorem 1.1.6. In fact, exploiting Theorem 1.1.6, we have the following result which is an embedding theorem. 1, p

np Corollary 1.1.5. If u ∈ W0 (), then u ∈ L q () with p ≤ q ≤ n− p if 1 ≤ p < n, and p ≤ q < +∞ if p = n. Moreover, if p > n, u coincides a.e. in with a (uniquely determined) d function of C(). Finally, u satisfies the following inequalities:

uq, ≤ Cu1, p, if 1 ≤ p < n, p ≤ q ≤ uq, ≤ Cu1, p, if p = n, p ≤ q < ∞, uC ≤ Cu1, p, if p > n, where C = C(n, p, q).

np , n−p

(1.1.20) (1.1.21) (1.1.22)

12

Chapter 1. Preliminary m, p

In fact, we can generalize Corollary 1.1.5 to functions from W0 following embedding theorem. m, p

Theorem 1.1.10. Let u ∈ W0

() to obtain the

(), p ≥ 1, m ≥ 0. Then

(1) if mp < n, then we have m, p

W0

()

→ L q (), ∀q ∈ p,

np n − mp

(1.1.23)

and there is a constant C1 > 0 depending only on m, p, q and n such that np ; (1.1.24) uq, ≤ C1 um, p, , ∀q ∈ p, n − mp (2) if mp = n, then we have m, p

W0

() →

L q (), ∀q ∈ [ p, ∞)

(1.1.25)

and there is a constant C2 > 0 depending only on m, p, q and n such that uq, ≤ C2 um, p, , ∀q ∈ [ p, ∞);

(1.1.26)

m, p

(3) if mp > n, each u ∈ W0 () is equal a.e. in to a unique function in C k (), for all k ∈ [0, m − n/ p) and there is a constant C3 > 0 depending only on m, p, q and n such that uC k ≤ C3 um, p, . (1.1.27) Remark 1.1.4. In case (2) in Theorem 1.1.10, the following exception case holds for m = n, p = 1, q = ∞: W n,1 () →

L ∞ (). (1.1.28) When is a smooth bounded domain in Rn , we have the following famous RellichKondrachov compactness theorem. Theorem 1.1.11. (Rellich-Kondrachov Compactness Theorem) Let ⊂ Rn be a m, p smooth bounded domain. Then, if mp < n and p ≥ 1, W0 () is compactly embedded m, p q in L () where q < np/(n − mp). If mp > n + kp, W0 () is compactly embedded in C k (). It is worth pointing out here that the above results in this subsection are stated in m, p terms of functions in W0 (). A natural and important question is to identify those domains for which the results are valid for functions in W m, p (). One answer can be formulated in terms of those domains of having the property that there exists a bounded linear operator L : W m, p () → W m, p (Rn ) such that L(u) | = u for all u ∈ W m, p (). We say that is an (m, p)-extension domain for W m, p () if there exists an extension operator for W m, p () with 1 ≤ p ≤ ∞, m a


13

non-negative integer. Clearly, the results before are also valid for u ∈ W m, p () when is a bounded extension domain. A fundamental result of Calderó´ n-Stein says that every Lipschitz domain is an extension domain. An open set is a Lipschitz domain if its boundary can be locally represented as the graph of a Lipschitz function defined on some open ball of Rn . This result was shown by Calderó´ n [47] for 1 < p < n and Stein [398] extended Calderó´ n’s result to p = 1, ∞. The following is an extension of the Rellich-Kondrachov compactness theorem. Theorem 1.1.12. If is a domain having the extension property, then W k+m, p ()

→ W k,q () is a compact embedding if mp < n, 1 ≤ q ≤ np/(n − mp) and m a non-negative integer. Another answer is formulated in terms of those domains of class C m whose definition is stated as follows. Definition 1.1.5. Let be a nonempty open subset of Rn . If for any point a of the boundary ∂ there is a neighborhood O of a and a homeomorphism of class C m from O to the open unit ball B(0, 1) = {x ∈ Rn : |x| < 1} of Rn such that (a) = 0, (O ∩ ) = {x ∈ Rn : |x| < 1, x n > 0}, (O ∩ ∂) = {x ∈ Rn : |x| < 1, x n = 0}, then is called an open set of class C m . If is an open set of class C m with bounded boundary, then there exist finite points a1 , . . . , a N on the boundary ∂ and a neighborhood Oi of ai , and a homeomorphism i of class C m from Oi to the open unit ball of Rn , i = 1, . . . , N satisfying the conditions of Definition 1.1.5 for a = ai , i = 1, . . . , N, and N ∪i=1 −1 ({x : |x| < 1/2}) ⊃ ∂.

When is not bounded, we may consider the uniformly regular open sets (see, e.g., Browder [45]). Definition 1.1.6. Assume that is a nonempty open subset of Rn whose boundary is not bounded. Then is said to be uniformly regular of class of C m if there are a family of open sets {Oi : i = 1, . . . } and of homeomorphisms {i } of Oi onto the unit ball B(0, 1) in Rn , and an integer N such that the following conditions are satisfied: ∞ n −1 neighbor(1) Let Oi = −1 i ({x ∈ R : |x| < 1/2}). Then ∪i=1 Oi contains the N hood of ∂. (2) For each i ,

i (Oi ∩ ) = {x : |x| < 1, x n > 0}, i (Oi ∩ ∂) = {x : |x| < 1, x n = 0}.

14


(3) Any N + 1 distinct sets of {Oi } have an empty intersection. (4) The family {Oi } is locally finite, i.e., only a finite number of Oi have a nonempty intersection with some neighborhood of each point of Rn . be the inverse mapping of i . Then for each i = 1, 2, . . . and (5) Let i = −1 i |x| < 1, |i (x) − i (0)| < M. Let ik (x), ik (y) be the kth components of i (x), i (y) respectively. Then |D α ik (x)| ≤ M, |D α ik (y)| ≤ M, |in (x)| ≤ M dist(x, ∂) for |α| ≤ m, x ∈ Oi , |y| < 1, k = 1, . . . , n and i = 1, 2, . . . . Note that when m ≥ 2 or also when m = 1, if ∂ik /∂ x i are equicontinuous, then (4) follows from (3) and (5). When is a smooth bounded domain of class C m , we have the following (compactness) embedding theorem. Theorem 1.1.13. (Embedding and Compactness Theorem) Assume that is a bounded domain of class C m . Then we have ∗

(i) If mp < n, then W m, p () is continuously embedded in L q () with ∗

W m, p () →

L q ().

1 q∗

=

1 p

−

m n:

(1.1.29)

Moreover, the embedding operator is compact for any q, 1 ≤ q < q ∗ . (ii) If mp = n, then W m, p () is continuously embedded in L q (), ∀q, 1 ≤ q < ∞: W m, p ()

→ L q ().

(1.1.30)

Moreover, the embedding operator is compact, ∀q, 1 ≤ q < ∞. If p = 1, m = n, then the above still holds for q = ∞. (iii) If k + 1 > m − np > k, k ∈ N, then writing m − np = k + α, α ∈ (0, 1), W m, p () is continuously embedded in C k,α (): W m, p ()

→ C k,α (),

(1.1.31)

where C k,α () is the space of functions in C k () whose derivatives of order k are Holder H continuous with exponent α. Moreover, if n = m − k − 1, and α = 1, p = ¨ 1, then (1.1.31) holds for α = 1, and the embedding operator is compact from W m, p () to C k,β (), ∀0 ≤ β < α. Remark 1.1.5. The embedding properties (i)–(iii) are still valid for smooth unbounded domains of R n provided that L q () in (1.1.29)–(1.1.30) and C k,β () in (1.1.31) are q replaced by L loc () and C k,α (B) for any bounded domain B ⊂ , respectively. m, p

Remark 1.1.6. The regularity assumption on can be weakened. When u ∈ W0 (), the above embedding properties are valid without any regularity assumptions on .


15

Theorem 1.1.14. (Density Theorem) If is a C m domain, m ≥ 1, 1 ≤ p < ∞, then C m () is dense in W m, p (). Note that in the previous embedding theorems, we always assume that m is a nonnegative integer in the definition of W m, p (). The embedding theorems may not be optimal. In fact, when = Rn , we may use the Bessel potential or the Riesz potential to define the Sobolev spaces of fractional order. In this situation, the Bessel potential Jα and the Riesz potential Pα are defined as Jα = (I − )−α/2 , Pα = (−)−α/2

(1.1.32)

with which we can define Banach spaces H α, p (Rn ) = Jα L p (Rn ), H˙ α, p (Rn ) = Pα L p (Rn ), α ∈ R. In particular, when α ∈ N,

H α, p (Rn ) = W α, p (Rn ).

(1.1.33)

(1.1.34)

However, when = Rn , we can still define the Sobolev spaces of fractional order as follows: Let s > 0 be a non-integer, s = [s] + λ, λ ∈ (0, 1). The Sobolev space W s, p () is defined as the completion space of the set of functions u ∈ C ∞ () : |∂ α (u(x)−u(y))|/|x − y|n/ p+λ ∈ L p (×), ∀α ∈ (Z∪{0})n , |α| = [s] (1.1.35) in terms of the norm 1/ p

|∂ α (u(x) − u(y))| p us, p, = u[s], p, + |α|=[s] d x d y . |x − y|n+ pλ

(1.1.36)

After having introduced the definition of Sobolev spaces of fractional order, we may restate the Sobolev embedding theorems in an exact and detailed manner. Theorem 1.1.15. Let ⊆ Rn . Then we have (1)

μ

Cb (), if s − n/ p > μ; W s, p () → W

s, p

() →

μ Cb (),

if s − n/ p = μ = non-negative i nteger ;

(1.1.37) (1.1.38)

(2) if p2 = ∞, then

W s2 , p2 () ⇐⇒ s1 − n/ p1 ≥ s2 − n/ p2 , 1/ p1 ≥ 1/ p2. (1.1.39) W s1 , p1 () → In particular case, p1 = 1, s1 = n, s2 = 0, p2 = ∞, we have W n,1 ()

→ L ∞ ().

(1.1.40)

16


Theorem 1.1.16. (Rellich-Kondrachov Compactness Embedding Theorem) Let ⊂ Rn be a bounded smooth domain. Then the following compactness embeddings hold: μ

W s, p () →

→

Cb () ⇐⇒ s − n/ p > μ;

(1.1.41)

→

W s2 , p2 () ⇐⇒ s1 − n/ p1 > s2 − n/ p2 , 1/ p1 ≥ 1/ p2 . W s1 , p1 () →

(1.1.42)

Remark 1.1.7. If m − n/ p is a non-negative integer, then we have μ

W m, p (Rn )

→ Cb (Rn ) ⇐⇒ m − n/ p > μ.

(1.1.43)

In particular, if 1 < p < ∞, then μ+

W m, p (Rn ) →

C0 (Rn ) where μ+

C0

(1.1.44)

= u : u(x) possesses the continuous derivatives up to order [μ] lim |∂ [μ] (u(x) − u(y))|/|x − y|α = 0, μ = [μ] + α, α ∈ [0, 1), lim D j u(x) = 0, j ≤ [μ] ,

and satisfies

x−y→0

|x|→∞

u

Cμ

= max

j

sup |D u(x)|, 0 ≤ j ≤ [μ]; sup |∂

x∈Rn

x = y

[μ]

(1.1.45) (1.1.46)

(u(x) − u(y))|/|x − y|

α

.

(1.1.47) When α = 0, the above condition implies that D k u(x) is uniformly continuous, and it is μ easy to verify that C0 (Rn ) is a separable space, while when μ is not an integer, C μ (Rn ) is not a separable space. If we assume that is a smooth bounded domain of class C m and u ∈ W m, p (). Then we can define the trace of u on which coincides with the value of u on when u is a smooth function of C m (). Theorem 1.1.17. (Trace Theorem) Let ν = (ν1 , . . . , νn ) be the unit outward normal on and ∂ ju γju = | , ∀u ∈ C m (), j = 0, . . . , m − 1. (1.1.48) ∂ν j Then the trace operator γ = (γ γ0 , . . . , γm −1 ) can be uniquely extended to a continuous m−1 m− j − 1 , p m, p p () to j =0 W (). operator from W γ0 u, . . . , γm −1 u) ∈ γ : u ∈ W m, p () → γ u = (γ

m−1

W

m− j − 1p , p

().

j =0

Moreover, it is a surjective mapping. Note that W

m− j − 1p , p

() are spaces with fractional order derivatives.

(1.1.49)


17

1.1.4 Interpolation Inequalities The following Gagliardo-Nirenberg interpolation inequalities (see, e.g., Nirenberg [299] and Friedman [108]) play a very important role in the theory and applications of partial differential equations. First we introduce some notation. For p > 0, |u| p, = u L p () . For p < 0, set −n/ p = h + α with h = [−n/ p] and α ∈ [0, 1). We define |u| p, = sup |D h u| ≡ sup |D β u|, if α = 0,

|β|=h

|u| p, = [D h u]α, ≡ ≡

sup[D β u]α

|β|=h

sup

|β|=h x,y∈,x = y

|D β u(x) − D β u(y)| , if α > 0. |x − y|α

If = Rn , we simply write |u| p instead of |u| p, . Theorem 1.1.18. Let j, m be any integers satisfying 0 ≤ j < m, and let 1 ≤ q, r ≤ ∞, and p ∈ R, j/m ≤ α ≤ 1 such that 1/ p − j/n = α[1/r − m/n] + (1 − α)/q. Then (i) For any u ∈ W m,r (Rn ) ∩ L q (Rn ), there is a positive constant C depending only on n, m, j, q, r, α such that the following inequality holds: |D j u| p ≤ C|D m u|rα |u|1−α q

(1.1.50)

with the following exception: if 1 < r < ∞ and m − j − n/ p is a non-negative integer, then (1.1.50) holds only for α satisfying j/m ≤ α < 1. (ii) For any u ∈ W m,r () ∩ L q () where is a bounded domain with smooth boundary, there are two positive constants C1 , C2 such that the following inequalities hold: α |D j u| p, ≤ C1 |D m u|r, |u|1−α (1.1.51) q, + C2 |u|q, with the same exception as in (i). m, p In particular, for any u ∈ W0 () ∩ L q (), the constant C2 in (1.1.51) can be taken as zero.

1.1.5 The Poincaré´ Inequality In this subsection, we shall recall the Poincaré´ inequality in different forms. Theorem 1.1.19. Let be a bounded domain in Rn and u ∈ H01(). Then there is a positive constant C depending only on and n such that u L 2 () ≤ C∇u L 2 () , ∀u ∈ H01().

(1.1.52)

18


Theorem 1.1.20. Let be a bounded domain of C 1 in Rn . There is a positive constant C depending only on , n such that for any u ∈ H 1(),

where u¯ =

1 || u(x)d x

u − u ¯ L 2 () ≤ C∇u L 2 ()

(1.1.53)

is the integral average of u over , and || is the volume of .

Theorem 1.1.21. Under assumptions of Theorem 1.1.20, for any u ∈ H 1(), we have

u L 2 () ≤ C ∇u L 2 () + | ud x| . (1.1.54)

1.2 Some Inequalities in Analysis In this section we shall recall integration inequalities such as the classical BellmanGronwall inequality, the generalized Bellman-Gronwall inequality and the uniform Bellman-Gronwall inequality. These inequalities furnish some powerful tools in establishing the global well-posedness and asymptotic behavior of solutions to nonlinear evolutionary differential equations arising from physics, fluid mechanics, and materials science, etc.

1.2.1 The Classical Bellman-Gronwall Inequality The following is the very famous Bellman-Gronwall inequality which plays a crucial role in analysis, especially in the study of existence, uniqueness and stability and estimates of solutions to differential equations (see, e.g., Bellman [34–37] and Gronwall [130]). Theorem 1.2.1. (The Classical Bellman-Gronwall Inequality) Let y(t) and g(t) be non-negative, continuous functions on 0 ≤ t ≤ τ , for which the inequality t y(t) ≤ η + g(s)y(s)ds, 0 ≤ t ≤ τ, (1.2.1) 0

holds, where η is a non-negative constant. Then

t g(s)ds , 0 ≤ t ≤ τ. y(t) ≤ η exp

(1.2.2)

0

Remark 1.2.1. In 1919, Gronwall [130] showed the case of g(t) = constant ≥ 0. Later in 1934, Bellman [35] (see, e.g., Kuang [206]) extended this result to the form of Theorem 1.2.1. Since this type of inequalities is a very powerful and useful tool in analysis, more and more improvements and generalizations of the classical Bellman-Gronwall inequality have been made. Remark 1.2.2. Bellman proved another inequality as follows (see, e.g., Kuang [206]): Let u(t), b(t) be continuous on (α, β), and b(t) be non-negative. If t u(t) ≤ u(t0 ) + b(s)u(s)ds, t0 , t ∈ (α, β), t0

1.2. Some Inequalities in Analysis

19

then for any t ≥ t0 , t t u(t0 ) exp − b(s)u(s)ds ≤ u(t) ≤ u(t0 ) exp b(s)u(s)ds . t0

t0

The above theorem provides bounds on solutions of (1.2.1) in terms of the solution of a related linear integral equation t v(t) = η + g(s)v(s)ds (1.2.3) 0

and is one of the basic tools in the theory of differential equations. On the basis of various motivations it has been extended and used considerably in various contexts. For instance, in the Picard-Cauchy type of iteration for establishing existence and uniqueness of solutions, this inequality and its variants play a significant role. Inequalities of this type (1.2.1) are also encountered frequently in the perturbation and stability theory of differential equations.

1.2.2 The Generalized Bellman-Gronwall Inequalities The following generalization can be found in Qin [315–321]. Theorem 1.2.2. (The Generalized Bellman-Gronwall Inequality) Assume that f (t), g(t) and y(t) are non-negative integrable functions in [τ, T ] (τ < T ) verifying the integral inequality y(t) ≤ g(t) + Then we have

y(t) ≤ g(t) +

t τ

t

exp

t

τ

f (s)y(s)ds, t ∈ [τ, T ].

f (θ )dθ

f (s)g(s)ds, t ∈ [τ, T ].

In addition, if g(t) is a nondecreasing function in [τ, T ], then we conclude t t exp f (θ )dθ f (s)ds , t ∈ [τ, T ], y(t) ≤ g(t) 1 + τ s t t f (s)ds exp f (θ )dθ , t ∈ [τ, T ]. ≤ g(t) 1 + If further T = +∞ and

where C = 1 +

+∞ τ

+∞ τ

(1.2.4)

s

τ

τ

(1.2.5) (1.2.6)

f (s)ds < +∞, then we conclude

y(t) ≤ Cg(t) +∞ f (s)ds exp{ τ f (θ )dθ } is a positive constant.

(1.2.7)

The following result can be regarded as a corollary of Theorem 1.2.2, which can be found in Racke [349].

20


Corollary 1.2.1. Let a > 0, φ, h ∈ C 0 ([0, a]), h ≥ 0 and g : [0, a] → R be increasing. If for any t ∈ [0, a], t φ(t) ≤ g(t) + h(s)φ(s)ds, (1.2.8) 0

then

t

φ(t) ≤ g(t) exp{

h(s)ds}, ∀t ∈ [0, a].

(1.2.9)

0

1.2.3

The Uniform Bellman-Gronwall Inequality

In this subsection we shall review some uniform Gronwall inequalities which provide some uniform bounds or some decay rates. This type of integral inequalities plays a very crucial role in the study of the global well-posedness and large-time behavior of solutions especially in the establishment of the existence of a (global) attractor for a semigroup or a semiflow. We begin with the following three theorems which can be found in Temam [407]. Theorem 1.2.3. (The Uniform Bellman-Gronwall Inequality) Let g(t), h(t) and y(t) be three positive locally integrable functions on (t0 , +∞) such that y (t) is locally integrable on (t0 , +∞) and the following inequalities are satisfied: dy ≤ gy + h, dt

t +r t

t +r

g(s)ds ≤ a1 ,

∀t ≥ t0 ,

t +r

h(s)ds ≤ a2 ,

t

y(s)ds ≤ a3 , ∀t ≥ t0

t

where r, ai (i = 1, 2, 3) are positive constants. Then we have a 3 y(t + r ) ≤ + a2 ea1 , ∀t ≥ t0 . r In the sequel, we shall review some uniform generalizations which may furnish some large time behavior of functions. This class of inequalities plays a very significant role in the study of the global well-posedness and asymptotic behavior of solutions to some evolutionary differential equations, and is a very convenient and powerful tool in establishment of the large-time behavior of solutions when we use energy methods to deal with the large-time behavior of global solutions. We begin with some familiar results in classical calculus for single real variable analysis. Lemma 1.2.1. (1) Let y(t) ∈ L 1 (0, +∞) with y(t) ≥ 0 for a.e. t ≥ 0, y (t) ∈ L 1 (0, +∞). Then we have lim y(t) = 0. t →+∞


21

(2) Let y(t) ∈ L 1 (0, +∞) with y(t) ≥ 0 for a.e. t ≥ 0, and limt →+∞ y(t) exist. Then we have lim y(t) = 0. t →+∞

(3) Let y(t) be uniformly continuous on [0, +∞), y(t) ∈ L 1 (0, +∞). Then we have lim y(t) = 0.

t →+∞

(4) Let y(t) be a monotone function on [0, +∞) and y(t) ∈ L 1 (0, +∞). Then lim y(t) = 0

t →+∞

and

y(t) = o(1/t) as t → +∞.

Note that the above lemma provides the asymptotic behavior of y(t) for a large time. The next theorem relating to the uniform Gronwall inequality was first established by Shen and Zheng [374] in 1993 (see, e.g., Zheng [448]) which is very useful and powerful in dealing with the global well-posedness and asymptotic behavior of solutions to some evolutionary differential equations. We W shall apply it frequently in the subsequent context of this book (see, e.g., Qin [315-321] and Chapters 2–6). Theorem 1.2.4. (The Shen-Zheng Inequality) Let T be given with 0 < T ≤ +∞. Suppose that y(t), h(t) are non-negative continuous functions defined on [0, T ] and satisfy the following conditions:

T 0

d y(t) ≤ A1 y 2 + A2 + h(t), dt T y(t)dt ≤ A3 , h(t)dt ≤ A4 0

where Ai (i = 1, 2, 3, 4) are given non-negative constants. Then for any r > 0 with 0 < r < T , the following estimate holds:

A3 y(t + r ) ≤ + A2r + A4 e A1 A3 , t ∈ (0, T − r ). r Furthermore, if T = +∞, then we have lim y(t) = 0.

t →+∞

Krejci and Sprekels [204] in 1998 extended the Shen-Zheng inequality when T = +∞ to the following result (see also Zheng [449, 450]), which can be also considered as a nonlinear generalization of the Bellman-Gronwall inequality in Theorem 1.2.1. Theorem 1.2.5. (The Krejci-Sprekels Inequality) Suppose that y(t) is continuous in [0, +∞), y(t) ≥ 0, y ∈ L 1loc (R+ ) and satisfies the following conditions: +∞ y(t)dt ≤ C1 < +∞, 0

y ≤ f (y) + h(t), ∀t ∈ (0, +∞)

22


where h(t) ≥ 0 with

+∞ 0

h(t)dt ≤ C2 < +∞

and f is a nondecreasing function from R+ into R+ . Then lim y(t) = 0.

t →+∞

Later on, Zheng [449] showed the strong version of the above inequality, namely Theorem 1.2.6. (The Zheng Inequality) Suppose that y(t) is a continuous non-negative function defined on [0, +∞), and satisfies the following conditions: +∞ y(t)dt ≤ C1 < +∞, 0

t

y(t) − y(s) ≤

( f (y) + h(τ ))dτ, ∀0 ≤ s < t < +∞

s

with f and h satisfying the same assumptions as in Theorem 1.2.5. Then lim y(t) = 0.

t →+∞

From the above context of this subsection, we only know that the non-negative function (y(t), say) goes to zero as time tends to infinity. We have no information on the decay rate of y(t). In fact, the decay rate of y(t) depends on some factors which include some terms in the inequality. This can be clearly seen from the following two theorems, which indicate that when the integral inequality involves a decay term h(t), the corresponding non-negative function y(t) also has a similar decay rate (see, e.g., Muñoz Rivera [275]). Theorem 1.2.7. Suppose that y(t) ∈ C 1 (R+ ), y(t) ≥ 0, ∀t > 0 and satisfies y (t) ≤ −C0 y(t) + C1 e−γ t where C0 , C1 and γ are positive constants. Then there exist some positive constants C and γ0 such that y(t) ≤ Ce−γγ0 t . Theorem 1.2.8. Suppose that y(t) ∈ C 1 (R+ ), y(t) ≥ 0, ∀t > 0 and satisfies y (t) ≤ −K 0 [y(t)]1+1/ p +

K1 (1 + t)1+ p

where K 0 > 0, K 1 > 0 and p > 1 are constants. Then there exists some constant K 2 > 0 such that K 2 [ py(0) + 2K 1 ] y(t) ≤ . (1 + t) p


23

1.2.4 The Nakao Inequalities In this subsection we shall introduce a series of Nakao inequalities (see, e.g., Nakao [288– 291]). These inequalities are connected with difference inequalities which are not only very important for the study of asymptotic behavior of global solutions, but also seem to be interesting in themselves. One advantage of the Nakao inequalities is that any form of the Nakao inequalities can furnish a decay rate. Theorem 1.2.9. Suppose that φ(t) is a bounded non-negative function on R+ satisfying max φ(s)1+α ≤ K 0 [φ(t) − φ(t + 1)] + g(t)

s∈[t,t +1]

where K 0 > 0 is a constant, g(t) a non-negative function, α a non-negative constant. Then we have (i) if limt →+∞ g(t) = 0, then limt →+∞ φ(t) = 0. Moreover, (ii) if we assume that α > 0 and g(t) ≤ K 1 |t|−θ−1 with constants θ > 1/α, K 1 ≥ 0, then φ(t) ≤ C3 t −1/α , f or t > 0 and (iii) if α = 0 and g(t) ≤ K 2 e−θt with constants θ > 0, K 2 ≥ 0, then φ(t) ≤ C4 e−θ1 t where θ1 = min θ, log other known constants.

K0

K 0 −1 ,

and C3 , C4 are positive constants depending on

The above Nakao inequality (see, e.g., Nakao [291]) has several generalizations which we shall state as follows. Theorem 1.2.10. Suppose that φ(t) is a non-negative continuous nonincreasing function on R+ satisfying the inequality φ(t + T ) ≤ C

2

(1 + t)θi [φ(t) − φ(t + T )]i , f or t ≥ 0

i=1

with some T > 0, C > 0, 0 < i ≤ 1 and θi ≤ i (i = 1, 2). Then φ(t) has the following decay properties: (i) If 0 < i < 1 with 1 + 2 < 1 and θi < i , i = 1, 2, then φ(t) ≤ C0 (1 + t)−γ with γ = mini=1,2 {(i − θi )/(1 − i )}, where we consider as (i − θi )/(1 − i ) = ∞ if i = 1.

24


(ii) If θ1 = 1 < 1 and θ2 < 2 ≤ 1, then φ(t) ≤ C0 {log(2 + t)}−1 /(1−1 ) . (iii) If θ1 = 1 < 1 and 2 = θ2 ≤ 1, then φ(t) ≤ C0 {log(2 + t)}−γ˜ with γ˜ = mini=1,2 {i /(1 − i )}. (iv) If 1 = 2 = 1, then φ(t) ≤ C0 exp{−λt 1−θ } if θ < 1, φ(t) ≤ C0 (1 + t)−λ if θ = 1 for some λ > 0, α > 0, where we set θ = min{θ1 , θ2 }. In the above, C0 denotes constants depending on φ(0) and other known constants. Remark 1.2.3. When 1 = 2 and θ1 = θ2 , more detailed results are proved in Nakao [289–290]. Remark 1.2.4. The above theorem can be easily generalized to the following difference inequality of the form φ(t + 1) ≤ C

m (1 + t)θi [φ(t) − φ(t + 1)]i .

(1.2.10)

i=1

For example, if 0 < i < 1 and θi < i , we obtain from (1.2.10) that φ(t) ≤ C0 (1 + t)−η with η = min1≤i≤m {(i − θi )/(1 − i )}. Theorem 1.2.11. Let φ(t) be a non-negative function on R+ ≡ [0, +∞) satisfying sup φ(s)1+γ ≤ K 0 (1 + t)γ {φ(t) − φ(t + 1)}

t ≤s≤t +1

for some constants K 0 > 0, γ > 0, β < 1. Then φ(t) has the decay property: φ(t) ≤ C0 (1 + t)

− (1−β) γ

;

and if γ = 0, then φ(t) ≤ C0 exp{−λt 1−β } where C0 > 0, λ > 0 are constants.

1.3. Some Differential Inequalities for Nonexistence of Global Solutions

25

Theorem 1.2.12. Let φ(t) be a non-negative function on R+ ≡ [0, +∞) satisfying sup

t ≤s≤t +T

φ(s)1+γ ≤ g(t)[φ(t) − φ(t + T )]

with constants T > 0, γ > 0 and g(t) is a non-decreasing function. Then φ(t) has the decay property: −1/γ t φ(t) ≤ φ(0)−γ + γ g(s)−1 ds

f or t ≥ T.

T

In particular, if γ = 0 and g(t) = constant in the above, then we have φ(t) ≤ Cφ(0) exp{−λt} for some constant λ > 0.

1.3 Some Differential Inequalities for Nonexistence of Global Solutions The following theorem (see, e.g., Ladyzhenskaya, Solonnikov and Uralceva [208], Levine [220–222]) is very useful to prove the nonexistence of global solutions to differential equations. Theorem 1.3.1. Assume that a twice differentiable, positive function (t) satisfies for all t > 0 the inequality (t) (t) − (1 + γ )( (t))2 ≥ −2C1 (t) (t) − C2 2 (t)

(1.3.1)

where γ > 0 and C1 , C2 ≥ 0. Then (1) if

(0) > 0, (0) + γ2 γ −1 (0) > 0, C1 + C2 > 0,

(1.3.2)

(t) → +∞

(1.3.3)

γ1 (0) + γ (0) , ln t → t1 ≤ t2 = γ2 (0) + γ (0) 2 C12 + γ C2

(1.3.4)

then we have as

1

where γ1 = −C1 + (2) if

C12 + γ C2 , γ2 = −C1 −

C12 + γ C2 ;

(0) > 0, (0) > 0, C1 = C2 = 0,

(1.3.5) (1.3.6)

26


then (t) → +∞ as t → t1 ≤ t2 =

(0) . γ (0)

(1.3.7)

(1.3.8)

Glassey [122, 123] used the following theorem to establish the blow-up of solutions to nonlinear wave equations. Theorem 1.3.2. Assume that φ(t) ∈ C 2 satisfies φ (t) ≥ h(φ) (t ≥ 0) and φ(0) = α > 0, φ (0) = β > 0. If for all s ≥ α, h(s) ≥ 0, then in the domain of φ (t), we have φ (t) > 0 and φ(t ) s t≤ [β 2 + 2 h(ξ )dξ ]−1/2 ds. α

α

The following two results are due to Friedman and Lacey [109] which were used to prove the nonexistence of global solutions. Theorem 1.3.3. Let c(t) and y(t) be two non-negative functions on [0, +∞) and α > 0. Assume that c(t) ∈ L 1 (0, T ) for any T > 0 and y(t) is absolutely continuous and satisfies d y(t) + c(t)y 1+α (t) ≤ 0, f or any t > 0. dt Then

t −/α y(t) ≤ C c(s)ds . 0

Theorem 1.3.4. Let α, C > 0. Let y(t) be a non-negative absolutely continuous function on [0, +∞) satisfying d y(t) + C y 1−α (t) ≤ 0, f or any t > 0. dt Then

1/α . y(t) ≤ y α (0) − αCt

1.4 Other Useful Inequalities In this section, we shall collect other useful inequalities which play very crucial roles in classical calculus. These inequalities include the Young inequality, the Hö¨ lder inequality, Minkowski inequality and the Jensen inequality.

1.4. Other Useful Inequalities

27

1.4.1 The Young Inequalities Theorem 1.4.1. Let f be a real-valued, continuous and strictly increasing function on [0, c] with c > 0. If f (0) = 0, a ∈ [0, c] and b ∈ [0, f (c)], then

a

b

f (x)d x +

0

f −1 (x)d x ≥ ab

(1.4.1)

0

with f −1 is the inverse function of f . Equality holds in (1.4.1) if and only if b = f (a). This is a classical result called “the Young inequality” whose proof can be found in Young [433]. If we take f (x) = x p−1 with p > 1 in the above theorem, then we conclude Corollary 1.4.1. There holds that ab ≤

ap bq + p q

(1.4.2)

where a, b ≥ 0, p > 1 and 1/ p + 1/q = 1. If 0 < p < 1, then ab ≥

ap bq + . p q

(1.4.3)

The equalities in (1.4.2) and (1.4.3) hold if and only if b = a p−1 . In Corollary 1.4.1, if we consider a and b as εa and ε −1 b respectively, we can get Corollary 1.4.2. For any ε > 0, there holds that ab ≤

ε pa p bq + q p qε

where a, b ≥ 0, p > 1 and 1/ p + 1/q = 1. The Young inequality has several variants in the following. Corollary 1.4.3. (1) Let a, b > 0, 1/ p + 1/q = 1, 1 0; (iii) a α b1−α ≤ αa + (1 − α)b, 0 < α < 1. (2) Let ak ≥ 0, pk > 0, m k=1 pk = 1. Then m k=1

p

ak k ≤

m k=1

pk ak .

28


¨ 1.4.2 The Holder Inequality The following is the discrete Hö¨ lder inequality which was proved by Hölder ¨ in 1889 (see, e.g., Holder ¨ [154]). However, as pointed out by Lech [217], in fact it should be called the Roger inequality or Roger-Hö¨ lder inequality since Roger established the inequality (1.4.4) in 1888 earlier than Hö¨ lder did in 1889. However, we will follow custom here to call it the Hö¨ lder inequality. Theorem 1.4.2. If ak ≥ 0, bk ≥ 0 for k = 1, 2, . . . , n, and 1/ p + 1/q = 1 with p > 1, then 1/ p n 1/q n n p q ak bk ≤ ak bk . (1.4.4) k=1

k=1

n

n

If 0 < p < 1, then ak bk ≥

k=1

k=1

1/ p p ak

k=1

n

1/q q bk

.

(1.4.5)

k=1 p

q

Here the equalities in (1.4.4)–(1.4.5) hold if and only if αak = βbk for k = 1, 2, . . . , n where α and β are real non-negative constants with α 2 + β 2 > 0. Remark 1.4.1. If p = 1 or p = ∞, we have the trivial case. n n ak bk ≤ ak sup bk , if p = 1; k=1 n

ak bk ≤

k=1

k=1 n

1≤k≤n

bk

k=1

sup ak , if p = ∞. 1≤k≤n

Remark 1.4.2. When p = q = 2, we call (1.4.4)–(1.4.5) the Cauchy inequality, or the Schwarz inequality or the Cauchy-Schwarz inequality or the Bunyakovskii inequality. By virtue of the discrete Hö¨ lder inequality (Theorem 1.4.2), we easily obtain the integral form of the Hö¨ lder inequality, namely Theorem 1.4.3. If f ∈ L p (), g ∈ L q () and ⊆ Rn is a smooth open set, then f g ∈ L 1 () and f g L 1 () ≤ f L p () g L q () with 1 ≤ p ≤ ∞, 1/ p + 1/q = 1 and 1/ p f L p () = | f (x)| p d x ;

If 0 < p < 1, then

(1.4.6)

f L ∞ () = esssupx∈ | f (x)|.

f g L 1 () ≥ f L p () g L q () .

(1.4.7)

1.4. Other Useful Inequalities

29

The equalities in (1.4.6) and (1.4.7) hold if and only if there exist β ∈ R and real numbers C1 , C2 which are not all zero such that C1 | f (x)| p = C2 |g(x)|q and arg( f (x)g(x)) = β a.e. on hold. Remark 1.4.3. We have the corresponding weighted Holder H¨ inequality of the integral form. Let 1 0 on . Then 1/ p

| f g|ω(x)d x ≤

1/q

p

| f (x)| ω(x)d x

q

|g(x)| ω(x)d x

.

1.4.3 The Minkowski Inequalities In 1896, Minkowski established the following famous inequality. Theorem 1.4.4. Let a = {a1 , . . . , an } or a = {a1 , . . . , an , . . . } be real or complex sequences. Define 1/ p p a p = |ak | if 1 ≤ p < ∞; k

a∞ = sup |ak | if p = ∞. k

Then for 1 ≤ p ≤ ∞, a + b p ≤ a p + b p .

(1.4.8)

a + b p ≥ a p + b p

(1.4.9)

If 0 = p < 1, then where when p < 0, we require that ak , bk , ak + bk = 0 (k = 1, 2, . . . ). Moreover, when p = 0, 1, the equality in (1.4.8) holds if the sequences a and b are proportional. When p = 1, the equality in (1.4.9) holds if and only if arg ak = arg bk , ∀k. Remark 1.4.4. If we replace p by 1/ p in (1.4.8), we can obtain the following assertion: (1) if 1 ≤ p < ∞, then there holds

|ak + bk |1/ p

p

≥

k

|ak |1/ p

p

+

k

|bk |1/ p

p

;

k

(2) if 0 < p < 1, then there holds k

|ak + bk |1/ p

p

≤

k

|ak |1/ p

p

+

|bk |1/ p

p

.

k

In the applications, the integral form of the Minkowski inequality is used frequently.

30


Theorem 1.4.5. Let be a smooth open set in R n and let ff, g ∈ L p () with 1 ≤ p ≤ +∞. Then f + g ∈ L p () and If 0 1, the equality in (1.4.10) holds if and only if there exist constants C1 and C2 which are not all zero such that C1 f (x) = C2 g(x) a.e. in . If p = 1, then the equality in (1.4.10) holds if and only if arg f (x) = arg g(x) a.e. in or there exists a non-negative measurable function h such that f h = g a.e. in the set A = x ∈ | f (x)g(x) = 0 .

1.4.4 The Jensen Inequality In this subsection, we shall recall the Jensen inequality and the generalized Jensen inequalities due to Steffensen [396] and Ciesielski [60]. Since these inequalities will involve the concept of a convex function on a line segment, we first give the definition of a convex function on a line segment. Definition 1.4.1. A function f is called convex on a line segment I ⊆ R if and only if f (λx + (1 − λ)y) ≤ λ f (x) + (1 − λ) f (y)

(1.4.12)

holds for all x, y ∈ I and all real numbers λ ∈ [0, 1]. A convex function f on I is said to be strictly convex if the strict inequality holds in (1.4.12) for x = y. If − f is convex on I , then f is said to be concave on I . Among all the inequalities for convex functions, the Jensen inequality should be the famous one which has the discrete form and integral form. The following is the discrete form (see, e.g., Jensen [163]). Theorem 1.4.6. Let φ(u) : [α, β] −→ R be a convex function. Suppose that ak ≥ 0 (k = 1, 2, . . . , n) are non-negative constants verifying ni=1 ak > 0, then for any x 1 , x 2 , . . . , x n ∈ [α, β], we have

n n ak φ(x k ) k=1 ak x k n φ n ≤ k=1 . k=1 ak k=1 ak The following is the integral form of Jensen’s inequality. Theorem 1.4.7. Let φ(u) : [α, β] → R be a convex function. Suppose that f : t ∈ [a, b] → [α, β], and p(t) are continuous functions with p(t) ≥ 0, p(t) ≡ 0. Then we have b b φ( f (t)) p(t)dt a f (t) p(t)dt φ ≤ a b . b p(t)dt p(t)dt a a

1.5. C0 -Semigroups of Linear Operators

31

1.5 C0 -Semigroups of Linear Operators In this section, we shall recall some basic results on C0 -semigroups of linear operators and global attractors. These results will be used in the following chapters.

1.5.1 C0 -Semigroups of Linear Operators In this subsection we always assume that (X, · ) is a Banach space. Definition 1.5.1. Let (X, .) (or simply X) be a Banach space. A one-parameter family T (t), 0 ≤ t < ∞, of bounded linear operators from X into X is a semigroup of bounded linear operators X if (i) T (0) = I , (I is the identity operator on X ); (ii) T (t + s) = T (t)T (s) for every t, s ≥ 0 (the semigroup property). Definition 1.5.2. The linear operator A defined by D(A) = x ∈ X : lim(T (t)x − x)/t exi sts t ↓0

(1.5.1)

and

d + T (t)x |t =0 f or x ∈ D(A) (1.5.2) t ↓0 dt is called the infinitesimal generator of the semigroup T (t), D(A) is called the domain of A. Ax = lim(T (t)x − x)/t =

Remark 1.5.1. In some other literature, instead of A in (1.5.2), the operator −A is defined as the infinitesimal generator of a C0 -semigroup (see, e.g., Zheng [450]). Definition 1.5.3. A semigroup T (t), 0 ≤ t < ∞, of bounded linear operators on X is a strongly continuous semigroup of bounded linear operators if lim T (t)x = x t ↓0

f or each x ∈ X,

(1.5.3)

that is, lim T (t)x − x = 0 t ↓0

f or each x ∈ X.

(1.5.4)

We call such a strongly continuous semigroup of bounded linear operators on X a semigroup of class C0 or a C0 -semigroup. The infinitesimal generator of T (t), 0 ≤ t < ∞ is the operator A which can be defined as in Definition 1.5.2, but T (t) is only a C0 semigroup of linear (not necessarily bounded) operators on a Banach space X . Definition 1.5.4. A semigroup T (t), 0 ≤ t < ∞ is called a semigroup of contraction (or a non-expansive semigroup) if there exists a constant α ∈ (0, 1] such that T (t)x − T (t)y ≤ αx − y f or all x, y ∈ X.

(1.5.5)

In particular, if α ∈ (0, 1) in (1.5.5), then we call T (t), 0 ≤ t < ∞ a semigroup of strict contraction.

32


The next theorem is a characterization for a C0 -semigroup, which plays a very crucial role in the study of partial differential equations (see, e.g., Pazy [308]). Theorem 1.5.1. Let T (t) be a C0 -semigroup and let A be its infinitesimal generator. Then a) For any x ∈ X, 1 h→0 h

t +h

lim

b) For any x ∈ X,

t 0

t

T (s)xds = T (t)x.

(1.5.6)

T (s)xds ∈ D(A) and

t

A 0

T (s)xds = T (t)x − x.

(1.5.7)

c) For x ∈ D(A), T (t)x ∈ D(A) and d T (t)x = AT (t)x = T (t)Ax. dt

(1.5.8)

d) For x ∈ D(A), T (t)x − T (s)x = s

t

t

AT (τ )xdτ =

T (τ )Axdτ.

(1.5.9)

s

After having established the above result, we easily derive that Theorem 1.5.2. If A is the infinitesimal generator of a C0 -semigroup T (t), then D(A), the domain of A, is dense in X and A is a closed linear operator. Now we recall the characterization of the infinitesimal generators of C0 -semigroups. To this end, we need the concepts of the spectrum and resolvent of an operator. It is well known that many mathematical problems can be reduced to the solvability of operator equations (λI − A)x = 0 and (λI − A)x = y (λ ∈ C) where A is an operator defined on a Banach space X. This means that we have to study the structure of solutions to these two operator equations and hence investigate the spectrum and resolvent of operator A. Moreover, in the theory of semigroups of linear operators we often need to investigate the properties of the spectrum of the infinitesimal generator. In what follows, we assume that X is a complex Banach space. R(A) = { Ax : x ∈ D(A) ⊆ X}, N(A) stands for the null space (or the kernel) of A, i.e., N(A) = {x ∈ X : Ax = 0}. Definition 1.5.5. Let A : X ⊇ D(A) → X be a closed operator, λ ∈ C. If there exists 0 = x ∈ D(A) such that Ax = λx, then we call λ an eigenvalue of A. The set of all eigenvalues of A is denoted by σ p (A). The nonzero vector x is called an eigenvector of A corresponding to λ. The set E λ = {x : Ax = λx} is called the characteristic space of A. The dimension of E λ , dim E λ , is called the multiplicity of eigenvalue λ.


33

Obviously, E λ = N(λI − A) and E λ is a linear space. If X is a finite-dimensional space, and if A : X → X is a linear map, then A certainly has some eigenvalues. To see that this is so, introduce a basis for X so that A can be identified with a square matrix. The following conditions on a complex number λ are then equivalent: (1) A − λI has a nontrivial null space; (2) A − λI is singular; (3) det(A − λI ) = 0 where det is the determinant function. Definition 1.5.6. Let A : X ⊇ D(A) → X be a closed linear operator, λ ∈ C. If λI − A : D(A) → X is a one-to-one correspondence, and (λI − A)−1 is a bounded linear operator, then we say that λ is a regular value of A. The set of all regular values of A is called the resolvent set of A, denoted by ρ(A). When λ ∈ ρ(A), R(λ; A) ≡ (λI − A)−1 is called the resolvent of A at λ. The set of complex numbers which are not regular values of A is called the spectral set of A, denoted by σ (A). Every point in σ (A) is called a spectral point. Obviously, any eigenvalue of A is a spectral point of A. Remark 1.5.2. In other literature, there is another definition of ρ(A): ρ(A) = λ ∈ C : D((λI − A)−1 ) is dense in X and (λI − A)−1 is bounded on its domain . However, it is easy to show that these two definitions of ρ(A) coincide if A ∈ C(X X ), the space of all operators which are continuous on X (see, e.g., Belleni-Morante and McBride [33]). Remark 1.5.3. If A is not necessarily closed, (λI − A)−1 may be extended to the whole space X. Definition 1.5.7. The operator defined by R(λ; A) ≡ (λI − A)−1 (whenever it exists) is called the resolvent operator. The resolvent operator plays a very crucial role in the study of the local and/or global well-posedness of solutions to differential equations. This can be seen from the following example. For any given λ ∈ ρ(A) and g ∈ X, the equation (λI − A) f = g,

f ∈ D(A)

(1.5.10)

has a unique solution f = (λI − A)−1 g = R(λ, A)g ∈ D(A). Furthermore, let fi (i = 1, 2) be two solutions to (1.5.10) corresponding to gi ∈ X (i = 1, 2) and g1 − g2 be

34


small, then f1 − f 2 D( A) is also small, in fact, due to the boundedness of R(λ, A), we have f 1 − f 2 D( A) = R(λ, A)(g1 − g2 ) ≤ R(λ, A)g1 − g2 where f1 D( A) = f 1 + A f 1 is the graph norm. This implies that if λ ∈ ρ(A), the equation (1.5.10) is well posed since we have obtained existence, uniqueness and stability of solutions. In such an ideal way our problem can be resolved. Just for this reason, we call the operator R(λ; A) (whenever it exists) the resolvent operator and the set ρ(A) the resolvent set. On the other hand, when λ ∈ σ (A), we encounter many difficulties in trying to solve equation (1.5.10). The situation now is rather complicated, requiring the spectrum σ (A) need to be subdivided. Further, we may investigate the structure in detail (see, e.g., Belleni-Morante and McBride [33], and Kato [186]). First, if there exists λ ∈ C such that (λI − A)−1 exists, then we have three cases for the range of λI − A in the following: (1) R(λI − A) = X : in this case, due to the closedness of A, (λI − A)−1 is a closed operator, i.e., it is a closed operator on the whole space X which implies, by the Closed Graph Theorem, that (λI − A)−1 is a linear bounded operator on X , hence λ ∈ ρ(A). (2) R(λI − A) = X, R(λI − A) = X; (3) R(λI − A) = X. We have the following definition for the complex number λ in the above cases (2) and (3). Definition 1.5.8. Let A : X ⊇ D(A) → X be a closed linear operator, λ ∈ C. If there exists λ such that (λI − A)−1 exists, but R(λI − A) = X, R(λI − A) = X , then we say that λ is a continuous spectrum of A, and the set of all continuous spectra of A is denoted by σc (A). If there exists λ such that (λI − A)−1 exists, but R(λI − A) = X, then we say that λ is a rest spectrum of A, and the set of all rest spectra of A is denoted by σr (A). It follows from the above definition that σ (A) = σ p (A) ∪ σc (A) ∪ σr (A). Generally speaking, for a linear operator A, there may exist three kinds of spectral points. Now we recall the characterization of the infinitesimal generators of C0 -semigroups. First, we have (see, e.g., Pazy [308]) Theorem 1.5.3. Let T (t), 0 ≤ t < ∞ be a C0 -semigroup on a Banach space X. Then there exist constants M > 0 and ω ≥ 0 such that T (t) ≤ Meωt f or all t ≥ 0.

(1.5.11)

Obviously, if M = 1 and ω = 0 in (1.5.11), then we obtain a C0 -semigroup of non-expansions or contractions.


35

Definition 1.5.9. For real numbers M > 0 and ω ≥ 0, let G(M, ω; X ) denote the set of generators of C0 -semigroups T (t), 0 ≤ t < ∞ on a Banach space X satisfying (1.5.11). With the above notation, we are now in a position to state necessary and sufficient conditions for an operator A to be in the class G(M, ω; X). We have (see, e.g., BelleniMorante and McBride [33], Chapter 3 and Pazy [308], Chapter 1). Theorem 1.5.4. (The Hille-Yoshida Theorem) A ∈ G(M, ω; X ) if and only if (i) A is a closed linear operator whose domain D(A) is dense in X; and (ii) for all real numbers λ > ω, λ ∈ ρ(A) (the resolvent set of A), and [R(λ; A)]n ≤

M (λ − ω)n

f or n = 1, 2, . . . .

(1.5.12)

Remark 1.5.4. If the condition (1.5.12) is replaced by the condition that for all complex numbers Re λ > ω, λ ∈ ρ(A) and [R(λ; A)]n ≤

M (Re λ − ω)n

f or n = 1, 2, . . . ,

(1.5.13)

then Theorem 1.5.4 still holds. It is well known that the C0 -semigroups of contractions occur frequently in practice, so in the following we shall pay attention to studying the characterization of their infinitesimal generators in more detail. To this end, let V (μ, A) = (I − μA)−1 = μ−1 R(μ−1 ; A) wherever the last expression is meaningful. For fixed A, we shall usually write V (μ, A) as V (μ). We shall state the Hille-Yosida Theorem for the infinitesimal generators of C0 -contraction semigroups. Theorem 1.5.5. (The Hille-Yosida Theorem for the Infinitesimal Generators of C0 Contraction Semigroups) The linear operator A generates a C0 -semigroup of contracX if and only if tions on X (i.e., A ∈ G(1, 0; X)) (i) A is a linear operator whose domain D(A) is dense in X, and (ii) for all real numbers μ > 0, μ ∈ ρ(A) with V (μ) ≤ 1. Remark 1.5.5. The assumption that D(A) is dense in X is not crucial, for if D(A) is not dense, A ∈ G(1, 0; D(A)).

36


Now we shall recall the Lumer-Phillips theorem which is very useful in the theory of linear semigroups. To state such a theorem, we now introduce the concept of dissipative operator which is borrowed from the case where X is a Hilbert space. We assume that X ∗ is the dual space of X . We denote the value of x ∗ ∈ X ∗ at ∗ x ∈ X by x ∗ or x ∗ , x. For each x ∈ X , we define x, the duality set F(x) ⊆ X by ∗ ∗ ∗ ∗ 2 ∗ 2 F(x) = x : x ∈ X and x , x = x = x . From the Hahn-Banach theorem it follows that F(x) = ∅ for every x ∈ X . Definition 1.5.10. A linear operator A is dissipative if for every x ∈ D(A) there is an x ∗ ∈ F(x) such that ReAx, x ∗ ≤ 0. The next theorem is a useful characterization of dissipative operators. Theorem 1.5.6. A linear operator A : X ⊃ D(A) −→ R(A) ⊆ X is dissipative if and only if (λI − A)x ≤ λx f or all x ∈ D(A), λ > 0. (1.5.14) Remark 1.5.6. The motivation for the use of the word “dissipative” comes from the case where X is a Hilbert space. Then, for a linear operator A, condition (1.5.14) is equivalent to the condition Re(x, Ax) ≤ 0 f or all f ∈ D(A) (1.5.15) where (., .) denotes the scalar inner product on X. Definition 1.5.11. A linear operator A : X ⊃ D(A) −→ R(A) ⊆ X is m-dissipative if A is dissipative and R(λI − A) = X f or all λ > 0 that is, for any given g ∈ X, there is f ∈ D(A) such that (λI − A) f = g. By virtue of the above two definitions, we readily conclude Corollary 1.5.1. Every m-dissipative operator is a dissipative operator. Now we state the Lumer-Phillips theorem as follows (see, e.g., Pazy [308]). Theorem 1.5.7. (The Lumer-Phillips Theorem of the Infinitesimal Generators of Contraction Semigroups) The linear operator A generates a C0 -semigroup of contracX if and only if tions on X (i.e., A ∈ G(1, 0; X)) (i) D(A) is dense in X, and (ii) A is m-dissipative. Corollary 1.5.2. Let A : X ⊇ D(A) −→ R(A) ⊆ X be a linear operator with dense domain D(A) in X. Then if A is dissipative and there is a λ0 > 0 such that the range, R(λ0 I − A), of λ0 I − A is X (i.e., R(λ0 I − A) = X ), then A is the infinitesimal generator of a C0 -semigroup of contractions on X. Moreover, for any x ∈ D(A) and every x ∗ ∈ F(x), ReAx, x ∗ ≤ 0.

1.6. Global Attractors

37

The next result indicates that we can judge that a densely defined closed linear operator A is the infinitesimal generator of a C0 -semigroup of contractions not only from this operator A itself but also from its adjoint operator A∗ . Theorem 1.5.8. Let A : X ⊇ D(A) −→ R(A) ⊆ X be a closed linear operator with dense domain D(A) in X. If both A and A∗ , the adjoint operator of A, are dissipative, then A is the infinitesimal generator of a C0 -semigroup of contractions on X. We conclude this subsection with some properties of dissipative operators. Theorem 1.5.9. Let A : X ⊇ D(A) −→ R(A) ⊆ X be a dissipative operator. Then (a) If for some λ0 > 0, R(λ0 I − A) = X, then R(λI − A) = X for all λ > 0. ¯ the closure of A, is also dissipative. (b) If A is closable, then A, (c) If D(A) = X, then A is closable. Theorem 1.5.10. Let A : X ⊇ D(A) −→ R(A) ⊆ X be a dissipative operator with R(I − A) = X. If X is reflexive, then D(A) = X.

1.6 Global Attractors For a given nonlinear evolution equation, once it is known that a solution exists for all time t > 0, a natural and interesting question is to ask about the asymptotic behavior of the solution as t → +∞. As stated in Zheng [450], the study of asymptotic behavior of the solution to a nonlinear evolutionary equation as time goes to infinity can be divided into two categories. The first category is to investigate the asymptotic behavior of solutions for any given initial datum. The second category is to investigate the asymptotic behavior of all solutions when initial data vary in any bounded set. There are essentially five apparently distinct properties that a semigroup may possess and such that each of them together with the existence of a bounded absorbing set leads to the existence of a global attractor. They are (uniform) compactness, asymptotic smoothness, asymptotic compactness, weak compactness and ω-limit compactness. The first condition is that a semigroup S(t) is such that S(t0 ) is a compact operator for some t0 ≥ 0; the second one is that for any closed, bounded, positively invariant set B there exists a compact set K = K (B) which attracts B; and the third one is the precompactness of the sequence S(tt j )u j j ∈N for every bounded sequence u j j ∈N in the phase space and every sequence {tt j } j ∈N of positive numbers with t j → ∞. The fourth one is induced by a sequence of two spaces, one of which can be embedded compactly into the other. The first condition is stronger than the other three, but the second and the third are, in fact, quite related. The main difference lies in the methods used in the applications in order to establish any one of those conditions. The choice of the proper method depends on the nature of each problem. The fifth one is to use the measure of non-compactness to show ω-limit compactness of the semigroup.

38


Compactness was the first one to be used. If the dynamical system is finite dimensional (corresponding to ordinary differential equations), this condition is a trivial consequence of the existence of an absorbing set in the phase, however for parabolic equations on bounded spatial domains this compactness property follows from a regularization of the solutions and some compact Sobolev embedding theorem (i.e., we can obtain the existence of a compact absorbing set). However, the solution semigroup fails to be compact for most of the infinite-dimensional dynamical systems arising from weakly damped hyperbolic equations or parabolic equations on unbounded domains, even if there is an absorbing set in the phase space. Therefore, this method breaks down here. Asymptotic smoothness and asymptotic compactness properties are needed to handle those non-compact semigroups. One approach is to show the so-called β-contraction property of the semigroup, which implies asymptotic smoothness. This condition has been successfully used by Hale [135] and many other authors. Another approach is to decompose the solution semigroup into two parts: a (uniformly) compact part and a part which decays (uniformly) to zero as time goes to infinite (see, e.g., Temam [407]). Then, the proof of the existence of a global attractor using this splitting amounts to (essentially or explicitly) proving either asymptotic smoothness or asymptotic compactness of the semigroup. It has been observed that splitting of the semigroup into a (uniformly) compact part and a (uniformly) decaying part is actually necessary and sufficient for the existence of a global attractor in the case where the phase space is a Hilbert space, and note that the same equivalence holds if the phase space is a uniformly convex Banach space. This means that a decomposition of the solution semigroup must exist if the global attractor exists. However, it may be difficult to find such a decomposition in applications. For example, no suitable decomposition has yet been found for the Kdv equation or for the 2D incompressible Navier-Stokes equations on unbounded domains when the forcing term does not belong to some weighted Sobolev space. Weak compactness was used by Ghidaglia [117] to establish the existence of global attractors for nonlinear damped Schrö¨ dinger equations. A new method called the energy equation method was recently established to derive the existence of a global attractor (see, e.g., Moise, Rosa and Wang [270]). This approach is relatively simple in that the assumptions are straightforward and may be verified directly from the equations. In most applications, the central part lies in establishing an energy-type equation (this may not be trivial, though, and is open for the Navier-Stokes equations in space dimension 3). For parabolic-type problems, the typical way is to establish enough regularity for the solutions, which then imply the energy equation. For hyperbolic-type problems, the typical way is to use time reversibility to establish the energy equation. Ball [22, 23] first used the energy equation method together with asymptotically compactness to prove the existence of global attractors, later on Ghidaglia [117], Ghidaglia and Temam [119], Goubet [125], Goubet and Moise [126], Moise and Rosa [269], Rosa [362], Wang [421] also used this approach to show the existence of global attractors. Ma, Wang and Zhong [246] devised a new method and used the measure of noncompactness to show ω-limit compactness of the semigroup.


39

Zhong, Yang and Sun [457] established the existence of global attractors for the norm-to-weak continuous semigroups. Recently, Pata and Zelik [307] proved a result of existence of global attractors for semigroups of closed operators which embrace the norm-to-weak continuous semigroups. In this section, we shall review some results on the global attractors, some of which will be used in the following context.

1.6.1 Compact Semigroups (Semiflows) for Autonomous Systems In this subsection, we recall some basic results on the global attractors of the compact semigroups (and semiflows). We consider the operator semigroups S(t) acting on a set E. Usually E is a complete metric space or a Banach space. In particular, E can be a closed subset of a Banach space. The concepts of semigroups and C0 -semigroups have been stated above. Here we only state the concept of semiflows. Definition 1.6.1. A semiflow S(t) on E is defined to be a mapping (t, w) = S(t)w, where : [0, +∞) × E −→ E satisfies the following three properties: (1) S(0)w = w, ∀w ∈ E. (2) The restricted mapping : (0, +∞) × E −→ E is continuous. (3) The following semigroup property holds: S(s)S(t)w = S(s+t)w, ∀w ∈ E, s, t ∈ [0, +∞). We assume that a semigroup (semiflow) S(t) acts on a metric space or a Banach space E. Let B(E) be the collection of all bounded sets in E with respect to the metric in E. Definition 1.6.2. The semigroup (semiflow) S(t) is called (E, E)-bounded for t ≥ 0 if S(t)B ∈ B(E) for any B ∈ B(E) and for all t ≥ 0. The semigroup (semiflow) S(t) is called (E, E)-bounded uniformly in t ≥ 0 if for any B ∈ B(E), there exists B1 ∈ B(E) such that S(t)B ⊆ B1 for all t ≥ 0. Definition 1.6.3. A set B0 ⊆ E is said to be absorbing if for every B ∈ B(E), there is a time t B = t (B) > 0 such that S(t)B ⊆ B0 for all t ≥ t B . An equivalent definition of an absorbing set in Hale [135] ] is the point dissipation which will be stated as follows. Definition 1.6.4. A semigroup (semiflow) S(t) on E is said to be point dissipative if there is a bounded set B0 in E with property that for every w ∈ E, there is a time tw = t (w) such that S(t)w ∈ B0 , for all t > tw . In this case, the set B0 is referred to as an absorbing set for the semigroup (semiflow) S(t). Definition 1.6.5. A set P ⊆ E is said to be an attracting set for a semigroup (semiflow) if for every B ∈ B(E), dist E (S(t)B, P) → 0 as t → +∞.

40


Here

dist E (X, Y ) = sup inf y − x E , X, Y ⊆ E. x∈X y∈Y

Definition 1.6.6. The semigroup (semiflow) S(t) on E is said to be a compact semigroup for t > 0 if for every B ∈ B(E) and every t > 0, the set S(t)B lies in a compact set in E, or equivalently, S(t) admits a compact absorbing set P, P →

→

E (i.e., the embedding from P into E is compact). It is said to be an asymptotically compact semigroup if there exists a compact attracting set K , K →

→

E. To study the asymptotic behavior of a solution, we have to investigate the structure of the ω-limit set ω(A) by ω(A) = ∩s≥0 ∪t ≥s S(t)A (1.6.1) where A is a set in E and the closure is taken in E. Equivalently, ω(A) can be also defined as ω(A) = φ : ∃ tn → +∞ and a sequence φn ∈ A

such that S(ttn )φn → φ as n → +∞ .

(1.6.2)

In the following theorem we collect some properties of ω(A). Theorem 1.6.1. Assume that S(t) is a nonlinear C0 -semigroup (semiflow) and B is a nonempty set in E. Then we have (1) ω(B) is positively invariant, i.e., for all t ≥ 0, S(t)ω(B) ⊆ ω(B); (2) if there is t0 > 0 such that

(1.6.3)

∪t ≥t0 S(t)B

is relatively compact in E, then ω(B) is a nonempty, compact invariant set. Furthermore, if B is connected, then ω(B) is also connected. In particular, if E is a complete metric space, B = {x} ⊂ E and there is t0 ≥ 0 such that ∪t ≥t0 S(t)x is relatively compact in E, then the ω-limit set ω(x) is a compact, connected invariant set. Now we begin with the discussion of a global attractor. Definition 1.6.7. Assume that E is a complete metric space, and S(t) is a nonlinear C0 semigroup (semiflow) of operators defined on E. A set A ⊆ E is called an attractor if the following conditions hold: (i) A is invariant under S(t), i.e., S(t)A = A, ∀t ≥ 0.

(1.6.4)

(ii) A possesses an open neighborhood U such that for any element u 0 ∈ U as t → +∞, S(t)u 0 converges to A A, i.e., dist(S(t)u 0 , A) = inf d(S(t)u 0 , y) → 0, as t → +∞. y∈A


41

If A is an attractor, then the maximal open set U satisfying (ii) is called the basin of attraction of A. According to the above definition, it can be also said that A attracts points of U. If a subset B ⊆ U satisfies dist(S(t)B, A) ≡ sup

inf d(x, y) → 0, as t → +∞,

x∈S(t )B y∈A

then A is said to uniformly attract B, or simply A attracts B. Definition 1.6.8. If A is a compact attractor, and it attracts bounded sets of E, then A is called a global attractor or universal attractor. Remark 1.6.1. It is easy to verify that a global attractor is maximal among all bounded attractors or bounded invariant sets in the sense of inclusion. The following two theorems characterize the existence of a global attractor of a semigroup (semiflow) S(t). Theorem 1.6.2. Assume that E is a metric space and S(t) is a nonlinear C0 -semigroup (semiflow) defined on E. Let the following conditions hold: (1) S(t) is a continuous (nonlinear) operator from E into itself, ∀t ≥ 0; (2) there exists a bounded absorbing set B0 ; (3) for any bounded set B, there is a time t0 (B) ≥ 0 depending on B such that ∪t ≥t0 (B) S(t)B is relatively compact in E, namely S(t) is a uniformly compact semigroup (semiflow). Then A = ω(B0 ) is a global attractor and it is connected. However, for some problems of evolutionary differential equations, the above condition (3) is very difficult or impossible to be verified. In this case, condition (3) can be weakened to some extent. More precisely, we have the following result. Theorem 1.6.3. Assume that E is a metric space and S(t) is a nonlinear C0 -semigroup (semiflow) defined on E. Let the following conditions hold: (1) S(t) is a continuous (nonlinear) operator from E into itself, ∀t ≥ 0; (2) there exists a bounded absorbing set B0 ; (3) for any t ≥ 0, S(t) can be written as S(t) = S1 (t) + S2 (t) where S1 (t) satisfies condition (3) in Theorem 1.6.2, and S2 (t) is a continuous mapping from E into E, and satisfies the following condition: γ K = sup S S2 (t)φ E → 0, as t → +∞ φ∈K

where K is any bounded set in E. Then A = ω(B0 ) is a global attractor and it is connected.

42


As a corollary of Theorem 1.6.2, we have the following result. Corollary 1.6.1. Let S(t) be a point dissipative, compact semiflow on a complete metric space E. Then S(t) has a global attractor A in E. Furthermore, the following conclusions hold: (1) A attracts all bounded sets in E; (2) A is maximal in the sense that every compact invariant set in E lies in A; (3) A is minimal in the sense that if B is any closed set in E that attracts each compact set in E, then one has A ⊂ B; (4) For each bounded set B in E, the ω-limit set ω(B) satisfies ω(B) ⊆ A; (5) A is a connected set in E; (6) A is Lyapunov stable, i.e., for every neighborhood V of A and every τ > 0, there is a neighborhood U of A with the property that S(t)U ⊆ V , for all t ≥ τ . (7)

A = ω(B0 ) = ∩s≥0 ∪t ≥s S(t)B0

is invariant under S(t), i.e., S(t)A = A, t ≥ 0. Here B0 is an absorbing set. (8) A is compact.

1.6.2 Weakly Compact Semigroups (Semiflows) for Autonomous Systems In this subsection, we introduce an abstract framework due to Ghidaglia [117] which is related to the existence of global attractors of weakly compact semigroups (semiflows). This framework is very useful for some non-compact semigroups (semiflows) generated by some partial differential equations such as the compressible Navier-Stokes equations and the nonlinear 1D thermoelastic systems which we shall discuss in Chapters 2–4 and Chapter 6, respectively. Theorem 1.6.4. Let H1 , H2, H3 be three Banach spaces verifying the following conditions: (1) the embeddings H3 →

H2 and H2 →

H1 are compact; (2) there are C0 -semigroups (semiflows) S(t) on H2 and H3 which map H2, H3 into H2 , H3 respectively and for any t > 0, S(t) are continuous (nonlinear) operators on H2, H3 respectively; (3) the semigroup (semiflow) S(t) on H3 possesses a bounded absorbing set in H3; then there is a weak universal attractor A3 in H3. If, further, the following conditions are valid, (4) the semigroup (semiflow) S(t) on H2 possesses a bounded absorbing set in H2; (5) for any t > 0, S(t) is continuous on bounded sets of H2 for the topology of the norm of H1, then there is a weak universal attractor A2 in H2.

1.7. Bibliographic Comments

43

One of the advantages of the above abstract framework is that we can obtain two universal attractors simultaneously. The other one is that we can use it to deal with a lot of problems that can not generate compact semigroups (semiflows). These models include the compressible Navier-Stokes equations to be studied in Chapters 2–4 and the nonlinear 1D thermoviscoelasticity in Chapter 6.

1.7 Bibliographic Comments For the basic theory of functional analysis, partial differential equations, distributions, inequalities, Sobolev spaces and semigroups, we consult the works by Adams [1], Agarwal [2], Agarwal and Pan [3], Alzer [5, 6], Aubin [15], Bassanini and Elcrat [24], Batchelor [25], Beckenbach and Bellman [29], Belleni-Morante and McBride [33], Bellman [34–37], Bihari [39], Bourgain [41], Brezis [42], Brezis and Wainger [43], Brokate and Sprekels [44], Brouwer [45], Calderó´ n [47], Cazenave [49], Chandra and Fleishman [50], Chen [53], Cheney [55], Chicone and Latushkin [58], Chu and Metcalf [59], Ciesielski [60], Courant and Hilbert [65], Crandall and Liggett [66], Daykin and Eliezer [81], Dieudonné [83], Dragomir [86], Dunford and Schwartz [89], Edwards [91], Evans [93], Everitt [94], Friedman [107, 108], Gagliardo [113, 114], Galdi [115], Gearhad [116], Goldstein [124], Gronwall [130], Guo [131], Gyö¨ ri [134], Hardy [139], Henry [140], Hille and Phillips [141], Hö¨ lder [154], James [162], Jensen [163], John [179], Jones [180], Kalantarov and Ladyzhenskaya [181], Kato [183–187], Komura [202, 203], Krylov [205], Kuang [206], Ladyzhenskaya, Solonnikov and Uralceva [208], Landau and Lifshitz [212], Lech [217], Levine [220–222], Levine, Park and Serrin [223], Levine, Pucci and Serrin [224], Levine and Sacks [225], Levine and Serrin [226], Li and Chen [227, 229], Li and Qin [230], Lieberman [231], Linz [232], Lions [233], Lions and Magenes [234], Liu and Zheng [240], Liu [241], Lunardi [244], Marcus and Mizel [247, 248], Maz’ja [258], Megginson [259], Miao [262, 263], Mitrinović´ and Vasić´ [268], Munoz ˜ Rivera [275], Nikol’skii [296], Nirenberg [297–300], Oleinik [304], Pao [306], Pazy [308], Pecaric and Svrtan [309], Pelczar [311], Poussin [312], Qin [315, 318, 319], Racke [349], Redheffer [358], Renardy, Hrusa and Nohel [361], Robinson [362], Runst and Sickel [364], Sansone and Conti [365], Schmaekeke and Sell [366], Schwartz [367], Segal [368], Smoller [380], Sobolev [381–384], Sogge [385], Stein [398], Takahashi [402], Tanabe [403], Taylor [405], Temam [406, 407], Teixeira [408], Torchinsky [409], Triebel [410], Troisi [411], Viswanatham [416], Walter [418], Wang [419, 420], Willet [426], Willet and Wong [427], Wong [428], Xia, Shu, Yan and Tong [430], Yang [436], Ye and Li [437], Yosida [438], Young [439], Zhang and Guo [442], Zhang and Lin [443], Zheng [450], Zhong, Fan and Chen [456], Zhou and Wang [458], Ziebur [461], Ziemer [462], Zmorovic [463]. For the basic theory of the study of the second category (i.e., for the basic theories of infinite-dimensional dynamical systems), we refer the readers to the works by Babin [16], Babin and Vishik [17, 18], Ball [22, 23], Bernard and Wang [38], Caraballo, Rubin and Valero [48], Chepyzhov, Gatti, Grasselli, Miranville and Pata [56], Chepyzhov and Vishik [57], Constantin and Foias [63], Constantin, Foias and Temam [64], Dlotko [84], Eden and Kalantarov [90], Edfendiev, Zelik and Miranville [92], Feireisl [97, 98, 100],

44


Feireisl and Petzeltova [101, 102], Ghidaglia [117, 118], Ghidaglia and Temam [119], Goubet [125], Goubet and Moise [126], Hale [135], Hale and Perissinotto [136], Haraux [138], Hoff and Ziane [150, 151], Ladyzhenskaya [207], Liu and Zheng [240], Lu, Wu and Zhong [242], Ma, Wang and Zhong [246], Miranville [265, 266], Miranville and Wang [267], Moise and Rosa [269], Moise, Rosa and Wang [270], Pata and Zelik [307], Qin [323], Qin and Fang [328], Qin, Liu and Song [333], Qin and Lü [334], Qin and Muñoz Rivera [337, 339], Qin and Schulze [342], Qin and Song [343], Robinson [362], Rosa [363], Sell [369], Sell and You [370, 371], Shen and Zheng [376], Sprekels and Zheng [391], Temam [407], Vishik and Chepyzhov [413, 414], Wang [421], Wang, Zhong and Zhou [422], Wu and Zhong [429], Zhao and Zhou [445], Zheng [450], Zheng and Qin [451, 452], Zhong, Yang and Sun [457], and references therein.

Chapter 2

A One-dimensional Nonlinear Viscous and Heat-conductive Real Gas This chapter is concerned with the global existence and asymptotic behavior of solutions with arbitrary initial data to a 1D nonlinear viscous heat-conductive real gas with two kinds of boundary conditions. More general constitutive relations will be studied and our assumptions on growth exponents of temperature include cases never before studied. For the results of Sections 2.1–2.2, we consult Qin [315, 318, 319, 321]; for the results of Section 2.3, we refer the readers to Qin [322]; for the results of Sections 2.5– 2.6, we consult Qin [323]. In Chapters 2, 3 and 6, we shall use the following notation: For non-negative integers λ, s and arbitrary T > 0, we define Dtλ Dxs = ∂ λ+s /∂t λ ∂ x s ,

= (0, 1), = [0, 1],

Q T = × (0, T ),

Q T = × [0, T ].

For a non-negative integer n and β ∈ (0, 1), we define |u|(0) = sup |u(x)|, ¯ x∈

u(n) =

n

|u|(β) =

|Dxi u|(0) ,

sup

¯ =x x,x ∈,x

|u(x) − u(x )/|x − x |β ,

u(n+β) = u(n) + |Dxn u|(β) ,

i=0 (0)

|u|T =

sup

(x,t )∈ Q¯ T

(β)

|u(x, t)|, |u|x,T =

sup

(x,t ),(x ,t )∈ Q¯ T ,x =x

|u(x, t) − u(x , t )|/|x − x |β ,

46

Chapter 2. A One-dimensional Nonlinear Viscous and Heat-conductive Real Gas (β/2)

|u|t,T

(β) |u|T

=

=

sup

(x,t ),(x ,t )∈ Q¯

(β) |u|x,T

T

,t =t

(β/2) + |u|t,T ,

|u(x, t) − u(x , t )|/|t − t |β/2 ,

u(n) T

n

=

|Dtλ Dxs u|(0) T ,

2λ+s=0 (n+β)

uT

(n)

= uT +

2λ+s=n

(n+β)

|||u|||T

=

n

(β)

|Dtλ Dxs u|x,T +

n

2λ+s=max(n−1,0)

|Dtλ Dxs u|(0) T +

λ+s=0

H n+β = u(x) :

λ+s=n

n+β

= u(x, t) :

(n+β)

|||u|||T

(β)

|Dtλ Dxs u|T ,

n+β u(n+β) < +∞ , HT = u(x, t) : BT

(β/2)

|Dtλ Dxs u|t,T ,

(n+β)

uT

< +∞ ,

< +∞ .

In Chapters 2, 3 and 6, in general and without danger of confusion we shall use the same symbol to denote state functions as well as their values along a thermodynamic process, e.g., p(u, θ ) and p(u(x, t), θ (x, t))., L p , 1 ≤ p ≤ +∞, H 1 = W 1,2 , H01 = W01,2 , denote the usual Lebesgue, Sobolev spaces on (0, 1); . B denotes the norm in the space d B, .: = . L 2 . Analogously, ∂t or dt or a subscript t and, likewise, ∂x or a subscript x, denote the partial derivatives with respect to t and x in the distribution sense, respectively. Letters C (sometimes C , C ) will denote universal constants depending only on the initial data, but independent of any length of time t. Other notation, not described above, will be explained where it appears. In this chapter, we study the global existence and asymptotic behavior, as time tends to infinity, of solutions to a 1D nonlinear viscous heat-conductive real gas for two types of boundary conditions. The system also consists of a hyperbolic equation and two parabolic equations. Some assumptions on the constitutive relations are more general than those in [163, 164] and [190]. The assumptions on exponents q and r include cases not studied before.

2.1 Fixed and Thermally Insulated Boundary Conditions 2.1.1 Main Results This subsection is concerned with global existence, uniqueness and asymptotic behavior, as time tends to infinity, of solutions to a system for a nonlinear viscous, heat-conductive, one-dimensional real gas with fixed and thermally insulated boundary conditions. The referential (Lagrangian) form of the conservation laws of mass, momentum, and energy

2.1. Fixed and Thermally Insulated Boundary Conditions

47

for a one-dimensional gas with reference density ρ0 = 1 is

(e +

u t − vx = 0, vt − σx = 0,

(2.1.1) (2.1.2)

v2 )t − (σ v)x + Q x = 0, 2

(2.1.3)

and the second law of thermodynamics is expressed by the Clausius-Duhem inequality Q ≥ 0. (2.1.4) ηt + θ x Here subscripts indicate partial differentiations, u, v, σ, e, Q, η and θ denote specific volume, velocity, stress, internal energy, heat flux, specific entropy and temperature, respectively. Note that u, θ and e may take only positive values. We consider the problem (2.1.1)–(2.1.3) in the region {0 ≤ x ≤ 1, t ≥ 0} under initial conditions u(x, 0) = u 0 (x), v(x, 0) = v0 (x), θ (x, 0) = θ0 (x) on [0, 1],

(2.1.5)

and boundary conditions of the form v(0, t) = v(1, t) = 0,

Q(0, t) = Q(1, t) = 0.

(2.1.6)

For a one-dimensional homogeneous real gas, e, σ, η and Q are given by the constitutive relations e = e(u, θ ), σ = σ (u, θ, vx ), η = η(u, θ ), Q = Q(u, θ, θ x ) (2.1.7) which in order to be consistent with (2.1.4) must satisfy σ (u, θ, 0) = u (u, θ ), η(u, θ ) = − θ (u, θ ), (σ (u, θ, w) − σ (u, θ, 0))w ≥ 0, Q(u, θ, g)g ≤ 0

(2.1.8) (2.1.9)

where = e − θ η is the Helmholtz free energy function. For the case of an ideal gas, i.e., e = cv θ, σ = −R

vx θx θ + μ , Q = −k , u u u

(2.1.10)

with suitable positive constants cv , R, μ and k, Kazhikhov [194, 195], Kazhikhov and Shelu-khin [196], Kawashima and Nishida [191], and Nagasawa [283–287] established the existence of global smooth solutions to the system (2.1.1)–(2.1.3). As it is known, the constitutive equations of a real gas are well approximated within moderate ranges of u and θ by the model of an ideal gas. However, under very high temperatures and densities, (2.1.10) becomes inadequate. Thus a more realistic model than (2.1.10) would be a linearly viscous gas (or Newtonian fluid) σ (u, θ, vx ) = − p(u, θ ) +

μ(u, θ ) vx u

(2.1.11)

48

Chapter 2. A One-dimensional Nonlinear Viscous and Heat-conductive Real Gas

satisfying Fourier’s law of heat flux Q(u, θ, θx ) = −

k(u, θ ) θx u

(2.1.12)

whose internal energy e and pressure p are coupled by the standard thermodynamical relation eu (u, θ ) = − p(u, θ ) + θ pθ (u, θ ), (2.1.13) to be consistent with (2.1.4). We assume that e, p, σ and k are twice continuously differential on 0 < u < +∞ and 0 ≤ θ < +∞, and there exist the exponents q and r satisfying one of the following relations: 0 ≤ r ≤ 1/3, 1/3 < q,

(2.1.14)

1/3 < r < 4/7, (2r + 1)/5 < q, 4/7 ≤ r ≤ 1, (5r + 1)/9 < q,

(2.1.15) (2.1.16)

1 < r ≤ 13/3, (9r + 1)/15 < q, 13/3 < r, (11r + 3)/19 < q;

(2.1.17) (2.1.18)

concerning growth of the temperature, we require that there be positive constants ν, p1 , p2, k0 and, for any u > 0, that there be positive constants N(u), p3 (u), p4 (u) and k1 (u) such that for any u ≥ u and θ ≥ 0 the following conditions hold: 0 ≤ e(u, 0), ν(1 + θ r ) ≤ eθ (u, θ ) ≤ N(u)(1 + θ r ), 0 < p1 ≤ up(u, θ ) ≤ p2(1 + θ − p3 (u)[l + (1 − l)θ + θ

r+1

r+1

(2.1.19)

),

(2.1.20)

] ≤ pu (u, θ )

≤ − p4(u)[l + (1 − l)θ + θ r+1 ], r

l = 0 or

1,

(2.1.21)

| pθ (u, θ )| ≤ p4 (u)(1 + θ ), k0 (1 + θ q ) ≤ k(u, θ ) ≤ k1 (u)(1 + θ q ),

(2.1.22) (2.1.23)

|ku (u, θ )| + |kuu (u, θ )| ≤ k1 (u)(1 + θ q ).

(2.1.24)

For the viscosity μ(u, θ ), we require that it be independent of θ , uniformly positive, and bounded (2.1.25) 0 < μ0 = μ(u, θ ). We are now in a position to state our main theorem. Theorem 2.1.1. In addition to assumptions (2.1.11)–(2.1.25), we assume that for α2 ∈ (0, 1) the initial data satisfy (u 0 , v0 , θ0 ) ∈ H 1+α2 × H 2+α2 × H 2+α2 and u 0 (x) > 0, θ0 (x) > 0 for any x ∈ [0, 1], and that the compatibility conditions hold. Then the problem (2.1.1)–(2.1.3) and (2.1.5)–(2.1.6) admits a unique global solution


49

(u(t), v(t), θ (t)) ∈ BT1+α2 × HT2+α2 × HT2+α2 for any 0 < T < +∞. Moreover, as t → +∞, we have u − u 0 H 1 → 0, v H 1 → 0, v L ∞ → 0, θθ x → 0, θ − θ¯ H 1 → 0, θ − θ¯ L ∞ → 0, p(u, θ ) − p(u 0 , θ¯ ) H 1 where u 0 =

1 0

σ ∗ (t) → 0, p∗ (t) → 0, → 0, σ (u, θ, vx ) + p(u 0 , θ¯ ) → 0

(2.1.26) (2.1.27) (2.1.28) (2.1.29)

u 0 d x and the constant θ¯ > 0 is uniquely determined by e(u 0 , θ¯ ) =

1 0

(e(u 0 , θ0 ) + v02 /2)(x)d x

and (u 0 , 0, θ¯ ) is the unique solution to the corresponding stationary problem to (2.1.1)– (2.1.3) and (2.1.5)–(2.1.6). Moreover, there exist positive constants t0 > 0, C1 and C1 such that for any t ≥ t0 , there holds u(t) − u 0 H 1 + v(t) H 1 + θ (t) − θ¯ H 1 ≤ C1 exp(−C1 t).

(2.1.30)

Remark 2.1.1. Theorem 2.1.1 is also valid under the assumptions in [163], i.e., (2.1.19), (2.1.22)–(2.1.24) and (2.7.1)–(2.7.2). So the results in Theorem 2.1.1 improve those in [163].

2.1.2 Uniform A Priori Estimates The proof of Theorem 2.1.1 is based on a priori estimates that can be used to continue a local solution globally in time. The existence and uniqueness of local solutions (with positive u and θ ) can be obtained by linearization of the problem (2.1.1)–(2.1.3) and (2.1.5)–(2.1.6), and by use of the Banach contraction mapping theorem. Theorem 2.1.2. Let (u, v, θ ) be a smooth solution as described in Theorem 2.1.1, then we have for any T > 0, (1+α2)

|||u|||T

(2+α2 )

+ vT

(2+α2 )

+ |||θ |||T

≤ C,

(2.1.31)

and 0 < C −1 ≤ u(x, t) ≤ C, 0 < θ (x, t) ≤ C, ∀(x, t) ∈ [0, 1] × [0, +∞). The proofs of Theorems 2.1.1–2.1.2 are divided into a series of lemmas. Lemma 2.1.1. The following estimates hold, 0

θ (x, t) > 0 on [0, 1] × [0, ∞), (2.1.32) 1 1 1 e(x, t) + v 2 (x, t) d x = e(x, 0) + v02 (x)2 d x = E 0 , ∀t > 0, (2.1.33) 2 2 0

1

50


1

(θ + θ 1+r )(x, t)d x ≤ C, ∀t > 0,

0

1 0

u(x, t)d x =

1 0

u 0 (x)d x = u 0 , ∀t > 0.

(2.1.34) (2.1.35)

Proof. Noting the positivity of θ0 , (2.1.19), (2.1.23)–(2.1.25) and applying the maximum principle [67] to the equation vx2 k(u, θ )θθ x eθ (u, θ )θt + θ pθ (u, θ )vx − μ0 = , (2.1.36) u u x which is equivalent to (2.1.3), it is easy to see that (2.1.32) holds. Integrating (2.1.1) over Q t := [0, 1] × [0, t] and noting (2.1.6), we have (2.1.33), the conservation law of total energy. Combining (2.1.33) with (2.1.19) leads to (2.1.34) and (2.1.35) is a direct result of (2.1.1) and (2.1.6). Lemma 2.1.2. There holds that for any t > 0, 1 t 1+r 2 [(θ −log θ −1)+θ +v ](x, t)d x + 0

0

1 0

(1 + θ q )θθx2 vx2 (x, s)d x ds ≤ C1 . + uθ uθ 2 (2.1.37)

Proof. Let E(u, θ ) = (u, θ ) − (1, 1) − u (1, 1)(u − 1) − θ (u, θ )(θ − 1). Then

(u, θ ) = e(u, θ ) − θ η(u, θ ) satisfies − θ (u, θ ) = η(u, θ ), u (u, θ ) = σ (u, θ, 0) = − p(u, θ ).

(2.1.38)

Thus, by (2.1.1)–(2.1.3), (2.1.11) and (2.1.38), and noting that eθ (u, θ ) = −θ θθ (u, θ ), we deduce after a direct calculation that

v2 μvx2 k(u, θ )θθ x2 (θ − 1)k(u, θ )θθ x ∂t E(u, θ ) + + + = σv + . (2.1.39) 2 θ θ θ2 x Integrating (2.1.39) over Q t and using (2.1.6) leads to t 1 2 1 v2 k(u, θ )θθ x2 μvx E(u, θ ) + d x ds (x, t)d x + + 2 θ θ2 0 0 0 1 v02 E(u 0 , θ0 ) + d x. (2.1.40) = 2 0 In view of (2.1.21), we have uu (u, 1) = − pu (u, 1) > 0 for u > 0. Therefore it follows from the Taylor theorem and the definition of E(u, θ ) that E(u, θ ) − (u, θ ) + (u, 1) + (θ − 1) θ (u, θ ) = (u, 1) − (1, 1) − u (1, 1)(u − η0 ) 1 (1 − ξ ) uu (1 + ξ(u − 1), 1)dξ ≥ 0. = (u − 1)2 0


51

Thus, E(u, θ ) ≥ (u, θ ) − (u, 1) − (θ − 1) θ (u, θ ) 1 2 = −(1 − θ ) (1 − τ ) θθ (u, θ + τ (1 − θ ))dτ

0 1

(1 − τ ){1 + [θ + τ (1 − θ )]r } dτ ≥ ν(1 − θ )2 θ + τ (1 − θ ) 0 ν(1−θ r ) ν(1−θ 1+r ) ν(θ − log θ − 1) + − , for r > 0, r r+1 = 2ν1(θ − log(θ ) − 1), for r = 0. ≥ ν(θ − log(θ ) − 1) + Cθ r+1 − C

which, combined with (2.1.21) and (2.1.40), yields (2.1.37).

Remark 2.1.2. It follows from the convexity of the function − ln y that there exist two positive constants r1 , r2 depending only on the initial data such that 1 θ d x ≤ r2 . 0 < r1 ≤ 0

Lemma 2.1.3. For any t ≥ 0, there exists one point x 1 = x 1 (t) ∈ [0, 1] such that the solution u(x, t) to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) possesses the following expression: t u(x, s) p(x, s) −1 u(x, t) = D(x, t)Z (t) 1 + μ0 ds (2.1.41) 0 D(x, s)Z (s) where

−1 D(x, t) = u 0 (x) exp μ0 + Z (t) = exp[−

Proof. Let

x 1 (t ) 1 1

u0

1 μ0 u 0

x

h(x, t) =

x

0

1

v0 (y)d y

0

u 0 (x)

t 0

x

v(y, t)d y − x

0

v0 (y)d yd x

(2.1.42)

,

(v 2 + up)(y, s)d yds].

(2.1.43)

0

t

v0 (y)d y +

0

σ (x, τ )dτ.

(2.1.44)

0

Then from (2.1.2), h(x, t) satisfies h x = v, h t = σ, and solves the equation ht = − p +

μ0 h x x u

(2.1.45)

52


with Hence, we have

x = 0, 1 : h x = v = 0.

(2.1.46)

(uh)t = hvx − up + μ0 h x x .

(2.1.47)

Integrating (2.1.47) over Q t , and using the boundary condition (2.1.46), we arrive at 1 t 1 1 uhd x = u0h0d x − (up + v 2 )d x dτ ≡ φ(t). (2.1.48) 0

0

0

0

Then for any t ≥ 0, there exists one point x 1 = x 1 (t) ∈ [0, 1] such that 1 1 uhd x = ud x · h(x 1 (t), t) = u 0 · h(x 1 (t), t), φ(t) ≡ 0

i.e.,

t

0

x 1 (t )

p(x 1 (t), τ )dτ =

0

v0 (y)d y + μ0 log

0

with φ(t) = −

t 0

1

1

(v 2 + up)(x, s)d x ds +

0

0

u(x 1 (t), t) φ(t) − u 0 (x 1 (t)) u0

u 0 (x)

x

v0 (y)d yd x.

(2.1.49)

(2.1.50)

0

On the other hand, (2.1.2) can be rewritten as vt − μ0 (log u)xt = − p x = − p ∗x .

(2.1.51)

Integrating (2.1.51) over [x 1 (t), x] × [0, t] for fixed t > 0, we get u(x, t) =

u 0 (x)u(x 1 (t), t) (2.1.52) u 0 (x 1 (t)) x t 1 × exp (v(y, t) − v0 (y))d y + ( p(x, τ ) − p(x 1(t), τ ))dτ . μ0 x1 (t ) 0

Inserting (2.1.49) into (2.1.52) and noting (2.1.42), (2.1.43) and (2.1.50), we have t 1 u −1 (x, t) exp p(x, s)ds = D −1 (x, t)Z −1 (t) (2.1.53) μ0 0 which implies that t 1 −1 d 1 exp p(x, s)ds = D (x, t)Z −1 (t)u(x, t) p(x, t), dt μ0 0 μ0 i.e.,

1 exp μ0

t

p(x, s)ds 0

1 =1+ μ0

t 0

D −1 (x, s)Z −1 (s)u(x, s) p(x, s)ds.

Thus, (2.1.41) follows from (2.1.53) and (2.1.54).

(2.1.54)


53

Lemma 2.1.4. For any t ≥ 0, there exists one point a(t) ∈ [0, 1] such that the solution u(x, t) to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) possesses the following expression: t 1 −1 −1 u(x, t) = B (x, t)Y (t) 1 + u(x, s) p(x, s)Y (s)B(x, s)ds (2.1.55) μ0 0 where

t 1 p(x, s)ds , Y (t) = u 0 (a(t)) exp μ0 0 x 1 1 B(x, t) = exp (v0 (y) − v(y, t))d y , u 0 (x)u 0 μ0 a (t ) u(a(t), t) = u 0 .

(2.1.56) (2.1.57) (2.1.58)

Proof. It follows from (2.1.35) that there exists a(t) ∈ [0, 1] such that (2.1.58) holds. Substituting x 1 (t) by a(t) in (2.1.49) yields t 1 p(x, s)ds u(x, t) = Y −1 (t)B −1 (x, t) exp μ0 0 and

t 1 1 d exp p(x, s)ds = p(x, t)u(x, t)Y (t)B(x, t) dt μ0 0 μ0

from which (2.1.55) follows. Lemma 2.1.5. There holds that 0 < C −1 ≤ u(x, t) ≤ C, ∀(x, t) ∈ [0, 1] × [0, +∞).

(2.1.59)

Proof. Let Mu (t) = max u(x, t), m u (t) = min u(x, t), x∈[0,1]

x∈[0,1]

Mθ (t) = max θ (x, t), m θ (t) = min θ (x, t). x∈[0,1]

x∈[0,1]

It follows from (2.1.37) and convexity of the function − log y that

1 0

1

θ d x − log

1

θdx − 1 ≤

0

(θ − log θ − 1)d x ≤ C1

0

which along with Remark 2.1.2 implies that there exist b(t) ∈ [0, 1] and two positive constants r1 , r2 such that 0 < r1 ≤

1 0

θ (x, t)d x = θ (b(t), t) ≤ r2

54


with r1 , r2 being two positive roots of the equation y − log y − 1 = C1 . Thus (2.1.20) and (2.1.37) yield 1 1 0 < a1 ≤ (up + v 2 )(x, s)d x ≤ a2 (2.1.60) μ0 u 0 0 with a1 = p1 /μ0 u 0 , a2 = ( p2C1 + C1 + p2 )/μ0 u 0 . On the other hand, we have x m1 m1 m 1 −1 θ θx d y |θ (x, t) − θ (b(t), t)| ≤ C

1

≤C 0

where

θx2 (1 + θ q ) dx uθ 2

V1 (t) = and

t

1 0

1 0

uθ 2m 1 dx 1 + θq

1 2

1/2

1/2

≤ C V1 (t)M Mu (t)

θx2 (1 + θ q )/uθ 2 d x, 0 ≤ m 1 ≤ m = (q + r + 1)/2

1

V1 (s)ds ≤ C,

0

Thus,

b(t )

1 2

1

θ 2m 1 /(1 + θ q )d x ≤ C

0

(1 + θ 1+r )d x ≤ C.

0

C −1 − C V1 (t)M Mu (t) ≤ θ 2m 1 (x, t) ≤ C + C V1 (t)M Mu (t).

(2.1.61)

Using Lemma 2.1.1–Lemma 2.1.3, (2.1.42), (2.1.61) and noticing that 0 < C −1 ≤ D(x, t) ≤ C and

u(x, s) p(x, s) ≤ p2 (1 + θ r+1 ) ≤ C(1 + θ 2m ) ≤ C + C V1 (s)M Mu (s),

we obtain

t u(x, t) ≤ C 1 + V1 (s) exp(−a1 (t − s))M Mu (s)ds 0 t V1 (s)M Mu (s)ds , ≤C 1+ 0

i.e.,

t Mu (t) ≤ C 1 + V1 (s)M Mu (s)ds . 0

Thus, it follows from Gronwall’s inequality and

t 0

(2.1.62)

V1 (s)ds ≤ C that

Mu (t) ≤ C

(2.1.63)

C −1 − C V1 (t) ≤ θ 2m 1 (x, t) ≤ C(1 + V1 (t)).

(2.1.64)

which with (2.1.61) leads to


55

Similarly, Lemma 2.1.1–Lemma 2.1.3, (2.1.43) and (2.1.60) yield exp(−a1 t) ≥ Z (t) ≥ exp(−a2 t) and

t u(x, t) ≥ C −1 e−a2 t + e−a2 (t −s)ds ≥ C −1 (1 − ea2 t ). 0

Thus there exists t0 > 0 such that for t ≥ t0 , we have u(x, t) ≥ C −1 .

(2.1.65)

Moreover, we obtain from Lemma 2.1.4 and (2.1.64) that 1 t 1 u 0 Y (t) = u(x, t)Y (t)d x ≤ C 1 + (1 + θ 2m )d x Y (s)ds 0 0 0 t (1 + V1 (s))Y (s)ds . ≤C 1+ 0

By Gronwall’s inequality and noting

t 0

V1 (s)ds ≤ C, we see that

t Y (t) ≤ C exp C (1 + V1 (s))ds ≤ C exp(Ct).

(2.1.66)

0

Thus,

u(x, t) ≥ B −1 (x, t)Y −1 (t) ≥ C −1 Y −1 (t) ≥ C −1

(2.1.67)

for 0 ≤ t ≤ t0 . By (2.1.63), (2.1.65) and (2.1.67), we complete the proof of (2.1.59).

Corollary 2.1.1. There holds that for any (x, t) ∈ [0, 1] × [0, +∞), C −1 − C V2 (t) ≤ θ 2m 1 (x, t) ≤ C + C V2 (t) with 0 ≤ m 1 ≤ m = (q + r + 1)/2 and V2 (t) = ∞.

1 0

(1+θ q )θθx2 dx θ2

satisfying

(2.1.68) ∞ 0

Lemma 2.1.6. The following estimates hold for any t > 0: t v(s)2L ∞ ds ≤ C, t 0 0 t 1

u x (t)2 + 0

0

with β = max(r + 1 − q, 0).

V2 (t)dt
0 in (2.1.74) and applying the generalized Bellman-Gronwall inequality (see, e.g., Theorem 1.2.2), we get 2

u x +

t 0

1 0

[l + (1 − l)θ + θ r+1 ]u 2x d x ds ≤ C(1 + sup θ (s) L ∞ )β 0≤s≤t

which (for l = 1) and (2.1.74) (for l = 0) yield the desired estimate (2.1.71).

Lemma 2.1.7. There holds that for any t > 0, t 0

1 0

(1 + θ )2m u 2x d x ds ≤ C(1 + sup θ (s) L ∞ )β .

(2.1.75)

0≤s≤t

Proof. The inequality (2.1.75) follows from Corollary 2.1.1 and Lemma 2.1.6. Lemma 2.1.8. The following estimates hold that for any t > 0, t vx 2 ds ≤ C(1 + sup θ (s) L ∞ )β/2 , 0 t

2

vx (t) + 0

vx x 2 ds ≤ C(1 + sup θ (s) L ∞ )β27 ,

(2.1.77)

0≤s≤t

vx (s)2L ∞ ds ≤ C(1 + sup θ (s) L ∞ )β28 ,

(2.1.78)

0≤s≤t

t

2

vx (t) +

(2.1.76)

0≤s≤t

0 t

0

vt 2 ds ≤ C(1 + sup θ (s) L ∞ )β1 0≤s≤t

with β27 = max(5β/2, β1 ), β28 = β/4 + β27 /2 and β1 = max(2 + 2r − q, 0).

(2.1.79)

58


Proof. Multiplying (2.1.2) by v, vx x and vt , respectively, and then integrating the resultants over Q t , using (2.1.69)–(2.1.71) and (2.1.75) and Lemmas 2.1.5–2.1.7, we get t 1 2 vx 2 v + 2μ0 d x ds u 0 0 t 1 ≤C+C pu u x + pθ θx )vd x ds 0 0 t 1 ≤C+C [(1 + θ 1+r )|u x v| + (1 + θ r )|θθ x v|]d x ds 0

≤C+C

0

+C

t

1 0

(1 + θ )r+1 u 2x d x ds

V2 (s)ds

0

≤C+C

0

t

t 0

1 0

1/2 t 0

1/2 t 0

t

(1 + θ )2m u 2x d x ds

+ C(1 + sup θ (s) L ∞ )

0

t 0

0≤s≤t

0

1/2 (1 + θ )r+1 v 2 d x ds

(1 + θ )2r θ 2 v 2 d x ds 1 + θq 1/2

1 0

δ/2

1

1 0

(1 + θ )

1

0

1/2

(1 + θ )

1/2 2m 2

v d x ds

1/2

2m 2

v d x ds

≤ C(1 + sup θ (s) L ∞ )β/2 , vx 2 +

(2.1.80)

0≤s≤t t

vx x 2 ds

0

≤C+C

t 0

1 0

[|u x vx vx x | + (1 + θ 1+r )|u x vx x | + (1 + θ r )|θθ x vx x |]d x ds

t 1 1 t vx x 2 ds + C [vx2 u 2x + (1 + θ 1+r )2 u 2x + (1 + θ )2r θx2 ]d x ds 4 0 0 0 t 1 t vx x 2 ds + C(1 + sup θ (s) L ∞ )β vx 2L ∞ ds ≤C+ 4 0 0 0≤s≤t t 1 (1 + θ )2m u 2x d x ds + C(1 + sup θ (s) L ∞ )β

≤C+

0

0≤s≤t

+ C(1 + sup θ (s) L ∞ )β1

0

t 0

0≤s≤t

1 4

V2 (s)ds

≤ C(1 + sup θ (s) L ∞ )2β + 0≤s≤t

+ C(1 + sup θ (s) L ∞ )β 0≤s≤t

0

t 0

t

vx x 2 ds

vx 2 ds

1/2

t 0

1/2 vx x 2 ds


59

+ C(1 + sup θ (s) L ∞ )β1 0≤s≤t

≤ C(1 + sup θ (s)

L∞

)

5β/2

+ C(1 + sup θ (s)

0≤s≤t

L∞

)

β1

0≤s≤t

i.e.,

t

2

vx (t) +

1 + 2

t

vx x 2 ds,

0

vx x 2 ds ≤ C(1 + sup θ (s) L ∞ )β27

0

(2.1.81)

0≤s≤t

and vx (t)2 +

t

0 t

vt 2 ds

1

p x 2 +

≤C+C 0

0

t

≤C+C

0

1 0

|vx |3 d x ds u2

[(1 + θ )2r+2 u 2x + (1 + θ )2r θx2 + |vx |3 ]d x ds

≤ C(1 + sup θ (s) L ∞ )2β + C(1 + sup θ (s) L ∞ )β1 0≤s≤t t

0≤s≤t

vx 5/2 vx x 1/2 ds

+C 0

≤ C(1 + sup θ (s) L ∞ )β1 0≤s≤t

t

+ C sup vx 0≤s≤t

3/4

t

vx 2 ds

0

1/4 vx x 2 ds

0

≤ C(1 + sup θ (s) L ∞ )β1 + C(1 + sup θ (s) L ∞ )3β/4+β27/2 0≤s≤t

0≤s≤t

1 sup vx 2 + 2 0≤s≤t ≤ C(1 + sup θ (s) L ∞ )β1 + 0≤s≤t

1 sup vx 2 2 0≤s≤t

which with (2.1.81) yield the estimates (2.1.78) and (2.1.79) with β1 ≥ 3β/4 + β27 /2. Corollary 2.1.2. The following estimates are valid for any t > 0, t 0

t 0

0

1 0

1

(1 + θ )2m vx2 d x ds ≤ C(1 + sup θ (s) L ∞ )β1 ,

(2.1.82)

0≤s≤t

(1 + θ )2m+1 vx2 d x ds ≤ C(1 + sup θ (s) L ∞ )β29 , 0≤s≤t

(2.1.83)

60


t 0

1

(1 + θ )q+1 |vx |3 d x ds ≤ C(1 + sup θ (s) L ∞ )β30 ,

0

t 0

(2.1.84)

0≤s≤t 1 0

(1 + θ )q−r vx4 d x ds ≤ C(1 + sup θ (s) L ∞ )β31

(2.1.85)

0≤s≤t

where β29 = min(1 + β1 , 2m + 1 + β/2), β30 = min[q1 + (5β1 + β27 )/4, q + 1 + β1 /2 + 3β/8 + β27 /4], β31 = min[max(q − r, 0) + β1 + β/4 + β27 /2, q2 + 3β1 /2 + β27 /2], q1 = max((q + 1 − 3r )/4, 0), q2 = max((q − 3r − 1)/2, 0). Proof. By Corollary 2.1.1 and Lemma 2.1.6–Lemma 2.1.8, using the same method as that in Lemma 2.1.9, we complete the proof of Corollary 2.1.2. Lemma 2.1.9. There holds that for any t > 0, θ (t) + θ 1+r (t)2 +

t 0

1 0

(1 + θ )q+r θx2 d x ds ≤ C(1 + sup θ (s) L ∞ )β32 (2.1.86) 0≤s≤t

where β9 = min[max(2r + 1 − 2q, 0), max(3r + 3 − 2q, 0)/2], β32 = max[β28 , 3β/2, β9 , β + 1]. Proof. The equation (2.1.3) can be rewritten as et − σ v x − (

kθθx )x = 0. u

(2.1.87)

Multiply (2.1.87) by e, integrate the resultant over Q t and use (2.1.6), Lemma 2.1.1– Lemma 2.1.8 to get t 1 θ + θ r+1 2 + (1 + θ )q+r θx2 d x ds 0 0 t 1 μ v2 e kθθx eu u x 0 x ≤C+C + v(ep)x − d x ds 0 0 u u t 1 t 1 ≤C+C vx 2L ∞ (1 + θ r+1 )d x ds + C (1 + θ )2r+2 |vu x |d x ds 0

+C

0

t 0

1 0

0

0

[(1 + θ )2r+1 |vθθx | + (1 + θ )q+r+1 |θθx u x |]d x ds

(2.1.88)


61

and t 0

1 0

≤C

(1 + θ )2r+2 |vu x |d x ds t 0

1 0

(1 + θ )2r+2 u 2x d x ds

≤ C(1 + sup θ (s) L ∞ )

β

t 0

0≤s≤t

t

×

0

1/2 t

1

0 1

0

1

1/2 (1 + θ )

2r+2 2

0

v d x ds

1/2

(1 + θ )2r+2 u 2x d x ds

1/2

(1 + θ )2m v 2 d x ds

0

≤ C(1 + sup θ (s) L ∞ )3β/2 .

(2.1.89)

0≤s≤t

Similarly, t 0

1

0

(1 + θ )2r+1 |vθθx |d x ds

≤

t 0

V2 (s)ds

1/2 t 0

1 0

θ 2 (1 + θ )4r+2 v 2 d x ds 1 + θq

t

≤ C(1 + sup θ (s) L ∞ )max(3r+3−2q,0)/2

0

0≤s≤t

1

0

1/2 1/2

(1 + θ )2m v 2 d x ds

≤ C(1 + sup θ (s) L ∞ )max(3r+3−2q,0)/2 0≤s≤t

and t 0

1

0

(1 + θ )2r+1 |vθθx |d x ds ≤

1 4

t 0

0

1

(1 + θ )q+r θx2 d x ds

+C(1 + sup θ (s) L ∞ )max(2r+1−2q,0) ≤

1 4

0

0≤s≤t t 1

0

0

t

1

(1 + θ )2m v 2 d x ds

0

(1 + θ )q+r θx2 d x ds + C(1 + sup θ (s) L ∞ )max(2r+1−2q,0) . 0≤s≤t

So t 0

0

1

(1 + θ )

2r+1

1 |vθθx |d x ds ≤ 4

t 0

1 0

(1 + θ )q+r θx2 d x ds + C(1 + sup θ (s) L ∞ )β9 . 0≤s≤t

(2.1.90)

62


On the other hand, by the Cauchy inequality and Lemma 2.1.5, we have t 0

1

(1 + θ )q+r+1 |θθx u x |d x ds

0

1 ≤ 4

t 0

1

0

(1 + θ )q+r θx2 d x ds + C(1 + sup θ (s) L ∞ )β+1 . (2.1.91) 0≤s≤t

Therefore, it follows from (2.1.88)–(2.1.91) and Lemmas 2.1.1–2.1.8 that θ + θ

t

1+r 2

+

0

1 0


≤ C(1 + sup θ (s) L ∞ )β28 + C(1 + sup θ (s) L ∞ )3β/2 0≤s≤t

0≤s≤t

+ C(1 + sup θ (s) L ∞ )β9 + C(1 + sup θ (s) L ∞ )β+1 0≤s≤t

≤ C(1 + sup θ (s)

0≤s≤t L∞

)

β32

0≤s≤t

which implies (2.1.86). Lemma 2.1.10. There holds that for any t > 0,

1 0

(1 + θ )2q θx2 d x +

t 0

1 0

(1 + θ )q+r θt2 d x ds ≤ C(1 + sup θ (s) L ∞ )β35 , ∀t > 0, 0≤s≤t

(2.1.92) where β36 = [max(3q + 2 − r, 0) + β27 + β32 ]/2, β33 = min[β36, (3q + 4 + β27 )/2], β37 = max 2 max(q − r, 0) + 2β + β32 , max(q − r, 0) + β + (β32 + β29 )/2, max(q − r, 0) + β + (β32 + β31 )/2 , β38 = max max(q − r, 0) + q + 2 + β, 2 max(q − r, 0) + r + 2 + 2β,

max(q − r, 0) + β + (β29 + r + 2)/2, max(q − r, 0) + β + (β31 + r + 2)/2 ,

β34 = min(β37 , β38 ), β35 = max[β29 , β30 , β31 , β33 , β34 ]. Proof. Let

θ

H (x, t) = H (u, θ ) = 0

X (t) =

t 0

1 0

k(u, ξ ) dξ, u

(1 + θ )q+r θt2 d x ds, Y (t)

=

1 0

(1 + θ )2q θx2 d x.


63

Then it is easy to verify that kθt , Ht = Hu v x + u kθθx k Hxt = + Hu vx x + Huu vx u x + u x θt . u t u u Multiply (2.1.3) by Ht and integrate the resultant over Q t to get t 1 t 1 μ0 vx2 kθθx Ht x Ht d x ds + d x ds = 0. eθ θ t + θ p θ v x − u u 0 0 0 0

(2.1.93)

But we know from (2.1.24) and (2.1.25) that Huu | ≤ C(1 + θ )q+1 . |H Hu | + |H

(2.1.94)

Now we estimate each term in (2.1.93) by using (2.1.20), (2.1.23)–(2.1.25), (2.1.94), Lemmas 2.1.1–2.1.9 and Corollary 2.1.2. It is easy to see from (2.1.20), (2.1.24) and Lemma 2.1.5 that t 1 eθ θt Ht d x ds ≥ C0 X (t) − C(1 + sup θ (s) L ∞ )β29 . (2.1.95) 0

0

0≤s≤t

Similarly, t 1 C μ0 vx2 0 θ pθ v x − Ht d x ds ≤ X (t) + C(1 + sup θ (s) L ∞ )β29 0 0 u 8 0≤s≤t +C(1 + sup θ (s) L ∞ )β30 + C(1 + sup θ (s) L ∞ )β31 , 0≤s≤t

t 0

and

(2.1.96)

0≤s≤t 1 0

kθθx kθθx ( )t d x ds ≥ CY (t) − C u u

(2.1.97)

t 1 kθθ x (H Hu vx x + Huu vx u x )d x ds ≤ C(1 + sup θ (s) L ∞ )β36 0 0 u 0≤s≤t

where β27 ≥ β + β28 . On the other hand, we know t 1 kθθ x (H Hu vx x + Huu vx u x )d x ds ≤ C(1 + sup θ (s) L ∞ )(3q+4+β27)/2 . 0 0 u 0≤s≤t Therefore, t 1 kθθ x (H Hu vx x + Huu vx u x )d x ds ≤ C(1 + sup θ (s) L ∞ )β33 . 0 0 u 0≤s≤t

(2.1.98)

64


By Lemmas 2.1.1–2.1.9, we get t 1 t 1 kθθ k kθθ x x u x θt |d x ds u x θt d x ds ≤ C (1 + θ )q | 0 0 u u u u 0 0 t 1 C0 kθθx 2 ≤ X (t) + C (1 + θ )q−r u 2x d x ds 8 u 0 0 t C0 kθθ x 2 X (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+β ≤ ds 8 u L∞ 0 0≤s≤t C0 X (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+β ≤ 8 0≤s≤t t 1 kθθ x 2 kθθx kθθx × + d x ds u u u x 0 0 C0 ≤ X (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+β 8 0≤s≤t t 1 max(q−r,0) ∞ × (1 + sup θ (s) L ) (1 + θ )q+r θx2 d x ds +

0

0≤s≤t 1

t 0

0


0

1/2 t 0

1

(1 + θ )q−r |

0

kθθx u

|2 d x ds

1/2

x

which along with (2.1.36) leads to t 1 kθθ k x u x θt d x ds 0 0 u u u ≤

C0 X (t) + C(1 + sup θ (s) L ∞ )2 max(q−r,0)+β+β32 8 0≤s≤t +C(1 + sup θ (s) L ∞ )max(q−r,0)+β+β32 /2

0≤s≤t

× X (t) + ≤

t 0

1 0

1/2 [(1 + θ )q+r+2 vx2

+ (1 + θ )q−r vx4 ]d x ds

C0 X (t) + C(1 + sup θ (s) L ∞ )2 max(q−r,0)+2β+β32 4 0≤s≤t +C(1 + sup θ (s) L ∞ )max(q−r,0)+β+(β32 +β29 )/2 0≤s≤t

+C(1 + sup θ (s) L ∞ )max(q−r,0)+β+(β32 +β31 )/2 0≤s≤t

C0 X (t) + C(1 + sup θ (s) L ∞ )β37 . ≤ 4 0≤s≤t

(2.1.99)


65

But we also know that t 1 kθθ k x u x θt d x ds 0 0 u u u t 1 C0 kθθx 2 ≤ X (t) + C (1 + θ )q−r u 2x d x ds 8 u 0 0 C0 X 1 (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+β ≤ 8 0≤s≤t t 1 t 1 kθθx 2q 2 q × (1 + θ ) θx d x ds + (1 + θ ) |θθx || |d x ds u x 0 0 0 0 ≤

C0 X (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+β 8 0≤s≤t t

× (1 + sup θ (s) L ∞ )q+2 0≤s≤t

t

+

V2 (s)ds

1/2 t

0

≤

0

V2 (s)ds

0 1 0

kθθx θ (1 + θ ) | u 2

q

2

| d x ds

1/2

x

C0 X 1 (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+q+2+β 8 0≤s≤t + C(1 + sup θ (s) L ∞ )max(q−r,0)+β+(r+2)/2 ×

t 0

0≤s≤t 1

(1 + θ )

0

kθθx u

q−r

x

2 1/2 d x ds

C0 X (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+q+2+β ≤ 4 0≤s≤t + C(1 + sup θ (s) L ∞ )2 max(q−r,0)+2β+r+2 0≤s≤t

+ C(1 + sup θ (s) L ∞ )max(q−r,0)+β+(r+2+β29 )/2 0≤s≤t


C0 X (t) + C(1 + sup θ (s) L ∞ )β38 ≤ 4 0≤s≤t which together with (2.1.99) yields t 1 kθθ k C x 0 X (t) + C(1 + sup θ (s) L ∞ )β34 . u x θt d x ds ≤ 0 0 u u u 4 0≤s≤t Therefore, (2.1.92) follows from (2.1.93)–(2.1.100).

(2.1.100)

66


Lemma 2.1.11. There holds that for any t > 0, θ (t) L ∞ ≤ C, 1 [θθ x2 + u 2x + vx2 ](t)d x 0

+

t 0

1 0

(2.1.101)

[u 2x + θx2 + θt2 + vt2 + vx2 + vx2 x ]d x ds ≤ C.

(2.1.102)

Proof. The embedding theorem, Lemmas 2.1.2–2.1.10, and Young’s inequality result in 1 1 q+(r+3)/2 θ d x L ∞ ≤C |θ q+(r+1)/2 θx |d x θ (t) − 0

0

≤ CY

1/2

(t)

1

1/2 θ

r+1

≤ CY 1/2 (t)

dx

0

which gives 2q+r+3

θ (t) L ∞

≤ C + CY (t) ≤ C(1 + sup θ (s) L ∞ )β35 .

(2.1.103)

0≤s≤t

Similarly, q+r+2 θ (t) L ∞

1

≤C+C

|θ

q+r+1

θx |d x ≤ C + CY

1/2

(t)

0

1

1/2 θ

2r+2

dx

0

which implies 2q+2r+4

θ (t) L ∞

1

≤ C + CY (t)

θ 2r+2 d x ≤ C(1 + sup θ (s) L ∞ )β32 +β35 . (2.1.104)

0

0≤s≤t

After a lengthy calculation, we deduce that assumptions (2.1.14)–(2.1.18) imply that β32 + β35 < 2q + 2r + 4 or β35 < 2q + r + 3. Therefore, by the Young inequality, we derive from (2.1.103) or (2.1.104) that θ (t) L ∞ ≤ C which, combined with Lemmas 2.1.6–2.1.10, yields the desired estimate (2.1.102).

To end this section, we shall study the asymptotic behavior of solutions. The method for proof of the asymptotic behavior used here is different from that in [165]. Lemma 2.1.12. The following estimates are valid for any t > 0, t ( p∗ 2 + σ ∗ 2 )(s)ds ≤ C,

(2.1.105)

0

d p∗ (t)2 ≤ C(vt (t)2 + θt (t)2 + 1), dt d σ ∗ (t)2 ≤ C(vt (t)2 + θt (t)2 + 1). dt

(2.1.106) (2.1.107)


67

Proof. The equation (2.1.2) can be rewritten as v x = − p ∗x (2.1.108) vt − μ0 u x 1 with p∗ = p − 0 pd x. Noting (2.1.21) and integrating by parts, we see that x v x x p ∗ 2 = − p∗x , p∗ d y = vt − μ0 , p∗ d y u x 0 0 x x vx ∗ ∂ ,p vt d y, p ∗ d y + μ0 = ∂x 0 u 0 x x v ∂ x , p∗ . vd y, p ∗ + vd y, pt∗ + μ0 =− ∂t u 0 0 Thus, 0

t

∗

∗

2

t

1/2 v2L 1 ds

t

pt∗ 2 ds

1/2

p (s) ds ≤ C + v L 1 p L 1 + 0 0 t 1 t ∗ 2 + p ds + C vx 2 ds 2 0 0 1/2 t 1 t ∗ 2 ≤C +C (θt 2 + vx 2 )ds + p ds 2 0 0 1 t ∗ p (s)2 ds, ≤C + 2 0

which implies that

t

p ∗ (s)2 ds ≤ C, t t v ∗ x p ∗ 2 + σ ∗ (s)2 ds ≤ C 2 ds u 0 0 t ≤C ( p∗ 2 + vx 2 )(s)ds ≤ C

(2.1.109)

0

(2.1.110) (2.1.111)

0

with σ ∗ = σ −

1

σ d x. On the other hand, we have from Lemmas 2.1.1–2.1.10 that x d ∗ 2 ∗ ∗ ∗ ∗ p (t) = 2( p , pt ) = 2 p x , − pt d y dt 0 x x vx ∗ ∗ p t d y − 2 μ0 , p dy = 2 vt , u x 0 t 0 x pt∗ d y + vx pt∗ ≤ C vt 0

0

2

≤ C(vt (t) + θt (t)2 + 1).

68


Similarly, noting the equalities σt = − pt + μ0

v x

u

t

= − p u v x − p θ θ t + μ0

x 0

vt vxt dy = + u u

x 0

v2 vxt − x2 u u

,

vt u x d y, u2

we easily get x x d ∗ 2 ∗ ∗ ∗ σ (t) = 2 σx , − σt d y = 2 vt , − σt d y dt 0 0 x σt∗ d y2 ≤ C vt 2 + 0

≤ C(vt (t)2 + vx (t)2 + θt (t)2 + vx (t)4 + vt (t)2 u x (t)2 ) ≤ C(vt (t)2 + θt (t)2 + 1).

Lemma 2.1.13. The following estimates are valid for any t > 0: d u x (t)2 ≤ vx x (t)2 + u x (t)2 , dt

1 d θθx (t)2 + C (1 + θ )q−r θx2x d x ≤ C(vx x (t)2 + 1), dt 0 t 1 (1 + θ )q−r θx2x d x ds ≤ C. θθx (t)2 + 0

(2.1.112) (2.1.113) (2.1.114)

0

Proof. Differentiating (2.1.1) with respect to x and multiplying the resultant by u x yields the estimate (2.1.112). Multiplying (2.1.3) by eθ−1 θx x and integrating the resultant on [0, 1] leads to 1 d k 2 2 θθx + 2 θ x x eθ d x dt 0 u

1 μvx2 (k/u)x θx θ pθ v x = θx x d x − − eθ eθ eθ 0 ≤ θθx x 2 + C(vx 2 + vx 4L 4 + θθx 4L 4 + u x θx 2 ) ≤ θθx x 2 + C(vx 2 + vx 3 vx x + vx 4 + θθx 3 θθx x + θθx 4 + θθx 2L ∞ ) ≤ 2θθx x 2 + C(vx 2 + vx x 2 + θθx 2 ).


69

Hence for small , we have d θθx (t)2 + C dt

1 0

(1 + θ )q−r θx2x d x ≤ C(vx (t)2 + vx x (t)2 + θθx (t)2 ) ≤ C(vx x (t)2 + 1)

which implies θθx (t)2 +

t 0

0

1

(1 + θ )q−r θx2x d x ds ≤ C + C

t

(vx 2 + vx x 2 + θθx 2 )(s)ds

0

≤ C.

Lemma 2.1.14. As t → +∞, we have u(t) − u 0 H 1 → 0, u x (t) → 0, u(t) − u 0 L ∞ → 0, v(t) H 1 → 0, vx (t) → 0,

(2.1.115) (2.1.116)

θθx (t) → 0,

(2.1.117)

θ (t) − θ¯ H 1 → 0, θ (t) − θ¯ L ∞ → 0, p (t) → 0, σ (t) → 0, σ (u, θ, vx )(t) + p(u 0 , θ¯ ) → 0.

(2.1.118)

∗

∗

(2.1.119)

Moreover, there exist positive constants t0 , C1 and C1 such that for any t ≥ t0 , u(t) − u 0 H 1 + v(t) H 1 + θ (t) − θ¯ H 1 ≤ C1 exp(−C1 t).

(2.1.120)

Proof. By Lemmas 2.1.11–2.1.13 and applying Theorem 1.2.4, we conclude as t → +∞, u x (t) → 0, θθx (t) → 0, p∗ (t) → 0, σ ∗ (t) → 0.

(2.1.121)

Thus (2.1.115) and (2.1.117) follow from the embedding theorem and (2.1.35). It is obvious from (2.1.121) that

v ∗ x

u

2 ≤ C(σ ∗ 2 + p∗ 2 ) → 0

as t → +∞ and

v ∗ 1 vu vx x x dx vx (t) ≤ C ≤ C + 2 u u 0 u v ∗ x ≤C + u x (t) → 0 u

(2.1.122)

70


as t → +∞. Thus,

v(t) H 1 ≤ Cvx (t) → 0.

By (2.1.33), Lemmas 2.1.1–2.1.13 and the Poincaré´ inequality, we can get e(u, θ ) − e(u, ¯ θ¯ ) ≤ e(u, θ ) −

1

e(u, θ )d x + v2 /2

0

≤ C(ex + v) ≤ C(u x (t) + θθx (t) + vx (t))

(2.1.123)

1 1 with u¯ = 0 ud x = 0 u 0 d x = u 0 . By the mean value theorem, there are u˜ and θ˜ with u˜ = λθ + (1 − λ)θ¯ and u˜ = λu + (1 − λ)u¯ such that e(u, θ ) − e(u, ¯ u) ¯ = eu (u, ˜ θ˜ )(u − u) ¯ + eθ (u, ˜ θ˜ )(θ − θ¯ ).

(2.1.124)

By Lemmas 2.1.1–2.1.13, we infer that 0 < C −1 ≤ mi n{u, u} ¯ ≤ u˜ ≤ C, 0 < mi n(θ, θ¯ ) ≤ θ˜ ≤ C which along with (2.1.121), (2.1.122) and (2.1.124) gives that, as t → +∞, θ − θ¯ ≤ eθ−1 (u, ˜ θ˜ )[e(u, θ ) − e(u, ¯ θ¯ )] + eθ−1 (u, ˜ θ˜ )eu (u, ˜ θ˜ )(u − u) ¯ −1 −1 ¯ ≤ ν e(u, θ ) − e(u, ¯ θ ) + Cν u − u ¯ ≤ C(u x + θθx + vx ) → 0.

(2.1.125)

Thus (2.1.118) follows from (2.1.121) and (2.1.125). Noting that σ (u, θ, vx ) + p(u, ¯ θ¯ ) = −[ p(u, θ ) − p(u, ¯ θ¯ )] + vx /u, we can derive (2.1.119) from (2.1.121)–(2.1.122) and the mean value theorem. By a similar method as that in Section 2.3 (see also Okada and Kawashima [303]), we can deduce (2.1.120). Proofs of Theorem 2.1.1 and Theorem 2.1.2 Lemmas 2.1.1–2.1.14 yield (2.1.31) by the standard argument (see, e.g., Tani [404]) from which with Lemma 2.1.14, we finish the proofs of Theorem 2.1.1 and Theorem 2.1.2. Remark 2.1.3. It follows from the proofs of Lemmas 2.1.1–2.1.13, Theorem 2.1.1 and Theorem 2.1.2 that all the constants in Lemmas 2.1.1–2.1.13 depend only on the H 1 norm of the initial data (u 0 , v0 , θ0 ). Therefore the following results of global existence, uniqueness and the same results of the asymptotic behavior as those in Theorem 2.1.1 hold: If (u 0 , v0 , θ0 ) ∈ H 1 × H01 × H 1, the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) admits a unique generalized solution (u(t), v(t), θ (t)) in the sense that u ∈ L ∞ (0, +∞; H 1), u t (t) ∈ L ∞ (0, +∞; L 2 ), (v, θ ) ∈ L ∞ (0, +∞; H 1)∩L 2 (0, +∞; H 2)∩H 1 (0, +∞; L 2 ). Moreover, (2.1.115)–(2.1.120) are also valid.

2.2. Clamped and Constant Temperature Boundary Conditions

71

2.2 Clamped and Constant Temperature Boundary Conditions This section is concerned with the global existence, uniqueness and asymptotic behavior, as time tends to infinity, of solutions to the system (2.1.1)–(2.1.3) under the initial conditions u(x, 0) = u 0 (x), v(x, 0) = v0 (x), θ (x, 0) = θ0 (x) on [0, 1],

(2.2.1)

and the clamped and constant temperature boundary conditions v(0, t) = v(1, t) = 0, θ (0, t) = θ (1, t) = T0

(2.2.2)

where T0 > 0 is a constant. We assume that e, p, σ and k are twice continuously differential on 0 < u < +∞ and 0 ≤ θ < +∞, and satisfy (2.1.11)–(2.1.25). We are now in position to state our main theorem. Theorem 2.2.1. In addition to the assumptions (2.1.11)–(2.1.25), we assume that for α2 ∈ (0, 1) the initial data satisfy that (u 0 (x), v0 (x), θ0 (x)) ∈ H 1+α2 × H 2+α2 × H 2+α2 and u 0 (x) > 0, θ0 (x) > 0 for any x ∈ [0, 1], and that the compatibility conditions hold. Then the problem (2.1.1)–(2.1.3) and (2.2.1)–(2.2.2) admits a unique global solution (u(t), v(t), θ (t)) ∈ BT1+α2 × HT2+α2 × HT2+α2 for any 0 < T < +∞. Moreover, as t → +∞, we have u(t) − u ¯ H 1 → 0, v(t) H 1 → 0, v(t) L ∞ → 0, θθx (t) → 0, θ (t) − θ¯ H 1 → 0, θ (t) − θ¯ L ∞ → 0,

(2.2.3)

σ (t) → 0, p (t) → 0, → 0, σ (u, θ, vx ) + p(u, ¯ θ¯ ) → 0

(2.2.5)

∗

p(u, θ ) − p(u, ¯ θ¯ ) H 1 and there exist positive constants

t0 , C1 , C1

∗

(2.2.4) (2.2.6)

such that for all t ≥ t0 , there holds that

(2.2.7) u(t) − u ¯ H 1 + v(t) H 1 + θ (t) − θ¯ H 1 ≤ C1 exp(−C1 t) 1 1 where u 0 = 0 u 0 d x = u¯ = 0 ud x, θ¯ = T0 and (u, ¯ 0, θ¯ ) is the solution to the corresponding stationary problem to (2.1.1)–(2.1.3) and (2.2.1)–(2.2.2). Remark 2.2.1. Theorem 2.2.1 is also valid under the assumptions in [163], i.e., (2.1.19), (2.1.22)–(2.1.24), (2.7.1)–(2.7.2). Thus the results in Theorem 2.2.1 improve those in [163]. In the sequel, we derive some uniform a priori estimates. Theorem 2.2.2. Let (u, v, θ ) be a smooth solution as described in Theorem 2.2.1, then we have for any T > 0, (1+α2)

|||u|||T

(2+α2 )

+ ||v||T

(2+α2 )

+ |||θ |||T

≤ C,

and 0 < C −1 ≤ u(x, t) ≤ C, 0 < θ (x, t) ≤ C, ∀(x, t) ∈ [0, 1] × [0, +∞).

(2.2.8)

72


Since the proofs of the following lemmas are basically the same as those in Section 2.1.1, we only sketch some lemmas whose proofs will be given if necessary . Lemma 2.2.1. The following estimates are valid: θ (x, t) > 0 on [0, 1] × [0, +∞), 1 1 u(x, t)d x = u 0 (x)d x = u 0 , 0

0

1

∀t > 0.

(2.2.10)

0

Lemma 2.2.2. There holds that for any t > 0,

(2.2.9)

[(θ/T T0 − log(θ/T T0 ) − 1) + θ

1+r

2

+ v ]d x +

t 0

1

0

(1 + θ q )θθx2 vx2 d x ds ≤ C. + uθ uθ 2 (2.2.11)

Proof. The proof is similar to that of Lemma 2.1.12. Here we only state some differences: (1) Similar to E(u, θ ), we define E(u, θ ) := (u, θ ) − (1, T0 ) − u (1, T0 )(u − 1) − θ (u, θ )(θ − T0 ). (2) Similar to (2.1.39), we have ∂t

k(u, θ )θθ x2 μ0 vx2 + T0 + uθ uθ 2 (θ − T0 )k(u, θ )θθ x = (σ v)x + p(1, T0 )vx + . uθ x

v2 E(u, θ ) + 2

(2.2.12)

(2.2.13)

(3) Similar to (2.1.40), we have t 1 1 v2 k(u, θ )θθ x2 μ0 vx2 E(u, θ ) + d x ds (x, t)d x + T0 + 2 uθ uθ 2 0 0 0 1 v02 E(u 0 , θ0 ) + d x. (2.2.14) = 2 0 (4) In view of (2.1.21), we have

uu (u, T0 ) = − pu (u, T0 ) > 0 f or u > 0

(2.2.15)

and it follows from the Taylor theorem and (2.2.12) that E(u, θ ) − (u, θ ) + (u, T0 ) + (θ − T0 ) θ (u, θ ) = (u, T0 ) − (1, T0) − u (1, T0 )(u − 1) 1 (1 − ξ ) uu (1 + ξ(u − 1), T0 )dξ ≥ 0. = (u − 1)2 0

(2.2.16)


73

Thus E(u, θ ) ≥ (u, θ ) − (u, T0 ) − (θ − T0 ) θ (u, θ ) 1 2 (1 − τ ) θθ (u, θ + τ (T T0 − θ ))dτ = −(T T0 − θ ) ≥ ν(T T0 − θ )2

0 1

0

(1 − τ ){1 + [θ + τ (T T0 − θ )]r } dτ, θ + τ (T T0 − θ )

i.e., T0 (θ/T T0 − log(θ/T T0 ) − 1) + E(u, θ ) ≥ νT T0 − log(θ/T T0 ) − 1), 2νT T0 (θ/T

νT T0 (T T0r −θ r ) r

− f or r = 0.

νT T0 (T T0r+1 −θ r+1 ) , r+1

f or r > 0,

≥ νT T0 (θ/T T0 − log(θ/T T0 ) − 1) + C θ r+1 − C.

(2.2.17)

Lemma 2.2.3. There holds that for any (x, t) ∈ [0, 1] × [0, +∞), 0 < C −1 ≤ u(x, t) ≤ C.

(2.2.18)

Lemma 2.2.4. We have that C − C V3 (t) ≤ θ 2m 1 (x, t) ≤ C + C V3 (t), ∀(x, t) ∈ [0, 1] × [0, +∞), with 0 ≤ m 1 ≤ m = (q + r + 1)/2 and V3 (t) = ∞.

1 0

(1+θ q )θθx2 dx θ2

satisfying

∞ 0

(2.2.19) V3 (t)dt
0: t 0 0 t 1

u x (t)2 + 0

0

t

v(s)2L ∞ ds ≤ C,

(2.2.20)

(1 + θ )2m v 2 d x ds ≤ C,

(2.2.21)

0 1

(1 + θ 1+r )u 2x d x ds ≤ C(1 + sup θ (s) L ∞ )β

(2.2.22)

0≤s≤t

with β = max(r + 1 − q, 0). Lemma 2.2.6. There holds that for any t > 0, t 0

1 0

(1 + θ )2m u 2x d x ds ≤ C(1 + sup θ (s) L ∞ )β . 0≤s≤t

(2.2.23)

74


Lemma 2.2.7. The following estimates hold for any t > 0:

t 0 t

0

vx (s)2 ds ≤ C(1 + sup θ (s) L ∞ )β/2 ,

vx x (s)2 ds ≤ C(1 + sup θ (s) L ∞ )β27 ,

t

vt (s)2 ds ≤ C(1 + sup θ (s) L ∞ )β1 ,

0 t 0

(2.2.25)

0≤s≤t

vx (t)2 +

(2.2.24)

0≤s≤t

(2.2.26)

0≤s≤t

vx (s)2L ∞ ds ≤ C(1 + sup θ (s) L ∞ )β28

(2.2.27)

0≤s≤t

where β27 = max(5β/2, β1 ), β1 = max(2r + 2 − q, 0) and β28 = β/4 + β27 /2. Lemma 2.2.8. The following estimates are valid for any t > 0: t 0

0

t 0

0

1

1


(2.2.28)

0≤s≤t

(1 + θ )2m+1 vx2 d x ds ≤ C(1 + sup θ (s) L ∞ )β29 ,

(2.2.29)


(2.2.30)

0

t

1

0≤s≤t

0

t 0

0≤s≤t 1

0

(1 + θ )q−r vx4 d x ds ≤ C(1 + sup θ (s) L ∞ )β31

(2.2.31)

0≤s≤t

where β29 = min(1 + β1 , 2m + 1 + β/2), β30 = min[q1 + (5β1 + β27 )/4, q + 1 + β1 /2 + 3β/8 + β27 /4], β31 = min[q2 + 3β1 /2 + β27 /2, max(q − r, 0) + β/4 + β1 + β27 /2], q1 = min[(q + 1 − 3r )/4, 0], q2 = max((q − 3r − 1)/2, 0). Lemma 2.2.9. The following estimates are valid for any t > 0: θ 1+r (t)2 +

t 0

1 0

[

(T T0 − θ )2 (1 + θ )q+r θx2 + (θ r−1 + θ q+r−1 )θθx2 ](x, s)d x ds θ2

≤ C(1 + sup θ (s) L ∞ )β39 ,

(2.2.32)

0≤s≤t

t 0

1 0

(1 + θ )q+r θx2 d x ds ≤ C(1 + sup θ (s) L ∞ )β40 , 0≤s≤t

(2.2.33)


75

where β9 = min[max(2r + 1 − 2q, 0), max(3r + 3 − 2q, 0)/2], β40 = max[β39 , r ], β39 = max[3β/2, β28, β9 , β + 1, max((3r + 3 − 2q)/2, 0)]. Proof. Let E 2 (u, θ ) = E(u, θ ) + C1 . Thus we know from (2.2.17) that E 2 (u, θ ) ≥ C1 θ r+1 > 0. First, we shall prove

|E 2 | ≤ C(1 + θ r+1 ).

In fact, it follows from Lemma 2.2.2 that = C1 + (u − 1)2 E 2 = C1 + E

1

−(T T0 − θ )2 0

0

1

(1 − ξ ) uu (1 + ξ(u − 1), T0 )dξ

(1 − τ ) θθ (u, θ + τ (T T0 − θ ))dτ

1

≤ C1 − (u − 1)2

(1 − ξ ) pu (1 + ξ(u − 1), T0 )dξ

0

T0 − θ ) +N(u)(T

2

1

(1 − τ ){1 + [θ + τ (T T0 − θ )]r } dτ θ + τ (T T0 − θ )

0 ⎧ ⎪ C + N(u)T T [θ/T T T0 ) − 1] 0 0 − log(θ/T ⎪ ⎨ N(u)T T0 (T T0r −θ r ) N(u)(T T01+r −θ 1+r ) + − , f or r > 0, ≤ r r+1 ⎪ ⎪ ⎩ C + 2N(u)T T0 [θ/T T0 − log(θ/T T0 ) − 1], f or r = 0.

≤ C(1 + θ 1+r ). Second, the equation (2.2.13) can be rewritten as

k(u, θ )θθ x2 μ0 vx2 (θ − T0 )k(u, θ )θθ x + = σ vx + p(1, T0 )vx + ∂t E 2 +T T0 . (2.2.34) uθ uθ uθ 2 x Multiplying (2.2.34) by E 2 , integrating the resultant over Q t , using (2.2.12)–(2.2.13), integrating by parts, and noting that eθ (u, θ ) = −θ θθ (u, θ ) and x = ( p(1, T0 ) − p(u, θ ))u x + pθ (u, θ )(θ − T0 )u x + eθ (u, θ ) (θ − T0 )θθx , E 2x = E θ we get t 1 (θ − T0 )2 (1 + θ )q+r θx2 r+1 2 r−1 q+r−1 2 (θ d x ds θ + +θ )θθx + θ2 0 0 t 1 v 2 |E 2 | ≤C +C + | p(1, T0)v E 2x | |v(E 2 p)x | + μ0 x u 0 0

76


|(θ − T0 )k(u, θ )θθ x | |( p(1, T0) − p(u, θ ))u x + pθ (u, θ )(θ − T0 )u x | d x ds θu t 1 ≤C +C (1 + θ )2r+2 |vu x | + (1 + θ )2r+1 |vθθx | +

0

+

(1 +

+C where t 0

t 0

t

1

0 2r+1 θ) |(θ

t 0

0

− T0 )vθθ x |

θ 1

t 1 d x ds + C vx 2L ∞ (1 + θ 1+r )d x ds 0

0

(1 + θ )q+r+1 |(θ − T0 )u x θx | d x ds θ

(2.2.35)

(1 + θ )2r+2 |vu x |d x ds ≤ C(1 + sup θ (s) L ∞ )3β/2 ,

0 1 0

(1 + θ )2r+1 |vθθx |d x ds ≤ C(1 + sup θ (s) L ∞ )max((3r+3−2q),0)/2,

(2.2.37)

0≤s≤t

(1 + θ )2r+1 |(θ − T0 )vθθ x | d x ds, θ 0 0 1 t 1 (T T0 − θ )2 (1 + θ )q+r θx2 ≤ d x ds + C(1 + sup θ (s) L ∞ )β9 , 8 0 0 θ2 0≤s≤t 1 t vx 2L ∞ (1 + θ r+1 )d x ds ≤ C(1 + sup θ (s) L ∞ )β28 , 1

0

t

(2.2.36)

0≤s≤t

0

(2.2.38) (2.2.39)

0≤s≤t

(1 + θ )q+r+1 |(θ − T0 )u x θx | d x ds θ 0 0 t 1 1 (T T0 − θ )2 (1 + θ )q+r θx2 ≤ d x ds + C(1 + sup θ (s) L ∞ )β+1 . (2.2.40) 8 0 0 θ2 0≤s≤t 1

Therefore, (2.2.32) follows from (2.2.35)–(2.2.40).

Lemma 2.2.10. There holds that for any t > 0, t 1 1 2q 2 (1 + θ ) θx d x + (1 + θ )q+r θt2 d x ds ≤ C(1 + sup θ (s) L ∞ )β41 , ∀t > 0, 0

0

0

0≤s≤t

(2.2.41) where β42 = max(3q + 2 − r, 0) + (β27 + β40 )/2, β43 = min max((3q + 2 − r )/2, 0) + (β27 + β40 )/2, (3q + 4 + β27 )/2 , β44 = max 2 max(q − r, 0) + 2β + β40 , max(q − r, 0) + β + (β40 + β29 )/2, max(q − r, 0) + β + (β40 + β31 )/2 ,


77

β45 = max max(q − r, 0) + β + q + 2, 2 max(q − r, 0) + 2β + r + 2,

max(q − r, 0) + (β29 + r + 2)/2 + β, max(q − r, 0) + (β31 + r + 2)/2 + β ,

β46 = min[β44 , β45 ], β47 = max (β28 + max(q − r, 0) + β40 )/2, (2β28 + β40 )/3, β28 /2 + (β40 + β29 )/4, β28 /2 + (β40 + β31 )/4 , β48 = max (β28 + q + 2)/2, (2β28 + r + 2)/3, β28/2 + (β29 + r + 2)/4, β28 /2 + (β31 + r + 2)/4 , β49 = min[β47 , β48 ], β41 = max[β29 , β30 , β31 , β43 , β46 , β49 ]. Proof. The proof is similar to that of Lemma 2.1.10. We shall give the outline of the proof. Similarly to Lemma 2.1.10, let θ k(u, ξ ) dξ, H (x, t) = H (u, θ ) = u 0 t 1 1 q+r 2 X (t) = (1 + θ ) θt d x ds, Y (t) = (1 + θ )2q θx2 d x. 0

It is easy to get t 0

where

1 0

t 0

1 0

0

0

t 1 μ0 vx2 kθθx Ht x eθ θ t + θ p θ v x − Ht d x ds + d x ds u u 0 0 t t kθθx Ht kθθx Ht (1, s)ds + (0, s)ds = 0, − u u 0 0

eθ θt Ht d x ds ≥ C0 X (t) − C(1 + sup θ (s) L ∞ )β5 ,

(2.2.42)

(2.2.43)

0≤s≤t

C t 1 μ0 v x 0 Ht d x ds ≤ X (t) + C(1 + sup θ (s) L ∞ )β29 θ pθ v x − 0 0 u 8 0≤s≤t +C(1 + sup θ (s) L ∞ )β30 + C(1 + sup θ (s) L ∞ )β31 , 0≤s≤t

t

(2.2.44)

0≤s≤t 1

kθθx u

kθθx u

d x ds ≥ CY (t) − C, (2.2.45) 0 0 t t 1 kθθ x (H Hu vx x + Huu vx u x )d x ds ≤ C(1 + sup θ (s) L ∞ )β43 , (2.2.46) 0 0 u 0≤s≤t

78


t 1 kθθ k C x 0 X (t) + C(1 + sup θ (s) L ∞ )β46 , u x θt d x ds ≤ 0 0 u u u 4 0≤s≤t t C0 kθθx Ht ( η, s)ds ≤ X (t) + C(1 + sup θ (s) L ∞ )β49 u 8 0 0≤s≤t where η = 0 or 1. Therefore, (2.2.41) follows from (2.2.42)–(2.2.48).

(2.2.47)

(2.2.48)

Lemma 2.2.11. The following estimates are valid for any t > 0: θ (t) L ∞ ≤ C, 1 [θθ x2 + u 2x + vx2 ](x, t)d x 0

+

t 0

1

0

(2.2.49)

[u 2x + θx2 + θt2 + vt2 + vx2 + vx2 x ]d x ds ≤ C.

(2.2.50)

Proof. Similarly to (2.1.103)–(2.1.104), we derive 2q+r+3

θ (t) L ∞

≤ C + CY (t) ≤ C(1 + sup θ (s) L ∞ )β41

(2.2.51)

0≤s≤t

and 2q+2r+4

θ (t) L ∞

≤ C + CY (t) ≤ C(1 + sup θ (s) L ∞ )β39 +β40 .

(2.2.52)

0≤s≤t

A lengthy calculation implies that (2.1.14)–(2.1.18) give β41 < 2q + r + 3 or β39 + β40 < 2q + 2r + 4. Hence by the Young inequality, we deduce (2.2.49) and (2.2.50) from (2.2.51)–(2.2.52). Since the proof of asymptotic behavior is basically the same as that of Theorem 2.2.1, therefore we will not repeat it here. The proofs of Theorem 2.2.1 and Theorem 2.2.2 are the same as those of Theorem 2.1.1 and Theorem 2.1.2. The similar statements in Remark 2.1.3 are true for the problem (2.1.1)–(2.1.3) and (2.2.1)–(2.2.2).

2.3 Exponential Stability in H 1 and H 2 2.3.1 Main Results Based on the results in Sections 2.1–2.2, in this section we shall further study existence and exponential stability in H+i (i = 1, 2), an incomplete metric subspace of H i × H i × H i (i = 1, 2), of a nonlinear C0 -semigroup S(t) for problem (2.1.1)–(2.1.3) of a nonlinear heat-conductive viscous real gas in bounded domain = (0, 1). We consider the problem (2.1.1)–(2.1.3) in the region {0 ≤ x ≤ 1, t ≥ 0} under the initial conditions (2.1.5) and the boundary conditions (2.1.6) or (2.2.2).

2.3. Exponential Stability in H 1 and H 2

79

In this section we assume that e, p, σ and k are C 2 or C 3 satisfying (2.1.19)– (2.1.25) (for the precision, see Theorem 2.3.1 and Theorem 2.3.2 below) functions on 0 < u < +∞ and 0 ≤ θ < +∞. Let q and r be two positive constants (exponents of growth) satisfying (2.1.14)–(2.1.18). We define two spaces as follows: H+1 = (u, v, θ ) ∈ H 1[0, 1] × H 1[0, 1] × H 1[0, 1] : u(x) > 0, θ (x) > 0, x ∈ [0, 1], v|x=0 = v|x=1 = 0, θ |x=0 = θ |x=1 = T0 for (2.2.2) and

H+2 = (u, v, θ ) ∈ H 2[0, 1] × H 2[0, 1] × H 2[0, 1] : u(x) > 0, θ (x) > 0, x ∈ [0, 1], v|x=0 = v|x=1 = 0, θx |x=0 = θx |x=1 = 0 for (2.1.6) or θ |x=0 = θ |x=1 = T0 for (2.2.2)

which become two metric spaces when equipped with the metrics induced from the usual norms. In the above, H 1, H 2 are the usual Sobolev spaces. We use Ci (sometimes use Ci ) (i = 1, 2) to denote the generic constant depending only on the H i norm of initial datum (u 0 , v0 , θ0 ), min u 0 (x) and min θ0 (x), but inx∈[0,1]

x∈[0,1]

dependent of t. Without danger of confusion we will use the same symbol to denote the state functions as well as their values along a thermodynamic process, e.g., p(u, θ ), and p(u(x, t), θ (x, t)) and p(x, t). We are now in a position to state our main theorems. Theorem 2.3.1. Assume that e, p, σ and k are C 2 functions satisfying (2.1.19)–(2.1.25) on 0 < u < +∞ and 0 ≤ θ < +∞, and q, r satisfy assumptions (2.1.14)–(2.1.18). Then the unique generalized global solution (u(t), v(t), θ (t)) in H 1 × H 1 × H 1 to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) defines a nonlinear C0 -semigroup S(t) on H+1 . Moreover, for any (u 0 , v0 , θ0 ) ∈ H+1 , there exists a constant γ1 = γ1 (C1 ) > 0 such that for any fixed γ ∈ (0, γ1 ] and for any t > 0, the following inequality holds: (u(t), v(t), θ (t))−(u, ¯ 0, θ¯ )2H 1 = S(t)(u 0 , v0 , θ0 )−(u, ¯ 0, θ¯ )2H 1 ≤ C1 e−γ t , (2.3.1) +

+

which means that the semigroup S(t) decays exponentially on H+1 . Here 1 u¯ = u 0 (x)d x, θ¯ = T0 for (2.2.2)

(2.3.2)

0

or for (2.1.6) θ¯ > 0 is uniquely determined by 1 v2 e(u 0 , θ0 ) + 0 (x)d x. e(u, ¯ θ¯ ) = 2 0

(2.3.3)

80


Remark 2.3.1. Theorem 3.3.1 is also valid under the assumptions in [163], i.e., (2.1.19), (2.1.22)–(2.1.24) and (2.7.1)–(2.7.2). Note that (2.1.14)–(2.1.18) imply 0 ≤ r, r + 1 ≤ q, so the results in Theorem 3.3.1 improve those in [165, 192]. Theorem 2.3.2. Assume that e, p, σ and k are C 3 functions satisfying (2.1.19)–(2.1.25) on 0 < u < +∞ and 0 ≤ θ < +∞, and q, r satisfy (2.1.14)–(2.1.18). Then there exists a unique generalized global solution (u(t), v(t), θ (t)) in H+2 to the problem (2.1.1)– (2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) which defines a nonlinear C0 semigroup S(t) (also denoted by S(t) by the uniqueness of solution in H+1 ) on H+2 . Moreover, for any (u 0 , v0 , θ0 ) ∈ H+2 , there exists a constant γ2 = γ2 (C2 ) > 0 such that for any fixed γ ∈ (0, γ2 ] and for any t > 0, the following inequality holds: (u(t), v(t), θ (t))−(u, ¯ 0, θ¯ )2H 2 = S(t)(u 0 , v0 , θ0 )−(u, ¯ 0, θ¯ )2H 2 ≤ C2 e−γ t , (2.3.4) +

+

which implies that the semigroup S(t) decays exponentially on

H+2 .

Remark 2.3.2. We know that the generalized global solution (u(t), v(t), θ (t)) ∈ H+2 obtained in Theorem 2.3.2 is not the classical one. Indeed, if (u 0 , v0 , θ0 ) ∈ H+2 , by the 1

embedding theorem, we have u 0 , v0 , θ0 ∈ C 1+ 2 (0, 1). If we impose on the higher regularities of v0 , θ0 ∈ C 2+α (0, 1), α ∈ (0, 1), the following results on the global existence of classical (smooth) solutions are obtained in Qin [315, 318, 319]: If in addition to the assumptions in Theorem 2.3.2, we further assume that u 0 ∈ C 1+α (0, 1), v0 , θ0 ∈ C 2+α (0, 1), α ∈ (0, 1) and the compatibility conditions u t |x=0,1 = vt |x=0,1 = θt |x=0,1 = 0 hold, then the generalized global solution (u(t), v(t), θ (t)) ∈ H+2 obtained in Theoα rem 2.3.2 is the classical one satisfying u(x, t) ∈ C 1+α,1+ 2 (Q T ), v(x, t), θ (x, t) ∈ α C 2+α,2+ 2 (Q T ) for any T > 0, Q T = (0, 1) × (0, T ). Moreover, if the initial data possess higher regularities, then the (generalized global) solutions also possess higher regularities. Therefore the generalized (global) solution (u(t), v(t), θ (t)) in H+2 can be understood as a generalized (global) solution between the classical (global) solution and the generalized (global) solution (u(t), v(t), θ (t)) in H+1 . Remark 2.3.3. The results in Theorem 2.3.2 were not obtained before.

2.3.2 Exponential Stability in H 1 In this subsection we shall complete the proof of Theorem 2.3.1 and assume that the assumptions in Theorem 2.3.1 are valid. We begin with the following lemma. Lemma 2.3.1. If (u 0 , v0 , θ0 ) ∈ H+1 , then there exists a unique generalized global solution (u(t), v(t), θ (t)) in H+1 to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)– (2.1.3), (2.2.1), (2.2.2) satisfies u t , vt , θt , θx , vx , u x , vx x , θx x ∈ L 2 ([0, +∞), L 2 ),

(2.3.5)

0 < θ (x, t) ≤ C1 on [0, 1] × [0, +∞),

(2.3.6)

0 < C1−1 ≤ u(x, t) ≤ C1

(2.3.7)

on [0, 1] × [0, +∞),


81

t u x 2 + v2H 2 + v2L ∞ u(t)2H 1 + θ (t)2H 1 + v(t)2H 1 + 0 + θθx 2H 1 + vt 2 + θt 2 (τ )dτ ≤ C1 , ∀t > 0

(2.3.8)

and there exist positive constants C1 , t0 , C1 , independent of t, such that (u(t) − u, ¯ v(t), θ (t) − θ¯ ) H 1 ≤ C1 e−C1 t , ∀t ≥ t0 .

(2.3.9)

Proof. See, e.g., Theorems 2.1.1 and 2.2.1.

Lemma 2.3.2. The unique generalized global solution (u(t), v(t), θ (t)) in H+1 defines a nonlinear C0 -semigroup S(t) on H+1 . Moreover, for any (u 0 , v0 , θ0 ) ∈ H+1 , the generalized global solution (u(t), v(t), θ (t)) to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) satisfies (u(t), v(t), θ (t)) = S(t)(u 0 , v0 , θ0 ) ∈ C([0, +∞), H+1 ), u(t) ∈ C

1/2

1

([0, +∞), H ), v(t), θ (t) ∈ C

1/2

2

([0, +∞), L ).

(2.3.10) (2.3.11)

Proof. For any t1 ≥ 0, t > 0, integrating (2.1.1) over (t1 , t) and using Lemma 2.3.1, we obtain t 1/2 u(t) − u(t1 ) H 1 ≤ C1 (vx 2 + vx x 2 )dτ |t − t1 |1/2 ≤ C1 |t − t1 |1/2 t1

which implies u(t) ∈ C 1/2 ([0, +∞), H 1). In the same manner we easily prove v(t), θ (t) ∈ C 1/2 ([0, +∞), L 2 ). Thus (2.3.11) follows. By Lemma 2.3.1, we know that for any t > 0, the operator S(t) : (u 0 , v0 , θ0 ) ∈ H+1 −→ (u(t), v(t), θ (t)) ∈ H+1 exists and, by the uniqueness of generalized global solutions, satisfies on H+1 that for any t1 , t2 ∈ [0, +∞), S(t1 + t2 ) = S(t1 )S(tt2 ) = S(tt2 )S(t1 ).

(2.3.12)

Moreover, by Lemma 2.3.1, S(t) is uniformly bounded on H+1 with respect to t > 0, i.e., S(t)L(H H 1 ,H H 1 ) ≤ C1 . +

+

(2.3.13)

We first verify the continuity of S(t) with respect to the initial data in H+1 for any fixed t > 0. To this end, we assume that (u 0 j , v0 j , θ0 j ) ∈ H+1 , (u j , v j , θ j ) = S(t)(u 0 j , v0 j , θ0 j ) ( j = 1, 2), and (u, v, θ ) = (u 1 , v1 , θ1 ) − (u 2 , v2 , θ2 ). Subtracting the corresponding

82


equations (2.1.1)–(2.1.3) satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), we obtain u t = vx ,

(2.3.14)

vt = − pu (u 1 , θ1 )u x − ( pu (u 1 , θ1 ) − pu (u 2 , θ2 ))u 2x − pθ (u 1 , θ1 )θθx vx u 1x vx x v2x u , − ( pθ (u 1 , θ1 ) − pθ (u 2 , θ2 ))θ2x + μ0 − − u1 u 1u 2 x u 21 (2.3.15) eθ (u 1 , θ1 )θt = −(eθ (u 1 , θ1 ) − (eθ (u 2 , θ2 ))θ2t − (eu (u 1 , θ1 ) − eu (u 2 , θ2 ))v2x − eu (u 1 , θ1 )vx − p(u 1 , θ1 )vx − ( p(u 1 , θ1 ) − p(u 2 , θ2 ))v2x + [k(u 1 , θ1 )θθx /u 1 + (k(u 1 , θ1 )/u 1 − k(u 2 , θ2 /u 2 ))θ2x ]x , (2.3.16) t = 0 : u = u 0 , v = v0 , θ = θ0 , x = 0, 1 : v = 0, θx = 0

or

θ = 0.

(2.3.17)

By Lemma 2.3.1, we know that for any t > 0 and j = 1, 2, (u j (t), v j (t), θ j (t))2H 1 t + (u j x 2 + v j 2H 2 + θθ j x 2H 1 + θθ j t (t)2 + v j t 2 )(τ )dτ ≤ C1 ,

(2.3.18)

0

here and hereafter in the proof of this lemma, C1 > 0 denotes the universal constant depending only on the H 1 norm of the initial data (u 0 j , v0 j , θ0 j ) and min u 0 j (x) and min θ0 j (x) ( j = 1, 2), but independent of t.

x∈[0,1]

x∈[0,1]

Multiplying (2.3.14), (2.3.15) and (2.3.16) by u, v and θ respectively, adding them up and integrating the result over [0, 1], and using Lemma 2.3.1, (2.3.17)–(2.3.18), the Cauchy inequality, the embedding theorem and the mean value theorem, we deduce that for any small > 0, 1 1 d μ0 vx2 2 2 2 2 (u(t) + v(t) + eθ (u 1 , θ1 )θ (t) ) + + k(u 1 , θ1 )θθx d x 2 dt u1 0 ≤ (vx (t)2 + θθx (t)2 ) + C1 H1(t)(u(t)2 + θ (t)2H 1 ) which, together with Lemma 2.3.1, leads to d (u(t)2 + v(t)2 + eθ (u 1 , θ1 )θ (t)2 ) + C1−1 (vx (t)2 + θθx (t)2 ) dt (2.3.19) ≤ C1 H1(t)(u(t)2 + θ (t)2H 1 ) where, by (2.3.18), H1(t) = θ1t (t)2 +θ2t (t)2 +v1x x (t)2 +v2x x (t)2 +θ1x x (t)2 + θ2x x (t)2 + 1 satisfies for any t > 0, t H1(τ )dτ ≤ C1 (1 + t). (2.3.20) 0


83

By Lemma 2.3.1, (2.3.15), the embedding theorem and the mean value theorem, we get vx x (t)2 ≤ C1 vt (t)2 + vx (t)2L ∞ + θ (t)2H 1 + (1 + v2x x (t)2 )u(t)2H 1 1 ≤ vx x (t)2 + C1 (vt (t)2 + θ (t)2H 1 ) 2 +C1 vx (t)2 + (1 + v2x x (t)2 )u(t)2H 1 which gives vx x (t)2 ≤ C1 vt (t)2 + C1 H1(t)(vx (t)2 + u(t)2H 1 + θ (t)2H 1 ).

(2.3.21)

Differentiating (2.3.14) with respect to x, multiplying the result by u x and integrating by parts, and using (2.3.21), we derive that for any small δ > 0, d u x (t)2 ≤ C1 δvt (t)2 + C1 (δ)H1(t)(vx (t)2 + u(t)2H 1 + θ (t)2H 1 ). (2.3.22) dt Multiplying (2.3.15) by vt , integrating it over [0, 1], and using Lemma 2.3.1, (2.3.17), the embedding theorem and the mean value theorem, we obtain d vx √ (t)2 + C1−1 vt (t)2 ≤ C1 H1 (t)(vx (t)2 + u(t)2H 1 + θ (t)2H 1 ). (2.3.23) dt u1 Similarly to (2.3.21), by (2.3.16), we infer that θθx x (t)2 ≤ C1 θt (t)2 + C1 H1(t)(vx (t)2 + u(t)2H 1 + θ (t)2H 1 ).

(2.3.24)

Similarly to (2.3.23), multiplying (2.3.16) by θt and using (2.3.17)–(2.3.24), we get d k (u 1 , θ1 )θθx (t)2 + C1−1 θt (t)2 ≤ C1 H1 (t)(vx (t)2 + u(t)2H 1 + θ (t)2H 1 ). dt (2.3.25) Adding up (2.3.19), (2.3.22), (2.3.23) and (2.3.25), and taking δ > 0 small enough, we finally conclude d M1 (t) ≤ C1 H1(t)(vx (t)2 + u(t)2H 1 + θ (t)2H 1 ) ≤ C1 H1(t)M1 (t) (2.3.26) dt where

2 vx M1 (t) = u(t) + u x (t) + v(t) + √ (t) u1 2 2 + eθ (u 1 , θ1 )θ (t) + k (u 1 , θ1 )θθx (t) 2

2

2

satisfies C1−1 (v(t)2H 1 + u(t)2H 1 + θ (t)2H 1 ) ≤ M1 (t) ≤ C1 (v(t)2H 1 + u(t)2H 1 + θ (t)2H 1 ).

(2.3.27)

84


Thus (2.3.26), combined with Gronwall’s inequality, (2.3.20) and (2.3.27), implies for any fixed t > 0, t u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 ≤ C1 M1 (0) exp(C1 H1 (τ )dτ ) 0 2 ≤ C1 exp(C1 t)(u 0 H 1

+ v0 2H 1 + θ0 2H 1 ).

That is, S(t)(u 01 , v01 , θ01 ) − S(t)(u 02 , v02 , θ02 ) H 1

+

≤ C1 exp(C1 t)(u 01 , v01 , θ01 ) − (u 02 , v02 , θ02 ) H 1

+

(2.3.28)

which leads to the continuity of S(t) with respect to the initial data in H+1 . By (2.3.12)– (2.3.13), in order to derive (2.3.10), it suffices to show that for any (u 0 , v0 , θ0 ) ∈ H+1 , S(t)(u 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 1 → 0, as t → 0+ , +

(2.3.29)

which also yields S(0) = I

(2.3.30)

H+1 .

with I being the unit operator on To derive (2.3.29), we choose a function sequence m , θ m ) which is smooth enough, for example, (u m , v m , θ m ) ∈ (C 1+α (0, 1) × , v (u m 0 0 0 0 0 0 C 2+α (0, 1) × C 2+α (0, 1)) ∩ H+1 for some α ∈ (0, 1), such that m m (u m 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 1 → 0, as m → +∞. +

(2.3.31)

By the regularity results (see also Remark 2.3.2), we conclude that for arbitrary but fixed T > 0, there exists a unique global smooth solution (u m (t),v m (t),θ m (t)) ∈ (C 1+α (Q T )× C 2+α (Q T ) × C 2+α (Q T )) ∩ H+1 , Q T = (0,1) × (0,T ). This gives for m = 1, 2, 3, . . . m m + (u m (t), v m (t), θ m (t)) − (u m 0 , v0 , θ0 ) H 1 → 0, as t → 0 . +

(2.3.32)

Fixing T = 1, by the continuity of the operator S(t), (2.3.28) and (2.3.31), for any t ∈ [0, 1], (u m (t), v m (t), θ m (t)) − (u(t), v(t), θ (t)) H 1

+

= ≤

m m S(t)(u m 0 , v0 , θ0 ) − S(t)(u 0 , v0 , θ0 ) H+1 m m C1 (u m 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H+1 → 0,

as m → +∞. This together with (2.3.31) and (2.3.32) implies S(t)(u 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 1 = (u(t), v(t), θ (t)) − (u 0 , v0 , θ0 ) H 1 +

+

≤ (u m (t), v m (t), θ m (t)) − (u(t), v(t), θ (t)) H 1

+

m

m

m

+ (u (t), v (t), θ (t)) −

m m (u m 0 , v0 , θ0 ) H+1

m m + + (u m 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 1 → 0, as m → +∞, t → 0 +


85

which gives (2.3.29) and (2.3.30). Thus S(t) is a C0 -semigroup on H+1 satisfying (2.3.10)– (2.3.11). The proof of Lemma 2.3.2 is complete. The next lemma concerns the uniform global (in time) positive lower boundedness (independent of t) of the absolute temperature θ , which was not obtained before. Lemma 2.3.3. If (u 0 , v0 , θ0 ) ∈ H+1 , then the generalized global solution (u(t), v(t), θ (t)) to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) satisfies 0 < C1−1 ≤ θ (x, t), ∀(x, t) ∈ [0, 1] × [0, +∞). (2.3.33) Proof. We prove (2.3.33) by contradiction. If (2.3.33) is not true, that is, inf

(x,t )∈[0,1]×[0,+∞)

θ (x, t) = 0,

then there exists a sequence (x n , tn ) ∈ [0, 1] × [0, +∞) such that as n → +∞, θ (x n , tn ) → 0.

(2.3.34)

If the sequence {ttn } has a subsequence, denoted also by tn , converging to +∞, then by the asymptotic behavior results in Lemma 2.3.1, we know that as n → +∞, θ (x n , tn ) → θ¯ > 0 which contradicts (2.3.34). If the sequence {ttn } is bounded, i.e., there exists a constant M > 0, independent of n, such that for any n = 1, 2, 3, . . . , 0 < tn ≤ M. Thus there exists a point (x ∗ , t ∗ ) ∈ [0, 1] × [0, M] such that (x n , tn ) → (x ∗ , t ∗ ) as n → +∞. On the other hand, by (2.3.34) and the continuity of solutions in Lemmas 2.3.1–2.3.2, we conclude that θ (x n , tn ) → θ (x ∗ , t ∗ ) = 0 as n → +∞, which contradicts (2.3.6). Thus the proof is complete. In what follows we shall prove the exponential stability of C0 -semigroup S(t), i.e., (2.3.1). We shall use a modified idea in Okada and Kawashima [303] to prove (2.3.1). Now we introduce the density of the gas, ρ = 1/u, then we know from (2.1.8)–(2.1.9) and (2.1.11)–(2.1.13) that the entropy η = η(1/ρ, θ ) satisfies ∂η/∂ρ = − pθ /ρ 2 ,

∂η/∂θ = eθ /θ.

(2.3.35)

We consider the transform A : (ρ, θ ) ∈ Dρ,θ = {(ρ, θ ) : ρ > 0, θ > 0} −→ (u, η) ∈ ADρ,θ

(2.3.36)

where u = 1/ρ and η = η(1/ρ, θ ). Owing to the Jacobian |∂(u, η)/∂(ρ, θ )| = −eθ /ρ 2 θ < 0 on Dρ,θ , there is a unique inverse function θ = θ (u, η) as a smooth function of (u, η) ∈ ADρ,θ . (In fact, Dρ,θ and ADρ,θ are bounded domains, e.g., Lemmas 2.3.1–2.3.2). Thus the functions e, p can be also regarded as smooth functions of (u, η). We denote by e = e(u, η) :≡ e(u, θ (u, η)) = e(1/ρ, θ ),

p = p(u, η) :≡ p(u, θ (u, η)) = p(1/ρ, θ ).

86


Then it is obvious from (2.1.8)–(2.1.9), (2.1.11)–(2.1.13) and (2.3.35)–(2.3.36) that e, p satisfy eu = − p, eη = θ, pη = θ pθ /eθ , θu = −θ pθ /eθ ,

pu = −(ρ 2 pρ + θ pθ2 /eθ ), θη = θ/eθ .

(2.3.37)

We define the energy form E(u, v, η) =

∂e ∂e v2 +e(u, η)−e(u, ¯ η) ¯ − (u, ¯ η)( ¯ u − u)− ¯ (u, ¯ η)(η ¯ − η), ¯ 2 ∂u ∂η

(2.3.38)

where ρ¯ = 1/u, ¯ η¯ = η(1/ρ, ¯ θ¯ ). The next two lemmas concern exponential decay of the generalized global solution (u(t), v(t), θ (t)) in H+1 (or equivalently, of C0 -semigroup S(t) on H+i ). Lemma 2.3.4. The unique generalized global solution (u(t), v(t), θ (t)) to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) satisfies the estimates v2 v2 + C1−1 (|u − u| + C1 (|u − u| ¯ 2 + |η − η| ¯ 2 ) ≤ E(u, v, η) ≤ ¯ 2 + |η − η| ¯ 2 ). (2.3.39) 2 2 Proof. By the mean value theorem, there exists a point ( u, η) between (u, η) and (u, ¯ η) ¯ such that v2 1 ∂ 2e ∂ 2e 2 E(u, v, η) = + ( u, η)(u − u)(η ¯ − η) ¯ ( u , η )( u − u) ¯ + 2 2 2 ∂u 2 ∂u∂η ∂ 2e 2 (2.3.40) + 2 ( u, η)(η − η) ¯ ∂η where

u = λ0 u¯ + (1 − λ0 )u,

η = λ0 η¯ + (1 − λ0 )η, 0 ≤ λ0 ≤ 1.

It follows from Lemmas 2.3.1–2.3.3 that 0 < C1−1 ≤ u ≤ C1 , | η| ≤ C1 which implies

∂ 2e ∂ 2e ∂ 2e ( u, η) + 2 ( u, η ) + u, η ) ≤ C 1 . 2 ( ∂u∂η ∂u ∂η Thus (2.3.40)–(2.3.41) and the Cauchy inequality give E(u, v, η) ≤

v2 + C1 [(u − u) ¯ 2 + (η − η) ¯ 2 ]. 2

(2.3.41)

(2.3.42)

On the other hand, we infer from (2.3.37) that euu = − pu = ρ 2 pρ + θ pθ2 /eθ , euη = − pη = θu = −θ pθ /eθ , eηη = θη = θ/eθ ,


87

which yields the Hessian of e(u, η) is positive definite for any u > 0 and θ > 0. Thus we deduce from (2.3.40), v2 v2 + λmin ( + C1−1 [(u − u) u, η )[(u − u) ¯ 2 + (η − η) ¯ 2] ≥ ¯ 2 + (η − η) ¯ 2] 2 2 (2.3.43) where λmin ( u, η)(≥ C1−1 ) is the smaller characteristic root of the Hessian of e( u, η ). Thus the combination of (2.3.42) and (2.3.43) gives the desired estimate (2.3.39). E(u, v, η) ≥

Lemma 2.3.5. There exists a positive constant γ1 = γ1 (C1 ) > 0 such that for any fixed γ ∈ (0, γ1 ], the generalized global solution (u(t), v(t), θ (t)) in H+1 to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) satisfies the following estimate eγ t (v(t)2 + u(t) − u ¯ 2 + θ (t) − θ¯ 2 + u x (t)2 + ρx (t)2 ) t + eγ τ (u x 2 + ρx 2 + θθx 2 + vx 2 )(τ )dτ ≤ C1 , ∀t > 0.

(2.3.44)

0

Proof. By equations (2.1.1)–(2.1.3), it is easy to verify that (ρ, v, η) satisfies

v2 e+ 2

= [− pv + μ0 ρvvx + ρkθθx ]x ,

(2.3.45)

t

k θx /θ )x + kρ(θ k θx /θ )2 + μ0 ρvx2 /θ. ηt = (kρθ

(2.3.46)

Owing to u¯ t = 0, θ¯t = 0, we infer from (2.3.45)–(2.3.46) and (2.1.1)–(2.1.2) that Et (1/ρ, v, η) + (θ¯ /θ )[μ0 ρvx2 + kρθ k θx2 /θ ] = [μ0 ρvvx + k(1 − θ¯ /θ )ρθθ x − ( p − p(1/ρ, ¯ θ¯ ))v]x , [μ20 (ρx /ρ)2 /2 +

μ0 ρx v/ρ]t +

(2.3.47)

μ0 pρ ρx2 /ρ

= −μ0 pθ ρx θx /ρ − μ0 (ρvvx )x + μ0 ρvx2 .

(2.3.48)

Multiplying (2.3.47), (2.3.48) by eγ t , βeγ t respectively, and then adding the results up, we get ∂ G(t) + eγ t [(θ¯ /θ )(μ0 ρvx2 + kρθ k θx2 /θ )/θ + β(μ0 pρ ρx2 /ρ − μ0 ρvx2 + μ0 pθ ρx θx /ρ)] ∂t = γ eγ t [E(1/ρ, v, η) + β(μ20 (ρx /ρ)2 /2 + μ0 ρx v/ρ)] + eγ t [(1 − β)μ0 ρvvx + k(1 − θ¯ /θ )ρθθ x − ( p − p(ρ, ¯ θ¯ ))v]x (2.3.49) where G(t) = eγ t [E(1/ρ, v, η) + β(μ20 (ρx /ρ)2 /2 + μ0 vρx /ρ)].

88


Integrating (2.3.49) over [0, 1]×[0, t], by Lemmas 2.3.1–2.3.3, Cauchy’s inequality and Poincaré´ ’s inequality, we deduce that for small β > 0 and for any γ > 0, eγ t [ρ(t) − ρ ¯ 2 + v(t)2 + η(t) − η ¯ 2 + ρx (t)2 ] t + eγ τ [ρx 2 + vx 2 + θθx 2 ](τ )dτ 0 t ≤ C1 + C1 γ eγ τ (v2 + ρ − ρ ¯ 2 + θ − θ¯ 2 + ρx 2 )(τ )dτ.

(2.3.50)

0

For the boundary conditions (2.2.2), we easily get x ¯ L∞ = θ (y, t)d y ≤ θθx (t), v(t) ≤ vx (t). θ (x, t) − θ y L∞

0

(2.3.51)

For the boundary conditions (2.1.6), integrating (2.1.3) over (0, 1) and using (2.3.3), we have 1 1 v02 v2 e(u, θ ) + e(u 0 , θ0 ) + dx = d x = e(u, ¯ θ¯ ) 2 2 0 0 which, together with Poincaré´ ’s inequality, Lemmas 2.3.1–2.3.3 and the mean value theorem, implies 1 e(u, θ ) − e(u, ¯ θ¯ ) ≤ e(u, θ ) − e(u, θ )d x + v(t)2 /2 0

≤ C1 (ex (t) + vx (t)) ≤ C1 (u x (t) + vx (t) + θθx (t)).

(2.3.52)

On the other hand, by Lemmas 2.3.1–2.3.3, (2.1.1), the mean value theorem and the Poincaré´ inequality, we have u(t) − u ¯ ≤ C1 u x (t), ¯ θ¯ ) + u(t) − u) ¯ θ (t) − θ¯ ≤ C1 (e(u, θ ) − e(u, ¯ θ¯ ) + u x (t)) ≤ C1 (e(u, θ ) − e(u,

(2.3.53)

which, combined with (2.3.5), gives θ (t) − θ¯ ≤ C1 (u x (t) + vx (t) + θθx (t)).

(2.3.54)

Similarly, we infer that ¯ ≤ ρ(t) − ρ ¯ ≤ C1 u(t) − u, ¯ C1−1 u(t) − u ¯ + u(t) − u). ¯ θ (t) − θ¯ ≤ C1 (η(t) − η

(2.3.55) (2.3.56)

It follows from (2.3.50)–(2.3.56) that there exists a constant γ1 = γ1 (C1 ) > 0 such that for any fixed γ ∈ (0, γ1 ], (2.3.44) holds. The the proof is complete.


89

Lemma 2.3.6. There exists a positive constant γ1 = γ1 (C1 ) ≤ γ1 such that for any fixed γ ∈ (0, γ1 ], the generalized global solution (u(t), v(t), θ (t)) in H+1 to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) satisfies the estimate eγ t vx (t)2 + θθx (t)2 t eγ τ vx x 2 + θθx x 2 + vt 2 + θt 2 (τ )dτ ≤ C1 , ∀t > 0. (2.3.57) + 0

Proof. By (2.1.2)–(2.1.3), Lemmas 2.3.1–2.3.3 and Poincaré´ ’s inequality, we get vx (t) ≤ C1 vx x (t), vt (t) ≤ C1 (u x (t) + θθx (t) + vx x (t)), (2.3.58) (2.3.59) θθx (t) ≤ C1 θθx x (t), θt (t) ≤ C1 (θθx x (t) + vx x (t)). Multiplying (2.1.2), (2.1.3) by −eγ t vx x , −eγ t θx x respectively, integrating the results over [0, 1] × [0, t], and adding them up, using Young’s inequality, the embedding theorem, Lemmas 2.3.1–2.3.3 and Lemma 2.3.5, we finally deduce that t √ √ −1 γt 2 2 e (vx (t) + eθ θx (t) ) + C1 eγ τ (vx x 2 + k θx x 2 )(τ )dτ 0 t ≤ C1 + C1 eγ τ (u x + θθx + u x vx 1/2 vx x 1/2 )vx x 0 +(vx + vx 3/2 vx x + u x θθx 1/2 θθx x 1/2 )θθx x dτ t +C1 eγ τ vx 2 + u x 2 + θθx 2 + (vx + θt )θθx 1/2 θθx x 1/2 (τ )dτ 0 t eγ τ (vx x 2 + θθx x 2 )(τ )dτ ≤ C1 + 1/(2C1 ) 0

which, with Lemmas 2.3.1–2.3.3, Lemma 2.3.5, equations (2.1.1)–(2.1.3) and (2.3.58)– (2.3.59), gives (2.3.57). Now we have completed the proof of Theorem 2.3.1.

2.3.3 Exponential Stability in H 2 In this subsection we will complete the proof of Theorem 2.3.2. We begin with the following lemma. Lemma 2.3.7. Under the assumptions in Theorem 2.3.2, the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) admits a unique generalized global solution (u(t), v(t), θ (t)) in H+2 , which defines a nonlinear C0 -semigroup S(t) (also denoted by S(t) by the uniqueness of solution in H+1 ) on H+2 such that for any (u 0 , v0 , θ0 ) ∈ H+2 , S(t)(u 0 , v0 , θ0 ) H 2 = (u(t), v(t), θ (t)) H 2 ≤ C2 , ∀t > 0, +

+

(2.3.60)

90


S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)) ∈ C([0, +∞), H+2 ), u(t) ∈ C

1/2

2

([0, +∞), H ), v(t), θ (t) ∈ C

1/2

1

([0, +∞), H ).

(2.3.61) (2.3.62)

The proof of Lemma 2.3.7 can be divided into the following five lemmas. The first of them is concerned with the uniform estimates of v, θ in H 2. Lemma 2.3.8. For any (u 0 , v0 , θ0 ) ∈ H+2 , the following estimates hold: t 2 2 θt (t) + vt (t) + (vxt 2 + θθxt 2 )(τ )dτ ≤ C2 , ∀t > 0, 0 t 2 2 (vx x x 2 + θθx x x 2 )(τ )dτ ≤ C2 , ∀t > 0. vx x (t) + θθx x (t) +

(2.3.63) (2.3.64)

0

Proof. Differentiating (2.1.2) with respect to t, multiplying the result by vt and integrating over (0, 1), we infer that 1 d vt (t)2 + C1−1 vxt (t)2 ≤ vxt (t)2 + C1 (vx (t)2 + vx (t)4L 4 + θt (t)2 ) dt 2C1 1 vxt (t)2 + C1 (vx x (t)2 + θt (t)2 ) ≤ 2C1 which, together with Lemma 2.3.1, yields t t 2 2 vt (t) + vxt (τ )dτ ≤ C2 + C1 (vx x 2 + θt 2 )(τ )dτ ≤ C2 . 0

(2.3.65)

0

On the other hand, using equation (2.1.2), Lemmas 2.3.1–2.3.3, (2.3.65), Sobolev’s embedding theorem and Young’s inequality, we have vx x (t) ≤ C1 (vt (t) + θθx (t) + u x (t) + vx (t)1/2 vx x (t)1/2 ) 1 ≤ vx x (t) + C1 (vt (t) + 1), 2

t

vx x x 2 (τ )dτ ≤ C2

0

which lead to Similarly,

vx x (t) ≤ C2 , vx (t) L ∞ ≤ C2 , ∀t > 0.

(2.3.66)

θθx x (t) ≤ C1 (θt (t) + 1), ∀t > 0.

(2.3.67)

Similarly to (2.3.65), by equation (2.1.3), we infer that for any δ1 > 0, d √ eθ θt (t)2 + C1−1 θθxt (t)2 dt ≤ δ1 θθxt (t)2 + C1 θθx (t)2 + vx (t)2 + θt (t)3L 3 + θt (t)2 + vxt (t)2 + (θt (t) + θt (t)1/2 θt x (t)1/2 )θt x (t) .

(2.3.68)


91

Integration of (2.3.68) gives

t

θt (t)2 +

θθxt 2 (τ )dτ 0 t t θt x 2 (τ )dτ + C1 (θt 5/2 θt x 1/2 + θt 3 )(τ )dτ ≤ C 2 + C 1 δ1 0 0 t 2 ≤ C 2 + C 1 δ1 θt x (τ )dτ + C1 sup θt (τ )4/3 ≤ C 2 + C 1 δ1

0≤τ ≤t

0 t 0

1 sup θt (τ )2 . θt x 2 (τ )dτ + 2 0≤τ ≤t

That is, 2

sup θt (τ ) +

0≤τ ≤t

t

t

2

θθxt (τ )dτ ≤ C2 + C1 δ1

0

θt x 2 (τ )dτ +

0

1 sup θt (τ )2 2 0≤τ ≤t

which, by taking δ1 > 0 small enough, implies

t

2

sup θt (τ ) +

0≤τ ≤t

0

θθxt 2 (τ )dτ ≤ C2 ,

∀t > 0.

(2.3.69)

By (2.1.3) and (2.3.8), we easily get θθ x x (t)2 +

t 0

θθx x x 2 (τ )dτ ≤ C2 , ∀t > 0.

(2.3.70)

Thus (2.3.63)–(2.3.64) follow from (2.3.65)–(2.3.70) and Lemma 2.3.1. The proof is complete. In what follows, our attention will be paid to the uniform estimate of the specific volume u in H 2. Lemma 2.3.9. For any (u 0 , v0 , θ0 ) ∈ H+2 , the following estimate holds: u(t) H 2 ≤ C2 , ∀t > 0.

(2.3.71)

Proof. Differentiating (2.1.2) with respect to x, using (2.1.1) (u t x x = vx x x ), we see that μ0

∂ uxx − pu u x x = vt x + E(x, t) ∂t u

(2.3.72)

with E(x, t) = ( puu u 2x + 2 pθu θx u x + pθθ θx2 ) + pθ θx x − 2μ0 vx u 2x /u 3 + 2μ0 u x vx x /u 2 .

92


Multiplying (2.3.72) by u x x /u, and by Young’s inequality, Lemmas 2.3.1–2.3.3 and (2.1.21), we can deduce that u d u x x 2 x x 2 (t) + C1−1 (t) dt u u 1 uxx 2 + C1 (θθx (t)4L 4 + u x (t)4L 4 + vxt (t)2 + θθx x (t)2 + vx u 2x (t)2 ) ≤ 4C1 u 1 u x x 2 (t) + C2 (θθ x x (t)2 + u x (t)2 + vxt (t)2 ) (2.3.73) ≤ 2C1 u which, combined with Lemma 2.3.1 and Lemma 2.3.8, gives u x x (t)2 +

t 0

u x x (τ )2 dτ ≤ C2 ,

∀t > 0.

(2.3.74)

Thus (2.3.71) follows from Lemma 2.3.1 and (2.3.74). The proof is complete.

The estimate (2.3.60) and the global existence of generalized solution (u(t), v(t), θ (t)) ∈ H+2 follow from Lemma 2.3.1 and Lemmas 2.3.8–2.3.9. Similarly to (2.3.11), we can prove that the relation (2.3.62) is valid. To complete the proof of Lemma 2.3.7, it suffices to prove (2.3.61) and the continuity of S(t) with respect to (u 0 , v0 , θ0 ) ∈ H+2 , which also leads to the uniqueness of the generalized global solutions in H+2 . This will be done in the next lemma. Lemma 2.3.10. The generalized global solution (u(t), v(t), θ (t)) in H+2 defines a nonlinear C0 -semigroup S(t) on H+2 . Proof. The uniqueness of generalized global solutions in H+2 follows from that in H+1 . Thus S(t) satisfies (2.3.12) on H+2 and by Lemmas 2.3.8–2.3.9, S(t)L(H H 2 ,H H 2 ) ≤ C2 . In +

+

the same manner as in the proof of Lemma 2.3.2, we assume that (u 0 j , v0 j , θ0 j ) ∈ H+2 , ( j = 1, 2), (u j , v j , θ j ) = S(t)(u 0 j , v0 j , θ0 j ), and (u, v, θ ) = (u 1 , v1 , θ1 ) − (u 2 , v2 , θ2 ). We denote by e j = e(u j , θ j ), p j = p(u j , θ j ), k j = k(u j , θ j ), ( j = 1, 2). Subtracting the corresponding equations (2.1.1)–(2.1.3) satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), we obtain equations (2.3.14)–(2.3.17). Similarly to (2.3.21), we have θθx x (t)2 ≤ C1 (θt (t)2 + H1(t)M1 (t)) ≤ C2 (θt (t)2 + u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 ).

(2.3.75)

Differentiating (2.3.15) with respect to x, we see that vt x = μ0 (vx x x /u 1 − 2vx x u 1x /u 21 ) + R(x, t)

(2.3.76)


93

where 1 1 R(x, t) = −( puu u 1x + puθ θ1x )u x − pu1 u x x − ( pu1 − pu2 )u 2x x 1 1 2 1 1 2 − puu u x + ( puu − puu )u 2x + puθ θx + ( puθ − puθ )θ2x u 2x 1 1 −( pθu u 1x + pθθ θ1x )θθx − pθ1 θx x − ( pθ1 − pθ2 )θ2x x 1 1 2 1 1 2 − pθu u x + ( pθu − pθu )u 2x + pθθ θx + ( pθθ − pθθ )θ2x θ2x

−μ0 (vx u 1x x /u 21 + 2vx u 21x /u 31 ) with p j = p(u j , θ j ) ( j = 1, 2). By Lemmas 2.3.1–2.3.3, Lemmas 2.3.7–2.3.9, the embedding theorem and the mean value theorem, we easily obtain R(t)2

≤ C2 (u x (t)2 + u x x (t)2 + u(t)2L ∞ + θ (t)2L ∞ + θθx (t)2 + θθx x (t)2 + vx (t)2L ∞ ) ≤ C2 (u(t)2H 2 + θ (t)2H 2 ).

(2.3.77)

Here and hereafter in the proof of this lemma, C2 > 0 denotes the universal constant depending only on the H 2 norm of the initial data (u 0 j , v0 j , θ0 j ) and min u 0 j (x) and min θ0 j (x) ( j = 1, 2), but independent of t.

x∈[0,1]

x∈[0,1]

By (2.3.76)–(2.3.77) and the embedding theorem, we infer that vx x x (t)2 ≤ C1 vt x (t)2 + C2 (vx x (t)2L ∞ + R(t)2 ) 1 ≤ vx x x (t)2 + C1 vt x (t)2 + C2 (vx x (t)2 + u(t)2H 2 + θ (t)2H 2 ) 2 which gives vx x x (t)2 ≤ C1 vt x (t)2 + C2 (vx x (t)2 + u(t)2H 2 + θ (t)2H 2 ). (2.3.78) Differentiating (2.3.14) twice with respect to x, multiplying the result by u x x , integrating the resulting equation over [0, 1], using (2.3.78) and the Cauchy inequality, we have d u x x (t)2 ≤ C1 vt x (t)2 + C2 (u(t)2H 2 + vx x (t)2 + θ (t)2H 2 ) dt ≤ C1 vt x (t)2 + C2 (u(t)2H 2 + v(t)2H 2 + θ (t)2H 2 ). (2.3.79) Differentiating (2.3.15) with respect to t, multiplying it by vt , integrating the resulting equation over [0, 1], and using Lemmas 2.3.1–2.3.3 and Lemmas 2.3.7–2.3.9, we deduce that d vt (t)2 + C1−1 vt x (t)2 ≤ C2 (1 + v2xt (t)2 )(vt (t)2 + θt (t)2 + u(t)2H 1 dt +v(t)2H 1 + θ (t)2H 1 ). (2.3.80)

94


Multiplying (2.1.3) by eθ−1 , differentiating the resulting equation with respect to t, we arrive at θt t = I1 (u, v, θ ) + I2 (u, v, θ ) + I3 (u, v, θ ) + I4 (u, v, θ ) + I5 (u, v, θ )

(2.3.81)

where I1 (u, v, θ ) = −(eθt /eθ2 )(kθθx /u)x , I2 (u, v, θ ) = (kθθ x /u)xt /eθ , I3 (u, v, θ ) = θ pθ vx eθt /eθ2 , I4 (u, v, θ ) = −(θt pθ vx + θ pθt vx + θ pθ vxt )/eθ , I5 (u, v, θ ) = μ0 [2vx vxt /eθ u − vx2 (eθt u + eθ vx )/eθ2 u 2 ]. We write j

Ii = Ii (u j , v j , θ j ),

j = 1, 2, i = 1, 2, 3, 4, 5.

By Lemmas 2.3.1–2.3.3, Lemmas 2.3.7–2.3.9, (2.3.21), the embedding theorem and the mean value theorem, we infer that for (u, v, θ ) = (u 1 − u 2 , v1 − v2 , θ1 − θ2 ), I11 − I12 2 ≤ C2 (1 + θ1x x x (t)2 )(u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 + θt (t)2 ), (2.3.82) II31 − I32 2 ≤ C2 (u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 + θt (t)2 + vx x (t)2 ) ≤ C2 (u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 + θt (t)2 + vt (t)2 ), (2.3.83) II41 − I42 2 ≤ C2 vxt (t)2 + C2 (1 + v1xt (t)2 )(u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 + θt (t)2 + vt (t)2 ), II51

−

I52 2

2

(2.3.84) 2

≤ C2 vxt (t) + C2 (1 + v2xt (t)

)(u(t)2H 1

+ v(t)2H 1

+ θt (t)2 + vt (t)2 ).

+ θ (t)2H 1 (2.3.85)

Subtracting the corresponding equation (2.3.81) satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), multiplying the resulting equation by θt = (θ1 − θ2 )t and using (2.3.82)–(2.3.85), we easily infer that d θt (t)2 ≤ dt

1 0

(II21 − I22 )θt d x + C2 vxt (t)2 + C2 (1 + θ1x x x (t)2 + v1xt (t)2

+ v2xt (t)2 )(u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 + θt (t)2 + vt (t)2 ). (2.3.86) In (2.3.86), using (2.3.17) and integration by parts, the first term on the right-hand side can be estimated as follows for any small δ2 > 0:

1 0

(II21

−

I22 )θt d x

=−

1 0

2 k 1 θxt /eθ1 u 1 d x + J1 + J2 + J3 ,

(2.3.87)


1

J1 = − 0

95

(θt x /eθ1 )[kt1 θx /u 1 − k 1 θx v1x /u 21 + ((u 2 θ2x (k 1 − k 2 ) − k 2 uθ2x )/u 1 u 2 )t ]d x

≤ δ2 θt x (t)2 + C2 (1 + θ2xt (t)2 )(u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 + θt (t)2 ), (2.3.88) 1 (eθ1x θt /(eθ1 )2 )[(k 1 θt x + kt1 θx )/u 1 + k 1 θx v1x /u 21 ]d x J2 = 0

≤ δ2 θt x (t)2 + C2 (θ (t)2H 1 + θt (t)2 ), 1 (eθ1x θt /(eθ1 )2 )[(u 2 θ2x (k 1 − k 2 ) − k 2 uθ2x )/u 1 u 2 ]t d x J3 =

(2.3.89)

0

≤ C2 (1 + θ2xt (t)2 )(u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 + θt (t)2 )

(2.3.90)

where k j = k(u j , θ j ), e j = e(u j , θ j ), j = 1, 2. Taking δ2 > 0 small enough in (2.3.88)– (2.3.89), using Lemmas 2.3.1–2.3.3 and inserting (2.3.87)–(2.3.90) into (2.3.86), we finally conclude that d θt (t)2 + C1−1 θt x (t)2 ≤ C1 vt x (t)2 + C2 H2(t)(vt (t)2 + θt (t)2 + u(t)2H 1 dt +v(t)2H 1 + θ (t)2H 1 ) (2.3.91) where, by Lemma 2.3.8, H2(t) = 1 + θ1x x x (t)2 + θ2xt (t)2 + v1xt (t)2 + v2xt (t)2 satisfies t

H2(τ )dτ ≤ C2 (1 + t),

∀t > 0.

(2.3.92)

0

Similarly to (2.3.21) and (2.3.75), we easily obtain from (2.3.15)–(2.3.16), vt (t)2 ≤ C2 (vx x (t)2 + u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 ), 2

2

θt (t) ≤ C2 (θθx x (t)

+ u(t)2H 1

+ v(t)2H 1

+ θ (t)2H 1 ).

(2.3.93) (2.3.94)

Now multiplying (2.3.80) by a large number N2 > 2C12 , then adding up the result, (2.3.79) and (2.3.91), we conclude d M2 (t) ≤ C2 H2(t)(vt (t)2 + θt (t)2 + u(t)2H 2 + θ (t)2H 1 + v(t)2H 1 ) dt (2.3.95) ≤ C2 H2(t)(M1 (t) + M2 (t)) where M2 (t) = u x x (t)2 + N2 vt (t)2 + θt (t)2 . Adding (2.3.81) to (2.3.95) gives d M3 (t) ≤ C2 H2(t)M3 (t) (2.3.96) dt where, by (2.3.21), (2.3.24) and (2.3.93)–(2.3.94), M3 (t) = M1 (t) + M2 (t) satisfies C2−1 (u(t)2H 2 + v(t)2H 2 + θ (t)2H 2 ) ≤ M(t) ≤ C2 (u(t)2H 2 + v(t)2H 2 + θ (t)2H 2 ).

(2.3.97)

96


Thus it follows from (2.3.96), Gronwall’s inequality, (2.3.92) and (2.3.97) that u(t)2H 2 + v(t)2H 2 + θ (t)2H 2 ≤ C2 M(t) ≤ C2 M(0) exp C2

t

H2(τ )dτ

0

≤ C2 exp(C2 t)(u 0 2H 2 + v0 2H 2 + θ0 2H 2 ), ∀t > 0 which implies the continuity of S(t) with respect to the initial data in H+2 . Similarly to the proof of (2.3.10), we can prove that (2.3.61) holds. Thus the proof is complete. From Lemmas 2.3.8–2.3.10, we know that the proof of Lemma 2.3.7 is complete.

The next two lemmas concern the exponential decay of generalized global solution (u(t), v(t), θ (t)) in H+2 (or equivalently, of semigroup S(t) on H+2 ). Lemma 2.3.11. There exists a positive constant γ2 = γ2 (C2 ) ≤ γ1 such that for any fixed γ ∈ (0, γ2 ], the generalized global solution (u(t), v(t), θ (t)) in H+2 to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) satisfies the estimate eγ t (θ (t) − θ¯ 2H 2 + v(t)2H 2 ) +

t 0

eγ τ (vxt 2 + θθxt 2 )(τ )dτ ≤ C2 , ∀t > 0. (2.3.98)

Proof. Differentiating equation (2.1.2) with respect to t, multiplying result by vt eγ t and integrating the resulting equation over [0, 1] ×[0, t], by Lemmas 2.3.1–2.3.6 and Young’s inequality, we easily conclude t √ 1 γt 2 e vt (t) + μ0 eγ τ vxt / u 2 (τ )dτ 2 0 t ≤ C2 + γ /2 eγ τ vt 2 (τ )dτ 0 t t √ γτ 2 2 4 e (vx + θt + vx L 4 )(τ )dτ + μ0 /2 eγ τ vxt / u2 (τ )dτ + C1 0 0 t √ ≤ C2 + (C2 γ + μ0 /2) eγ τ vxt / u 2 (τ )dτ 0 t eγ τ (θt 2 + vx 2 + vx x 2 )(τ )dτ + C1 0

which, combined with Lemma 2.3.1, Lemmas 2.3.6–2.3.7 and (2.3.21), implies that there exists a constant γ2 = γ2 (C2 ) ≤ γ1 such that for any fixed γ ∈ (0, γ2 ], γt

2

t

2

e (vt (t) + vx x (t) ) +

0

eγ τ vxt 2 (τ )dτ ≤ C2 , ∀t > 0.

(2.3.99)

2.4. Exponential Stability in H 4

97

In the same manner, multiplying (2.3.81) by θt eγ t , integrating the result over [0, 1]×[0, t] and using Lemmas 2.3.1–2.3.6 and (2.3.99), we infer that t eγ t (θt (t)2 + θθx x (t)2 ) + eγ τ θθxt 2 (τ )dτ ≤ C2 0

which, together with (2.3.99) and Lemmas 2.3.4–2.3.5, yields (2.3.98). The proof is complete. Lemma 2.3.12. There exists a positive constant γ2 = γ2 (C2 ) ≤ γ2 such that for any fixed γ ∈ (0, γ2 ], the generalized global solution (u(t), v(t), θ (t)) in H+2 to the problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) satisfies the estimate u(t) − u ¯ 2H 2 ≤ C2 e−γ t .

(2.3.100)

Proof. Multiplying (2.3.18) by et /2C1 and choosing γ so small that γ ≤ min(γ γ2 , 1/4C1 ) = γ2 (C2 ), and using Lemmas 2.3.5–2.3.6 and Lemma 2.3.11, we conclude that u x x (t)2 ≤ C2 e−t /2C1 + C2 e−γ t ≤ C2 e−γ t which, together with Lemmas 2.3.5–2.3.6, gives the estimate (2.3.100). The proof of Lemma 2.3.12 is complete. Now we have completed the proof of Theorem 2.3.2.

2.4 Exponential Stability in H 4 In this section we further prove the regularity and exponential stability of solutions in H 4 to problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2). In this section, we always assume that all assumptions (2.1.11)–(2.1.25) hold. Now let us state some new ingredients in this section. First, based on the results in H i (i = 1, 2) in Sections 2.3.1–2.3.3, we establish the regularity and exponential stability of global solutions in H 4 (or associated C0 -semigroup), which are two of the new ingredients of this section. As a result, by the embedding theorem, the global solutions obtained in H 4 is actually a classical one in C 3+1/2 when it is subjected to corresponding compatibility conditions. Thus the exponential stability of classical solutions is obtained, which is a new result for this model. This is the third new ingredient. Note that the global existence and exponential stability of solutions in H+i (i = 1, 2) were established in Sections 2.3.1–2.3.3. Chen, Hoff and Trivisa [52] obtained global existence, asymptotic behavior and regularity in H 1 of weak solutions to the compressible Navier-Stokes equations (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) with discontinuous initial data, which were assumed to be that u 0 ∈ BV , v0 ∈ L 4 (0, 1), θ0 ∈ L 2 (0, 1), v0 L 4 + θ0 + T V (u 0 ) ≤ c0 ,

98


c0−1 ≤ u 0 (x) ≤ c0 , θ0 ≥ c0−1 , while in our case, we have established global existence and exponential stability in H 4 for both solutions and the associated nonlinear C0 -semigroup. This is the fourth new ingredient of this section. On the other hand, it is well known that continuous dependence of solutions on initial data is very important, especially when we study infinite-dimensional dynamics (which is in fact equivalent to the fact that the associated semigroup is continuous with respect to initial data or the semigroup as an operator is continuous for any but fixed time t). For example, we refer to the works by Hoff [145], Hoff [146], Hoff and Serre [147] and Hoff and Zarnowski [149] and the references therein. Hoff [145] established continuous dependence on initial data in L 2 for the Cauchy problem of the Navier-Stokes equations of one-dimensional isentropic compressible flow with discontinuous initial data, while we show that the associated C0 -semigroup is continuous with respect to initial data in H 4 which implies continuous dependence on initial data in H 4. This is the fifth new ingredient. It is noteworthy that since our arguments will involve more general constitutive relations in (2.1.11)–(2.1.13) and (2.1.19)–(2.1.25), the higher nonlinearities and partial derivatives of higher order, more delicate estimates are needed. Besides H+1 and H+2 defined as in Section 2.3, we further define H+4 = (u, v, θ ) ∈ H 4[0, 1] × H 4[0, 1] × H 4[0, 1] : u(x) > 0, θ (x) > 0, x ∈ [0, 1], v|x=0 = v|x=1 = 0, θx |x=0 = θx |x=1 = 0 for (2.1.6) or θ |x=0 = θ |x=1 = T0 for (2.2.2) which becomes a metric space when equipped with the metrics induced from the usual norm. In the above, H 4 is the usual Sobolev space. We use Ci (i = 1, 2, 4) to denote the universal positive constant depending only on the H i norm of initial data, min u 0 (x) and min θ0 (x). Without danger of confusion x∈[0,1]

x∈[0,1]

we will use the same symbol to denote the state functions as well as their values along a thermodynamic process, e.g., p(u, θ ), and p(u(x, t), θ (x, t)) and p(x, t). Our main results read as follows: Theorem 2.4.1. Assume that e, p, σ and k are C 5 functions satisfying (2.1.11)–(2.1.13), (2.1.19)–(2.1.25) on 0 < u < +∞ and 0 ≤ θ < +∞, and q, r satisfy assumptions (2.1.14)–(2.1.18). Then for any (u 0 , v0 , θ0 ) ∈ H+4 , there exists a unique global solution (u(t), v(t), θ (t)) ∈ C([0, +∞); H+4 ) to problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) verifying that for any (x, t) ∈ [0, 1] × [0, +∞), 0 < C1−1 ≤ θ (x, t) ≤ C1 , 0 < C1−1 ≤ u(x, t) ≤ C1

(2.4.1)

and for any t > 0, ¯ 2W 3,∞ + u t (t)2H 3 + u t t (t)2H 1 + v(t)2H 4 u(t) − u ¯ 2H 4 + u(t) − u + v(t)2W 3,∞ + vt (t)2H 2 + vt t (t)2 + θ (t) − θ¯ 2H 4 + θ (t) − θ¯ 2W 3,∞ + θt (t)2H 2 + θt t (t)2 ≤ C4 ,

(2.4.2)


99

t u − u ¯ 2H 4 + u − u ¯ 2W 3,∞ + u t 2H 4 + u t t 2H 2 + u t t t 2 + v2H 5 0

+ v(t)2W 4,∞ + vt 2H 3 + vt t 2H 1 + θ − θ¯ 2H 5 + θ (t) − θ¯ 2W 4,∞ +θt 2H 3 + θt t 2H 1 (τ )dτ ≤ C4 .

(2.4.3)

Moreover, the global solution (u(t), v(t), θ (t)) ∈ H+4 defines a nonlinear C0 -semigroup S(t) on H+4 which maps H+4 into itself and satisfies that for any (u 0 , v0 , θ0 ) ∈ H+4 , S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)) ∈ C([0, +∞); H+4 )

(2.4.4)

and S(t) is continuous with respect to initial data, i.e., S(t)(u 01 , v01 , θ01 ) − S(t)(u 02 , v02 , θ02 ) H 4 ≤ C4 (u 01 , v01 , θ01 ) − (u 02 , v02 , θ02 ) H 4 + + (2.4.5) where (u j (t), v j (t), θ j (t)) ( j = 1, 2) is the unique global solution with initial datum (u 0 j , v0 j , θ0 j ) ∈ H+4 ( j = 1, 2). Finally, for any (u 0 , v0 , θ0 ) ∈ H+4 , there are constants C4 > 0 and γ4 = γ4 (C C4 ) > 0 such that for any fixed γ ∈ (0, γ4 ], the following estimates hold for any t > 0: u(t) − u ¯ 2H 4 + u(t) − u ¯ 2W 3,∞ + u t (t)2H 3 + u t t (t)2H 1 + v(t)2H 4 ¯ 2 3,∞ + v(t)2W 3,∞ + vt (t)2H 2 + vt t (t)2 + θ (t) − θ¯ 2H 4 + θ (t) − θ W + θt (t)2H 2 + θt t (t)2 ≤ C4 e−γ t , t ¯ 2H 4 + u − u eγ τ u − u ¯ 2W 3,∞ + u t 2H 4 + u t t 2H 2 + u t t t 2

(2.4.6)

0

+ v2H 5 + v(t)2W 4,∞ + vt 2H 3 + vt t 2H 1 + θ − θ¯ 2H 5 + θ (t) − θ¯ 2W 4,∞ + θt 2H 3 + θt t 2H 1 (τ )dτ ≤ C4 (2.4.7) where

1

u¯ =

u 0 (x)d x, θ¯ = T0 for (2.1.5),

(2.4.8)

0

or for (2.1.6), θ¯ > 0 is uniquely determined by 1 e(u, ¯ θ¯ ) = (e(u 0 , θ0 ) + v02 /2)(x)d x.

(2.4.9)

0

Corollary 2.4.1. Under assumptions of Theorem 2.4.1, estimate (2.4.6) implies that semigroup S(t) is exponentially stable on H+4 , i.e., for any fixed γ ∈ (0, γ4 ] and any t > 0, (u(t), v(t), θ (t)) − (u, ¯ 0, θ¯ )2H 4 = S(t)(u 0 , v0 , θ0 ) − (u, ¯ 0, θ¯ )2H 4 +

≤ C4 e

−γ t

+

.

(2.4.10)

100


Moreover, (u(t), v(t), θ (t)) is the classical solution verifying that for any fixed γ ∈ (0, γ4 ] and for any t > 0, (u(t) − u, ¯ v(t), θ (t) − θ¯ )2C 3+1/2 ×C 3+1/2 ×C 3+1/2 ≤ C4 e−γ t .

(2.4.11)

Remark 2.4.1. Similar results to those in Theorem 2.4.1 and Corollary 2.4.1 also hold for the thermoviscoelastic model in Qin [320] (see also Chapter 6). Remark 2.4.2. Similar results to those in Theorem 2.4.1 also hold for the models of a nonlinear polytropic viscous ideal gas between two horizontal parallel plates in R3 in Qin and Munoz ˜ Rivera [339]. Remark 2.4.3. Similar results to those in Theorem 2.4.1 also hold for the model in [165] under assumptions of the constitutive relations in [165].

2.4.1 Global Existence in H 4 This subsection concerns the existence of global solutions in H+4 , while the existence of global solutions and nonlinear C0 -semigroup S(t) on H+i (i = 1, 2) have been established in Theorems 2.3.1–2.3.2. In what follows, we establish estimates in H+4 . Lemma 2.4.1. Under assumptions of Theorem 2.4.1, for any (u 0 , v0 , θ0 ) ∈ H+4 and for any t > 0, and ∈ (0, 1) small enough, we have vt x (x, 0) + θt x (x, 0) ≤ C3 , vt t (x, 0) + θt t (x, 0) + vt x x (x, 0) + θt x x (x, 0) ≤ C4 ,

t

2

t

θt x x 2 (τ )dτ, (2.4.14) t t θt t x 2 (τ )dτ ≤ C4 −3 + C2 −1 θt x x 2 (τ )dτ θt t (t)2 + 0 0 t +C1 (vt t x 2 + vt x x 2 )(τ )dτ. (2.4.15)

vt t (t) +

2

(2.4.12) (2.4.13)

vt t x (τ )dτ ≤ C4 + C4

0

0

0

Proof. We easily infer from (2.1.2) and Theorems 2.3.1–2.3.2 that vt (t) ≤ C1 (u x (t) + θθx (t) + vx (t) L ∞ u x (t) + vx x (t)) ≤ C2 (vx (t) H 1 + u x (t) + θθx (t)).

(2.4.16)

We differentiate (2.1.2) with respect to x, and use Theorems 2.3.1–2.3.2 to get vt x (t) ≤ C2 (vx (t) H 2 + u x (t) H 1 + θθx (t) H 1 )

(2.4.17)

or vx x x (t) ≤ C2 (v(t) H 2 + u x (t) H 1 + θθx (t) H 1 + vt x (t)).

(2.4.18)


101

Differentiating (2.1.2) with respect to x twice, using Theorems 2.3.1–2.3.2 and the embedding theorem, we conclude vt x x (t) ≤ C2 (u x (t) H 2 + vx (t) H 3 + θθx (t) H 2 + vx (t) L ∞ u x x x (t) +u x (t) L ∞ vx x x (t) + vx x (t) L ∞ u x x (t)) ≤ C2 (u x (t) H 2 + θθx (t) H 2 + vx (t) H 3 )

(2.4.19)

or vx x x x (t) ≤ C2 (u x (t) H 2 + vx (t) H 2 + θθx (t) H 2 + vt x x (t)).

(2.4.20)

Analogously, we infer from (2.1.3), θt (t) ≤ C1 vx (t) + vx (t) L ∞ vx (t)

+(u x (t) + θθx (t))θθx (t) L ∞ + θθx x (t) ≤ C1 (θθx (t) H 1 + vx (t) H 1 ),

(2.4.21)

θt x (t) ≤ C2 (θt (t) + θθx (t) H 2 + u x (t) H 1 + vx x (t)) ≤ C2 (u x (t) H 1 + vx (t) H 1 + θθx (t) H 2 )

(2.4.22)

or θθx x x (t) ≤ C2 (θθx (t) H 1 + vx (t) H 1 + u x (t) H 1 + θt x (t))

(2.4.23)

and θt x x (t) ≤ C2 (u x (t) H 2 + vx (t) H 2 + θθx (t) H 3 )

(2.4.24)

or θθx x x x (t) ≤ C2 (u x (t) H 2 + vx (t) H 2 + θθx (t) H 2 + θt x x (t)).

(2.4.25)

Differentiating (2.1.2) with respect to t, using (2.4.17), (2.4.19) and (2.4.21)–(2.4.22), we have vt t (t) ≤ C2 (vx (t) H 1 + u x (t) + θt (t) + θt x (t) +vt x (t) + vt x x (t)) ≤ C2 (u x (t) H 2 + vx (t) H 3 + θθx (t) H 2 ).

(2.4.26) (2.4.27)

Similarly, we get θt t (t) ≤ C2 (vx (t) H 1 + u x (t) + θt (t) + θt x (t) +vt x (t) + θθx (t) H 2 + θt x x (t)) ≤ C2 (u x (t) H 2 + vx (t) H 2 + θθx (t) H 3 ).

(2.4.28) (2.4.29)

Thus estimates (2.4.12)–(2.4.13) follows from (2.4.17), (2.4.19), (2.4.22), (2.4.24), (2.4.27) and (2.4.29).

102


Differentiating (2.1.2) with respect to t twice, multiplying the resulting equation by vt t in L 2 (0, 1), performing an integration by parts and using Theorems 2.3.1–2.3.2, (2.4.12)–(2.4.13), we obtain 1 d vt t (t)2 = − 2 dt

1 0

σt t vt t x d x

≤ −μ0

vt2t x d x + C2 ( pt t (t) + vt x (t) + vx (t))vt t x (t) u

1 0

≤ −C1−1 vt t x (t)2 + C2 (vx (t)2 + θt (t)2 + θt x (t)2 +vt x (t)2 + θt t (t)2 ).

(2.4.30)

Thus, by Theorems 2.3.1–2.3.2,

t

2

vt t (t) +

2

vt t x (τ )dτ ≤ C4 + C2

0

t

θt t 2 (τ )dτ

0

which along with (2.4.28) gives estimate (2.4.14). Similarly, differentiating (2.1.3) with respect to t twice, multiplying the resulting equation by θt t in L 2 (0, 1) and integrating by parts, we arrive at 1 d 2 dt

1 0

1 kθ θ

eθ θt2t d x = − 3 − 2

−2 +

u

0

0

1 0 1

0 1

x

1

θt t x d x − tt

eθt θt2t d x

0

−

1 0

(eθt t θt + eut t vx )θt t d x

(eu + p − μ0 vx /u)vt t x θt t d x

[eut − (− p + μ0 vx /u)t ]vt x θt t d x

(− p + μ0 vx /u)t t vx θt t d x

= A1 + A2 + A3 + A4 + A5 + A6 .

(2.4.31)

By virtue of Theorems 2.3.1–2.3.3 and (2.4.12)–(2.4.13), and using the embedding theorem, we deduce for any ∈ (0, 1), A1 ≤ −C1−1 θt t x (t)2 + C2 θt x (t) L ∞ (vx (t) + θt (t))θt t x (t) k + C2 ( )t t (t)θθx (t) L ∞ θθx x (t) u ≤ −(2C1 )−1 θt t x (t)2 + C2 (vx (t)2H 1 + θt (t)2 + θt x (t)2 + vt x (t)2 + θt t (t)2 + θt x x (t)2 ), 1 [(|vx | + |θt |)2 + |vt x | + |θt t |](|θt | + |vx |)|θt t |d x A 2 ≤ C1 0

(2.4.32)


103

≤ C1 θt t (t) L ∞ (θt (t) + vx (t)) (vx (t) L ∞ + θt (t) L ∞ ) × (vx (t) + θt (t)) + vt x (t) + θt t (t) ≤ C2 (θt t (t) + θt t x (t))(vx (t) H 1 + θt (t) + θt x (t) + vt x (t) + θt t (t)) ≤ θt t x (t)2 + C2 −1 (vx (t)2H 1 + θt (t)2 + θt x (t)2 + vt x (t)2 + θt t (t)2 ), 1 (|vx | + |θt |)θt2t d x A 3 ≤ C1

(2.4.33)

0

≤ C1 (θt t (t) + θt t x (t))(vx (t) + θt (t))θt t (t) ≤ θt t x (t)2 + C2 −1 θt t (t)2 , 2

A4 ≤ vt t x (t) + C2

−1

(2.4.34)

2

θt t (t) , A5 ≤ C2 vx (t) L ∞ θt t (t) (vx (t) L ∞ + θt (t) L ∞ )(vx (t) + θt (t)) + vt x (t) + θt t (t) + vt t x (t) + vx (t)

(2.4.35)

≤ C2 θt t (t)(vx (t) H 1 + θt (t) + θt x (t) + vt x (t) + θt t (t) + vt t x (t)) ≤ vt t x (t)2 + C2 −1 (θt t (t)2 + vx (t)2H 1 + θt (t)2 + vt x (t)2 + θt x (t)2 )

(2.4.36)

and A 6 ≤ C1

1 0

(|vx | + |θt | + |vt x | + |vx |2 )|vt x ||θt t |d x

≤ C2 vt x (t)1/2 vt x x (t)1/2 (vx (t) + θt (t) + vt x (t))θt t (t)

(2.4.37)

which implies 1/4 1/4 t t t 2 2 A6 dτ ≤ C2 sup θt t (τ ) vt x x (τ )dτ vt x (τ )dτ 0≤τ ≤t

0

0

0

1/2 t 2 2 2 × (vx + θt + vt x )(τ )dτ 0 t ≤ sup θt t (τ )2 + vt x x 2 (τ )dτ + C2 −3 . 0≤τ ≤t

(2.4.38)

0

Thus we infer from (2.4.31)–(2.4.38) that for ∈ (0, 1) small enough, t t θt t x 2 (τ )dτ ≤ C1 sup θt t (τ )2 + (vt x x 2 + vt t x 2 )(τ )dτ θt t (t)2 + 0

0≤τ ≤t

+C C4 −3 + C2 −1

0

t 0

(θt t 2 + θt x x 2 )(τ )dτ. (2.4.39)

104


Thus taking supremum in t on the left-hand side of (2.4.39), picking ∈ (0, 1) small enough, and using (2.4.14), we can derive estimate (2.4.15). The proof is now complete. Lemma 2.4.2. Under assumptions of Theorem 2.4.1, for any (u 0 , v0 , θ0 ) ∈ H+4 , the following estimates hold for any t > 0 and for ∈ (0, 1) small enough: t t vt x (t)2 + vt x x 2 (τ )dτ ≤ C3 −6 + C1 2 (θt x x 2 + vt t x 2 )(τ )dτ, 0

θt x (t)2 +

0 t

0

θt x x 2 (τ )dτ ≤ C3 −6 + C2 2

(2.4.40)

t

0 2

(vt x x 2 + θt t x 2

+ θθx x x 2 θt x )(τ )dτ.

(2.4.41)

Proof. Differentiating (2.1.2) with respect to x and t, multiplying the resulting equation by vt x in L 2 (0, 1), and integrating by parts, we arrive at 1 d vt x (t)2 = B0 (x, t) + B1 (t) 2 dt with

B0 (x, t) = σt x vt x |x=1 x=0 ,

B1 (t) = −

0

1

(2.4.42)

σt x vt x x d x.

We employ Theorems 2.3.1–2.3.2, the interpolation inequality and Poincaré´ ’s inequality to get B0 ≤ C1 (vx (t) L ∞ + θt (t) L ∞ )(u x (t) L ∞ + θθx (t) L ∞ ) +vx x (t) L ∞ + θt x (t) L ∞ + vt x x (t) L ∞ + u x (t) L ∞ vt x (t) L ∞ +vx (t) L ∞ vx x (t) L ∞ + vx (t)2L ∞ )vt x (t) L ∞ ≤ C2 (B01 + B02 )vt x (t)1/2 vt x x (t)1/2

(2.4.43)

where B01 = vx (t) H 2 + θt (t) + θt x (t) and B02 = θt x (t)1/2 θt x x (t)1/2 + vt x x (t)1/2 vt x x x (t)1/2 +vt x x (t) + vt x (t)1/2 vt x x (t)1/2 . Applying Young’s inequality several times, we have that for any ∈ (0, 1), C2 B01vt x (t)1/2 vt x x (t)1/2 ≤

2 vt x x (t)2 + C2 −2/3 (vt x (t)2 2 + vx (t)2H 2 + θt (t)2 + θt x (t)2 )

(2.4.44)


105

and C2 B02vt x (t)1/2 vt x x (t)1/2 ≤

2 vt x x (t)2 + 2 (θt x x (t)2 + vt x x x (t)2 ) 2 (2.4.45) + C2 −6 (θt x (t)2 + vt x (t)2 ).

Thus we infer from (2.4.43)–(2.4.45), Theorems 2.3.1–2.3.2 and Lemma 2.4.1, B0 ≤ 2 (vt x x (t)2 + vt x x x (t)2 + θt x x (t)2 ) +C2 −6 (vx (t)2H 2 + θt (t)2 + θt x (t)2 + vt x (t)2 )

(2.4.46)

which together with Theorems 2.3.1–2.3.2 further leads to

t 0

B0 dτ ≤ 2

t 0

(vt x x 2 + vt x x x 2 + θt x x 2 )(τ )dτ + C2 −6 , ∀t > 0. (2.4.47)

Similarly, by Theorems 2.3.1–2.3.2 and the embedding theorem, we get that for any ∈ (0, 1),

vt2x x d x + C1 (vx (t) + θt (t))(u x (t) L ∞ + θθx (t) L ∞ ) u 0 + vx x (t) + θt x (t) + u x (t) L ∞ vt x (t) + vx (t) L ∞ vx x (t) + vx (t)2L ∞ u x (t) vt x x (t)

B1 ≤ −μ0

1

≤ −(2C1 )−1 vt x x (t)2 + C2 (vx (t)2H 1 + θt (t)2H 1 + vt x (t)2 + u x (t)2 )

(2.4.48)

which combined with (2.4.42), (2.4.47) and Theorems 2.3.1–2.3.2 gives that for ∈ (0, 1) small enough,

t

2

vt x (t) +

2

vt x x (τ )dτ ≤ C3

0

−6

+ C1

t

2

(θt x x 2 + vt x x x 2 )(τ )dτ.

0

(2.4.49) On the other hand, differentiating (2.1.2) with respect to x and t, and using Theorems 2.3.1–2.3.2 and Lemma 2.4.1, we derive vt x x x (t) ≤ C1 vt t x (t) + C2 (vx x (t) H 1 + θθx (t) H 1 + u x (t) H 1 + θt (t) H 2 ). (2.4.50) Thus inserting (2.4.50) into (2.4.49) implies estimate (2.4.40). Analogously, we get from (2.1.3), 1 d 2 dt

1 0

eθ θt2x d x = D0 (x, t) + D1 (t) + D2 (t) + D3 (t)

(2.4.51)

106


where

D0 (x, t) = D2 (t) = −

kθθx u 1

tx

θt x |x=1 x=0 , D1 (t) = −

0

1 kθ θ

u

x

θt x x d x, tx

(eu vx + σ vx )t x θt x d x, 1 1 eθt x θt + eθt + eθ x θt t θt x d x. D3 (t) = − 2 0 0

Similarly to (2.4.43)–(2.4.46), we infer D0 ≤ C2 (vx (t) L ∞ + θt (t) L ∞ + vx x (t) L ∞ + θt x (t) L ∞ +θt (t) L ∞ θθx x (t) L ∞ + θt x x (t) L ∞ + θθx x (t) L ∞ )θt x (t) L ∞ ≤ C2 (D01 + D02 )(D03 + D04 ) where 2 θt x x (t)2 + C2 −2 (vx (t)2H 2 + θθx (t)2H 2 + θt (t)2H 1 ), 3 2 C2 D02 D03 ≤ (θt x x (t)2 + θt x x x (t)2 ) + C2 −6 θt x (t)2 , 3 C2 D01 D04 ≤ C2 (vx (t)2H 2 + θt (t)2H 1 + θθx (t)2H 2 ),

C2 D01 D03 ≤

and C2 D02 D04 ≤

2 (θt x x (t)2 + θt x x x (t)2 ) + C2 −2 θt x (t)2 . 3

That is, D0 ≤ 2 (θt x x (t)2 + θt x x x (t)2 ) + C2 −6 (vx (t)2H 2 + θθx (t)2H 2 + θt (t)2H 1 ). (2.4.52) Similarly, D1 ≤ −(2C1 )−1 θt x x (t)2 + C2 (vx (t)2H 1 + θθx (t)2H 2 + θt (t)2H 1 ),

(2.4.53)

D2 ≤ 2 vt x x (t)2 + C2 −2 (vx (t)2H 2 + θt (t)2H 1 + vt x (t)2 ),

(2.4.54)

2

2

D3 ≤ θt x x (t) + C2

−2

(vx (t)2H 1 2

+ vt x (t)2 + u x (t) ).

+ θt (t)2H 1

+ θθx (t)2H 2 (2.4.55)

Using Lemma 2.4.1 and Theorems 2.3.1–2.3.2 and the embedding theorem, we easily deduce that k (t) ≤ C2 (u x (t) H 1 + vx (t) H 2 + θθx (t) H 1 + θt (t) H 2 ), u txx k k (t) + (t) ≤ C2 (vx (t) H 1 + θt (t) H 1 ) u t u tx


and

107

k k (t) + (t) ≤ C2 (u x (t) H 1 + θθx (t) H 1 ) ≤ C2 ∞ L u x u xx

which imply k k θx (t) ≤ C2 (t) u txx u txx ≤ C2 (u x (t) H 1 + vx (t) H 2 + θθx (t) H 1 + θt (t) H 2 ), k k θx x (t) ≤ C2 u tx u t x L∞ ≤ C2 (u x (t) H 1 + vx (t) H 2 + θθx (t) H 1 + θt (t) H 2 ), k k θx x x (t) ≤ C1 θθx x x (t) u t u t L∞ ≤ C2 (1 + θt x (t))θθx x x (t), k k θt x (t) + θt x x (t) ≤ C2 θt x (t) H 1 . u xx u x

(2.4.56)

(2.4.57)

(2.4.58) (2.4.59)

Differentiating (2.1.3) with respect to x and t, using Lemma 2.4.1 and Theorems 2.3.1– 2.3.2 and (2.4.46)–(2.4.49), we conclude kθθ x k k θt x x x (t) ≤ C1 (t) + θx (t) + θx x (t) u txx u txx u tx k k k + θt x (t) u θx x x (t) + u + u θt x x (t) t xx x ≤ C2 (u x (t) H 1 + vx (t) H 2 + θθx (t) H 2 + θt (t) H 2 + θt t (t) H 1 + vt x (t) H 1 + θt x (t)θθx x x (t)).

(2.4.60)

Hence inserting (2.4.60) into (2.4.52), using (2.4.28) and Theorems 2.3.1–2.3.2 and Lemma 2.4.1, and choosing ∈ (0, 1) small enough, we can derive estimate (2.4.41) from (2.4.51)–(2.4.55) and (2.4.60). The proof is complete. Lemma 2.4.3. Under assumptions of Theorem 2.4.1, for any (u 0 , v0 , θ0 ) ∈ H+4 and for any t > 0, we have 2

2

2

vt t (t) + vt x (t) + θt t (t) + θt x (t) + + θt t x 2 + θt x x 2 )(τ )dτ ≤ C4 , u x x x (t)2H 1 + u x x (t)2W 1,∞ t + (u x x x 2H 1 + u x x 2W 1,∞ )(τ )dτ ≤ C4 , 0

t

2

0

(vt t x 2 + vt x x 2 (2.4.61)

(2.4.62)

108


vx x x (t)2H 1 + vx x (t)2W 1,∞ + θθx x x (t)2H 1 + θθx x (t)2W 1,∞ + u t x x x (t)2 t 2 2 + vt x x (t) + θt x x (t) + (vt t 2 + θt t 2 + vx x 2W 2,∞ + θθx x 2W 2,∞ 0

+ θt x x 2H 1 + vt x x 2H 1 + θt x 2W 1,∞ + vt x 2W 1,∞ + u t x x x 2H 1 )(τ )dτ ≤ C4 , (2.4.63)

t 0

(vx x x x 2H 1 + θθx x x x 2H 1 )(τ )dτ ≤ C4 .

(2.4.64)

Proof. Adding up (2.4.40) and (2.4.41), picking ∈ (0, 1) small enough, we arrive at t vt x (t)2 + θt x (t)2 + (vt x x 2 + θt x x 2 )(τ ) ≤ C3 −6 0 t 2 +C C2 (vt t x 2 + θt t x 2 + θθx x x 2 θt x 2 )(τ )dτ.

(2.4.65)

0

Now multiplying (2.4.14) and (2.4.15) by and 3/2 respectively; then adding the resultant to (2.4.65), and choosing ∈ (0, 1) small enough, we obtain 2

2

2

t

2

vt x (t) + θt x (t) + vt t (t) + θt t (t) +

(θt x x 2 + vt x x 2

0

2

2

+vt t x + θt t x )(τ )dτ ≤ C4

−6

+ C2

2

t

θθx x x 2 θt x 2 (τ )dτ

0

which, by Lemma 2.3.2 and Gronwall’s inequality, gives estimate (2.4.61). Differentiating (2.3.72) with respect to x, and using (2.1.1), we get μ0

∂ uxxx − pu u x x x = E 1 (x, t) ∂t u

with E 1 (x, t) = vt x x + E x (x, t) + pux u x x + μ0

(2.4.66)

u u xx x . u2 t

Obviously, we can infer from Theorems 2.3.1–2.3.2 and Lemmas 2.4.1–2.4.2 that E 1 (t) ≤ C2 (u x (t) H 1 + vx (t) H 2 + θθx (t) H 2 + vt x x (t)) leading to

t

E 1 2 (τ )dτ ≤ C4 , ∀t > 0.

(2.4.67)

(2.4.68)

0

Multiplying (2.4.66) by

ux x x u

in L 2 (0, 1), we obtain

uxxx d uxxx (t)2 + C1−1 (t)2 ≤ C1 E 1 (t)2 dt u u

(2.4.69)


109

which combined with (2.4.68) and Theorems 2.3.1–2.3.2 and Lemmas 2.4.1–2.4.2 gives

t

u x x x (t)2 +

u x x x 2 (τ )dτ ≤ C4 , ∀t > 0.

(2.4.70)

0

By (2.4.18), (2.4.20), (2.4.23), (2.4.25), (2.4.61), (2.4.70) and Lemmas 2.4.1–2.4.2, Theorems 2.3.1–2.3.2, and using the embedding theorem, we get that for any t > 0, vx x x (t)2 + θθx x x (t)2 + vx x (t)2L ∞ + θθx x (t)2L ∞ (2.4.71) t + (vx x x 2H 1 + θθx x x 2H 1 + vx x 2W 1,∞ + θθx x 2W 1,∞ )(τ )dτ ≤ C4 . 0

Differentiating (2.1.2)–(2.1.3) with respect to t, using (2.4.61), we infer that for any t > 0, vt x x (t) ≤ C1 vt t (t) + C2 (vx (t) H 1 + vt x (t) + θt (t) H 1 ) ≤ C4 , (2.4.72) θt x x (t) ≤ C1 θt t (t) + C2 (vx (t) H 1 + vt x (t) + θt (t) H 1 +θθx (t) H 1 ) ≤ C4

(2.4.73)

which combined with (2.4.20) and (2.4.25) imply vx x x x (t)2 + θθx x x x (t)2 t + (θt x x 2 + θθx x x x 2 + vt x x 2 + vx x x x 2 )(τ )dτ ≤ C4 , ∀t > 0.

(2.4.74)

0

Therefore it follows from (2.4.71), (2.7.74) and the embedding theorem that vx x x (t)2L ∞ + θθx x x (t)2L ∞ +

t 0

(vx x x 2L ∞ + θθx x x 2L ∞ )(τ )dτ ≤ C4 , ∀t > 0. (2.4.75)

Now differentiating (2.4.66) with respect to x, we find μ0

∂ uxxxx − pu u x x x x = E 2 (x, t) ∂t u

where E 2 (x, t) = E 1x (x, t) + pux u x x x + μ0

(2.4.76)

∂ uxxxux ( ). ∂t u2

Using the embedding theorem, Lemmas 2.4.1–2.4.2 and Theorems 2.3.1–2.3.2, (2.4.61) and (2.4.72)–(2.4.75), we can deduce that E x x (t) ≤ C4 (θθx (t) H 3 + u x (t) H 2 + vx (t) H 2 ), u u xx x (t) E 1x (t) ≤ C1 E x x (t) + vt x x x (t) + ( pux u x x )x (t) + u2 t x ≤ C1 vt x x x (t) + C4 (θθ x (t) H 3 + u x (t) H 2 + vx (t) H 3 )

110


whence E 2 (t) ≤ C1 vt x x x (t) + C4 (θθ x (t) H 3 + u x (t) H 2 + vx (t) H 3 ). We infer from (2.4.26)–(2.4.29) that t (vt t 2 + θt t 2 )(τ )dτ ≤ C4 , ∀t > 0

(2.4.77)

(2.4.78)

0

which together with (2.4.50) and (2.4.60)–(2.4.61) gives t (vt x x x 2 + θt x x x 2 )(τ )dτ ≤ C4 ,

∀t > 0.

(2.4.79)

0

Thus it follows from (2.4.61), (2.4.77), (2.4.79) and Lemmas 2.4.1–2.4.2 and Theorems 2.3.1–2.3.2 that t

0

E 2 2 (τ )dτ ≤ C4 ,

∀t > 0.

Multiplying (2.4.76) by u xux x x in L 2 (0, 1), we get u d u x x x x 2 x x x x 2 (t) + C1−1 (t) ≤ C1 E 2 (t)2 dt u u

(2.4.80)

(2.4.81)

whence, by (2.4.80), u x x x x (t)2 +

t 0

u x x x x 2 (τ )dτ ≤ C4 , ∀t > 0.

(2.4.82)

Differentiating (2.1.2) with respect to x three times, using Lemmas 2.4.1–2.4.2 and Theorems 2.3.1–2.3.2 and Poincaré´ ’s inequality, we infer vx x x x x (t) ≤ C1 vt x x x (t) + C2 (u x (t) H 3 + vx (t) H 3 + θθx (t) H 3 ).

(2.4.83)

Thus we conclude from (2.1.1), (2.4.79), (2.4.82) and (2.4.83) that t (vx x x x x 2 + u t x x x 2H 1 )(τ )dτ ≤ C4 , ∀t > 0.

(2.4.84)

Similarly, we can deduce from (2.1.3) that t θθx x x x x 2 (τ )dτ ≤ C4 ,

(2.4.85)

0

0

∀t > 0

which with (2.4.84) and (2.4.71) gives t (vx x 2W 2,∞ + θθx x 2W 2,∞ )(τ )dτ ≤ C4 , ∀t > 0.

(2.4.86)

0

Finally, using (2.1.1), (2.4.70)–(2.4.75), (2.4.80), (2.4.82), (2.4.84)–(2.4.86) and Sobolev’s interpolation inequality, we can derive the desired estimates (2.4.62)–(2.4.64). The proof is complete.


111

Lemma 2.4.4. Under assumptions of Theorem 2.4.1, for any (u 0 , v0 , θ0 ) ∈ H+4 and for any t > 0, we have u(t) − u ¯ 2H 4 + u t (t)2H 3 + u t t (t)2H 1 + v(t)2H 4 + vt (t)2H 2 + vt t (t)2 t +θ (t) − θ¯ 2H 4 + θt (t)2H 2 + θt t (t)2 + (u − u ¯ 2H 4 + v2H 5 + vt 2H 3 0 2 2 2 2 ¯ +vt t H 1 + θ − θ H 5 + θt H 3 + θt t H 1 )(τ )dτ t (u t 2H 4 + u t t 2H 2 + u t t t 2 )(τ )dτ ≤ C4 . 0

≤ C4 ,

(2.4.87) (2.4.88)

Proof. Exploiting (2.1.1) and Lemmas 2.4.1–2.4.3 and Theorems 2.3.1–2.3.2, we easily obtain estimates (2.4.87)–(2.4.88). The proof is complete. By Lemmas 2.4.3–2.4.4, we have proved the global existence of solutions to (2.1.1)– (2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) in H+4 with arbitrary initial datum (u 0 , v0 , θ0 ) ∈ H+4 and the uniqueness of solutions in H+4 follows from that of solutions in H+1 or in H+2 .

2.4.2 A Nonlinear C0 -Semigroup S(t) on H 4 In this subsection we establish the existence of a nonlinear C0 -semigroup S(t) on H+4 . Lemma 2.4.5. The global solution (u(t), v(t), θ (t)) in H+4 to problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) defines a nonlinear C0 -semigroup S(t) on H+4 (also denoted by S(t) by the uniqueness of solution in H+1 and H+2 ) such that for any (u 0 , v0 , θ0 ) ∈ H+4 , we have S(t)(u 0 , v0 , θ0 ) H 4 = (u(t), v(t), θ (t)) H 4 ≤ C4 , ∀t > 0, +

+

S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)) ∈ C([0, ∞); H+4 ), ∀t > 0.

(2.4.89) (2.4.90)

Proof. We conclude from Lemmas 2.4.3–2.4.4 that for any t > 0, the operator S(t) : (u 0 , v0 , θ0 ) ∈ H+4 −→ (u(t), v(t), θ (t)) ∈ H+4 exists and (2.4.89) holds, where (u(t), v(t), θ (t)) is the unique solution to problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) with the initial datum (u 0 , v0 , θ0 ) ∈ H+4 , and by the uniqueness of global solution in H+4 , it verifies on H+4 that for any t1 , t2 ∈ [0, +∞), S(t1 + t2 ) = S(t1 )S(tt2 ) = S(tt2 )S(t1 ).

(2.4.91)

We know from Lemmas 2.4.3–2.4.4, S(t) is uniformly bounded on H+4 with respect to t > 0, i.e., S(t)L(H (2.4.92) H 4 ,H H 4 ) ≤ C4 , ∀t > 0. +

+

First of all, we verify the continuity of S(t) with respect to the initial data in H+4 for any t > 0. To this end, we assume that (u 0 j , v0 j , θ0 j ) ∈ H+4 , ( j = 1, 2), (u j , v j , θ j ) =

112


S(t)(u 0 j , v0 j , θ0 j ), and (u, v, θ ) = (u 1 , v1 , θ1 )−(u 2 , v2 , θ2 ). Subtracting the corresponding equations (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), we obtain (2.3.14)–(2.3.17). Clearly, we know from Lemmas 2.4.3–2.4.4 that for any t > 0 and j = 1, 2, (u j (t) − u, ¯ v j (t), θ j (t) − θ¯ )2H 4 + u j t (t)2H 3 + u j t t (t)2H 1 + v j t (t)2H 2 + v j t t (t)2 + θθ j t (t)2H 2 + θθ j t t (t)2 t ¯ 2 5 + v j t 2 3 + v j t t 2 1 + θθ j t 2 3 u j − u + ¯ 2H 4 + v j 2H 5 + θθ j − θ H H H H 0 +θθ j t t 2H 1 + u j t 2H 4 + u j t t 2H 2 + u j t t t 2 (τ )dτ ≤ C4 . (2.4.93) Here and hereafter in the proof of this lemma, C4 > 0 denotes the universal constant depending only on the H 4 norm of the initial data (u 0 j , v0 j , θ0 j ) and min u 0 j (x) and min θ0 j (x) ( j = 1, 2), but independent of t.

x∈[0,1]

x∈[0,1]

We easily know that (2.3.29) and (2.3.98)–(2.3.100) hold with H2(t)| ≤ C4 , ∀t > 0, |H1(t)| + |H t [H1(τ ) + H2(τ )]dτ ≤ C4 , ∀t > 0.

(2.4.94) (2.4.95)

0

By virtue of Lemmas 2.4.1–2.4.4, Theorems 2.4.1–2.4.2, the embedding theorem and the mean value theorem, we easily obtain ∂xi R(t)2 ≤ C4 (u(t)2H 2+i + θ (t)2H 2+i + v(t)2H 1+i ), i = 0, 1, 2. It follows from (2.1.21) and (2.3.79) that uxx − pu1 u x x = R1 (x, t) μ0 u1 t with R1 (x, t) = vt x − μ0

(2.4.96)

(2.4.97)

v1x u x x − R(x, t) − pu1 u x x u 21

verifying ∂xi R1 (t) ≤ C1 ∂xi+1 vt (t) + C4 (u(t) H i+2 + θ (t) H i+2 + v(t) H i+1 ), i = 0, 1, 2. (2.4.98) Differentiating (2.4.97) with respect to x, we arrive at μ0

uxxx u1

t

− pu1 u x x x = R2 (x, t)

(2.4.99)


113

where, by Lemmas 2.4.1–2.4.4, Theorems 2.3.1–2.3.2 and (2.4.98), the mean value theorem and the embedding theorem,

u 1x u x x 1 R2 (x, t) = R1x + μ0 + pux uxx u 21 t verifies ∂xi R2 (t)2 ≤ C1 ∂xi+2 vt (t)2 + C4 (u(t)2H 3+i + v(t)2H 3+i + θ (t)2H 3+i ), i = 0, 1. (2.4.100) Multiplying (2.4.99) by uux 1x x in L 2 (0, 1) and using Lemmas 2.4.1–2.4.4 and Theorems 2.3.1–2.3.2, we get uxxx d uxxx (t)2 + C1−1 (t)2 ≤ C2 R2 (t)2 . dt u 1 u1 Differentiating (2.4.99) with respect to x, we see that uxxxx − pu1 u x x x x = R3 (x, t) μ0 u1 t

(2.4.101)

(2.4.102)

where, by Lemmas 2.4.1–2.4.4 and (2.4.100),

u 1x u x x x 1 + pux u x x x + R2 (x, t) R3 (x, t) = μ0 u 21 t satisfies R3 (t) ≤ C4 (vt x x x (t) + u(t) H 4 + v(t) H 4 + θ (t) H 4 ).

(2.4.103)

On the other hand, we differentiate (2.3.76) with respect to t, and use the embedding theorem to get vt x x x (t) ≤ C4 (vt t x (t) + vt x x (t) + θt x x (t) + u(t) H 2 + v(t) H 2 +θ (t) H 2 + θt (t) + vt x (t) + θt x (t))

(2.4.104)

which with (2.4.103) implies R3 (t) ≤ C4 (vt t x (t) + vt x x (t) + θt x x (t) + u(t) H 4 + v(t) H 4 +θ (t) H 4 + θt (t) + vt x (t) + θt x (t)). (2.4.105) Analogously, we get u d u x x x x 2 x x x x 2 (t) + C1−1 (t) ≤ C1 R3 (t)2 . dt u1 u1

(2.4.106)

114


We easily deduce from (2.3.15)–(2.3.16) vt (t) ≤ C2 (u(t) H 1 + v(t) H 2 + θ (t) H 1 ), θt (t) ≤ C2 (θ (t) H 2 + u(t) H 1 + v(t) H 1 )

(2.4.107) (2.4.108)

vx x (t) ≤ C2 (u(t) H 1 + v(t) H 1 + vt (t) + θ (t) H 1 ), θθx x (t) ≤ C2 (θt (t) + θ (t) H 1 + u(t) H 1 + v(t) H 1 ).

(2.4.109) (2.4.110)

or

We differentiate (2.3.15)–(2.3.16) with respect to x respectively and use Lemmas 2.4.1– 2.4.4 and Theorems 2.3.1–2.3.2 to derive vt x (t) ≤ C4 (θ (t) H 2 + u(t) H 2 + v(t) H 3 ),

(2.4.111)

θt x (t) ≤ C4 (θ (t) H 3 + u(t) H 2 + v(t) H 2 )

(2.4.112)

or, by (2.4.109) and (2.4.110), vx x x (t) ≤ C4 (u(t) H 2 + v(t) H 2 + θ (t) H 2 + vt x (t)), ≤ C4 (u(t) H 2 + v(t) H 1 + θ (t) H 1 + vt (t) +θt (t) + vt x (t)),

(2.4.113)

θθx x x (t) ≤ C4 (θ (t) H 2 + θt x (t) + u(t) H 2 + v(t) H 2 ) ≤ C4 (θt x (t) + u(t) H 2 + v(t) H 1 + vt (t) +θ (t) H 1 + θt (t)).

(2.4.114)

Similarly, we differentiate (2.3.15) and (3.3.16) with respect to t respectively and use (2.4.109), (2.4.110), (2.4.113)–(2.4.114) to get vt x x (t) ≤ C4 (vt t (t) + u(t) H 1 + v(t) H 2 + θ (t) H 1 +vt x (t) + θt (t) + θt x (t)) ≤ C4 (vt t (t) + u(t) H 1 + v(t) H 1 + θ (t) H 1 +vt x (t) + θt (t) + θt x (t)), θt x x (t) ≤ C4 (θt t (t) + u(t) H 1 + v(t) H 1 + θ (t) H 1 +vt x (t) + θt (t) + θt x (t))

(2.4.115) (2.4.116)

or vt t (t) ≤ C4 (vt x x (t) + u(t) H 1 + v(t) H 1 + θ (t) H 1 +θt (t) + vt x (t) + θt x (t)), θt t (t) ≤ C4 (θt x x (t) + u(t) H 1 + v(t) H 1 + θ (t) H 1 +θt (t) + vt x (t) + θt x (t)).

(2.4.117) (2.4.118)


115

We differentiate (2.3.15)–(2.3.16) with respect to x twice and use the mean value theorem to get vt x x (t) ≤ C4 (u(t) H 3 + v(t) H 4 + θ (t) H 3 ), (2.4.119) θt x x (t) ≤ C4 (u(t) H 3 + v(t) H 3 + θ (t) H 4 + θt (t) + θt x (t)) ≤ C4 (u(t) H 3 + v(t) H 3 + θ (t) H 4 )

(2.4.120)

or, by (2.4.109)–(2.4.110) and (2.4.113)–(2.4.116), vx x x x (t) ≤ C4 (u(t) H 3 + v(t) H 3 + θ (t) H 3 + vt x x (t) ≤ C4 (u(t) H 3 + v(t) H 1 + θ (t) H 1 + θt (t) +vt x (t) + vt t (t) + θt x (t)), θθx x x x (t) ≤ C4 (u(t) H 3 + v(t) H 3 + θ (t) H 3 + θt (t)

(2.4.121)

+θt x (t) + θt x x (t)) ≤ C4 (u(t) H 3 + v(t) H 1 + θ (t) H 1 + θt (t) +vt x (t) + θt x (t) + θt t (t)).

(2.4.122)

On the other hand, inserting (2.4.107)–(2.4.108), (2.4.111)–(2.4.112) and (2.4.119)– (2.4.120) into (2.4.117)–(2.4.118) gives (2.4.117)–(2.4.118) that vt t (t) ≤ C4 (u(t) H 3 + v(t) H 4 + θ (t) H 3 ),

(2.4.123)

θt t (t) ≤ C4 (u(t) H 3 + v(t) H 3 + θ (t) H 4 ).

(2.4.124)

Now differentiating (2.3.16) with respect to t twice, multiplying the resulting equation by vt t in L 2 (0, 1), integrating by parts, and employing Lemmas 2.4.1–2.4.4 and Theorems 2.3.1–2.3.2, estimates (2.4.107)–(2.4.124), the mean value theorem and the embedding theorem, we finally deduce d vt t (t)2 + C1−1 vt t x (t)2 ≤ C4 (1 + v2t t x (t)2 + θ2t t x (t)2 ) dt ×(u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 + vt (t)2 + θt (t)2 +vt x (t)2 + θt x (t)2 + vt t (t)2 + θt t (t)2 ).

(2.4.125)

Similarly, differentiating (2.3.81) with respect to t, subtracting the corresponding equations satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), multiplying the resulting equation by θt t = (θ1 − θ2 )t t , using Lemmas 2.4.1–2.4.4, the embedding theorem and the mean value theorem, we infer that for (u, v, θ ) = (u 1 − u 2 , v1 − v2 , θ1 − θ2 ), d θt t (t)2 + C1−1 θt t x (t)2 ≤ C4 (1 + θ2t t x (t)2 )(u(t)2H 1 + v(t)2H 1 dt +θ (t)2H 1 + θt (t)2 + vt x (t)2 + θt x (t)2 + θt t (t)2 ). (2.4.126) Differentiating (2.3.16) with respect to t and x respectively, perfuming an integration by parts, we arrive at 1 d vt x (t)2 = h 0 + h 1 + h 2 + h 3 2 dt

(2.4.127)

116


where x=1 v2x u vx h 0 = μ0 − vt x , u1 u1u2 t x x=0 1 v2x u vx h 1 = −μ0 − vt x x d x, u1 u1u2 t x 0 1 h 2 = −μ0 ( pu1 u x + pθ1 θx )t x vt x d x,

0

1

h3 = 0

[( pu1 − pu2 )u 2x + ( pθ1 − pθ2 )θ2x ]t x vt x d x.

Using Sobolev’s interpolation inequality, we infer from Lemmas 2.4.1–2.4.4, Theorems 2.3.1–2.3.2, (2.4.109)–(2.4.110), (2.4.113)–(2.4.114) and (2.4.121)–(2.4.122) that for any ∈ (0, 1), h 0 ≤ C4 vt x x (t) L ∞ + vt x (t) L ∞ + vx (t) L ∞ + vx x (t) L ∞ +(1 + v2t x x (t) L ∞ )u(t) L ∞ + u x (t) L ∞ vt x (t) L ∞ ≤ C4 vt x x (t)1/2 vt x x x (t)1/2 + vt x x (t) + vt x (t) + v(t) H 3 +v2t x x x (t)u t + u(t) H 2 (vt x (t)1/2 vt x x 1/2 + vt x (t)) ≤ (vt x x x (t)2 + vt x x (t)2 ) + C4 ()(1 + v2t x x x (t)2 )(u(t)2H 2 +v(t)2H 1 + θ (t)2H 1 + θt (t)2 + vt (t)2 + vt x (t)2 ), (2.4.128) 1 2 vt x x d x + vt x x (t)2 + C4 ()(vt x (t)2 + vx (t)2H 1 ) h 1 ≤ −μ0 0 u1 ≤ −(C1−1 − )vt x x (t)2 + C4 ()(vt x (t)2 + u(t)2H 1 + v(t)2H 1 +θ (t)2H 1 + vt (t)2 ).

(2.4.129)

Similarly, we conclude for any ∈ (0, 1), h 2 ≤ (vt x x (t)2 + θt x x (t)2 ) + C4 ()(u(t)2H 2 + v(t)2H 1 + θ (t)2H 1 +vt (t)2 + θt (t)2 + vt x (t)2 + θt x (t)2 ), 2

h 3 ≤ vt x x (t) + C4 ()(u(t)2H 1 +θt (t)2 + θt x (t)2 ).

+

v(t)2H 1

+

θ (t)2H 1

(2.4.130) 2

+ vt (t)

(2.4.131)

Thus the combination of (2.4.127)–(2.4.131) gives 1 d vt x (t)2 ≤ −(C1−1 − C4 )vt x x (t)2 + C4 (vt t x (t)2 + θt x x (t)2 ) 2 dt +C C4 ()(1 + v2t x x x (t)2 )(u(t)2H 2 + v(t)2H 1 +θ (t)2H 1 + θt (t)2 + vt x (t)2 + θt x (t)2 ).

(2.4.132)


117

Similarly, differentiating (2.3.81) with respect to x, subtracting the resulting equations satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), multiplying the corresponding equation by θt x = (θ1 − θ2 )t x , we finally conclude 1 d θt x (t)2 ≤ −(C1−1 − C4 )θt x x (t)2 + C4 (θt t x (t)2 + vt x x (t)2 ) 2 dt +C C4 ()(1 + θ2t x x x (t)2 )(u(t)2H 2 + v(t)2H 1 +θ (t)2H 1 + θt (t)2 + vt x (t)2 + θt x (t)2 ).

(2.4.133)

Put 1 1 M4 (t) = vt t (t)2 + vt x (t)2 + θt t (t)2 + θt x (t)2 2 2 u x x x 2 u x x x x 2 + (t) + (t) . u1 u1 Then multiplying (2.4.101) and (2.4.106) by respectively, adding up the resulting equations, (2.4.125)–(2.4.126) and (2.4.132)–(2.4.133), and using (2.4.100), (2.4.105), and picking > 0 small enough, we get d M4 (t) + C4−1 (vt t x (t)2 + vt x x (t)2 + θt t x (t)2 + θt x x (t)2 dt +u x x x (t)2 + u x x x x (t)2 ) (2.4.134) ≤ C4 H3(t)G(t) where G(t) = u(t)2H 4 + v(t)2H 4 + θ (t)|2H 4 + vt (t)2 + θt (t)2 + vt x (t)2 +θt x (t)2 + vt t (t)2 + θt t (t)2 and, by Lemmas 2.4.3–2.4.4, H3(t) = 1 + v1t t x (t)2 + v2t t x (t)2 + θ2t t x (t)2 + θ2t x x x (t)2 + v2t x x x (t)2 verifies

t 0

H3(τ )dτ ≤ C4 (1 + t),

∀t > 0.

(2.4.135)

On the other hand, we derive from (2.4.107)–(2.4.108), (2.4.111)–(2.4.112), (2.4.123)– (2.4.124) u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ≤ G(t) ≤ C4 (u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ). (2.4.136)

118


If we put M(t) = M3 (t) + M4 (t), then we easily infer from (2.4.109)–(2.4.110), (2.4.113)–(2.4.114) and (2.4.121)–(2.4.122) that G(t) ≤ C4 (u(t)2H 4 + v(t)2H 1 + θ (t)2H 1 + vt (t)2 + θt (t)2 +vt x (t)2 + θt x (t)2 + vt t (t)2 + θt t (t)2 ) ≤ C4 M(t).

(2.4.137)

Also it follows from the definition of M(t), (2.3.27) and (2.4.137) that M(t) ≤ C4 (u(t)2H 4 + v(t)2H 1 + θ (t)2H 1 + vt (t)2 + θt (t)2 +vt x (t)2 + θt x (t)2 + vt t (t)2 + θt t (t)2 ) ≤ C4 G(t) which along with (2.4.137) gives C4−1 (u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ) ≤ M(t) ≤ C4 (u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ).

(2.4.138)

Thus adding up (2.3.96) and (2.4.134) yields d M(t) ≤ C4 H3(t)M(t) dt which with (2.4.138) and Gronwall’s inequality that for any t > 0, 2 2 2 u(t) H 4 + v(t) H 4 + θ (t) H 4 ≤ C4 M(t) ≤ C4 M(0) exp C4

t 0

H3(τ )dτ

C4 t)[u 0 2H 4 + v0 2H 4 + θ0 2H 4 ], ≤ C4 exp(C ∀t > 0. That is, S(t)(u 10 , v10 , θ10 ) − S(t)(u 20 , v20 , θ20 ) H 4

+

≤ C4 exp(C C4 t)(u 10 , v10 , θ10 ) − (u 20 , v20 , θ20 ) H 4

+

which implies the continuity of semigroup S(t) with respect to the initial data in H+4 (and also the uniqueness of global solutions in H+4 ). In order to prove (2.4.90), by (2.4.91)– (2.4.92), it suffices to show that for any (u 0 , v0 , θ0 ) ∈ H+4 , S(t)(u 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 4 −→ 0 +

as t → 0+ , which also gives

S(0) = I

(2.4.139)

(2.4.140)


119

with I being the unit operator (i.e., identity operator) on H+4 . To show (2.4.139) and (2.4.140), we choose a function sequence which is smooth enough, for example, m m 6 6 6 (u m 0 , v0 , θ0 ) ∈ H × H × H

such that

m m (u m 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 4 −→ 0

(2.4.141)

+

as m → +∞. By the local regularity result, we conclude that there is a small t0 > 0 such that there exists a unique smooth solution (u m (t), v m (t), θ m (t)) ∈ H 6 × H 6 × H 6 (∀t ∈ (0, t0 )). This implies that for m = 1, 2, . . . , m m (u m (t), v m (t), θ m (t)) − (u m 0 , v0 , θ0 ) H 4 −→ 0

(2.4.142)

+

as t → 0+ . By the continuity of the operator S(t), we conclude that for any t ∈ (0, t0 ), (u m (t), v m (t), θ m (t)) − (u(t), v(t), θ (t)) H 4

+

m m = S(t)(u m 0 , v0 , θ0 ) − S(t)(u 0 , v0 , θ0 ) H 4

+

m m ≤ C4 (t0 )(u m 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 4 −→ 0 +

as m → +∞, this along with (2.4.141)–(2.4.142) leads to S(t)(u 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 4 = (u(t), v(t), θ (t)) − (u 0 , v0 , θ0 ) H 4 +

+


+

m m m m m +(u m (t), v m (t), θ m (t)) − (u m 0 , v0 , θ0 ) H 4 + (u 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 4 +

−→ 0

+

as m → +∞ and t → 0+ , which implies (2.4.139) and (2.4.140). The proof is now complete.

2.4.3 Exponential Stability in H 4 In this subsection, we shall use estimates established in Sections 2.4.1–2.4.2 to show the exponential stability of a solution or of the nonlinear C0 -semigroup S(t) on H+4 . Lemma 2.4.6. Under assumptions of Theorem 2.4.1, for any (u 0 , v0 , θ0 ) ∈ H+4 , there exists a positive constant γ4(1) = γ4(1) (C C4 ) ≤ γ2 (C2 ) such that for any fixed γ ∈ (0, γ4(1) ], the following estimates hold for any t > 0 and ∈ (0, 1) small enough: t t eγ t vt t (t)2 + eγ τ vt t x 2 (τ )dτ ≤ C4 + C4 eγ τ θt x x 2 (τ )dτ, (2.4.143) 0 0 t t eγ τ θt t x 2 (τ )dτ ≤ C1 eγ τ (vt x x 2 + vt t x )(τ )dτ eγ t θt t (t)2 + 0 0 t −3 −1 γτ 2 + C4 + C2 e θt x x (τ )dτ. (2.4.144) 0

120


Proof. The proofs of (2.4.143)–(2.4.144) are basically the same as those of (2.4.14)– (2.4.15). The difference here is to estimate (2.4.143)–(2.4.144) with weighted exponential function eγ t . Multiplying (2.4.30) by eγ t and using (2.4.26) and Theorem 2.3.2, we have 1 γt e vt t (t)2 ≤ C4 − (C1−1 − C1 γ ) 2 ≤ C4 −

(C1−1

− C1 γ )

t

e

γτ

vt t x (τ )dτ + C2

0 t

e

γτ

0

t

2

t

2

vt t x (τ )dτ + C4

eγ τ θt t 2 (τ )dτ

0

0

which gives (2.4.143) if we take γ > 0 so small that 0 < γ ≤ min[

eγ τ θt x x 2 (τ )dτ

1 , γ2 (C2 )]. 4C 12

Similarly, multiplying (2.4.31) by eγ t and using (2.4.28), (2.4.32)–(2.4.37) and Theorem 2.3.2, we derive that for any ∈ (0, 1) small enough, 1 γt √ γ t γτ e eθ θt t (t)2 ≤ C4 + e θt t 2 (τ )dτ 2 2 0 t + eγ τ (A1 + A2 + A3 + A4 + A5 + A6 )(τ )dτ 0 t t ≤ C4 −3 − (C1−1 − 2) eγ τ θt t x 2 (τ )dτ + C2 −1 eγ τ θt x x 2 (τ )dτ 0

t

+

0

eγ τ vt t x 2 (τ )dτ + C2 eγ t /2

0

t

×

2

2

2

1/4 t γτ 2 sup θt t (τ ) e vt x x (τ )dτ

0≤τ ≤t 1/2 t

0 γτ

1/4 2

(vx + θt + vt x )(τ )dτ e vt x (τ )dτ 0 t t −1 −3 γτ 2 −1 ≤ C4 − (C1 − 2) e θt t x (τ )dτ + C2 eγ τ θt x x 2 (τ )dτ 0 0 t eγ τ (vt t x 2 + vt x x 2 )(τ )dτ + eγ t sup θt t (τ )2 + 0

0≤τ ≤t

0

which, by taking supremum on the right-hand side and choosing ∈ (0, 1) small enough, implies (2.4.144). The proof is complete. Lemma 2.4.7. Under assumptions of Theorem 2.4.1, for any (u 0 , v0 , θ0 ) ∈ H+4 , there is a positive constant γ4(2) ≤ γ4(1) such that for any fixed γ ∈ (0, γ4(2) ], the following estimates hold for any t > 0 and ∈ (0, 1) small enough: eγ t vt x (t)2 +

t

eγ τ vt x x 2 (τ )dτ 0 t −6 ≤ C3 + C2 2 eγ τ (θt x x 2 + vt t x 2 )(τ )dτ, 0

(2.4.145)


eγ t θt x (t)2 +

121 t

eγ τ θt x x 2 (τ )dτ 0 t −6 eγ τ (vt x x 2 + θt t x 2 )(τ )dτ, ≤ C3 + C2 2 0 t eγ t (vt x (t)2 + θt x (t)2 ) + eγ τ (vt x x 2 + θt x x 2 )(τ )dτ 0 t eγ τ (θt t x 2 + vt t x 2 )(τ )dτ. ≤ C3 −6 + C2 2

(2.4.146)

(2.4.147)

0

Proof. Multiplying (2.4.42) by eγ t , using (2.4.46), (2.4.48) and Theorem 2.3.2, we infer that for any ∈ (0, 1) small enough, t eγ t vt x (t)2 ≤ C3 −6 − [(2C1 )−1 − 2 − C1 γ ] eγ τ vt x x 2 (τ )dτ 0 t eγ τ (vt x x x 2 + θt x x 2 )(τ )dτ + 2 0

which with (2.4.50) gives (2.4.145) if we take γ > 0 and ∈ (0, 1) so small that 0 < (1) (2) < min[1, 1/(8C1 )] and 0 < γ ≤ min[γ γ4 , 1/(8C12 )] ≡ γ4 . In the same manner, we easily derive (2.4.146) from (2.4.51)–(2.4.55) and (2.4.60). Adding (2.4.145) to (2.4.146) and picking ∈ (0, 1) small enough give (2.4.147). The proof is complete. Lemma 2.4.8. Under assumptions of Theorem 2.4.1, for any (u 0 , v0 , θ0 ) ∈ H+4 , there is (2) a positive constant γ4 ≤ γ4 such that for any fixed γ ∈ (0, γ4 ], the following estimates hold for any t > 0: t eγ t vt t (t)2 + vt x (t)2 + θt t (t)2 + θt x (t)2 + eγ τ vt t x 2 + vt x x 2 0 2 2 + θt t x + θt x x (τ )dτ ≤ C4 , (2.4.148) t eγ t u x x x (t)2H 1 + u x x (t)2W 1,∞ + eγ τ u x x x 2H 1 + u x x 2W 1,∞ (τ )dτ ≤ C4 , 0

(2.4.149)

eγ t vx x x (t)2H 1 + vx x (t)2W 1,∞ + θθx x x (t)2H 1 + θθx x (t)2W 1,∞ + u t x x x (t)2 t + vt x x (t)2 + θt x x (t)2 + eγ τ vt t 2 + vx x x x 2H 1 + vt x x 2H 1 2

+ θt t +

θθx x x x 2H 1

0 2 + θt x x H 1

+ vx x 2W 2,∞ + vt x 2W 1,∞ + θθx x 2W 2,∞

+ θt x 2W 1,∞ + u t x x x 2H 1 (τ )dτ ≤ C4 .

(2.4.150)

Proof. Multiplying (2.4.143) and (2.4.144) by and 3/2 respectively, adding the resulting inequality to (2.4.147), and then taking > 0 small enough, we can obtain the desired estimate (2.4.148).

122


Multiplying (2.4.69) by eγ t , using (2.4.67), (2.4.148) and Theorem 2.3.2, and (2) choosing γ > 0 so small that 0 < γ ≤ γ4 ≡ min[ 2C1 1 , γ4 ], we conclude that for any t > 0, t t u u 2 1 x x x 2 xxx (t) + eγ t eγ τ (τ )dτ ≤ C + C eγ τ E 1 (τ )2 dτ ≤ C4 3 1 u 2C1 0 u 0 whence γt

t

2

e u x x x (t) +

eγ τ u x x x 2 (τ )dτ ≤ C4 ,

∀t > 0.

(2.4.151)

0

Similarly to (2.4.71), (2.4.74)–(2.4.75), (2.4.78)–(2.4.79), using (2.4.148), (2.4.151) and Theorem 2.3.2, we have that for any fixed γ ∈ (0, γ4 ], eγ t (vx x x (t)2H 1 + vx x (t)2W 1,∞ + vt x x (t)2 + θθx x x (t)2H 1 t +θθx x (t)2W 1,∞ + θt x x (t)2 ) + eγ τ (vx x x 2H 1 + vx x 2W 1,∞ 0

+θθx x x 2H 1 + θθx x 2W 1,∞ + vt x x 2 + θt x x 2 )(τ )dτ ≤ C4

(2.4.152)

and

t 0

eγ τ (vt t 2 + θt t 2 + vt x x x 2 + θt x x x 2 )(τ )dτ ≤ C4 ,

∀t > 0.

(2.4.153)

Similarly to (2.4.151), multiplying (2.4.81) by eγ t , using (2.4.77), (2.4.148), (2.4.151)– (2.4.153) and Theorem 2.3.2, we get that for any fixed γ ∈ (0, γ4 ], t t 2 2 1 γ t uxxxx γ τ uxxxx (t) + e eγ τ E 2 2 (τ )dτ ≤ C4 . e (τ )dτ ≤ C4 + C1 u 2C1 0 u 0 That is, eγ t u x x x x (t)2 +

t

eγ τ u x x x x 2 (τ )dτ ≤ C4 ,

∀t > 0.

(2.4.154)

0

Similarly to (2.4.84)–(2.4.86), we easily derive that for any fixed γ ∈ (0, γ4 ], t eγ τ vx x x x x 2 + θθx x x x x 2 + u t x x x 2H 1 + vx x 2W 2,∞ 0 +θθx x 2W 2,∞ (τ )dτ ≤ C4 , ∀t > 0. (2.4.155) Finally, we combine estimates (2.4.148), (2.4.151)–(2.4.155) with the interpolation inequality to derive the required estimates (2.4.149)–(2.4.150). The proof is now complete.

2.5. Attractors in H 1 and H 2

123

Now we can use (2.1.1), Theorem 2.3.2, Lemmas 2.4.6–2.4.8 to prove the following lemma. Lemma 2.4.9. Under assumptions of Theorem 2.4.1, for any (u 0 , v0 , θ0 ) ∈ H+4 and for any fixed γ ∈ (0, γ4 ], the following estimates hold for any t > 0: ¯ 2 4 + u t (t)2 3 + u t t (t)2 1 + vt (t)2 2 eγ t (u(t) − u ¯ 2H 4 + v(t)2H 4 + θ (t) − θ H H H H 0

+ vt t (t)2 + θt (t)2H 2 + θt t (t)2 ) ≤ C4 , t

(2.4.156)

¯ 2 5 + vt 2 3 + vt t 2 1 + θt 2 3 + θt t 2 1 eγ τ (u − u ¯ 2H 4 + v2H 5 + θ − θ H H H H H

+ u t 2H 4 + u t t 2H 2 + u t t t 2 )(τ )dτ ≤ C4 .

(2.4.157)

2.5 Attractors in H 1 and H 2 In this section, we are concerned with the existence of universal (maximal) attractors for problems (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) and (2.1.1)–(2.1.3), (2.2.1), (2.2.2) with more general constitutive relations which are the same as those in Theorems 2.3.1–2.3.2. Let us first explain some mathematical difficulties in studying the dynamics of our problems. Firstly, from physical reasons, the special volume u and the absolute temperature θ should be positive for all time. These constraints give rise to some severe mathematical difficulties. For instance, we must work on incomplete metric spaces H+1 and H+2 , H+2 ⊂ H+1 which are usual Sobolev spaces with these constraints. Secondly, the nonlinear semigroup S(t) defined by problem (2.1.1)–(2.1.3), (2.1.5)– (2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2), maps each H+1 and H+2 into itself, as proved in Section 2.3. It is clear from equations (2.1.2) and (2.1.3) that we cannot continuously extend the semigroup S(t) to the closure of H+1 and H+2 . Notice the following significant differences between the study of global existence and the study of existence of a (maximal) universal attractor: for the study of global existence, the initial datum is given while for the study of existence of a (maximal) universal attractor in certain metric space, the initial data are varying in that space. Since the (maximal) universal attractor is just the ω − li mi t set of an absorbing set in weak topology, the requirement on completeness of spaces is needed. To overcome this severe mathematical difficulty, we restrict ourselves to a sequence of closed subspaces of H+1 and H+2 . It turns out that it is very crucial to prove that the orbit starting from any bounded set of this closed subspace will re-enter this subspace and stay there after a finite time, which should be uniform with respect to all orbits starting from a bounded set; otherwise, there is no ground to talk about existence of an absorbing set and a maximal universal attractor in this subspace. The proof of the above fact becomes an essential part of this section and it will be done by use of delicate a priori estimates. Thirdly, the total mass with (2.1.6) and (2.2.2) and the total energy with (2.1.6) are conserved. Indeed, if we integrate the equation (2.1.1) with respect to x and t and exploit

124


the boundary conditions (2.1.6) and (2.2.2), we will end up with

1

1

u(x, t)d x =

0

u 0 (x)d x,

∀t > 0.

(2.5.1)

0

Next, for (2.1.6) if we integrate the equation (2.1.3) with respect to x and t and use (2.1.6), we finally get

1 0

(e(x, t) + v 2 (x, t)/2)d x =

0

1

(e(x, 0) + v02 (x)/2)d x.

(2.5.2)

These conservations indicate that there can be no absorbing set for initial data varying in the whole space. Instead, we should rather consider the dynamics in a sequence of closed subspaces defined by some parameters. In this regard, the situation is quite similar to those encountered for the single Cahn-Hilliard equation in the isothermal case (see, Temam [407]), and for the coupled Cahn-Hilliard equations (see, Shen and Zheng [375]) and for a one-dimensional polytropic viscous ideal gas (2.1.10) with (2.1.6) (see, Zheng and Qin [451]). Therefore, one of the key issues is how to choose these closed subspaces. Fourthly, (2.1.1)–(2.1.3) is a hyperbolic-parabolic coupled system. It turns out that in general the orbit is not compact. In order to prove the existence of a maximal attractor by the theory presented by Temam in [407], we have either to show the uniform compactness of the orbit of semigroup S(t) for large time or to show that one can decompose S(t) into two parts, S1 (t) and S2 (t), with S1 (t) being uniformly compact for large time and S2 (t) going to zero uniformly. Since equations (2.1.1)–(2.1.3) constitute a hyperbolic-parabolic coupled system, the orbit is not compact. Moreover, since our system is quasilinear, the usual way of decomposition of S(t) into two parts for a semilinear system does not seem feasible. To overcome this difficulty, we will adopt an approach motivated by the ideas in [117] (see also, Theorem 1.6.4) and Zheng and Qin [451, 452]. Finally, unlike the one-dimensional polytropic viscous ideal gas (2.1.10) (the special case of q = r = 0), equations (2.1.1)–(2.1.3) look more complicated than that of the special case of q = r = 0. It turns out that much more delicate estimates are needed. Let 1 i i Hδ := (u, v, θ ) ∈ H+ : (E(u, θ ) + v 2 /2)d x ≤ δ1 ,

0

0

1

δ6 ≤ δ2 ≤

1 0

(e(u, θ ) + v 2 /2)d x ≤ δ7 for (2.1.6), ud x ≤ δ3 , δ4 ≤ θ ≤ δ5 , δ2 /2 ≤ u ≤ 2δ3 , i = 1, 2, 4

where E(u, θ ) =: (u, θ ) − (1, ) − u (1, )(u − 1) − θ (u, θ )(θ − )

(2.5.3)


125

with = 1 for (2.1.6) or = T0 for (2.2.2), while δi (i = 1, . . . , 7) are any given constants satisfying δ1 ∈ R, 0 < δ2 < δ3 , 0 < δ4 < δ5 , 0 < δ6 < δ7

(2.5.4)

with the constraints 0 < δ4 < T0 < δ5 for

(2.2.2),

θˆ (ξ, e),

θˆ (ξ, e) < δ5 for (2.1.6); (2.5.6)

(2.5.5)

or 0 < δ4
0:

1

δ2 ≤

0

u(x, t)d x =

1

u 0 (x)d x ≤ δ3 ,

0 1

δ6 ≤

0

0

1

0

(E(u, θ ) + v 2 /2)(x, t)d x + 1

= 0

(e(u 0 , θ0 ) + v02 /2)(x)d x ≤ δ7 , for (2.1.6),

t 0

1

(e(u, θ ) + v 2 /2)(x, t)d x =

(2.5.7)

1

0

μ0 vx2 k(u, θ )θθ x2 + uθ uθ 2

(2.5.8)

d x dτ

(E(u 0 , θ0 ) + v02 /2)(x)d x ≤ δ1 .

(2.5.9)

Proof. Estimates (2.5.7)–(2.5.8) have already been obtained in (2.5.1) and (2.5.2). Note that (u, θ ) = e(u, θ )−θ η(u, θ ) is the Helmholtz free energy function. Recalling (2.5.3), the definition of E = E(u, θ ), noting that eθ (u, θ ) = −θ θθ (u, θ ), by (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) and (2.1.11)–(2.1.13), we deduce after a direct calculation that

k(u, θ )θθ x2 μ0 vx2 ∂t [E(u, θ ) + v /2] + + uθ uθ 2 (θ − )k(u, θ )θθ x = (σ v)x + p(1, )vx + . uθ x

2

(2.5.10)


127

Integrating (2.5.10) over Q t := (0, 1) × (0, t) and using (2.1.2)–(2.1.3), we obtain

1 t 1 k(u, θ )θθ x2 μ0 vx2 2 + d x ds (E(u, θ ) + v /2)(x, t)d x + uθ uθ 2 0 0 0 1 = (E(u 0 , θ0 ) + v02 /2)d x 0

which gives (2.5.9).

Lemma 2.5.2. If (u 0 , v0 , θ0 ) ∈ Hδ1, then the following estimates hold for any t > 0: t 1 1 q )θ 2 2 v θ (1 + θ x (θ r+1 + v 2 )(x, t)d x + + x (x, τ )d x dτ ≤ Cδ , (2.5.11) uθ uθ 2 0 0 0 1 −1 0 < Cδ ≤ θ (x, t)d x ≤ Cδ . (2.5.12) 0

Proof. In view of (2.1.8) and (2.1.11), we have uu (u, ) = − pu (u, ) > 0 for any u > 0. Therefore it follows from the Taylor theorem and (2.5.3) that E(u, θ ) − (u, θ ) + (u, ) + (θ − ) θ (u, θ ) = (u, ) − (1, ) − u (1, )(u − 1) 1 (1 − ξ ) uu (1 + ξ(u − 1), )dξ ≥ 0. = (u − 1)2 0

Thus E(u, θ ) ≥ (u, θ ) − (u, ) − (θ − ) θ (u, θ ) 1 = −( − θ )2 (1 − τ ) θθ (u, θ + τ ( − θ ))dτ ≥ ν( − θ )2 i.e.,

E(u, θ ) ≥

0 1

0

(1 − τ ){1 + [θ + τ ( − θ )]r } dτ, θ + τ ( − θ )

ν(θ/ − log(θ/) − 1) + 2ν(θ/ − log(θ/) − 1),

ν[()r −θ r ] r

−

ν[()r+1 −θ r+1 ] , r+1

for r > 0, for r = 0

≥ ν(θ/ − log(θ/) − 1) + Cδ θ r+1 − Cδ which, combined with (2.5.9) and (2.1.23), gives 1 [(θ/ − log(θ/) − 1) + θ r+1 + v 2 ]d x 0 t 1 vx2 (1 + θ q )θθx2 + (x, τ )d x dτ ≤ Cδ , ∀t > 0. + uθ 2 uθ 0 0

(2.5.13)

128


On the other hand, by (2.5.13) and Jensen’s inequality, we have

1 1 θ/d x − log θ/d x − 1 ≤ Cδ , ∀t > 0 0

0

which leads to

1

r1 ≤

θ (x, t)/d x ≤ r2

(2.5.14)

0

where ri = ri (δ) (i = 1, 2) are two positive roots of the equation y − log y − 1 = Cδ . Thus (2.5.11) and (2.5.12) follow from (2.5.13) and (2.5.14). The proof is complete. Lemma 2.5.3. If (u 0 , v0 , θ0 ) ∈ Hδ1, then the following inequalities hold: θ∗ ≤ θ¯ ≤ θ ∗ ,

0< T0 , where θ∗ = min[T

Cδ−1

≤ θ (x, t),

min

u∈[δ2 ,δ3 ],e∈[δ6 ,δ7 ]

(2.5.15) ∀(x, t) ∈ [0, 1] × [0, +∞)

T0 , θˆ (u, e)] and θ ∗ = max[T

(2.5.16)

max

u∈[δ2 ,δ3 ],e∈[δ6 ,δ7 ]

θˆ (u, e)].

Proof. We first show that for (2.1.6), min

u∈[δ2 ,δ3 ],e∈[δ6 ,δ7 ]

θˆ (u, e) ≤ θ¯ ≤

max

u∈[δ2 ,δ3 ],e∈[δ6 ,δ7 ]

θˆ (u, e).

(2.5.17)

In fact, it follows from (2.5.7)–(2.5.8) that δ6 ≤ ¯ := e(u, ¯ θ¯ ) ≤ δ7 , δ2 ≤ u¯ ≤ δ3 which implies that θ¯ = θˆ (u, ¯ e) ¯ and (2.5.17). Thus (2.5.15) follows. In fact, if the assertion in (2.5.16) is not true, then there exists a sequence of solutions (u n , vn , θn ) with the initial data (u n0 , vn0 , θn0 ) ∈ Hδ1 converging weakly in H 1, strongly in C[0, 1] to (u 0 , v0 , θ0 ) ∈ inf θ = 0. Hδ1 such that for the corresponding solution (u, v, θ ) to (u 0 , v0 , θ0 ), Thus there is (x n , tn ) ∈ [0, 1] × [0, +∞) such that as n → +∞, θ (x n , tn ) → 0.

x∈[0,1],t ≥0

(2.5.18)

If the sequence {ttn } has a subsequence, denoted also by tn , converging to +∞, then by Theorem 2.3.1 and (2.5.15), we know that as n → +∞, θ (x n , tn ) → θ¯ ≥ θ∗ > 0 which contradicts (2.5.18). If the sequence {ttn } is bounded, i.e., there exists a constant M > 0, independent of n, such that for any n = 1, 2, 3, . . . , 0 < tn ≤ M, then there exists a point (x ∗ , t ∗ ) ∈ [0, 1] × [0, M] such that (x n , tn ) → (x ∗ , t ∗ ) as n → +∞. On the other hand, by (2.5.18) and the continuity of solutions in Theorem 2.3.1, we conclude that θ (x n , tn ) → θ (x ∗ , t ∗ ) = 0 as n → +∞, which contradicts (2.1.32). Thus the proof is complete. In what follows we shall estimate the point-wise positive lower bound and upper bound for u. To this end, we need Lemma 2.1.3.


129

Lemma 2.5.4. If (u 0 , v0 , θ0 ) ∈ Hδ1, then the following estimate holds: 0 < Cδ−1 ≤ u(x, t) ≤ Cδ , ∀(x, t) ∈ [0, 1] × [0, +∞).

(2.5.19)

Proof. We use a similar idea as in the proof of Lemma 2.1.5. But we should note the dependence of the constants Cδ depending only on the parameters δ1 , . . . , δ7 . Let Mu (t) = max u(x, t), m u (t) = min u(x, t), x∈[0,1]

x∈[0,1]

Mθ (t) = max θ (x, t), m θ (t) = min θ (x, t). x∈[0,1]

x∈[0,1]

By (2.1.20), (2.1.41)–(2.1.43), (2.5.7), (2.5.11), (2.5.16) and Lemma 2.1.3, we have 0 < Cδ−1 ≤ D(x, t) ≤ Cδ , exp(−Cδ (t − s)) ≤ Z (t)Z

−1

0 < Cδ−1 ≤

∀(x, t) ∈ [0, 1] × [0, +∞),

(s) ≤ exp(−Cδ−1 (t 1 1 2

μ0 u¯

0

|θ (x, t) − θ (a(t), t)| ≤ C

x

m

a (t )

≤C ≤ where

V (t) =

1 0

θ

m−1

1 0

(2.5.22)

1 2

1 0

uθ 2m dx 1 + θq

1 2

1/2 C V 1/2 (t)M Mu (t)

m = (q + r + 1)/2

θ

(2.5.21)

θx d y

θx2 (1 + θ q ) dx uθ 2

θx2 (1 + θ q )/uθ 2 d x,

and, by Lemma 2.5.2, t V (s)ds ≤ Cδ , 0

1 0

t ≥ s ≥ 0,

(up + v )(x, s)d x ≤ Cδ .

On the other hand, we have m

− s)),

(2.5.20)

2m

1

q

/(1 + θ )d x ≤ C

0

(1 + θ 1+r )d x ≤ Cδ

and for any t ≥ 0, there is a point a(t) ∈ [0, 1] such that 1 −1 Cδ ≤ θ (a(t), t) = θ (x, t)d x ≤ Cδ . 0

Thus

Cδ − Cδ V (t)M Mu (t) ≤ θ 2m (x, t) ≤ Cδ (1 + V (t)M Mu (t)).

Using Lemmas 2.5.1–2.5.3, (2.5.20)–(2.5.22) and noticing that u(x, s) p(x, s) ≤ p1 (1 + θ r+1 ) ≤ C(1 + θ 2m ) ≤ C(1 + V (s)M Mu (s)),

(2.5.23)

130


we obtain t V (s) exp(−Cδ (t − s))M Mu (s)ds u(x, t) ≤ Cδ 1 + 0 t V (s)M Mu (s)ds , ≤ Cδ 1 + 0

i.e., t Mu (t) ≤ Cδ 1 + V (s)M Mu (s)ds .

(2.5.24)

0

Thus it follows from Gronwall’s inequality and Lemma 2.5.2 that Mu (t) ≤ Cδ .

(2.5.25)

By Lemma 2.1.3 and (2.5.20)–(2.5.21), we have u(x, t) ≥ D(x, t)Z (t) ≥ Cδ−1 exp(−Cδ t).

(2.5.26)

Similarly to the proof of (2.5.16) in Lemma 2.5.3, using (2.5.24), we can prove u(x, t) ≥ Cδ−1 , ∀(x, t) ∈ [0, 1] × [0, +∞).

The proof is complete.

Lemma 2.5.5. For initial data belonging to an arbitrary fixed bounded set B of there is t0 > 0 depending only on boundedness of this bounded set B such that for all t ≥ t0 , x ∈ [0, 1], δ4 ≤ θ (x, t) ≤ δ5 , δ2 /2 ≤ u(x, t) ≤ 2δ3 . (2.5.27) Hδ1

Proof. Suppose that the assertion in Lemma 2.5.5 is not true. Then there is a sequence tn → +∞ such that for all x ∈ [0, 1], sup θ (x, tn ) > δ5

(2.5.28)

where sup is taken for all initial data in a given bounded set B of Hδ1. In the same manner as for the proof of Lemma 2.5.3, there exists (u 0 , v0 , θ0 ) ∈ B such that for the corresponding solution (u, v, θ ), we have θ (x, tn ) ≥ δ5 , ∀x ∀ ∈ [0, 1] which with (2.3.4) yields

θ¯ ≥ δ5 .

(2.5.29)

This contradicts (2.5.5) or (2.5.6) and (2.5.15). Similarly, we can prove other parts of (2.5.27).


131

Remark 2.5.4. It follows from Lemma 2.5.1 and Lemma 2.5.5 that for initial data belonging to a given bounded set B of Hδ1, the orbit will re-enter Hδ1 and stay there after a finite time. In the sequel, we shall prove the existence of an absorbing ball in Hδ1. Since we assume that the initial data (u 0 , v0 , θ0 ) belong to an arbitrarily bounded set B of Hδ1, , C there is a positive constant B such that (u 0 , v0 , θ0 ) H 1 ≤ B. We use Cδ,B or Cδ,B δ,B to denote generic positive constants depending on B and δi , (i = 1, . . . , 7). Lemma 2.5.6. For any initial data (u 0 , v0 , θ0 ) ∈ Hδ1, the unique generalized global solution (u(t), v(t), θ (t)) to problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) satisfies the estimates v2 v2 + C B,δ (|u − u| + C −1 (|u − u| ¯ 2 + |η − η| ¯ 2 ) ≤ E(u, v, η) ≤ ¯ 2 + |η − η| ¯ 2 ). (2.5.30) B,δ 2 2 Proof. By virtue of Lemmas 2.5.1–2.5.5, we can use similar argumentation to the proof of Lemma 2.4.3 to show this lemma. Lemma 2.5.7. There exists a positive constant γ1 = γ1 (C B,δ ) > 0 such that for any fixed γ ∈ (0, γ1 ], the following estimate holds: ¯ 2 + θ (t) − θ¯ 2 + u x (t)2 + ρx (t)2 ) (2.5.31) eγ t (v(t)2 + u(t) − u t + eγ τ (u x 2 + ρx 2 + θθx 2 + vx 2 )(τ )dτ ≤ C B,δ , ∀t > 0. 0

Proof. By virtue of Lemmas 2.5.1–2.5.6, we can use similar argumentation to the proof of Lemma 2.4.3 to show this lemma. Lemma 2.5.8. There exists a positive constant γ1 = γ1 (C B,δ ) ≤ γ1 such that for any fixed γ ∈ (0, γ1 ], the following estimate holds: eγ t (vx (t)2 + θθx (t)2 ) t + eγ τ (vx x 2 + θθx x 2 + vt 2 + θt 2 )(τ )dτ ≤ C B,δ , 0

(2.5.32) ∀t > 0

which with Lemma 2.5.7 implies that for any fixed γ ∈ (0, γ1 ], (u(t) − u, ¯ v(t), θ (t) − θ¯ )2H 1 ≤ C B,δ e−γ t , ∀t > 0. +

(2.5.33)

Proof. By virtue of Lemmas 2.5.1–2.5.7, we can use similar argumentation to the proof of Lemma 2.4.3 to show this lemma.

132


Thus the following results on the existence of an absorbing set in Hδ1 follow from Lemma 2.5.8. Lemma 2.5.9. Let

" R1 = R1 (δ) = 2 δ32 + (θ ∗ )2

and

B1 = {(u, v, θ ) ∈ Hδ1, (u, v, θ ) H 1 ≤ R1 }. +

Then B1 is an absorbing ball in Hδ1, i.e., there exists some t1 = t1 (C B,δ ) = max{−γ1−1 log[2(δ32 + (θ ∗ )2 )/C B,δ ], t0 } ≥ t0 such that when t ≥ t1 , (u(t), v(t), θ (t))2

H+1

≤ R12 .

2.5.2 An Absorbing Set in H 2 In this subsection we are going to prove the existence of an absorbing set in Hδ2. Throughout this subsection we always assume that the initial data belong to an arbitrarily fixed bounded set B in Hδ2, i.e., (u 0 , v0 , θ0 ) H 2 ≤ B with B being a given positive constant. The next two lemmas concern the existence of an absorbing set in Hδ2. Lemma 2.5.10. There exists a positive constant γ2 = γ2 (C B,δ ) ≤ γ1 such that for any fixed γ ∈ (0, γ2 ], the following estimate holds: eγ t (θt (t)2 + vt (t)2 + θ (t) − θ¯ 2H 2 + v(t)2H 2 ) t eγ τ (vxt 2 + θθxt 2 )(τ )dτ ≤ C B,δ , ∀t > 0. +

(2.5.34)

0

Proof. The proof is similar to that of Lemma 2.3.11.

Lemma 2.5.11. There exists a positive constant γ2 = γ2 (C B,δ ) ≤ γ2 such that for any fixed γ ∈ (0, γ2 ], the following estimate holds: u(t) − u ¯ 2H 2 ≤ C B,δ e−γ t

(2.5.35)

which together with Lemma 2.5.10 implies that for any fixed γ ∈ (0, γ2 ] and for all t > 0, u(t)2H 2 + θ (t)2H 2 + v(t)2H 2 ≤ 2(δ32 + (θ ∗ )2 ) + C B,δ e−γ t .

(2.5.36)

Proof. The proof is similar to that of Lemma 2.3.12. then

Now if we define t2 = t2 (C B,δ ) ≥ max(t1 (C B,δ ), −γ γ2−1 log(2(δ32 estimate (2.5.36) implies that for any t ≥ t2 (C B,δ ), u(t)2H 2 + θ (t)2H 2 + v(t)2H 2 ≤ 4(δ32 + (θ ∗ )2 ).

+

" Taking R2 = 2 δ32 + (θ ∗ )2 , we immediately infer the following theorem.

(θ ∗ )2 )/C

B,δ ),


133

Theorem 2.5.2. The ball B2 = {(u, v, θ ) ∈ Hδ2, (u(t), v(t), θ (t))2

H+2

absorbing ball in Hδ2, i.e., when t ≥ t2 , we have

≤ R22 } is an

(u(t), v(t), θ (t))2H 2 ≤ R22 . +

In this sequel we finish the proof of Theorem 2.5.1. Having proved the existence of absorbing balls in Hδ2 and Hδ1, we can use the abstract framework established in [117] by Ghidaglia (see also Theorem 1.6.4) to conclude that Lemma 2.5.12. The set ω(B2 ) =

#!

S(t)B2

(2.5.37)

s≥0t ≥s

where the closures are taken with respect to the weak topology of H+2 , is included in B2 and is nonempty. It is invariant by S(t), i.e., S(t)ω(B2 ) = ω(B2 ),

∀t > 0.

(2.5.38)

Remark 2.5.5. If we take B a bounded set in Hδ2, we can also define ω(B) by (2.5.37) and when B is nonempty, ω(B) is also included in B2 , nonempty and invariant. Since B2 is an absorbing ball, it is clear that ω(B) ⊆ ω(B2 ). This shows that ω(B2 ) is maximal in the sense of inclusion. Theorem 2.5.3. The set satisfies

A2,δ = ω(B2 )

(2.5.39)

A2,δ is bounded and weakly closed in Hδ2,

(2.5.40)

S(t)A2,δ = A2,δ , for every bounded set B in

∀t ≥ 0,

(2.5.41)

lim d w (S(t)B, A2,δ ) = 0.

(2.5.42)

Hδ2, t −→+∞

Moreover, it is the maximal set in the sense of inclusion that satisfies (2.5.40), (2.5.41) and (2.5.42). Proof. The proofs of Lemma 2.5.12 and Theorem 2.5.3 follow from Theorem 1.6.4, using the facts that S(t) is continuous on Hδ1 and Hδ2, respectively, Hδ2 is compactly imbedded in Hδ1, B2 and B1 are absorbing balls in Hδ2 and Hδ1, respectively. Following [117], we also call A2,δ the universal attractor of S(t) in Hδ2. In order to discuss the existence of a universal attractor in Hδ1, we need to prove the following lemma. Lemma 2.5.13. For every t ≥ 0, the mapping S(t) is continuous on bounded sets of Hδ1 for the topology induced by the norm in L 2 × L 2 × L 2 .

134


Proof. We suppose that (u 0 j , v0 j , θ0 j ) ∈ Hδ1, (u 0 j , v0 j , θ0 j ) H 1 ≤ R, (u j , v j , θ j ) = S(t)(u 0 j , v0 j , θ0 j ) ( j = 1, 2), and (u, v, θ ) = (u 1 , v1 , θ1 ) − (u 2 , v2 , θ2 ). Subtracting the corresponding equations (2.1.1)–(2.1.3) satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), we obtain (2.5.43) u t = vx , vt = − pu (u 1 , θ1 )u x − ( pu (u 1 , θ1 ) − pu (u 2 , θ2 ))u 2x − pθ (u 1 , θ1 )θθx vx u 1x vx x v2x u , −( pθ (u 1 , θ1 ) − pθ (u 2 , θ2 ))θ2x + μ0 − − u1 u1u2 x u 21 (2.5.44) eθ (u 1 , θ1 )θt = −(eθ (u 1 , θ1 ) − (eθ (u 2 , θ2 ))θ2t − (eu (u 1 , θ1 ) − eu (u 2 , θ2 ))v2x −eu (u 1 , θ1 )vx − p(u 1 , θ1 )vx − ( p(u 1 , θ1 ) − p(u 2 , θ2 ))v2x +[k(u 1 , θ1 )θθx /u 1 + (k(u 1 , θ1 )/u 1 − k(u 2 , θ2 )/u 2 )θ2x ]x ,

(2.5.45)

t = 0 : u = u 0 , v = v0 , θ = θ0 , x = 0, 1 : v = 0, θx = 0

or

θ = 0.

(2.5.46)

By Lemma 2.3.1, Lemmas 2.5.1–2.5.4, we know that for any t > 0 and j = 1, 2, (u j (t), v j (t), θ j (t))2H 1 t + (u j x 2 + v j 2H 2 + θθ j x 2H 1 + θθ j t (t)2 + v j t 2 )(τ )dτ ≤ C R,δ

(2.5.47)

0

where C R,δ > 0 is a constant depending only on R and δ. Multiplying (2.5.43), (2.5.44) and (2.5.45) by u, v and θ respectively, adding them up and integrating the result over [0, 1], and using (2.1.23), (2.5.47), the Cauchy inequality, the embedding theorem, the mean value theorem and inequalities θ 2L ∞ ≤ C(θ θθx + θ 2 ), v L ∞ ≤ vx , we deduce that for any small > 0, 1 d (u(t)2 + v(t)2 + eθ (u 1 , θ1 )θ (t)2 ) + 2 dt 2

2

1 0 2

[

μ0 vx2 + k(u 1 , θ1 )θθx2 ]d x u1

≤ (vx (t) + θθx (t) ) + C R,δ ()H (t)(u(t) + eθ (u 1 , θ1 )θ (t)2 + v(t)2 ) which, together with Lemmas 2.5.1–2.5.4 and (2.1.23), gives d (u(t)2 + v(t)2 + eθ (u 1 , θ1 )θ (t)2 ) + Cδ−1 (vx (t)2 + θθx (t)2 ) dt (2.5.48) ≤ C B,δ H (t)(u(t)2 + eθ (u 1 , θ1 )θ (t)2 + v(t)2 )

2.6. Universal Attractor in H 4

135

where, by (2.5.47), H (t) = θ1t (t)2 +θ2t (t)2 +v1x x (t)2 +v2x x (t)2 +θ1x x (t)2 + θ2x x (t)2 + 1 satisfies for any t > 0,

t

H (τ )dτ ≤ C R,δ (1 + t).

(2.5.49)

0

Therefore the assertion of this lemma follows from Gronwall’s inequality, (2.5.48)– (2.5.49) and (2.1.19). The proof is complete. Now we can again use Theorem 1.6.4 to obtain the following result on existence of a universal attractor in Hδ1. Theorem 2.5.4. The set A1,δ =

#!

S(t)B1

(2.5.50)

s≥0 t ≥s

where the closures are taken with respect to the weak topology of H+1 , is the (maximal) universal attractor in Hδ1. Remark 2.5.6. Since A2,δ is bounded in H+2 , it is bounded in H+1 and by the invariance property (2.5.41), we have A2,δ ⊆ A1,δ . (2.5.51) On the contrary if we knew that A1,δ is bounded in H+2 then the opposite inclusion would hold.

2.6 Universal Attractor in H 4 In this section we shall establish the existence of a universal attractor in Hδ4 for problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2). We have the following main result. Theorem 2.6.1. Assume that e, p, σ and k are C 5 functions satisfying (2.1.11)–(2.1.13) and (2.1.19)–(2.1.25) on 0 < u < +∞ and 0 ≤ θ < +∞, and q, r satisfy assumptions (2.1.14)–(2.1.18). Then semigroup S(t) defined on H+4 by the solution to problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) maps H+4 into itself. Moreover, for any δi (i = 1, . . . , 7) satisfying (2.5.4)–(2.5.6), it possesses in Hδ4 a universal (maximal) attractor A4,δ . Remark 2.6.1. The set A4 =

!

A4,δ is a global noncompact attractor in

δ1 ,...,δ5 or δ1 ,...,δ7

the metric space H+4 in the sense that it attracts any bounded sets of H+4 with constraints u ≥ u, θ ≥ θ with u, θ being any given positive constants.

136


In what follows, we will establish the existence of an absorbing set in Hδ4. To this end, from now on, we always assume that the initial data belong to a given bounded set B4 in Hδ4, and so there exists a sufficiently large positive constant B4 such that (u 0 , v0 , θ0 ) H 4 ≤ B4 . By virtue of Lemma 2.3.11–2.3.12, Lemmas 2.5.1–2.5.5 and + repeating the same argument as the proof of Lemma 2.4.9, we easily derive the following lemma which yields the existence of an absorbing set in H+4 . Lemma 2.6.1. There exists a positive constant γˆ4 = γˆ4 (Cδ,B4 ) ≤ γˆ3 (Cδ,B4 ) such that for any fixed γ ∈ (0, γˆ4 ], it holds that for any t > 0, ¯ 2 4 + u t (t)2 3 + u t t (t)2 1 ¯ 2H 4 + v2H 4 + θ − θ eγ t u − u H H H 2 2 2 2 + vt (t) H 2 + vt t (t) + θt (t) H 2 + θt t (t) t ¯ 2H 4 + v2H 5 + θ − θ¯ 2H 5 + vt 2H 3 + vt t 2H 1 eγ τ u − u + 0 + θt 2H 3 + θt t 2H 1 + u t 2H 4 + u t t 2H 2 + u t t t 2 (τ )dτ ≤ Cδ,B4 which implies u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ≤ 2(u¯ 2 + θ¯ 2 ) + Cδ,B4 e−γ t ≤ R 2 (δ) + Cδ,B4 e−γ t

(2.6.1)

with R 2 (δ) = 2[δ32 + (θ ∗ )2 ]. It follows from Lemma 2.6.1 that for any bounded set Bi ∈ Hδi (i = 1, 2) and any initial datum (u 0 , v0 , θ0 ) ∈ Bi (i = 1, 2) with (u 0 , v0 , θ0 ) ≤ Bi (i = 1, 2) where Bi (i = 1, 2) are positive constants, there exists some time ti = ti (δ, Bi , t0 ) ≥ tˆ0 (i = 1, 2), t2 ≥ t1 ≥ tˆ0 such that as t ≥ ti , (u(t), v(t), θ (t))2H i ≤ 2R 2 (δ), (i = 1, 2), δ

i.e., the ball Bˆ i = (u, v, θ ) ∈ Hδi , (u, v, θ )2

Hδi

with Ai,δ = ω( Bˆ i ) =

#!

≤ 2R 2 (δ) is an absorbing ball in Hδi

S(t) Bˆ i

(i = 1, 2)

s≥0t ≥s

where the closures are taken with respect to the weak topology of H i (i = 1, 2). Now if we take 2 (δ) R , t4 = t4 (Cδ,B4 ) = max t2 Cδ,B2 , −γˆ4−1 ln Cδ,B4 then we readily derive the following lemma from Lemma 2.6.1.

2.6. Universal Attractor in H 4

137

Lemma 2.6.2. The ball Bˆ 4 = (u, v, θ ) ∈ Hδ4, (u, v, θ )2

Hδ4

set in Hδ4, i.e, when t ≥ t4 (Cδ,B4 ), we have

≤ 2R 2 (δ) is an absorbing

(u(t), v(t), θ (t))2H 4 ≤ 2R 2 (δ). δ

Since we have proved the existence of absorbing balls Bˆ 1 , Bˆ 2 and Bˆ 4 in Hδ1, Hδ2 and Hδ4, we can use Theorem 1.6.4 to conclude that Lemma 2.6.3. The set

ω( Bˆ 4 ) =

#!

S(t) Bˆ 4

(2.6.2)

s≥0t ≥s

where the closures are taken with respect to the weak topology of H+4 , is included in Bˆ 4 and is nonempty. It is invariant by S(t), i.e., S(t)ω( Bˆ 4 ) = ω( Bˆ 4 ),

∀t > 0.

(2.6.3)

Remark 2.6.2. If we take B a bounded set in Hδ4, we can also define ω(B) by (2.6.2) and when B is nonempty, ω(B) is also included in Bˆ 4 , nonempty and invariant. Since Bˆ 4 is an absorbing ball, we know that ω(B) ⊆ ω( Bˆ 4 ). This shows that ω( Bˆ 4 ) is maximal in the sense of inclusion. Moreover, by Theorem 1.6.4, we can also conclude Lemma 2.6.4. The set satisfies

A4,δ = ω( Bˆ 4 )

(2.6.4)

A4,δ is bounded and weakly closed in Hδ4,

(2.6.5)


∀t ≥ 0,

(2.6.6)

lim d w (S(t)B, A4,δ ) = 0.

(2.6.7)

Hδ4, t −→+∞

Moreover, it is the maximal set in the sense of inclusion that satisfies (2.6.5), (2.6.6) and (2.6.7). Remark 2.6.3. Since we have obtained three attractors A4,δ , A2,δ and A1,δ which satisfy that A4,δ is bounded in Hδ4(⊆ Hδ2 ⊆ Hδ1) and A2,δ is bounded in Hδ2(⊆ Hδ1), so A4,δ is bounded in both Hδ1 and Hδ2 and A2,δ is bounded in Hδ1, and by the invariance property (2.6.6), we have A4,δ ⊆ A2,δ ⊆ A1,δ . (2.6.8) On the contrary if we knew that A1,δ is bounded in Hδ2 or/and A2,δ is bounded in Hδ4, then we know that A1,δ = A2,δ or/and A2,δ = A4,δ . Now we have finished the proof of Theorem 2.6.1.

138


2.7 Bibliographic Comments Under the assumptions (2.1.19), (2.1.22)–(2.1.24) and −

p2 (l + (1 − l)θ + θ 1+r ) p1(l + (1 − l)θ + θ 1+r ) ≤ pu (u, θ ) ≤ − , l = 0 or l = 1; 2 u u2 (2.7.1)

0 ≤ p(u, θ ), p(u, θ ) → 0 as u → +∞,

r ∈ [0, 1], q ≥ r + 1,

(2.7.2)

Jiang [165] also established the results on asymptotic behavior of global large solutions for the boundary conditions (2.1.6) and (2.2.2). Under the assumptions (2.1.19), and − p2(1 + θ 1+r )u −2 ≤ pu (u, θ ) ≤ − p1 (1 + θ 1+r )u −2 , | pθ (u, θ )| ≤ p3 (u)u

−1/2

r

(1 + θ ), up(u, θ ) ≤ p4 (1 + θ

1+r

0 < p(u, θ ) ≤ N(u)(1 + θ k0 (1 + θ ) ≤ k(u, θ ) ≤ k2 (1 + θ q ),

),

r+1

(2.7.3) (2.7.4)

),

q

|ku (u, θ )| + |kuu (u, θ )| ≤ k2 (1 + θ q )

(2.7.5) (2.7.6) (2.7.7)

and for some γ < 2, η0 > 0 (constant), η(u, θ ) ≤ ((Mu)γ + η0 )e(u, θ )

(2.7.8)

u

) with Mu := 1 μ(ξ ξ dξ, r ∈ [0, 1], q ≥ 2r + 2, Kawohl [192] succeeded in globally solving the system (2.1.1)–(2.1.3) with the boundary conditions (2.1.6) or

q(0, t) = q(1, t) = 0, σ (0, t) = σ (1, t) = 0.

(2.7.9)

Under assumptions (2.1.19), (2.1.26) (for (2.7.15)–(2.7.16)), (2.1.23), (2.1.25) and | pθ (u, θ )| ≤ p3 (u)u −1 (1 + θ r ), up(u, θ ) ≤ p4 (1 + θ r+1 ), pu (u, T0 ) ≤ 0, for (2.2.2),

(2.7.10) (2.7.11)

| pθ (u, θ )| ≤ N(u)(1 + θ r ), 0 < μ0 ≤ μ(u) ≤ μ1 , for (2.7.15)–(2.7.16),

(2.7.12) (2.7.13)

μ(u) = μ0 ,

for

(2.2.2),

(2.7.14)

with the exponents r ∈ [0, 1], q ≥ r + 1, Jiang [166] also established the global existence with the basically same constitutive relations as those in Kawohl [192] for the boundary conditions (2.1.6) or (2.2.2) or q(0, t) = q(1, t) = 0, σ (0, t) = v(0, t), σ (1, t) = −v(1, t)

(2.7.15)

θ (0, t) = θ (1, t) = T0 , σ (0, t) = v(0, t), σ (1, t) = −v(1, t).

(2.7.16)

or


139

Here the boundary conditions σ (0, t) = v(0, t), σ (1, t) = −v(1, t) indicate that the endpoints of the interval [0,1] are connected to some sort of dashpot. It should be noted that our assumptions (2.1.19) and (2.1.21)–(2.1.22) are weaker than (2.7.3)–(2.7.25) in [192] and [165]. In [165] the asymptotic behavior was obtained for the case of r ∈ [0, 1], q ≥ r + 1. To the author’s knowledge, the case of q = r = 0 and the cases of (2.1.14)–(2.1.18) with the restriction q < r + 1 on their right-hand sides, were not studied before. In this chapter, we establish the results on both global existence and asymptotic behavior for the special case of q = r = 0 (see Chapter 3) and the cases mentioned above. Moreover, in this chapter we also discuss the case which improves the results in [165]. Hoff and Ziane [150, 151] obtained the existence of a compact global attractor for a one-dimensional isentropic compressible viscous flow in a finite interval. Moreover, the (global) attractors were obtained in [97] for the isentropic compressible viscous flow in a bounded domain in R3 . Sell [369] established the existence of uniform attractors for the non-autonomous incompressible Navier-Stokes equations in a bounded domain in R3 . Concerning an ideal gas (2.1.10), Zheng and Qin [452] proved the existence of maximal attractors in H i (i = 1, 2) for problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6). Recently Qin and Muñ˜ oz Rivera [337, 339] established the existence of universal attractors in H i (i = 1, 2) for a one-dimensional heat-conductive real gas of problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) (see also Section 2.5) and for a compressible flow between two horizontal parallel plates in R3 . Hoff and Ziane [150, 151] (see also Sections 2.6) proved the existence of a compact (global) attractor for the onedimensional isentropic compressible viscous flow in a finite interval. Qin [323] proved the existence of a universal attractor in H 4 for the compressible heat-conductive viscous non-isentropic real gas whose equations (2.1.1)–(2.1.3) are more complicated than those in [150, 151]. Moreover, the isentropic compressible viscous flows in a bounded domain in R3 was studied in [97–102, 369]. Since they were based on the fundamental results on global existence of weak solutions by Lions [235] and the uniqueness is not known, it is not possible to exploit the usual solution semigroup approach. As a result, the authors adopted a quite different method, i.e., they replaced the usual solution semigroup approach by a simple time shift, in other words, they worked on the space of “short” trajectories. Therefore, besides some differences above between [151, 152] and Sections 2.5–2.6, there also exist some differences between Sections 2.5–2.6 and references [97102, 369] in the following aspects: non-isentropic via isentropic; one-dimensional heatconductive viscous real gas via three-dimensional compressible flows; solution semigroup approach via simple time shift. In this direction, based on the results on the existence of universal attractors in H i (i = 1, 2) in Sections 2.1–2.5 and the abstract framework in Theorem 1.6.4, we have established the existence of a universal attractor in H 4. For the basic theory of the associated infinite-dimensional dynamics, we still refer to works by Babin [16], Babin and Vishik [17, 18], Ball [22, 23], Bernard and Wang [38], Caraballo, Rubin and Valero [48], Chepyzhov, Gatti, Grasselli, Miranville and Pata [56], Chepyzhov and Vishik [57], Dlotko [84], Eden and Kalantarov [90], Edfendiev, Zelik and Miranville [92], Feireisl [97, 98, 100], Feireisl and Petzeltova [101, 102], Ghidaglia [118],

140


Ghidaglia and Temam [119], Goubet [125], Goubet and Moise [126], Hale [135], Hale and Perissinotto [136], Haraux [138], Hoff and Ziane [150, 151], Ladyzhenskaya [207], Ma, Wang and Zhong [246], Pata and Zelik [307], Robinson [363], Sell [369], Temam [407], Zheng [450] and references therein. For the Navier-Stokes equations, we also consult the works by Babin and Vishik [17], Beirao da Veiga [37], Constantin and Foias [63], Constantin, Foias and Temam [64], Duan, Yang and Zhu [87], Ducomet and Zlotnik [88], Feireisl and Petzeltova [103], Feireisl, Novotny and Petzeltova [104], Foias and Temam [105], Frid and Shelukhin [106], Fujita-Yashima and Benabidallah [110, 111], Fujita-Yashima, Padula and Novotny [112], Galdi [115], Ghidaglia and Temam [119], Hoff [142–146], Hoff and Serre [147], Hoff and Smoller [148], Hoff and Zarnowski [149], Hsiao and Luo [158], Huang, Matsumura and Xin [160], Itaya [161], Jiang [164–167, 169–171], Jiang and Zhang [174– 177], Jiang and Zlotnik [178], Kanel [182], Kawashima [188, 189], Kawashima, Nishibata and Zhu [190], Kawashima and Nishida [191], Kawohl [192], Kazhikhov [193–195], LeFloch and Shelukhin [219], Lions [235], Matsumura [252], Matsumura and Nishida [253–257], Nagasawa [283–287], Novotny and Stra˘s˘ kraba [301, 302], Okada and Kawashima [303], Padula [305], Qin [323, 325, 326], Qin and Hu [329], Qin, Huang and Ma [330], Qin and Jiang [331], Qin and Kong [332], Qin, Ma, Cavalcanti and Andrade [335], Qin, Ma and Huang [336], Qin, Muñ˜ oz Rivera [337, 339], Qin and Song [343], Qin and Wen [344], Qin, Wu and Liu [345], Qin and Zhao [346], Rosa [363], Sell [369], Sell and You [370, 371], Shen and Zheng [375], Temam [406], Valli and Zajaczkowski [412]. For the global well-posedness of solutions to some evolutionary equations, we consult the works by Amann [7, 8], Amosov and Zlotnik [10,11], Andrews [12], Andrews and Ball [13], Antontsev, Kazhikhov and Monakhov [14], Ball [19], Beirao da Veiga [32], Bourgain [41], Brezis [42], Cazenave [49], Chen, Hoff and Trivisa [52], Chen and Hoffmann [54], Dafermos [67–76], Dafermos and Hsiao [77, 78], Dafermos and Nohel [79, 80], Duan, Yang and Zhu [87], Ducomet and Zlotnik [88], Feireisl [99], Feireisl and Petzeltova [103], Feireisl, Novotny and Petzeltova [104], Frid and Shelukhin [106], Hoff [142–146], Hoff and Serre [147], Hoff and Smoller [148], Hoff and Zarnowski [149], Hoffmann and Zheng [152], Hoffmann and Zochowski [153], Huang, Matsumura and Xin [160], Jiang [164–171], Jiang and Muñ˜ oz Rivera [172], Jiang and Racke [173], Kato [185–187], Kawashima [188, 189], Kawashima and Nishida [191], Kawohl [192], Kazhikhov [193–195], Kazhikhov and Shelukhin [196], Kim [197, 198], Krejci and Sprekels [204], Lagnese [209], Lakshmikanthan [210], Lax [215], Lebeau and Zuazua [216], LeFloch and Shelukhin [219], Li and Chen [227–229], J.L. Lions [233, 244], P.L. Lions [235], Liu and Zeng [237], Liu and Zheng [238–240], Luo [245], Matsumura [250– 253], Matsumura and Nishida [253–257], Messaoudi [261], Muñ˜ oz Rivera [274, 275], Munoz ˜ Rivera and Andrade [276], Muñoz ˜ Rivera and Barreto [277], Muñoz ˜ Rivera and Oliveira [278], Muñ˜ oz Rivera and Qin [279, 280], Muñoz ˜ Rivera and Racke [281, 282], ´ and Sprekels [293], Niezgoddka, ´ Nagasawa [283-287], Nakao [288–292], Niezgoddka Zheng and Sprekels [294], Nikolaev [295], Novotny and Stra˘s˘ kraba [301, 302], Okada and Kawashima [303], Oleinik [304], Padula [305], Pego [310], Qin [314–322, 324–326], Qin, Deng and Su [327], Qin and Hu [329], Qin, Huang and Ma [330], Qin and Jiang


141

[331], Qin and Kong [332], Qin, Ma, Cavalcanti and Andrade [335], Qin, Ma and Huang [336], Qin and Muñ˜ oz Rivera [338–340], Qin and Wen [344], Qin, Wu and Liu [345], Qin and Zhao [346], Quitanilla and Racke [347], Racke [348–351], Racke and Shibata [352], Racke, Shibata and Zheng [353], Racke and Wang [354], Racke and Zheng [355, 356], Rauch, Zhang and Zuazua [357], Reissig and Wang [360], Renardy, Hrusa and Nohel [361], Shen and Zheng [373, 374], Shen, Zheng and Zhu [376], Shibata [377], Slemrod [378], Sogge [385], Sprekels [388, 389], Sprekels and Zheng [390], Sprekels, Zheng and Zhu [392], Stra˘s˘kraba [399], Stra˘skraba and Zlotnik [400, 401], Tani [404], Valli and Zajaczkowski [412], Vong, Yang and Zhu [417], Wang [419, 420], Watson [423, 424], Weissler [425], Yang [433], Yang and Zhao [435], Yamada [432], Zhang and Fang [440, 441], Zhang and Zuazua [444], Zheng [446–450], Zheng and Shen [453, 454], Zuazua [464, 465], and the references therein.

Chapter 3

A One-dimensional Polytropic Viscous and Heat-conductive Gas In this chapter we shall investigate the global existence and asymptotic behavior in time of solutions to initial boundary value problems and the Cauchy problem (initial value problem) of compressible Navier-Stokes equations of a polytropic viscous and heatconductive gas. The results of this chapter come from Qin [315,316,345].

3.1 Initial Boundary Value Problems In this section we shall discuss some initial boundary value problems of compressible Navier-Stokes equations of a polytropic viscous and heat-conductive gas.

3.1.1 Global Existence and Asymptotic Behavior of Solutions In Chapter 2, we discussed problem (2.1.1)–(2.1.3), (2.1.5)–(2.1.6) or (2.1.1)–(2.1.3), (2.2.1), (2.2.2) with more general constitutive relations and with the exponents q, r satisfying (2.1.14)–(2.1.18). However, assumptions (2.1.14)–(2.1.18) cannot cover the important case of q = r = 0, to which the compressible Navier-Stokes equations of a polytropic viscous and heat-conductive gas belong, see, e.g., (2.1.10). In this subsection, we discuss the special case of q = r = 0. We make the following assumptions: (i) For any 0 < u < +∞ and 0 ≤ θ < +∞, p(u, θ ), e(u, θ ), Q(u, θ, θ x ) and k(u, θ ) satisfy e(u, θ ) = C V θ + F2 (u), F2 (u) ≥ 0, σ (u, v, vx ) = − p(u, θ ) + μ0 vx /u ≡ σ1 (u, θ ) + μ0 vx /u,

(3.1.1) (3.1.2)

Q(u, θ, θx ) = −K 0 θx /u,

(3.1.3)

144

Chapter 3. A One-dimensional Polytropic Viscous and Heat-conductive Gas

where σ1 (u, θ ) ≡ − p(u, θ ) = f 1 (u)θ + f 2 (u), K (u, θ ) ≡ K 0 > 0 and C V , μ0 , K 0 are positive constants. (ii) f1 (u), f 2 (u) ∈ C 1 (0, +∞) and we assume that there exist constants c1 > 0, d1 > 0, c2 ≥ 0, d2 ≥ 0 such that for any u > 0, there holds that −ci ≤ u f i (u) ≤ −di (i = 1, 2),

f 1 (u) > 0,

f 2 (u) ≥ 0

(3.1.4)

with Fi (u) = f i (u) (i = 1, 2). (iii) We assume that there exists a constant α3 ∈ (0, 1) such that the initial data (u 0 , v0 , θ0 ) ∈ H 1+α3 ×H 2+α3 ×H 2+α3 and u 0 (x) > 0, θ0 (x) > 0 for any x ∈ [0, 1], and the compatibility conditions hold. In this chapter we use C to denote the generic positive constants independent of time t. Put · L 2 = · . Theorem 3.1.1. Under assumptions (3.1.1)–(3.1.4), the conclusions of Theorem 2.1.1 are valid. Remark 3.1.1. Our assumptions include a model of ideal gas whose constitutive relations take the form of (2.1.10), i.e., f1 (u) = − Ru , f 2 (u) = F2 (u) = 0. Moreover, the results in Theorem 3.1.1 can cover those in Kazhikhov and Shelukhin [196]. Remark 3.1.2. The assumptions (i)–(iii) correspond to the case of q = r = 0. Under the assumptions (i)–(iii), problem (2.1.1)–(2.1.3) and (2.1.5)–(2.1.6) reads u t − vx = 0, v x vt − μ0 − σ1x = 0, u x μ0 vx2 θx C V θt − θ f 1 (u)vx − − K0 = 0, u u x

(3.1.5) (3.1.6) (3.1.7)

x = 0, 1 : v = 0, θx = 0,

(3.1.8)

t = 0 : u = u 0 (x), v = v0 (x), θ = θ0 (x).

(3.1.9)

Theorem 3.1.2. Under assumptions (i)–(iii), the results in Theorem 2.1.1 are also valid 1 1 with θ¯ = C V−1 (E 0 − 0 F2 (u 0 )d x), E 0 ≡ 0 (C V θ0 + F2 (u 0 (x)) + v02 (x)/2)d x. We derive the uniform a priori estimates in the following. Lemma 3.1.1. The following estimates are valid for any t > 0: θ (x, t) > 0, ∀(x, t) ∈ [0, 1] × [0, +∞), 1 1 u(x, t)d x = u 0 (x)d x ≡ u 0 , ∀t > 0, 0 0 1 1 1 C V θ + v 2 (x, t)d x + F2 (u(x, t))d x = E 0 , 2 0 0

(3.1.10) (3.1.11) (3.1.12)

3.1. Initial Boundary Value Problems

145

1

C V (θ − log θ − 1) + F1 (u) + F2 (u) + v 2 /2 (x, t)d x 0 t 1

μ0 vx2 K 0 θx2 + d x ds + uθ uθ 2 0 0 t 1 = C V (θ0 − log θ0 − 1) + F1 (u 0 ) + F2 (u 0 ) + v02 /2 d x ≡ E 1 , 0

0

1

0 < C1α ≤ 0

θ α (x, t)d x ≤ C2α , ∀α ∈ (0, 1].

(3.1.13) (3.1.14)

Proof. See, e.g., Lemmas 2.1.1–2.1.2. Lemma 3.1.2. There holds that for any (x, t) ∈ [0, 1] × [0, +∞), 0 < C −1 ≤ u(x, t) ≤ C.

(3.1.15)

Proof. See,e.g., Lemmas 2.1.3–2.1.5. Lemma 3.1.3. There holds that for any t > 0, t 2 4 θ (t) + v(t) L 4 + (vvx 2 + θθx 2 + vx 2 + v2L ∞ )(s)ds ≤ C.

(3.1.16)

0

Proof. By (3.1.8) and Lemmas 3.1.1–3.1.2, we get 2 t t 1 t 2 v(τ ) L ∞ dτ ≤ |vx |d x dτ ≤ 0

0

0

0

Using (3.1.5)–(3.1.6), (3.1.7) can be rewritten as 2

(C V θ + v /2)t + f 2 vx − (σ v)x −

1 0

θdx

K 0 θx u

1 0

vx2 d x dτ ≤ C. θ (3.1.17)

= 0.

(3.1.18)

x

Multiplying (3.1.18) by C V θ + v 2 /2 and integrating with respect to x yields 1 1 d C V K 0 θx2 2 2 Cv θ + v /2 + + (C V θ + v 2 /2) f 2 vx + (C V θx + vvx ) 2 dt u 0

μ0 vx K 0 vvx θx × v f 1 θ + f2 + + d x = 0. (3.1.19) u u Thus it follows from (3.1.5) and Lemmas 3.1.1–3.1.2 that 1 1 1 d d Cv θ + v 2 /22 + C −1 θθx 2 ≤ C V (θ − θ˜ ) f 2 vx d x + C V θ˜ F2 (u)d x 2 dt dt 0 0 1 2 +C v |vx | + (|θθx | + |vvx |)|v|(|θ | + 1)|vx | + |vvx θx | d x (3.1.20) 0

where θ˜ =

1 0

θ d x.

146


We easily deduce that for small ε > 0, 1 v 2 |vx |d x ≤ C(v2 + vvx 2 ), 1 0

1 0

1

0

(3.1.21)

0

|vθ θ x |d x ≤ εθθx 2 + Cv2L ∞ θ 2 ,

(3.1.22)

(|v||θθ x | + |v||vx ||θθx |)d x ≤ εθθx 2 + C(v2 + vvx 2 ),

(3.1.23)

(v 2 |vx ||θ | + v 2 |vx | + v 2 vx2 )d x ≤ C(vvx 2 + v2L ∞ θ 2 + v2 ), (3.1.24)

1

|C V

(θ − θ˜ ) f 2 vx d x| ≤ C

0

1

|θ − θ˜ |θ 1/2

0

|vx | dx θ 1/2

1

≤ Cθ − θ˜ L ∞ θ L 1 ≤ εθθx 2 + C

1 0

vx2 θ

0

vx2 dx θ

1/2

dx

(3.1.25)

and using (3.1.12), 1 1 d 2 E0 − F2 d x − v F2 (u)d x 2 dt 0 0 0

2 1 1 1 1 d 1 d F2 (u)d x − F2 (u)d x − v2 f 2 vx d x = E0 dt 0 2 dt 2 0 0

C V θ˜

1

d F2 (u)d x = dt

1

(3.1.26)

where 1 | v2 2

1

1

f 2 vx d x| ≤ Cv2

0

0

|vx |d x ≤ Cv2 θ L 1

≤ Cv4 + C

1 0

vx2 θ

d x.

1 0

vx2 dx θ

1/2

(3.1.27)

Inserting (3.1.21)–(3.1.27) into (3.1.20), picking ε small enough, integrating with respect to t and using Lemmas 3.1.1–3.1.2, we arrive at t t 2 4 2 θ (t) +v(t) L 4 + θθx (τ ) dτ ≤ C 1+ (vvx 2 +v2L ∞ θ 2 )dτ . (3.1.28) 0

0

On the other hand, multiplying (3.1.6) by v 3 and using Lemmas 3.1.1–3.1.2, we derive t t 4 2 v(t) L 4 + vvx dτ ≤ C[1 + v2L ∞ θ 2 dτ ]. (3.1.29) 0

0


147

Multiplying (3.1.29) by a large constant and adding it to (3.1.28) yield t t 2 4 2 2 (θθ x + vvx )(τ )dτ ≤ C[1 + v2L ∞ θ 2 dτ ] θ (t) + v(t) L 4 + 0

0

which along with (3.1.17) and the Gronwall inequality implies t θ (t)2 + v(t)4L 4 + (θθ x 2 + vvx 2 + v2L ∞ )(τ )dτ ≤ C.

(3.1.30)

0

Multiplying (3.1.7) by θ˜ −1 and noting the bound of θ˜ in (3.1.14) of Lemma 3.1.1, we obtain t t 1 C θ − K (θθ /u) − θ f (u)v v t 0 x x 1 x vx (τ )2 dτ ≤ C d x dτ . (3.1.31) ˜ θ 0 0 0 Noting that

t 1C θ V t d x dτ = C V | log θ˜ (t) − log θ˜ (0)| ≤ C, 0 0 θ˜ t 1 K (θθ /u) 0 x x d x dτ = 0, ˜ 0 0 θ

we infer from (3.1.30)–(3.1.31) and Lemmas 3.1.1–3.1.2 that t t 1 f t 1 1 vx (τ )2 dτ ≤ C 1 + f1 vx d x dτ (θ − θ˜ )vx d x dτ + ˜ 0 0 0 θ 0 0 t 1 t 1 t d 2 2 ≤ vx (τ ) dτ + C 1 + θθx (τ ) dτ + F2 (u)d x dτ 0 dt 0 2 0 0 t 1 ≤ vx (τ )2 dτ + C 2 0 which together with (3.1.30) gives (3.1.16). Lemma 3.1.4. The following estimates hold for any t > 0: t 1 2 vx2 θx + α (x, s)d x ds ≤ C, ∀t > 0, α ∈ (0, 1], θ θ 1+α 0 0 t θ α (s) − θ˜α 2L ∞ ds ≤ C, ∀t > 0, α ∈ [0, 1), 0 t t 2δ θ (s) − θ˜ L ∞1 ds ≤ C θθx (s)2 ds ≤ C, ∀t > 0, δ1 ∈ [1, 2] 0

where θα =

0

1 0

θ α d x.

(3.1.32) (3.1.33) (3.1.34)

148


Proof. If α = 1, then (3.1.32) is the direct result of Lemmas 3.1.1–3.1.2. If 0 < α < 1, 1 then multiplying (3.1.7) by θ −α ( 0 θ 1−α d x)−1 and integrating the resultant over Q t ≡ [0, 1] × [0, t], yields t 0

1 0

−1 θ

1−α

≤C+C

1

dx 0

t 0

1

μ0 vx2 αK 0 θx2 + uθ 1+α uθ α

t

≤C+C

θ

1−α

0

t 0

d x ds

1−α )|d x ds | f 1 (u)vx (θ 1−α − θ

0

≤C+C

1 0

θ

1−α 2 ∞ ds − θ L

−α

2 θx d x

)ds

1/2 t 0

1/2 t 0

0

1

1/2

2 |vx |d x

ds

0 1

vx2 d x ds θα

1/2 .

(3.1.35)

(1) When 1/2 ≤ α < 1, it follows from (3.1.35), the Young inequality and Lemmas 3.1.1–3.1.2 that vx2 + α d x ds θ θ 1+α 0 0 t 1 2 t 1 2 1 1 vx θx 2−2α ≤C+ d x ds + C dx θ d x ds 2 2 0 0 θα 0 0 θ 0 1 t 1 vx2 ≤C+ d x ds 2 0 0 θα

t

1

θx2

which gives (3.1.32). (2) When 0 < α < 1/2, we shall use the induction argument. Assume that when α ∈ 1 1 ](n = 2, 3, . . . ), (3.1.32) is valid. Now we suppose that α ∈ [ 2n+1 , 21n ], then by [ 21n , 2n−1 (3.1.35) and induction assumption, we have vx2 + α d x ds θ θ 1+α 0 0 t 1 2 t 1 1 1 vx θx2 1+ 21n −2α ≤C+ d x ds + C dx θ d x ds 1 2 0 0 θα 0 0 θ 1+ 2n 0 1 t 1 vx2 ≤C+ d x ds 2 0 0 θα

t

1

θx2

which yields (3.1.32).


149

1 Since 0 (θ α − θ˜α )d x = 0, for any t > 0 there is a point b(t) ∈ [0, 1] such that θ α (b(t), t) = θ˜α (t) which implies

t 0

2 t x α α ˜ (θ − θ ) y d y ds 0 b(t ) t 1 2 1 θx ≤C d x θ α d x ds ≤ C 2−α θ 0 0 0

θ − θ˜α 2L ∞ ds ≤ α

if 0 < α < 1. Similarly, θ (t) − θ˜ δL1∞ ≤ C

1

˜ δ1 −1 θx |d x |(θ − θ)

0

≤C

1

1/2 ˜ (θ − θ)

2(δ1 −1)

dx

θθx ≤ Cθθx (t)

0

which leads to (3.1.34). The proof is complete.

Lemma 3.1.5. There holds that for any t > 0,

t

u x (t)2 +

(u x 2 + θ 1/2 u x 2 )(s)ds

0

≤ C(1 + sup θ (s) L ∞ )α , ∀t > 0, α ∈ (0, 1]. 0≤s≤t

Proof. Similar to (2.1.74), we have t 1 u2 ux 1 v − μ0 2 + μ0 ( f 1 (u)θ + f 2 (u)) x d x ds 2 u u 0 0 t 1 u x d x ds ≤C+ ( f 1 (u)θ + f 2 (u))u x v + f1 (u)θθ x v − μ0 u 0 0

t 1 μ0 u 2x 2 ≤C+ + Cuv d x ds ( f1 (u)θ + f 2 (u)) 2u 0 0 t 1 +C |θθx |(|v| + |u x |)d x ds 0

0

0

0

u2 μ0 t 1 ( f 1 (u)θ + f 2 (u)) x d x ds ≤C+ 2 0 0 u t 1 +C [(1 + θ )v 2 + |θθx |(|v| + |u x |)]d x ds

(3.1.36)

150


i.e., u x (t)2 +

t 0

1 0

θ u 2x d x ds ≤ C + C

t 0

1/2 θx2 +C d x ds 0 0 0 0 θ t 1 2 1 t 1 2 θx ≤C+ θ u x d x ds + C sup θ αL ∞ d x ds 1+α 2 0 0 θ 0 0 0≤s≤t 1 t 1 2 θ u x d x ds. ≤ C(1 + sup θ (s) L ∞ )α + 2 0 0 0≤s≤t

t

1

θ u 2x d x ds

1/2 t

[v2L ∞ + θθx 2 ]ds

1

(3.1.37)

On the other hand, we easily know 1 ≤ Cθ + C V2 (t) which results in

t 0

u x (s)2 ds ≤ C

t 0

1 0

θ u 2x + C

t

u x 2 V2 (s)ds

0

≤ C(1 + sup θ (s) L ∞ )α 0≤s≤t

which, together with (3.1.37), completes the proof.

Remark 3.1.3. It is easy to see that β = 1 if q = r = 0 in (2.1.71) in Lemma 2.1.6. So this lemma has reduced the order of θ and later we shall see β = α = 1 does not work for the case discussed in this section (q = r = 0). This is why we have to establish the estimate (3.1.32). Lemma 3.1.6. There holds that for any t > 0, t vx (t)2 + (vt 2 + vx x 2 )(s)ds ≤ C(1 + sup θ (s) L ∞ )1+α , ∀α ∈ (0, 1]. 0

0≤s≤t

(3.1.38) Proof. Multiplying (3.1.6) by vt and integrating the resultant over Q t , and using the Nirenberg inequality and Lemmas 3.1.1–3.1.5, we get t 2 vx (t) + vt (s)2 ds 0 t ≤C +C [ px 2 + vx 3L 3 ](s)ds 0 t (θ u x 2 + θθx 2 + u x 2 + vx 5/2 vx x 1/2 )(s)ds ≤C +C 0


151

≤ C(1 + sup θ (s) L ∞ )α+1 0≤s≤t t

+C

2

4/3

vx x (s) ds + C sup vx (s)

0

0≤s≤t

vx (s)2 ds

0

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )α+1 +

t

1 sup vx (s)2 + C 2 0≤s≤t

t

vx x (s)2 ds.

(3.1.39)

0

Multiplying (3.1.6) by vx x , then integrating the resultant over Q t , we deduce t vx (t)2 + vx x (s)2 ds 0

≤C +C ≤C+

1 4

t

0

0

0 t

t

+C 0

≤C+

1 4

1

t

(|vx u x vx x | + | p x vx x |)(s)d x ds

vx x (s)2 ds (vx 2L ∞ u x 2 + θ L ∞ θ 1/2 u x 2 + θθx 2 + u x 2 )(s)ds vx x (s)2 ds

0 t

+C 0

vx (s)vx x (s)u x (s)2 ds + C sup θ (s)α+1 L∞ 0≤s≤t

≤ C(1 + sup θ (s) L ∞ ) 1 + 2

0≤s≤t t

α+1

0

2

4

vx x (s) ds + C sup u x (s) 0≤s≤t

≤ C(1 + sup θ (s) L ∞ )α+1 + 0≤s≤t

1 2

which, together with (3.1.39), gives (3.1.38).

t

0

t

vx (s)2 ds

vx x (s)2 ds

0

Lemma 3.1.7. The following estimate holds for any t > 0: t θθx x (s)2 ds ≤ C(1 + sup θ (s) L ∞ )2(1+α), ∀0 < α ≤ 1. (3.1.40) θθx (t)2 + 0

0≤s≤t

Proof. Multiply (3.1.7) by θ x x , then integrate the resultant over Q t , use Lemmas 3.1.1– 3.1.6 to get t t 1 2 2 θθx (t) + θθx x (s) ds ≤ C + C (θθx2 u 2x + θ 2 vx2 + vx4 )(x, s)d x ds 0 0 0 t [θθx 2L ∞ u x 2 + θ 2L ∞ vx 2 + vx 4L 4 ](s)ds ≤C +C 0

152


t

[(θθ x θθx x + θθx 2 )u x 2 + vx 3 vx x ](s)ds 0 t vx (s)2 ds + C sup θ (s)2L ∞

≤C +C

0

0≤s≤t

1 ≤ C(1 + sup θ (s) L ∞ ) + 2 0≤s≤t

t

2

θθx x (s) ds + C

0

t

4

+ C sup vx (s)

t 0

θθx (s)2 ds

vx (s)2 ds

0

0≤s≤t

vx x (s)2 ds

0

+ C sup (u x (s)2 + u x (s)4 ) 0≤s≤t

t

2

≤ C(1 + sup θ (s) L ∞ )2(1+α) + 0≤s≤t

1 2

t

θθx x (s)2 ds

0

which implies (3.1.40). Lemma 3.1.8. The following estimates hold for any t > 0: θ (t) L ∞ ≤ C,

(3.1.41)

θθx (t)2 + vx (t)2 + u x (t)2 +

t 0 2

(θθx 2 + vvx 2 + vx 2

+u x 2 + θt 2 + vt 2 + vx x + θθx x 2 )(s)ds ≤ C.

(3.1.42)

Proof. By virtue of the Nirenberg inequality and Lemmas 3.1.3–3.1.7, we get θ (t) L ∞ ≤ Cθ (t)1/2 θθx (t)1/2 + Cθ (t) ≤ C + Cθθx (t)1/2 , (3.1.43) sup θθx (s)2 ≤ C(1 + sup θ (s) L ∞ )2(1+α) 0≤s≤t

0≤s≤t

≤ C(1 + sup θθx (s)1+α ) ≤ C + 0≤s≤t

1 sup θθx (s)2 2 0≤s≤t

which, combined with (3.1.43) and α ∈ (0, 1), gives (3.1.41). On the other hand, multiply (3.1.7) by θt and integrate the resultant over Q t to get 2

t

t

1

θt (s) ds ≤ C + C (θθx2 |vx | + θ |vx θt | + vx2 |θt |)(x, s)d x ds 0 0 0 t 1 t 2 θt (s) ds + C (vx 2 + θθx 4L 4 + θ 2L ∞ vx 2 + vx 4L 4 )(s)ds ≤C+ 2 0 0 1 t ≤C+ θt (s)2 ds 2 0 t (θθx 3 θθx x + θθ x 4 + θ 2L ∞ vx 2 + vx 3 vx x )(s)ds +C

θθx (t) +

0

2


≤C+

1 2

t

1 2

t

θt (s)2 ds + C sup (θθx (s)2 + θθx (s)4 )

0

0≤s≤t

+ C sup vx (s)4 ≤C+

153

0≤s≤t t

t 0

vx (s)2 ds + C

0

θθx (s)2 ds

0 t

(θθx x 2 + vx 2 + vx x 2 )(s)ds

θt (s)2 ds

0

which, along with (3.1.41) and Lemmas 3.1.1–3.1.7, implies (3.1.42). The proof of asymptotic behavior is similar to that of Theorem 2.1.1.

3.1.2 Exponential Stability In this section we establish the exponential stability of solutions to the problem of the compressible Navier-Stokes equations of one-dimensional viscous polytropic ideal gas, which takes the form (3.1.5)–(3.1.9) with f1 (u) = −R/u, F2 (u) = f 2 (u) = 0, i.e., vt = σx ,

u t = vx , σ := μvx /u − Rθ/u,

(3.1.44) (3.1.45)

Cv θt = [K 0 θx /u]x + σ vx ,

(3.1.46)

x = 0, 1 : v = 0, θx = 0, t = 0 : u = u 0 , v = v0 , θ = θ0 .

(3.1.47) (3.1.48)

Now let us consider the spaces H+1 = (u, v, θ ) ∈ H 1[0, 1] × H 1[0, 1] × H 1[0, 1] : u(x) > 0, θ (x) > 0, x ∈ [0, 1], v|x=0 = v|x=1 = 0 and

H+i = (u, v, θ ) ∈ H i [0, 1] × H i [0, 1] × H i [0, 1] : u(x) > 0, θ (x) > 0, x ∈ [0, 1], v|x=0 = v|x=1 = θx |x=0 = θx |x=1 = 0 , i = 2, 4

which become three metric spaces when equipped with the metrics induced from the usual norms. In the above, H 1, H 2, H 4 are the usual Sobolev spaces. Repeating the same reasoning as those in the proofs of Theorems 2.3.1–2.3.2 and 2.4.1, we conclude Theorem 3.1.3. For any (u 0 , v0 , θ0 ) ∈ H+i (i = 1, 2, 4), there exists a unique (generalized) d global solution (u(t), v(t), θ (t)) ∈ H+i which defines a C0 -semigroup S(t) on H+i (i = 1, 2, 4). Moreover, there exists a constant γi = γi (Ci ) > 0 (i = 1, 2, 4) such that for any fixed γ ∈ (0, γi ] and for any t > 0, the following inequality holds: (u(t), v(t), θ (t)) − (u, ¯ 0, θ¯ )2H i = S(t)(u 0 , v0 , θ0 ) − (u, ¯ 0, θ¯ )2H i +

≤ Ci e

−γ t

+

(3.1.49)

154


which means that semigroup S(t) decays exponentially on H+i . Here

1

u¯ = 0

u 0 (x)d x, θ¯ = C V−1

1 0

(C V θ0 + v02 /2)d x.

3.1.3 Universal Attractors In this subsection, we shall establish the existence of universal attractors in H+i for problem (3.1.44)–(3.1.48). Without loss of generality we always assume that C V = R = μ = K 0 = 1. β Let βi (i = 1, . . . , 5) be any given constants such that β1 ∈ R, β2 > 0, β4 ≥ eβ21 > β3 > 0, 0 < β5 < β2 be arbitrarily given constants, and let 1 1 Hβi := (u, v, θ ) ∈ H+i : (log(θ ) + log(u))d x ≥ β1 , β5 ≤ (θ + v 2 /2)d x ≤ β2 , 0

1

β3 ≤

0

ud x ≤ β4 , β5 /2 ≤ θ ≤ 2β2 , β3 /2 ≤ u ≤ 2β4 , i = 1, 2, 4.

0

Clearly,

Hβi

is a sequence of closed subspaces of H+i (i = 1, 2, 4).

Our main result reads as follows. Theorem 3.1.4. The nonlinear semigroup S(t) defined by the solution to problem (3.1.44)–(3.1.48) maps H+i (i = 1, 2, 4) into itself. Moreover, for any βi (i = 1, . . . , 5) with β1 < 0, β2 > 0, β4 ≥ attractor Ai,β (i = 1, 2, 4).

eβ1 β2

> β3 > 0, 0 < β5 < β2 , it possesses in Hβi a maximal

Proof. Based on the results in Theorem 3.1.3, repeating the same reasoning as those in the proofs of Theorems 2.5.1 and 2.6.1, we can easily prove this theorem. Remark 3.1.4. The set Ai = Ai,β (i = 1, 2, 4) is a global noncompact β1 ,β2 ,β3 ,β4 ,β5

H+i

attractor in the metric space in the sense that it attracts any bounded sets of H+i with constraints u ≥ η1 , θ ≥ η2 with η1 , η2 being any given positive constants.

3.2 The Cauchy Problem 3.2.1 Global Existence in H 2 (R) In this subsection we study the regularity, continuous dependence on initial data and largetime behavior of H i -solutions (i = 1, 2, 4) solutions to the Cauchy problem (3.1.44)– (3.1.46) for the compressible Navier-Stokes equations of a one-dimensional viscous polytropic ideal gas in Lagrangian coordinates with the initial conditions (u(x, 0), v(x, 0), θ (x, 0)) = (u 0 (x), v0 (x), θ0 (x)), ∀x ∀ ∈ R.

(3.2.1)

3.2. The Cauchy Problem

155

The equations (3.1.44)–(3.1.46) describe the motion of a one-dimensional viscous polytropic ideal gas, where u, v, θ are the specific volume, velocity, and absolute temperature, respectively; σ is the stress, μ, C V and K 0 are positive constants. We introduce the following definition of H i -solutions (i = 2, 4). Definition 3.2.1. For a fixed constant T > 0 and some positive constants u¯ and θ¯ , we call (u(t), v(t), θ (t)) an H 2-generalized solution to the Cauchy problem (3.1.44)–(3.1.46), (3.2.1) if it is in the following set of functions: u − u, ¯ v, θ − θ¯ ∈ L ∞ ([0, T ], H 2(R)), ∞

1

2

(3.2.2)

2

u t ∈ L ((0, T ), H (R)) ∩ L ((0, T ), H (R)),

(3.2.3)

vt , θt ∈ L ∞ ((0, T ), L 2 (R)) ∩ L 2 ((0, T ), H 1(R)),

(3.2.4)

2

1

2

2

u x ∈ L ((0, T ), H (R)), vx , θx ∈ L ((0, T ), H (R)).

(3.2.5)

Furthermore, in addition to (3.2.2)–(3.2.5), if u − u, ¯ v, θ − θ¯ ∈ L ∞ ([0, T ], H 4(R)), ∞

3

2

(3.2.6)

2

u t ∈ L ((0, T ), H (R)) ∩ L ((0, T ), H (R)), ∞

2

2

3

vt , θt ∈ L ((0, T ), H (R)) ∩ L ((0, T ), H (R)), ∞

1

2

2

(3.2.7) (3.2.8)

u t t ∈ L ((0, T ), H (R)) ∩ L ((0, T ), H (R)),

(3.2.9)

vt t , θt t ∈ L ∞ ((0, T ), L 2 (R)) ∩ L 2 ((0, T ), H 1 (R)),

(3.2.10)

2

3

u x ∈ L ((0, T ), H (R)), 2

4

(3.2.11)

2

2

vx , θx ∈ L ((0, T ), H (R)), u t t t ∈ L ((0, T ), L (R)),

(3.2.12)

then we call (u(t), v(t), θ (t)) an H 4-solution to the Cauchy problem (3.1.44)–(3.1.46), (3.2.1). ¯ θ¯ , u 0 − Kazhikhov and Shelukhin [196] proved that if for some positive constants u, 1 ¯ u, ¯ v0 , θ0 − θ ∈ H (R) and u 0 (x), θ0 (x) > 0 on R, then there exists a unique global (large) solution (u(t), v(t), θ (t)) with positive u(x, t) and θ (x, t) to the Cauchy problem (3.1.44)–(3.1.46), (3.2.1) on R × [0, +∞) such that for any T > 0, u − u, ¯ v, θ − θ¯ ∈ L ∞ ((0, T ), H 1(R)), u t ∈ L ∞ ((0, T ), L 2 (R)), 2

2

vt , u x , θt , u xt , vx x , θx x ∈ L ((0, T ), L (R)).

(3.2.13) (3.2.14)

Now we call (u(t), v(t), θ (t)) verifying (3.2.13)–(3.2.14) an H 1-generalized solution to the Cauchy problem (3.1.44)–(3.1.46), (3.2.1). It is noteworthy that there is no result on asymptotic behavior given in [194]. The aim of this section is to prove the global existence and continuous dependence on initial data of H i (R) (i = 1, 2, 4) (global) solutions for large initial data and then further to show the large-time behavior of this H i (R) (i = 2, 4) solution for small initial data.

156


We put · = · L 2 (R) and denote by Ci (i = 1, 2, 3, 4) the universal constant depending only on min u 0 (x), min θ0 (x), the H i (R) (i = 1, 2, 3, 4) norm of (u 0 − x∈R

x∈R

¯ θ¯ ) and e0 or E 0 , E 1 (see, e.g., Theorem u, ¯ v0 , θ0 − θ¯ ) (for some positive constants u, 3.2.3), but independent of any length of time T > 0. We are now in a position to state our main theorems. Theorem 3.2.1. Assume that for some positive constants u, ¯ θ¯ , u 0 − u, ¯ v0 , θ0 − θ¯ ∈ H 2(R) and u 0 (x) > 0, θ0 (x) > 0 on R and the compatibility conditions hold. Then for any but fixed constant T > 0, the Cauchy problem (3.1.44)–(3.1.46), (3.2.1) admits a unique H 2-generalized global solution (u(t), v(t), θ (t)) on Q T verifying (3.2.2)–(3.2.5) and the following estimates hold for any t ∈ [0, T ], 0 < C1−1 (T ) ≤ θ (x, t) ≤ C1 (T ) on R × [0, T ],

(3.2.15)

0 < C1−1 (T ) ≤ u(x, t) ≤ C1 (T ) u(t) − u ¯ 2H 2 + u(t) − u ¯ 2W 1,∞

(3.2.16)

on R × [0, T ], + u t (t)2H 1

+ v(t)2H 2

+ v(t)2W 1,∞ t u x 2H 1 +vt (t)2 + θ (t) − θ¯ 2H 2 + θ (t) − θ¯ 2W 1,∞ + θt (t)2 + 0 +u t 2H 2 + vx 2H 2 + vx 2W 1,∞ + vt 2H 1 + θθx 2H 2 + θθx 2W 1,∞ +θt 2H 1 (τ )dτ ≤ C2 (T ).

+ u x 2L ∞

(3.2.17)

Moreover, the H i -generalized global solutions (i = 1, 2) are continuously dependent on initial data in the sense that (u 1 (t) − u 2 (t), v1 (t) − v2 (t), θ1 (t) − θ2 (t)) H i (3.2.18) ≤ Ci (T )(u 01 (t) − u 02 (t), v01 (t) − v02 (t), θ01 (t) − θ02 (t)) H i , i = 1, 2 where (u j (t), v j (t), θ j (t)) ( j = 1, 2) is the H i -generalized global solution (i = 1, 2) to the Cauchy problem (3.1.44)–(3.1.46), (3.2.1) with the initial datum (u 0 j , v0 j , θ0 j ) ∈ H i (R)×H i (R)×H i (R) satisfying u 0 j −u, ¯ v0 j , θ0 j −θ¯ ∈ H i (R), u 0 j (x) > 0, θ0 j (x) > 0 on R and the compatibility conditions ( j = 1, 2). This property implies the uniqueness of H i -generalized global solution (i = 1, 2). ¯ v0 , θ0 − θ¯ ∈ H 4(R) ¯ θ¯ , u 0 − u, Theorem 3.2.2. Assume that for some positive constants u, and u 0 (x) > 0, θ0 (x) > 0 on R and the compatibility conditions hold. Then for any but fixed constant T > 0, the Cauchy problem (3.1.44)–(3.1.46), (3.2.1) admits a unique H 4global solution (u(t), v(t), θ (t)) on Q T verifying (3.2.6)–(3.2.12) and (3.2.15)–(3.2.16), and the following estimates hold for any t ∈ [0, T ], u(t) − u ¯ 2H 4 + u(t) − u ¯ 2W 3,∞ + u t (t)2H 3 + u t t (t)2H 1 + v(t)2H 4 + v(t)2W 3,∞ + vt (t)2H 2 + vt t (t)2 + θ (t) − θ¯ 2H 4

+ θ (t) − θ¯ 2W 3,∞ + θt (t)2H 2 + θt t (t)2 ≤ C4 (T ),

(3.2.19)


t 0

157

(u x 2H 3 + u t 2H 4 + u t t 2H 2 + u t t t 2 + u x 2W 2,∞ + vx 2H 4 + vt 2H 3

+ vt t 2H 1 + vx 2W 3,∞ + θθx 2H 4 + θt 2H 3 + θt t 2H 1 + θθx 2W 3,∞ )(τ )dτ ≤ C4 (T ). (3.2.20) Moreover, the H 4-global solution is continuously dependent on initial data in the sense of (3.2.18) with i = 4. The proofs of Theorems 3.2.1–3.2.2 are similar to those of Theorem 2.3.2 and Theorem 2.4.1, but the difference is that now the constant depends on T , any given length of time. Remark 3.2.1. We know that H 2-generalized global solution (u(t), v(t), θ (t)) obtained in Theorem 3.2.1 is not the classical one. By the embedding theorem (the Morrey theorem), we have 1 u 0 − u, ¯ v0 , θ0 − θ¯ ∈ C 1+ 2 (R). If we impose on the higher regularities of v0 , θ0 − θ¯ ∈ C 2+γ (R), γ ∈ (0, 1), then the global existence of classical solutions was obtained in [194]. Remark 3.2.2. From Remark 3.2.1 we know that the H 2-generalized global solution (u(t), v(t), θ (t)) obtained in Theorem 3.2.1 can be understood as a generalized (global) solution between the classical (global) solution and the H 1-generalized (global) solution. Remark 3.2.3. Similar results in Theorems 3.2.1–3.2.2 with θ¯ = 0 hold for the initial boundary value problem (3.1.44)–(3.1.46) with the boundary conditions v|x=0,1 = θ |x=0,1 = 0. ¯ v0 , θ0 − θ¯ ∈ Theorem 3.2.3. Assume that for some positive constants u, ¯ θ¯ , u 0 − u, H i (R) (i = 2, 4) and u 0 (x) > 0, θ0 (x) > 0 on R and the compatibility conditions hold. Define e0 := u 0 − u ¯ 2L ∞ + with α >

1 2

+∞ −∞

1(1 + x 2 )α [(u 0 (x) − u) ¯ 2 + v02 (x) + (θ0 (x) − θ¯ )2 + v04 (x)]d x

being an arbitrary but fixed constant, and

El = (log(ρ0 /ρ), ¯ log(v0 ), log(θ0 /θ¯ )) H l , (l = 0, 1), ρ0 = 1/u 0 , ρ¯ = 1/u. ¯ Then there exists a constant 0 ∈ (0, 1] such that if e0 ≤ 0 or E 0 E 1 ≤ 0 , then the H i -global solution (u(t), v(t), θ (t)) (i = 2, 4) obtained in Theorems 3.2.1–3.2.2 to the Cauchy problem (3.1.44)–(3.1.46), (3.2.1) verifying 0 < C1−1 ≤ θ (x, t) ≤ C1 on R × [0, +∞), 0
0,

(3.2.23)

and for i = 4, besides (3.2.21)–(3.2.23) and (3.2.6)–(3.2.12) with T = +∞, we have ¯ 2W 3,∞ + u t (t)2H 3 + u t t (t)2H 1 + v(t)2H 4 + vt (t)2H 2 u(t) − u ¯ 2H 4 + u(t) − u + vt t (t)2 + vx (t)2W 3,∞ + θ (t) − θ¯ 2H 4 + θ (t) − θ¯ 2W 3,∞ + θt (t)2H 1 + θt t (t)2 ≤ C4 , ∀t > 0,

t 0

(3.2.24)

u x 2H 3 + u t 2H 4 + u t t 2H 2 + u t t t 2 + u x 2W 2,∞ + vx 2H 4 + vt 2H 3 + vt t 2H 1 + vx 2W 3,∞ + θθx 2H 4 + θt 2H 3 + θt t 2H 1 + θθx 2W 3,∞ (τ )dτ ≤ C4 , ∀t > 0.

(3.2.25)

Moreover, the H i -(generalized) d global solutions (i = 1, 2, 4) are continuously dependent on initial data in the sense that (u 1 (t) − u 2 (t), v1 (t) − v2 (t), θ1 (t) − θ2 (t)) H i

(3.2.26)

≤ Ci (u 01 (t) − u 02 (t), v01 (t) − v02 (t), θ01 (t) − θ02 (t)) H i , i = 1, 2, 4 where (u j (t), v j (t), θ j (t)) ( j = 1, 2) has the same sense as in (3.2.18). Finally, for the H 2-global solution (u(t), v(t), θ (t)), as t → +∞, u t (t) H 1 + u t (t) L ∞ + vt (t) + θt (t) → 0, (u(t), v(t), θ (t)) − (u, ¯ 0, θ¯ )W 1,∞ + (u x (t), vx (t), θx (t)) H 1 → 0

(3.2.27) (3.2.28)

and for the H 4-global solution (u(t), v(t), θ (t)), as t → +∞, (u x (t), vx (t), θx (t)) H 3 + u t (t) H 3 + u t (t)W 2,∞ + vt (t) H 2 +vt (t)W 1,∞ + θt (t) H 2 + θt (t)W 1,∞ → 0, (3.2.29) u t t (t) H 1 + vt t (t) + θt t (t) + (u x (t), vx (t), θx (t))W 2,∞ → 0. (3.2.30) Corollary 3.2.1. The H 4-global solution (u(t), v(t), θ (t)) obtained in Theorem 3.2.2 is a classical one. Moreover, under assumptions in Theorem 3.2.3, we have the following large-time behavior of classical solution (u(t), v(t), θ (t)): as t → +∞, (u x (t), vx (t), θx (t))C 2+1/2 + u t (t)C 2+1/2 + (vt (t), θt (t))C 1+1/2 + u t t (t)C 1/2 → 0. (3.2.31)


159

3.2.2 Large-Time Behavior of Solutions In this subsection, we finish the proof of Theorem 3.2.3. In order to study the large-time behavior of the H i -global solutions (i = 2, 4), obviously all the estimates in the proofs of Theorems 3.2.1–3.2.2 will no longer work because those estimates depend heavily on T > 0, any given length of time. Thus we have to derive the uniform estimates in H i (R) (i = 1, 2, 4) in which all the constants depend only on min u 0 (x), min θ0 (x), the x∈R

x∈R

¯ (and e0 or E 0 , E 1 (see, e.g., Theorem H i (R) (i = 1, 2, 4) norm of (u 0 − u, ¯ v0 , θ0 − θ) 3.2.3)), but independent of any length of time T > 0. Since for any unbounded domain, the Poincaré´ inequality will not be valid and hence, unlike the corresponding initial boundary value problems in bounded domains (see, e.g., Section 3.1; see also, Amosov and Zlotnik [10, 11], Chen [51], Chen, Hoff and Trivisa [52], Ducomet and Zlotnik [88], Fujita-Yashima and Benabidallah [110, 111], Fujita-Yashima, Padula and Novotny [112], Hsiao and Luo [159], Jiang [165, 166], Luo [245], Matsumura and Nishida [255, 257], Nagasawa [283–287], Novotny and Stra˘s˘ kraba [301, 302], Okada and Kawashima [303], Qin [315–326],Qin and Fang [328], Qin and Hu [329], Qin, Huang and Ma [330], Qin and Jiang [331], Qin and Kong [332], Qin, Ma, Cavalcanti and Andrade [335], Qin and Muñ˜ oz Rivera [337, 338], Qin and Song [343], Qin and Wen [344], Qin and Zhao [346], Zheng and Qin [451, 452]), the exponential decay of solutions will not be anticipated (see, e.g., Deckelnick [82], Hoff [143], Itaya [161], Jiang [167, 169–171], Jiang and Zlotnik [178], Kanel [182], Kawashima and Nishida [191], Kazhikhov [193, 194], Matsumura [251], Matsumura and Nishida [253, 254], Okada and Kawashima [303], Qin, Wu and Liu [345], Valli and Zajaczkowski [412], Zheng and Shen [453, 454]). Now we first use some H 1-estimates given in [170, 196, 303] to establish uniform 1 H -estimates in the following lemma. Lemma 3.2.1. Assume that some constants u¯ > 0, θ¯ > 0, u 0 − u, ¯ v0 , θ0 − θ¯ ∈ H 1(R) and u 0 (x) > 0, θ0 (x) > 0 on R, and the compatibility conditions hold. Then there exists a constant 0 ∈ (0, 1] such that (I) if E 0 E 1 ≤ 0 , then, besides (3.2.13)–(3.2.14) with T = +∞, the H 1-generalized global solution (u(t), v(t), θ (t)) to the Cauchy problem (3.1.44)–(3.1.46), (3.2.1) satisfies that for any (x, t) ∈ R × [0, +∞), 0 < C1−1 ≤ θ (x, t) ≤ C1 , 0
0, u(t) − u ¯ 2H 1 + v(t)2H 1 + θ (t) − θ¯ 2H 1 t + (vx 2H 1 + θθx 2H 1 + u x 2 + vt 2 + θt 2 )(τ )dτ ≤ C1 , (3.2.34) 0

160


¯ 2L ∞ u(t) − u ¯ 2L ∞ + v(t)2L ∞ + θ (t) − θ t + (u t 2H 1 + vx 2L ∞ + θθx 2L ∞ )(τ )dτ ≤ C1

(3.2.35)

0

and as t → +∞, (u(t) − u, ¯ v(t), θ (t) − θ¯ ) L ∞ + (u x (t), vx (t), θx (t)) → 0

(3.2.36)

or (II) if e0 ≤ 0 , then, besides (3.2.13)–(3.2.14) with T = +∞ and (3.2.32)–(3.2.36), the H 1-generalized global solution (u(t), v(t), θ (t)) satisfies that for any (x, t) ∈ R × [0, +∞), 1 |u(x, t) − u| ¯ + φ(t)|θ (x, t) − θ¯ | < min(u, ¯ θ¯ ) 3

(3.2.37)

where φ(t) = min(1, t). Proof. Case I: From Okada and Kawashima [303] (see, e.g, Theorem 2.1) it follows that there exists a constant 1 ∈ (0, 1] such that if E 0 E 1 ≤ 1 , then H 1-generalized global solution (u(t), v(t), θ (t)) to the Cauchy problem (3.1.44)–(3.1.46), (3.2.1) satisfies estimates (3.2.32)–(3.2.34) and (3.2.36). Using the interpolation inequality: f L ∞ ≤ C f 1/2 f x 1/2 for any f ∈ H 1(R) where C > 0 is a positive constant independent of any length of time, we easily deduce (3.2.35) from (3.2.34). Case II: We know from Jiang [170] (see, e.g., Theorem 1.1 (ii) or [193,194]) there is a constant 2 ∈ (0, 1] such that if e0 ≤ 2 , then estimates (3.2.36)–(3.2.37) and u(t)− u ¯ 2 +v(t)2 +θ (t)− θ¯ 2 +

t

(vx 2 +θθx 2 )(τ )dτ ≤ C1 , ∀t > 0 (3.2.38)

0

hold. Clearly, (3.2.33) is the direct result of (3.2.37). By (3.2.37), we get that for any t ≥ 1, 0 < C1−1 ≤ θ (x, t) ≤ C1 , ∀x ∀ ∈ R.

(3.2.39)

Moreover, we find from the proofs in Kazhikhov and Shelukhin [196] that C1−1 e−C1 t ≤ θ (x, t) ≤ C1 eC1 t ,

∀(x, t) ∈ R × [0, +∞).

Note that this estimate is not enough to derive (3.2.32), but combining it with (3.2.39) can yield estimate (3.2.32). In view of (3.1.44), we can write (3.1.45) in the form μ(

ux θ )t = vt + R( )x . u u

(3.2.40)


161

Multiplying (3.2.40) by u x /u in L 2 (R), using (3.2.13)–(3.2.33) and (3.2.38), integrating by parts, and noting that (u x /u)t = (u t /u)x = (vx /u)x , we deduce that μ 2

t 1 2 ux 2 θ ux ) dx + R d x dτ 3 u 0 0 0 u t 1 1 t 1 2 ux vx θx u x d x dτ + R v |t0 d x + d x dτ ≤ C1 + u u u2 0 0 0 0 0 R t 1 θ u 2x μ 1 ux 2 d x dτ + ( ) dx ≤ C1 + 2 0 0 u3 4 0 u 1

(

which, together with (3.2.32)–(3.2.33), gives t u x (t)2 + u x 2 (τ )dτ ≤ C1 , ∀t > 0.

(3.2.41)

0

Multiplying (3.1.45) by vx x in L 2 (R), using (3.2.32)–(3.2.33), (3.2.38) and (3.2.41), the interpolation inequality and integrating by parts, we have t t vx (t)2 + vx x 2 (τ )dτ ≤ C1 + C1 (vx vx x u x 2 + θθx 2 + u x 2 )(τ )dτ 0 0 1 t vx x 2 (τ )dτ ≤ C1 + 2 0 whence

vx (t)2 +

t 0

vx x 2 (τ )dτ ≤ C1 ,

∀t > 0.

(3.2.42)

Analogously, from (3.1.46) we get t 2 θθ x (t) + θθx x 2 (τ )dτ 0 t ≤ C1 + C1 (θθx θθx x u x 2 + vx 3 vx x + vx 2 )(τ )dτ 0 1 t θθx x 2 (τ )dτ ≤ C1 + 2 0 implying

t

2

θθ x (t) +

0

θθx x 2 (τ )dτ ≤ C1 ,

∀t > 0.

(3.2.43)

By (3.1.44)–(3.1.46), (3.2.32)–(3.2.33), (3.2.38) and (3.2.41)–(3.2.43), using the interpolation inequality, we derive vt (t) ≤ C1 (vx x (t) + vx (t)1/2 vx x (t)1/2 u x + θθ x (t) + u x (t)), ≤ C1 (vx x (t) + vx (t) + u x (t) + θθx (t)),

(3.2.44)

162


θt (t) ≤ C1 θθ x x (t) + θθx (t)1/2 θθx x (t)1/2 u x (t) + vx (t)3/2 vx x (t)1/2 + vx (t) ≤ C1 (θθx x (t) + vx (t) + θθx (t) + vx x (t)) which, combined with (3.2.38) and (3.2.41)–(3.2.44), implies estimate (3.2.34). Therefore taking 0 = min[ 1 , 2 ] ends the proof.

Since we have established uniform H 1-estimates in Lemma 3.2.1, we only need to repeat the same argumentations as the proof of Theorem 2.3.1 (see also Lemma 2.3.1) to be able to reach estimates (3.2.21)–(3.2.25) in Theorem 3.2.3. Now all constants in these estimates will no longer depend on T > 0, any length of time, i.e., Ci (+∞) = Ci (i = 1, 2, 4). In order to finish the proof of Theorem 3.2.3, it suffices to prove the results on the large-time behavior of the H i (i = 2, 4)-global solutions in Theorem 3.2.3. The next two lemmas concern the large-time behavior of H 2 and H 4 global solutions respectively. Lemma 3.2.2. Under the assumptions in Theorem 3.2.3 with i = 2, if e0 ≤ 0 or E 0 E 1 ≤ 0 , then the H 2-generalized global solution (u(t), v(t), θ (t)) obtained in Theorem 3.2.1 to the Cauchy problem (3.1.44)–(3.1.46), (3.2.1) satisfies (3.2.27)–(3.2.28). Proof. We start from Lemma 3.2.1, repeat the same reasoning as the derivation of (2.3.65)–(2.3.68), (2.3.70), (2.3.73)–(2.3.74) in Lemmas 2.3.7–2.3.9 to obtain d vt (t)2 + (2C1 )−1 vt x (t)2 ≤ C2 (vx (t)2 + vx x (t)2 + θt (t)2 ), dt d θt (t)2 + (2C1 )−1 θt x (t)2 dt ≤ C2 (vx (t)2 + θθx (t)2 + θt (t)2 + vt x (t)2 ), d uxx (t)2 + (2C1 )−1 u x x (t)2 dt u ≤ C2 (θθx (t)2 + u x (t)2 + vx x (t)2 + θθx x (t)2 + vt x (t)2 ),

(3.2.46)

vx x (t) ≤ C1 (vt (t) + vx (t) + u x (t)) ≤ C2 , θθx x (t) ≤ C1 (θt (t) + θθx (t) + vx (t) + vx x (t)) ≤ C2 ,

(3.2.48) (3.2.49)

(3.2.45)

(3.2.47)

vx (t)2L ∞ ≤ Cvx (t)vx x (t) ≤ C2 , θθx (t)2L ∞ ≤ Cθθx (t)θθx x (t) ≤ C2 , (3.2.50) u x (t)2L ∞ ≤ Cu x (t)u x x (t) ≤ C2 .

(3.2.51)

Applying Theorem 1.2.4 to (3.2.45)–(3.2.47) and using estimate (3.2.25), we obtain that as t → +∞, vt (t) → 0, θt (t) → 0, u x x (t) → 0

(3.2.52)


163

which, with (3.2.36) and (3.2.48)–(3.2.51), implies that as t → +∞, vx x (t) + θθx x (t) + u t (t) H 1 → 0, u t (t)

L∞

+ (u x (t), vx (t), θx (t))

L∞

→ 0.

(3.2.53) (3.2.54)

Thus (3.2.27)–(3.2.28) follows from (3.2.36) and (3.2.52)–(3.2.54). The proof is complete. Lemma 3.2.3. Under the assumptions in Theorem 3.2.3 with i = 4, if e0 ≤ 0 or E 0 E 1 ≤ 0 , then the H 4-global solution (u(t), v(t), θ (t)) obtained in Theorem 3.2.2 to the Cauchy problem (3.1.44)–(3.1.46), (3.2.1) satisfies (3.2.29)–(3.2.30). Proof. Similarly to (2.4.30), (2.4.31), (2.4.42)–(2.4.49), (2.4.51), (2.4.69), (2.4.81) and using (3.2.25), we derive d vt t (t)2 + (2C1 )−1 vt t x (t)2 (3.2.55) dt ≤ C2 (θθx x (t)2 + θt x (t)2H 1 + vx (t)2H 1 + vt x (t)2 + θt (t)2 + u x (t)2 ), d θt t (t)2 + C1−1 θt t x (t)2 (3.2.56) dt ≤ C4 (θt x (t)2 + vt x (t)2H 1 + vx (t)2 + θt (t)2 + vt t x (t)2 + θt t (t)2 ), d vt x (t)2 + C1−1 vt x x (t)2 (3.2.57) dt ≤ C2 (θt x (t)2 + vt x (t)2 + θt (t)2 + vx x (t)2 + θθx (t)2 + u x (t)2 ), d θt x (t)2 + C1−1 θt x x (t)2 (3.2.58) dt ≤ C2 (θt x (t)2 + θθx x (t)2 + vx x (t)2 + u x (t)2 ), u d u x x x 2 x x x 2 (t) + C1−1 (t) ≤ C1 E 1 (t)2 , (3.2.59) dt u u d u x x x x 2 u x x x x 2 (t) + C1−1 (t) ≤ C1 E 2 (t)2 (3.2.60) dt u u where, by (3.2.25),

t

(E 1 2 + E 2 2 )(τ )dτ ≤ C4 , ∀t > 0

(3.2.61)

0

with E 1 (x,t) = μ

θ x u x x 2Rθ u x u x x vx x x u x + u x x vx x 2u x u x x vx − 2 + − + vt x x + E x (x,t), u2 u3 u u3

Rθθ x x 2μvx x u x − 2Rθθ x u x 2Rθ u 2x − 2μvx u 2x + + , 3 u2 u u 2Rθ u x u x x x Rθθ x u x x x vx x u x x x + u x vx x x x 2u x vx u x x x + − − + E 1x (x,t). E 2 (x,t) = μ 2 3 u u u3 u2 E(x,t) =

164


Applying Theorem 1.2.4 to (3.2.55)–(3.2.60) and using estimates (3.2.25) and (3.2.61), we infer that as t → +∞, vt t (t) → 0, θt t (t) → 0, vt x (t) → 0,

(3.2.62)

θt x (t) → 0, u x x x (t) → 0, u x x x x (t) → 0.

(3.2.63)

In the same manner as the proofs of Lemmas 2.4.1–2.4.4 and using the interpolation inequality, we deduce that vx x x (t) ≤ C2 (vx (t) + u x (t) H 1 + θθx (t) H 1 + vt x (t)), (3.2.64) vt x x (t) ≤ C1 vt t (t) + C2 (vx x (t) + u x (t) + vt x (t) + θθx (t) + θt (t) + θt x ), vx x x x (t) ≤ C2 (vx (t) H 2 + u x (t) H 2 + θθx (t) H 2 + vt x x (t)),

(3.2.65) (3.2.66)

vt x (t)2L ∞ ≤ Cvt x (t)vt x x (t), vt (t)2L ∞ ≤ Cvt (t)vt x (t),

(3.2.67)

vx x (t)2L ∞

≤ Cvx x (t)vx x x (t),

vx x x (t)2L ∞

≤ Cvx x x (t)vx x x x (t), u x x (t)2L ∞

≤ Cu x x (t)u x x x (t),

(3.2.68)

u x x x (t)2L ∞

≤ Cu x x x (t)u x x x x (t).

(3.2.69)

Thus it follows from (3.1.44), (3.2.62)–(3.2.69) and Theorem 1.2.4 that as t → +∞, (u x (t), vx (t)) H 3 + vt (t) H 2 + u t (t) H 3 + u t (t)W 2,∞ +u t t (t) H 1 + (u x (t), vx (t))W 2,∞ → 0.

(3.2.70)

Analogously, we can derive that as t → +∞, θθx (t) H 3 + θt (t) H 2 + θt (t)W 1,∞ + θθx (t)W 2,∞ → 0 which together with Theorem 1.2.4 and (3.2.70) implies estimates (3.2.29)–(3.2.30). The proof is complete. Now we have finished the proof of Theorem 3.2.3.

Proof of Corollary 3.2.3. Applying the embedding theorem, we readily get estimate (3.2.31) and complete the proof from Theorem 3.2.3

3.3 Bibliographic Comments For the one-dimensional Cauchy problem (3.1.44)–(3.1.46), (3.2.1), Itaya [161], Kanel [182], Kazhikhov [193–195] obtained the global existence and large-time behavior (only for v, θ ) of H 1-solutions. In this case, Okada and Kawashima [303] established the global existence and large-time behavior of classical (or H 1-) solution with small initial data and Jiang [170] proved the large-time behavior of H 1-solution with weighted small initial


165

data. Qin, Wu and Liu [345] established the existence and asymptotic behavior of global H i -solutions (i = 2, 4). For the one-dimensional initial (boundary) value problems, we still refer to the works by Amosov and Zlotnik [10, 11], Chen [51], Chen, Hoff and Trivisa [52], Ducomet and Zlotnik [88], Hoff and Ziane [150, 151], Hsiao and Luo [158], Huang, Matsumura and Xin [160], Jiang [165, 166, 170, 171], Jiang and Zhang [177], Jiang and Zlotnik [178], Kawashima and Nishida [191], Kawohl [192], Kazhikhov [193, 194], Nagasawa [283–287], Nikolaev [295], Okada and Kawashima [303], Qin [315–326], Qin and Fang [328], Qin and Hu [329], Qin, Huang and Ma [330], Qin and Jiang [331], Qin and Kong [332], Qin, Ma, Cavalcanti and Andrade [335], Qin, Ma and Huang [336], Qin and Muñoz Rivera [337, 338], Qin, Wu and Liu [345], Qin and Zhao [346], Vong, Yang and Zhu [417], Yang [433], Yang, Yao and Zhu [434], Yang and Zhao [435], Zhang and Fang [440, 441], Zheng and Qin [451, 452], Zheng and Shen [453, 454], and the references therein. For multi-dimensional initial (boundary) value problems, the global existence and large-time behavior of smooth solutions have been investigated for general domains only in case of “small initial data”. We also consult the works by Deckelnick [82], Feireisl [97– 100], Feireisl and Petzeltova [101–103], Feireisl, Novotny and Petzeltova [104], Foias and Temam [105], Frid and Shelukhin [106], Fujita-Yashima and Benabidallah [110, 111], Fujita-Yashima, Padula and Novotny [112], Galdi [115], Ghidaglia and Temam [119], Hoff [142–146], Hoff and Serre [147], Jiang [167, 169, 170], Jiang and Zhang [174–176], Kawashima [188, 189], Matsumura [251], Matsumura and Nishida [253–257], Novotny and Stra˘s˘kraba [301, 302], Padula [305], Qin [325, 326], Qin, Deng and Su [327], Qin and Hu [329], Qin, Huang and Ma [330], Qin and Jiang [331], Qin and Ma, Cavalcanti and Andrade [335], Qin and Song [343], Qin and Wen [344], Rosa [363], Sell [369], Valli and Zajaczkowski [412], Vishik and Chepyzhov [413, 414], Wu ana Zhong [429], Xin [431], Zhang and Fang [441], Zheng [449], Zheng and Qin [452], and the references therein. For aspects of infinite-dimensional dynamical systems, we consult Section 2.7.

Chapter 4

A Polytropic Ideal Gas in Bounded Annular Domains in Rn In this chapter we shall establish the global existence and exponential stability of solutions in H i (i = 1, 2, 4) for the compressible Navier-Stokes equations of a polytropic ideal gas in bounded annular domains in Rn (n = 2, 3). Since the versions of these equations are more complicated than (2.1.1)–(2.1.3) in Chapter 2, it is necessary for us to further study this model. The results of this chapter are adopted from Qin [325, 335, 452].

4.1 Global Existence and Asymptotic Behavior in H 1 and H 2 In this section we study the global existence and exponential stability of solutions of a nonlinear C0 -semigroup S(t) in H+i (i = 1, 2), the subspaces of H i ×H H0i ×H i (i = 1, 2), for the compressible Navier-Stokes equations of a viscous polytropic ideal gas in Eulerian coordinates in the bounded annular domain G n = {x ∈ Rn : 0 < a < |x| < b}, (see, e.g., [110, 111, 170, 171, 325, 331, 335, 343, 452]) (n − 1) ρv = 0, (4.1.1) r (n − 1) (n − 1) ∂r v − v + R∂r (ρθ ) = 0, r ∈ G n , t > 0, ρ(∂t v + v∂r v) − β ∂r2 v + r r2 (4.1.2)

∂t ρ + ∂r (ρv) +

Chapter 4. A Polytropic Ideal Gas in Bounded Annular Domains in Rn

168

C V ρ(∂t θ + v∂r θ ) − κ∂r2 θ − κ

(n − 1)v (n − 1) ∂r θ + Rρθ ∂r v + r r

(n − 1)v 2 2μ(n − 1)v 2 − 2μ(∂r v)2 − = 0. − λ ∂r v + r r2

(4.1.3)

Here subscripts denote partial differentiations, and R, μ, C V , κ and λ are constants satisfying R, C V , κ, μ > 0, λ + 2μ/n ≥ 0 for n = 2, 3 and β = λ + 2μ > 0. We shall consider problem (4.1.1)–(4.1.3) in the region {r ∈ G n , t ≥ 0} subject to the initial boundary conditions ρ(r, 0) = ρ0 (r ), v(r, 0) = v0 (r ), θ (r, 0) = θ0 (r ), r ∈ G n , v(a, t) = v(b, t) = 0, θr (a, t) = θr (b, t) = 0, t ≥ 0.

(4.1.4) (4.1.5)

The equations (4.1.1)–(4.1.3) describe the spherically symmetric motion of a viscous polytropic ideal gas in the annular domain G n in the cases of n = 2, 3 (see, e.g., [110], [111], [170], [171], [325], [331], [335], [343], [452]), where ρ, v, θ are the density, velocity, and absolute temperature, respectively; λ and μ are the constant viscosity coefficients, R, C V , and κ are the gas constant, specific heat capacity, and thermal conductivity, respectively. The aim of this section is to prove that for the compressible Navier-Stokes equations (4.1.1)–(4.1.5) of a viscous polytropic ideal gas in bounded annular domains in Rn (n = 2, 3), the generalized global (spherically symmetric for n = 2, 3) solutions define a nonlinear C0 -semigroup S(t) on two incomplete metric subspaces H+i (i = 1, 2) of H i × H0i × H i (i = 1, 2); then we show that the semigroup S(t) is exponentially stable on H+i (i = 1, 2), which further leads to the exponential convergence to a steady constant state in H+i (i = 1, 2) of the generalized global solutions as time goes to infinity. In what follows we first transfer problem (4.1.1)–(4.1.5) to that in Lagrangian coordinates and obtain the results on exponential stability of C0 -semigroup S(t). Then we go back to the Eulerian coordinates and draw the corresponding conclusions. It is known that the Eulerian coordinates (r, t) are connected to the Lagrangian coordinates (ξ, t) by the relation

t

r (ξ, t) = r0 (ξ ) +

v (ξ, τ )dτ

(4.1.6)

0

where v (ξ, t) = v(r (ξ, t), t) and r0 (ξ ) = η−1 (ξ ),

η(r ) =

r a

s n−1 ρ0 (s)ds, r ∈ G n .

(4.1.7)

By equation (4.1.1), (4.1.6) and (4.1.7), we obtain ∂t

r(ξ,t ) a

s n−1 ρ(s, t)ds = δn1 v(0, t)ρ(0, t) = 0

(4.1.8)

4.1. Global Existence and Asymptotic Behavior in H 1 and H 2

169

with δi j being the Kronecker delta, which implies

r(ξ,t )

s

n−1

a

ρ(s, t)ds =

r0 (ξ ) a

s n−1 ρ0 (s)ds = ξ.

(4.1.9)

Thus under the assumption inf{ f ρ(s, t); s ∈ G¯n , t ≥ 0} > 0 (which we need to justify), G n is transformed to n = (0, L) with

b

L=

b

s n−1 ρ0 (s)ds =

a

s n−1 ρ(s, t)ds

a

which, with (4.1.6)–(4.1.7) and (4.1.9), implies that L is invariant along the trajectory {ρ(s, t) : a ≤ s ≤ b, t ≥ 0}. Moreover, differentiating (4.1.9) with respect to ξ , we have ∂ξ r (ξ, t) = [r (ξ, t)n−1 ρ(r (ξ, t), t)]−1 .

(4.1.10)

φ (ξ, t) = φ(r (ξ, t), t). Then by virtue of (4.1.6)–(4.1.7) We denote a function φ(r, t) by and (4.1.10), we finally arrive at ∂t φ(ξ, t) = ∂t φ(r, t) + v∂r φ(r, t), ∂r φ(r, t) . ∂ξ φ (ξ, t) = ∂r φ(r, t)∂ξ r (ξ, t) = n−1 r ρ(r, t)

(4.1.11) (4.1.12)

We denote ( ρ , v, θ ) still by (ρ, v, θ ) and (ξ, t) by (x, t) if there is no danger of confusion. We use u = 1/ρ to denote the specific volume. Thus, by virtue of (4.1.6)–(4.1.7) and (4.1.11)–(4.1.12), equations (4.1.1)–(4.1.3) in the new variables (x, t) are u t = (r n−1 v)x , (4.1.13) n−1 θ v)x β(r −R , x ∈ n , t > 0, (4.1.14) vt = r n−1 u u x 2n−2 1 θx r C V θt = κ + β(r n−1 v)x − Rθ (r n−1 v)x − 2μ(n − 1)(r n−2 v 2 )x , u u x (4.1.15) with the initial boundary conditions u(x, 0) = u 0 (x), v(x, 0) = v0 (x), θ (x, 0) = θ0 (x), x ∈ n , v(0, t) = v(L, t) = 0, θ x (0, t) = θ x (L, t) = 0, t ≥ 0, and, by (4.1.6) and (4.1.10), r (x, t) is determined by t r (x, t) = r0 (x) + v(x, τ )dτ, r0 (x) = (a n + n 0

x 0

u 0 (y)d y)1/n

(4.1.16) (4.1.17)

if n = 2, 3. (4.1.18)


170

For the case of n = 2 or n = 3, we assume that λ and μ satisfy nλ + 2μ > 0.

(4.1.19)

Our first task is to study problem (4.1.13)–(4.1.17) with fixed L > 0. We define two spaces,

H+1 = (u, v, θ ) ∈ H 1[0, L] × H 1[0, L] × H 1[0, L] : u(x) > 0, θ (x) > 0, x ∈ [0, L], v|x=0 = v|x=L = 0 and

H+2 = (u, v, θ ) ∈ H 2[0, L] × H 2[0, L] × H 2[0, L] : u(x) > 0, θ (x) > 0, x ∈ [0, L], v|x=0 = v|x=L = θx |x=0 = θx |x=L = 0

which become two metric spaces when equipped with the metrics induced from the usual norms. In the above, H 1, H 2 are the usual Sobolev spaces. We put · = · L 2 , Q T := n × (0, T ). We use Ci (i = 1, 2) to denote the universal constant depending only on the H i norm of initial data, min u 0 (x) and x∈[0,L]

min θ0 (x), but independent of t.

x∈[0,L]

Theorem 4.1.1. Assume that (4.1.19) is valid. If (u 0 , v0 , θ0 ) ∈ H+1 , then there exists a unique generalized global solution (u(t), v(t), θ (t)) in H+1 to problem (4.1.13)–(4.1.17), which defines a nonlinear C0 -semigroup S(t) on H+1 such that S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)) ∈ C([0, +∞); H+1 ), u − u, ¯ v, θ − θ¯ , u t , vt , θt , θx , vx , u x , vx x , θx x , r − r, ¯ (r − r¯ )x , (r − r¯ )x x , rt , rt x , rt x x ∈ L 2 ([0, +∞); L 2) and the following estimates hold: 0 < C1−1 ≤ θ (x, t) ≤ C1 on [0, L] × [0, +∞), 0
0 and γ1 = γ1 (C1 ) > 0 such that for any fixed γ ∈ (0, γ1 ], we have that for any t > 0,

eγ t rt (t)2H 1 + r (t) − r ¯ 2H 2 + u(t) − u ¯ 2H 1 + u(t) − u ¯ 2L ∞ + u t (t)2 t

¯ 2H 1 +θ (t) − θ¯ 2H 1 + v(t)2H 1 + v(t)2L ∞ + θ (t) − θ¯ 2L ∞ + eγ τ u − u 0

+u

− u ¯ 2L ∞

+ + θ − θ¯ 2H 2 + vt 2 + θt 2 + r − r¯ 2H 2 + rt 2H 2 (τ )dτ ≤ C1

+ u t 2H 1

+θ − θ¯ 2W 1,∞

v2H 2

+ v2W 1,∞

(4.1.24)

where 1 u¯ = L

1

u 0 (x)d x,

0

¯ 1/n , r¯ (x) = (a n + n ux)

θ¯ =

1 v02 (x)d x, C V θ0 + CV L 0 2 1

if n = 2, 3.

(4.1.25) (4.1.26)

Moreover, (4.1.24) means that the semigroup S(t) is exponentially stable on H+1 , i.e., for any fixed γ ∈ (0, γ1 ], the following inequality holds for any t > 0, S(t)(u 0 , v0 , θ0 ) − (u, ¯ 0, θ¯ )2H 1 ≤ C1 e−γ t . +

(4.1.27)

Theorem 4.1.2. Assume that (4.1.19) is valid. If (u 0 , v0 , θ0 ) ∈ H+2 , then there exists a unique generalized global solution (u(t), v(t), θ (t)) in H+2 to problem (4.1.13)–(4.1.17), which defines a nonlinear C0 -semigroup S(t) on H+2 such that S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)) ∈ C([0, +∞); H+2 ). In addition to Theorem 4.1.1, we have u x x , u t x , u t x x , vx x x , vt x , θx x x , θt x , (r − r¯ )x x x , rt x x x ∈ L 2 ([0, +∞); L 2) and the following estimates hold: rt (t)2H 2 + r (t) − r¯ 2H 3 + u(t) − u ¯ 2H 2 + u(t) − u ¯ 2W 1,∞ + u t (t)2H 1 + v(t)2H 2 + v(t)2W 1,∞ + vt (t)2 + θ (t) − θ¯ 2H 2 + θ (t) − θ¯ 2W 1,∞ t

u − u ¯ 2H 2 + u − u + θt (t)2 + ¯ 2W 1,∞ + u t 2H 2 + v2H 3 + v2W 2,∞ 0

+ vt 2H 1 + θ − θ¯ 2H 3 + θ − θ¯ 2W 2,∞ + θt 2H 1 + r − r¯ 2H 3 + rt 2H 3 (τ )dτ ≤ C2 , ∀t > 0

(4.1.28)

172


and there exist constants C2 > 0 and γ2 = γ2 (C2 )(≤ γ1 ) > 0 such that for any fixed γ ∈ (0, γ2 ], we have that for any t > 0,

eγ t rt (t)2H 2 + r (t) − r¯ 2H 3 + u(t) − u ¯ 2H 2 + u(t) − u ¯ 2W 1,∞ + u t (t)2H 1 + v(t)2H 2 + v(t)2W 1,∞ + vt (t)2 + θ (t) − θ¯ 2H 2 + θ (t) − θ¯ 2W 1,∞ + θt (t)2 t

¯ 2H 2 + u(t) − u ¯ 2W 1,∞ + u t 2H 2 + v2H 3 + v2W 2,∞ + θ − θ¯ 2H 3 + eγ τ u − u 0 + θ − θ¯ 2W 2,∞ + vt 2H 1 + θt 2H 1 + r − r¯ 2H 3 + rt 2H 3 (τ )dτ ≤ C2 . (4.1.29) Moreover, (4.1.29) means that the semigroup S(t) is exponentially stable on H+2 , i.e., for any fixed γ ∈ (0, γ2 ], the following inequality holds for any t > 0, S(t)(u 0 , v0 , θ0 ) − (u, ¯ 0, θ¯ )2H 2 ≤ C2 e−γ t . +

(4.1.30)

Remark 4.1.1. We know that the generalized global solution (u(t), v(t), θ (t)) ∈ H+2 obtained in Theorem 4.1.2 is not the classical one. Indeed, if (u 0 , v0 , θ0 ) ∈ H+2 , by the 1

embedding theorem, we have u 0 , v0 , θ0 ∈ C 1+ 2 ( n ). If we impose on the higher regularities of v0 , θ0 ∈ C 2+α ( n ), α ∈ (0, 1), by the same method as that in [194] for n = 1, the following results on the global existence of classical solutions are readily obtained. Theorem 4.1.3. We assume that u 0 ∈ C 1+α ( n ), v0 , θ0 ∈ C 2+α ( n ), α ∈ (0, 1) and the compatibility conditions u t |x=0,L = vt |x=0,L = θt |x=0,L = 0 hold, then the generalized global solution (u(t), v(t), θ (t)) ∈ H+2 obtained in Theorem 4.1.2 is the α classical one satisfying u(x, t) ∈ C 1+α,1+ 2 (Q T ), α v(x, t), θ (x, t) ∈ C 2+α,2+ 2 (Q T ) for any T > 0. Remark 4.1.2. We know from Theorems 4.1.1–4.1.2 that the generalized (global) solution (u(t), v(t), θ (t)) in H+2 can be understood as a generalized (global) solution between the classical (global) solution and the generalized (global) solution (u(t), v(t), θ (t)) in H+1 . Remark 4.1.3. The results for i = 1 in Theorem 4.1.1 improve those in [170] for large initial data. Remark 4.1.4. Theorems 4.1.1–4.1.2 also hold when the boundary conditions (4.1.17) are replaced by v(0, t) = v(L, t) = 0, θ (0, t) = θ (L, t) = θ˜ where θ˜ > 0 is a constant and θ¯ is replaced by θ˜ . Now we go back to the Eulerian coordinates and consider problem (4.1.1)–(4.1.5) with G n = (a, b) being fixed. Let b H L1 ,G = (ρ, v, θ ) ∈ H 1[a, b] × H 1[a, b] × H 1[a, b] : s n−1 ρds a

= L, ρ(x) > 0, θ (x) > 0, x ∈ [a, b], v|x=a = v|x=b = 0


and

173

H L2 ,G = {(ρ, v, θ ) ∈ H 2[a, b] × H 2[a, b] × H 2[a, b] : = L, ρ(x) > 0, θ (x) > 0,

b a

s n−1 ρds

x ∈ [a, b], v|x=a = v|x=b = θx |x=a = θx |x=b = 0}

where L > 0 is any given positive number. We now state the corresponding results in Theorems 4.1.1–4.1.2 in Eulerian coordinates. Theorem 4.1.4. Assume that (4.1.19) is valid. For any given constant L > 0, if (ρ0 , v0 , θ0 ) ∈ H L1 ,G , then there exists a unique generalized global solution (ρ(t), v(t), θ (t)) in H L1 ,G ˜ on H 1 such to problem (4.1.1)–(4.1.5), which defines a nonlinear C0 -semigroup S(t) L ,G that ˜ S(t)(ρ 0 , v0 , θ0 ) = (ρ(t), v(t), θ (t)) ∈ C([0, +∞); 1 H L ,G ), ρ − ρ, ¯ v, θ − θ¯ , ρt , vt , θt , θx , vx , ρx , vx x , θx x ∈ L 2 ([0, +∞); L 2 ) and the following estimates hold: 0 < C1−1 ≤ θ (x, t) ≤ C1 on [a, b] × [0, +∞), 0
0,

(4.1.33)

and there exist constants C1 > 0 and γ˜1 = γ˜1 (C1 ) > 0 such that for any fixed γ˜ ∈ (0, γ˜1 ], we have that for any t > 0,

¯ 2L ∞ eγ˜ t ρ(t) − ρ ¯ 2H 1 + ρ(t) − ρ ¯ 2L ∞ + ρt (t)2 + θ (t) − θ¯ 2H 1 + θ (t) − θ t

¯ 2H 1 + ρ − ρ + v(t)2H 1 + v(t)2L ∞ + eγ˜ τ ρ − ρ ¯ 2L ∞ + ρt 2H 1 + v2H 2 0 + v2W 1,∞ + θ − θ¯ 2H 2 + θ − θ¯ 2W 1,∞ + vt 2 + θt 2 (τ )dτ ≤ C1 (4.1.34) where ρ¯ =

nL , bn − a n

θ¯ =

1 CV L

b a

C V θ0 +

v02 n−1 r ρ0 (r )dr. 2

(4.1.35)


174

˜ is exponentially stable on H 1 , i.e., Moreover, (4.1.34) means that the semigroup S(t) L ,G for any fixed γ˜ ∈ (0, γ˜1 ], the following inequality holds for any t > 0, ˜ S(t)(ρ ¯ 0, θ¯ )2H 1 0 , v0 , θ0 ) − (ρ,

L ,G

≤ C1 e−γ˜ t .

(4.1.36)

Theorem 4.1.5. Assume that (4.1.19) is valid. For any given constant L > 0, if (ρ0 , v0 , θ0 ) ∈ H L2 ,G , then there exists a unique generalized global solution (ρ(t), v(t), θ (t)) in H L2 ,G ˜ on H 2 such to problem (4.1.1)–(4.1.5), which defines a nonlinear C0 -semigroup S(t) L ,G 2 ˜ that S(t)(ρ 0 , v0 , θ0 ) = (ρ(t), v(t), θ (t)) ∈ C([0, +∞); H L ,G ). In addition to Theorem 4.1.4, we have ρx x , ρt x , ρt x x , vx x x , vt x , θx x x , θt x ∈ L 2 ([0, +∞); L 2) and the following estimates hold for any t > 0: ρ(t) − ρ ¯ 2H 2 + ρ(t) − ρ ¯ 2W 1,∞ + ρt (t)2H 1 + v(t)2H 2 + v(t)2W 1,∞ + vt (t)2 t

ρ − ρ ¯ 2H 2 + ρ − ρ + θ (t) − θ¯ 2H 2 + θ (t) − θ¯ 2W 1,∞ + θt (t)2 + ¯ 2W 1,∞ 0

+ ρt 2H 2 + v2H 3 + v2W 2,∞ + vt 2H 1 + θ − θ¯ 2H 3 + θ − θ¯ 2W 2,∞ + θt 2H 1 (τ )dτ ≤ C2 ,

(4.1.37)

and there exist constants C2 > 0 and γ˜2 = γ˜2 (C2 )(≤ γ˜1 ) > 0 such that for any fixed γ˜ ∈ (0, γ˜2 ], we have that for any t > 0,

eγ˜ t ρ(t) − ρ ¯ 2H 2 + ρ(t) − ρ ¯ 2W 1,∞ + ρt (t)2H 1 + v(t)2H 2 + v(t)2W 1,∞ t

¯ 2H 2 + vt (t)2 + θ (t) − θ¯ 2H 2 + θ (t) − θ¯ 2W 1,∞ + θt (t)2 + eγ˜ τ ρ − ρ 0

+ ρ

− ρ ¯ 2W 1,∞

+ ρt 2H 2

+ v2H 3

+ vt 2H 1 + θt 2H 1 (τ )dτ ≤ C2 .

+ v2W 2,∞

+ θ − θ¯ 2H 3 + θ − θ¯ 2W 2,∞ (4.1.38)

˜ is exponentially stable on H 2 , i.e., Moreover, (4.1.38) means that the semigroup S(t) L ,G for any fixed γ˜ ∈ (0, γ˜2 ], the following inequality holds for any t > 0, ˜ S(t)(ρ ¯ 0, θ¯ )2H 2 0 , v0 , θ0 ) − (ρ,

L ,G

≤ C2 e−γ˜ t .

(4.1.39)

Remark 4.1.5. Theorems 4.1.4–4.1.5 also hold when the boundary conditions (4.1.5) are replaced by v(a, t) = v(b, t) = 0, θ (a, t) = θ (b, t) = θ˜ where θ˜ > 0 is a constant and θ¯ is replaced by θ˜ .


175

4.1.1 Uniform A Priori Estimates in H 1 In this subsection we complete the proof of Theorem 4.1.1. We begin with the following lemma. Lemma 4.1.1. If (u 0 , v0 , θ0 ) ∈ H+1 , then there exists a unique generalized global solution (u(t), v(t), θ (t)) in H+1 to problem (4.1.13)–(4.1.17) satisfying u t , vt , θt , θx , vx , u x , vx x , θx x ∈ L 2 ([0, +∞); L 2), 0 < θ (x, t) ≤ C1 on [0, L] × [0, ∞), 1 ≤ n ≤ 3, 0 < C1−1 ≤ u(x, t) ≤ C1

(4.1.40) (4.1.41)

on [0, L] × [0, ∞), 1 ≤ n ≤ 3,

(4.1.42)

0 < a ≤ r (x, t) ≤ b, 0 < C1−1 ≤ r x (x, t) = r 1−n (x, t)u(x, t) ≤ C1 , on [0, L] × [0, ∞), n = 2, 3, (4.1.43) 2 2 2 2 2 ¯ 1 + v(t) 1 ¯ H 1 + θ (t) − θ rt (t) H 1 + r (t) − r¯ H 2 + u(t) − u H H t

u − u ¯ 2H 1 + u − u +θ (t) − θ¯ 2L ∞ + v(t)2L ∞ + u t (t)2 + ¯ 2L ∞ + v2H 2 0

+v2W 1,∞ + θ − θ¯ 2H 2 + θ − θ¯ 2W 1,∞ + u t 2H 1 + vt 2 + θt 2 +(n − 1)(r − r¯ 2H 2 + rt 2H 2 ) (τ )dτ ≤ C1 , ∀t > 0

(4.1.44)

and there exist positive constants T0 , C1 , independent of t, such that −1

(u(t) − u, ¯ v(t), θ (t) − θ¯ ) H 1 ≤ C1 e−C1 t , ∀t ≥ T0 .

(4.1.45)

Proof. The existence of generalized global solutions in H+1 and the estimates (4.1.40)– (4.1.43) and (4.1.45) were obtained in [170] for n = 2, 3. By the results in [170] for n = 2, 3, we know that 2

2

2

u x (t) + θθx (t) + vx (t) +

0

t

[u x 2 + vx x 2 + θθx 2 + θθx x 2

+vt 2 + θt 2 ](τ )dτ ≤ C1 , ∀t > 0.

(4.1.46)

By (4.1.13)–(4.1.17), we have that for any t > 0,

L 0

L

CV θ +

0

L

u(x, t)d x =

u 0 (x)d x,

0

L v2 v2 (x, t)d x = C V θ0 + 0 (x)d x 2 2 0

which, along with the Poincaré´ inequality, (4.1.13), (4.1.25), (4.1.46) and the interpolation


176

inequality, imply u(t) − u ¯ L ∞ ≤ C1 u x (t), v(t)W i,∞ ≤ C1 vx (t) H i , i = 0, 1,

(4.1.47)

u t (t) H i ≤ C1 vx (t) H i , i = 0, 1, θθx (t) L ∞ ≤ C1 (θθx (t) + θθx x ), (4.1.48) 1 L θ (x, t)d x L ∞ + C1 v(x, t)2L ∞ θ (t) − θ¯ L ∞ ≤ θ (t) − L 0 ≤ C1 (θθx (t) + vx (t)). (4.1.49) On the other hand, we infer from (4.1.10), (4.1.13) and (4.1.26) that for n = 2, 3, rt (t) H 1 = v(t) H 1 ≤ C1 vx (t)

and

x

r n (t) − r¯ n = n

(4.1.50)

(u − u)d ¯ y

0

which gives r (t) − r¯ =

n

x 0

(u − u)d ¯ y ˆ r¯ ) d(r,

(4.1.51)

ˆ r¯ ) = r (t) + ¯ for n = 2 or d(r, ˆ r¯ ) = r 2 (t) + r (t)¯r + ¯ 2 for n = 3. Using with d(r, −1 (4.1.43), (4.1.46) and noting that 0 < C1 ≤ ¯ ≤ C1 , we get ˆ r¯ ) ≤ C1 , dˆx (r, r¯ L ∞ + dˆx x (r, r¯ ) ≤ C1 0 < C1−1 ≤ d(r, which together with (4.1.48) and (4.1.51) gives r (t) − r¯ ≤ C1 u(t) − u ¯ L 1 ≤ C1 u x (t), (r (t) − r¯ )x ≤ C1 (u(t) − u ¯ + dˆx L ∞ u(t) − u ¯ L1 )

(4.1.52)

¯ ≤ C1 u x (t), (4.1.53) ≤ C1 u(t) − u ˆ ˆ ˆ ∞ ¯ d x L + (dx + d x x )u(t) − u ¯ L1 ) (r (t) − r¯ )x x ≤ C1 (u x (t) + u(t) − u (4.1.54) ≤ C1 u x (t). The combination of (4.1.46)–(4.1.50) and (4.1.52)–(4.1.54) implies (4.1.44). The proof is complete. In what follows we shall derive the uniform estimates in H+1 in Theorem 4.1.1 of generalized global solution (u(t), v(t), θ (t)). The next lemma concerns the uniform global (in time) positive lower boundedness (independent of t) of the absolute temperature θ , which was not obtained before. Lemma 4.1.2. If (u 0 , v0 , θ0 ) ∈ H+1 , then 0 < C1−1 ≤ θ (x, t),

∀(x, t) ∈ [0, L] × [0, +∞).

(4.1.55)


177

Proof. Let w = θ1 . By virtue of (4.1.19) and taking δ > 0 such that 2(n − 1)μ 2(n − 2)μ < < δ < 1, if n = 2, 3, 0 ≤ (n − 1)β nβ then after a straightforward calculation, equation (4.1.15) can be transformed to 2n−2 wx )x − 2κρr 2n−2 θ w2x + ρw2 [(n − 1)δβ − 2(n − 2)μ] C V wt = κ(ρr × r −1 uv +

2 (βδ − 2μ)r n−1 vx 2μ[nδβ − 2(n − 1)μ]ρw2r 2n−2 vx2 + (n − 1)δβ − 2(n − 2)μ (n − 1)δβ − 2(n − 2)μ 2 R2 ρ Rθ . (4.1.56) + +β(1 − δ)ρw2 (r n−1 v)x − 2β(1 − δ) 4(1 − δ)β

Multiplying (4.1.56) by 2mw2m−1 with m being an arbitrary natural number, and integrating the resultant over n = (0, L), by the fact that the expression in the bracket {·} is non-negative and by Hö¨ lder’s inequality, we get L R2 2m−1 d C V w(t) L 2m w(t) L 2m ≤ ρw2m−1 d x dt 4(1 − δ)β 0 ≤ C1 w(t)2m−1 ≤ C1 ρ L 2m w(t)2m−1 L 2m L 2m which implies, by taking m −→ +∞, that w(t) L ∞ ≤ 1/θ0 L ∞ + C1 t ≤ C1 (1 + t). Thus, for all x ∈ [0, L], t ≥ 0, 1 . (4.1.57) C1 (1 + t) By (4.1.45) and the imbedding theorem, there exists t0 ≥ T0 such that for any t ≥ t0 , we have that for all x ∈ [0, L], t ≥ 0, θ (x, t) ≥

θ (x, t) ≥ θ¯ /2 > 0 which, together with (4.1.57), implies (4.1.55). The proof is complete.

Let v2 ∂e ∂e ¯ ¯ ¯ ¯ +e(u, S)−e(u, ¯ S)− (u, ¯ S)(u − u)− ¯ (u, ¯ S)(S − S), 2 ∂u ∂S S = C V log θ + R log u, (entropy), C V exp(S/C V ) , (internal energy), e(u, S) = C V θ = C V θ (u, S) = u R/C V where u¯ and θ¯ are the same as those in (4.1.25) and S¯ is defined as E(u, v, S) =

¯ S¯ = C V log θ¯ + R log u. The next three lemmas concern the exponential stability in H+1 of the generalized global solution (u(t), v(t), θ (t)).

178


Lemma 4.1.3. The following inequalities hold: 2 v2 ¯ 2 ) ≤ E(u, v, S) ≤ v + C1 (|u − u| ¯ 2 ). (4.1.58) + C1−1 (|u − u| ¯ 2 + |S − S| ¯ 2 + |S − S| 2 2

¯ Proof. By the mean value theorem, there exists a point ( u, S) between (u, S) and (u, ¯ S) such that E(u, v, S)

1 ∂ 2e ∂ 2e v2 ∂ 2e 2 2 ¯ ¯ + ( u , S)(u − u)(S ¯ − S) + 2 ( = ( u , S)(u − u) ¯ +2 u , S)(S − S) 2 2 ∂u 2 ∂u∂ S ∂S 1 R(R + C V ) −2 v2 2R −1 θ 2 2 ¯ ¯ + = u θ (u − u) u θ (u − u)(S (S − S) ¯ − ¯ − S) + 2 2 CV CV CV (4.1.59)

where u = λ0 u¯ + (1 − λ0 )u,

θ = e S/C V / u R/C V , 0 ≤ λ0 ≤ 1,

S = λ0 S¯ + (1 − λ0 )S,

0 ≤ λ0 ≤ 1.

It follows from Lemmas 4.1.1–4.1.2 that u ≤ C1 , 0 < C1−1 ≤ θ ≤ C1 , | S| ≤ C1 . 0 < C1−1 ≤

(4.1.60)

Thus by (4.1.59)–(4.1.60) and the Cauchy inequality, we have E(u, v, S) ≤

v2 ¯ 2 ]. + C1 [(u − u) ¯ 2 + (S − S) 2

(4.1.61)

On the other hand, Young’s inequality and (4.1.59) yield v2 1 R −2 1 ¯ 2 E(u, v, S) ≥ + u θ (u − u) ¯ 2+ θ (S − S) 2 2 2 2R + C V 2 v ¯ 2 + C1−1 (u − u) ≥ ¯ 2 + (S − S) 2 which, combined with (4.1.61), completes the proof of the lemma.

Lemma 4.1.4. There exists a positive constant γ1 = γ1 (C1 ) > 0 such that for any fixed γ ∈ (0, γ1 ], the following estimate holds: ¯ 2H 1 + θ (t) − θ¯ 2 eγ t v(t)2 + u(t) − u t eγ τ u − u ¯ 2H 1 + θθx 2 + vx 2 (τ )dτ + 0

t eγ τ v2L ∞ θθx 2 (τ )dτ , ∀t > 0. (4.1.62) ≤ C1 1 + 0


179

Proof. By equations (4.1.13)–(4.1.15), it is easy to verify that (ρ, v, S) satisfies 2n−2 v2 r θx n−1 n−2 2 + σr = κ v − 2(n − 1)μr v , CV θ + 2 t u x

(4.1.63)

St − (κρr 2n−2 θx /θ )x − κρ(r n−1 θx /θ )2 − βρ(r n−1 v)2x /θ +2(n − 1)μ(r n−2 v 2 )x /θ = 0

(4.1.64)

with σ = β(r n−1 v)x /u − Rθ/u. Since u¯ t = 0, θ¯t = 0, we have, by (4.1.63) and (4.1.64), that ρ θ¯ κ(r n−1 θx )2 n−1 2 E t (ρ , v, S) + β(r = βρ(r n−1 v)(r n−1 v)x v)x + (4.1.65) θ θ ¯ n−1 v − 2(n − 1)μ(1 − θ¯ /θ )(r n−2 v 2 )x , +κ(1 − θ¯ /θ )ρr 2n−2 θx − R(ρθ − ρ¯ θ)r x β(ρx /ρ)2 /2 + ρx r 1−n v/ρ t + (n − 1)r −n v 2 ρx /ρ + Rθρx2 /ρ = ρ(r 1−n v)x (r n−1 v)x −1

−Rρx θx /ρ − [ρr 1−n v(r n−1 v)x ]x

(4.1.66)

with ρ¯ = 1/u. ¯ Multiplying (4.1.65), (4.1.66) by eγ t , ηeγ t respectively, and then adding the results up, we get

∂ M(t) + eγ t θ¯ ρ β(r n−1 v)2x + κ(r n−1 θx )2 /θ /θ ∂t

(4.1.67)

+ ηeγ t Rθρx2 /ρ + Rρx θx /ρ − ρ(r 1−n v)x (r n−1 v)x + (n − 1)r −n v 2 ρx /ρ

= γ eγ t E + ηβ(ρx /ρ)2 /2 + ηρx r 1−n v/ρ + eγ t ρ(βr n−1 v − ηr 1−n v)(r n−1 v)x ¯ n−1 v − 2(n − 1)μ(1 − θ¯ /θ )(r n−2 v 2 )x eγ t + κ(1 − θ¯ /θ )ρr 2n−2 θx − R(ρθ − ρ¯ θ)r x

where M(t) = eγ t E + ηβ(ρx /ρ)2 /2 + ηr 1−n vρx /ρ . Integrating (4.1.67) over [0, L] × [0, t], by Lemmas 4.1.1–4.1.3, Young’s inequality, and Poincaré´ ’s inequality, integrating by parts in the last term, we deduce that

L

M1 (t) ≡ +η

t 0 0

M(t)d x +

0 L

t 0 0

L

eγ τ θ¯ ρ[β(r n−1 v)2x + κ(r n−1 θx )2 /θ )]/θ (x,τ )d x dτ

eγ τ (Rθρx2 /ρ + Rρx θx − ρ(r 1−n v)x (r n−1 v)x + (n − 1)r −n v 2 ρx /ρ)(x,τ )d x dτ

t

¯ 2 ) d x dτ eγ τ (1/2 + ηr 2−2n /2β)v 2 + ηβ(ρx /ρ)2 + C1 (|u − u| ¯ 2 + |S − S| 0 0 t t + C1 η eγ τ vx 2 (τ )dτ + C1 η−1 eγ τ v2L ∞ θθx 2 (τ )dτ. (4.1.68)

≤ C1 + γ

0

L

0


180

On the other hand, by Lemmas 4.1.1–4.1.3, Young’s inequality and keeping in mind that (r 1−n v)x ≤ C1 (r n−1 v)x , v4 ≤ C1 (r n−1 v)x 2 , we deduce that

¯ 2 ) + ηβρx /ρ2 /4 M1 (t) ≥ eγ t (1/2 − ηa 2−2n /β)v2 + C1−1 (u − u ¯ 2 + S − S t L

eγ τ C1−1 (θ¯ /θ − C1 η)(r n−1 v)2x + (κ θ¯ /θ − Rη/a 2n−2 )ρ(r n−1 θx )2 /θ + 0 0 +ηθρx2 /2ρ d x dτ ¯ 2 + ρx 2 + u x 2 ) ¯ 2 + S(t) − S ≥ C1−1 eγ t (v(t)2 + u(t) − u t +C1−1 eγ τ (vx 2 + θθx 2 + ρx 2 + u x 2 )dτ

(4.1.69)

0

where we take η so small that 0 < 1/2 − ηa 2−2n /β, C1−1 − C1 η > 0, θ¯ /θ − C1 η ≥ ¯

C1 − C1 η > 0 and κθθ − Rη/a 2n−2 ≥ C1 − Rη/a 2n−2 > 0. By the mean value theorem, (4.1.47)–(4.1.49), Poincaré´ ’s inequality and Lemmas 4.1.1–4.1.3, we have

and

¯ ≤ C1 (u − u S − S ¯ + θ − θ¯ ) ≤ C1 (u x + θθ x + vx )

(4.1.70)

¯ + u − u). θ − θ¯ ≤ C1 (S − S ¯

(4.1.71)

Thus it follows from (4.1.68)–(4.1.71) that ¯ 2 + u x (t)2 ¯ 2 + S(t) − S eγ t v(t)2 + u(t) − u t + eγ τ vx 2 + θθx 2 + u x 2 (τ )dτ 0 t t ≤ C1 + C1 γ eγ τ vx 2 + θθx 2 + u x 2 (τ )dτ + C1 v2L ∞ eγ τ θθx 2 (τ )dτ 0

0

which with (4.1.47) implies that there exists a positive constant γ1 = γ1 (C1 ) = such that for any fixed γ ∈ (0, γ1 ] (4.1.62) holds. Thus the proof is complete.

1 2C 1

>0

Lemma 4.1.5. For any (u 0 , v0 , θ0 ) ∈ H+1 , there exists a positive constant γ1 = γ1 (C1 ) ≤ γ1 such that for any fixed γ ∈ (0, γ1 ], the estimate (4.1.24) holds. Proof. System (4.1.13)–(4.1.15) can be rewritten as ρt + ρ 2 (r n−1 v)x = 0, vt − βr

n−1

[(ρ(r

C V θt − κ(ρr

n−1

2n−2

(4.1.72)

v)x ]x + Rr

n−1

(ρθ )x = 0,

θx )x + Rρθ (r

n−1

v)x

− βρ(r n−1 v)2x + 2μ(n − 1)(r n−2 v 2 )x = 0.

(4.1.73) (4.1.74)


181

Multiplying (4.1.73), (4.1.74) by −eγ t vx x , −eγ t θx x respectively, then integrating them over [0, L] × [0, t], and adding the results up, by Young’s inequality, the imbedding theorem and keeping in mind that βρr n−1 vx x (r n−1 v)x x ≥ C1−1 vx2 x − C1 (v 2 + vx2 + u 2x ), ρ(r 2n−2 θx )x θx x ≥ C1−1 θx2x − C1 θx2 , we finally deduce that t 1 γt e (vx (t)2 + C V θθx (t)2 ) + C1−1 eγ τ (vx x 2 + θθx x 2 )(τ )dτ 2 0 t eγ τ (vx 2 + C V θθx 2 )(τ )dτ ≤ C1 + C1 (γ ) t

0

eγ τ r n−1 [R(ρθ )x − βρx (r n−1 v)x ]vx x + [2μ(n − 1)(r n−2 v 2 )x + 0 0 −Rρθ (r n−1 v)x − βρ(r n−1 v)2x − κρx r 2n−2 θx ]θθ x x d x dτ t eγ τ [(ρx + θθ x + ρx vx 1/2 vx x 1/2 )vx x ≤ C1 + C1 L

0

+(vx + vx 3/2 vx x + ρx θθx 1/2 θθx x 1/2 + ρx θθx )θθx x ]dτ t eγ τ (vx 2 + u x 2 + θθx 2 )(τ )dτ +C1 (γ ) 0 t t 1 eγ τ (vx x 2 + θθx x 2 )(τ )dτ + C1 eγ τ v2L ∞ θθx 2 (τ )dτ ≤ C1 + 2C1 0 0 which, together with Gronwall’s inequality, Lemma 4.1.1 and equations (4.1.13)–(4.1.14), gives eγ t vx (t)2 + θθx (t)2 t + eγ τ vx x 2 + θθx x 2 + vt 2 + θt 2 (τ )dτ ≤ C1 , ∀t > 0. (4.1.75) 0

By (4.1.75), Lemma 4.1.1 and Lemma 4.1.4, we have ¯ 2H 1 + θ (t) − θ¯ 2H 1 eγ t v(t)2H 1 + u(t) − u t eγ τ u − u ¯ 2H 1 + θθx 2 + vx 2 (τ )dτ ≤ C1 , ∀t > 0. +

(4.1.76)

0

Thus the estimate (4.1.24) follows from (4.1.47)–(4.1.50), (4.1.52)–(4.1.54) and (4.1.75)– (4.1.76). The proof is complete. Lemma 4.1.6. For any (u 0 , v0 , θ0 ) ∈ H+1 , the generalized global solution (u(t), v(t), θ (t)) in H+1 to problem (4.1.13)–(4.1.17) satisfies (u(t), v(t), θ (t)) ∈ C([0, +∞), H+1 ), u(t) ∈ C 1/2 ([0, +∞), H 1), v(t), θ (t) ∈ C 1/2 ([0, +∞), L 2 ).

(4.1.77) (4.1.78)


182

Moreover, this generalized global solution (u(t), v(t), θ (t)) defines a nonlinear C0 -semigroup S(t) on H+1 . Proof. For any t1 ≥ 0, t > 0, integrating (4.1.13) over (t1 , t) and using Lemmas 4.1.1– 4.1.2, we obtain t 1/2 u(t) − u(t1 ) H 1 ≤ C1 (r n−1 v)x x 2 dτ |t − t1 |1/2 ≤ C1 |t − t1 |1/2 t1

which implies u(t) ∈ C 1/2 ([0, +∞), H 1). In the same manner we easily prove v(t), θ (t) ∈ C 1/2 ([0, +∞), L 2 ). By Lemma 4.1.1, we know that for any t > 0, the operator S(t) : (u 0 , v0 , θ0 ) ∈ H+1 −→ (u(t), v(t), θ (t)) ∈ H+1 exists, where (u(t), v(t), θ (t)) is the unique generalized global solution to problem (4.1.13)–(4.1.17) with the initial datum (u 0 , v0 , θ0 ), by the uniqueness of generalized global solutions, and S(t) satisfies on H+1 that, for any t1 , t2 ∈ [0, +∞), S(t1 + t2 ) = S(t1 )S(tt2 ) = S(tt2 )S(t1 ).

(4.1.79)

Moreover, by Lemma 4.1.1, S(t) is uniformly bounded on H+1 with respect to t > 0, i.e., S(t)L(H H 1 ,H H 1 ) ≤ C1 . +

+

(4.1.80)

We first verify the continuity of S(t) with respect to the initial data in H+1 for any t > 0. To this end, we assume that (u 0 j , v0 j , θ0 j ) ∈ H+1 , ( j = 1, 2),

(u j , v j , θ j ) = S(t)(u 0 j , v0 j , θ0 j ),

and (u, v, θ ) = (u 1 , v1 , θ1 ) − (u 2 , v2 , θ2 ). Subtracting the corresponding equations (4.1.13)–(4.1.17) satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), we obtain u t = (r1n−1 v)x + [(r1n−1 − r2n−1 )v2 ]x , (4.1.81) (r n−1 v2 )x u ((r n−1 − r2n−1 )v2 )x (r1n−1 v)x vt = βr1n−1 − 1 + 1 + β(r1n−1 − r2n−1 ) u1 u1u2 u2 x (r2n−1 v2 )x u − θ u θ θ2 2 2 + Rr1n−1 − R(r1n−1 − r2n−1 ) , × u2 u1u2 u2 x x x

(4.1.82)


183

r22n−2 θ2x u (r12n−2 − r22n−2 )θ2x r12n−2 θx 1 β(r1n−1 v)x C V θt = κ − + + u1 u 1u 2 u1 u1 x

+ β((r1n−1 − r2n−1 )v2 )x − Rθ (r1n−1 v1 )x + [β(r2n−1 v2 )x − Rθ2 ]

×

u 2 (r1n−1 v)x − (r2n−1 v2 )x u + u 1 ((r1n−1 − r2n−1 )v2 )x u 1u 2

− 2μ(n − 1)[r1n−2 (v1 + v2 )v + (r1n−2 − r2n−2 )v22 ]x , (4.1.83) t = 0 : u = u 0 := u 01 − u 02 , v = v0 := v01 − v02 , θ = θ0 := θ01 − θ02 , x = 0, L : v = θx = 0 where r j (x, t) = r0 j (x) +

t 0

v j (x, τ )dτ, r0 j (x) = a n + n

x 0

1/n u 0 j (y)d y

(4.1.84)

and r n−1 (x, t)rr j x (x, t) = u j (x, t), j

j = 1, 2,

∀(x, t) ∈ [0, L] × [0, +∞).

(4.1.85)

By Lemma 4.1.1, we know that for any t > 0 and for j = 1, 2, t ¯ 2 1 + (u j x 2 +v j 2 2 +θθ j x 2 +θθ j x x 2 )(τ )dτ ≤ C1 . ¯ v j (t), θ j (t)− θ) (u j (t)− u, H H 0

(4.1.86) Here and hereafter in the proof of this lemma, C1 > 0 denotes the universal constant depending only on the H 1 norm of the initial data (u 0 j , v0 j , θ0 j ) and min u 0 j (x) and x∈[0,L]

min θ0 j (x) ( j = 1, 2), but independent of t.

x∈[0,L]

By (4.1.17) and (4.1.85), we have n n r1n (0, t) − r2n (0, t) = r01 (0) − r02 (0) = 0

which, together with (11.1.85), implies that for any (x, t) ∈ [0, L] × [0, +∞), x r1n (x, t) − r2n (x, t) = n u(y, t)d y, (r1n (x, t) − r2n (x, t))x = nu(x, t). (4.1.87) 0

For any integer k ≥ 1, we know that r1k (x, t) − r2k (x, t) = (r1n (x, t) − r2n (x, t))d(x, t)

(4.1.88)

where d(x, t) = d(r1 (x, t), r2 (x, t)) is a C 2 function satisfying |d(x, t)| + |ddx (x, t)| + |d dt (x, t)| ≤ C1 ,

∀(x, t) ∈ [0, L] × [0, +∞).

(4.1.89)


184

Thus by (4.1.87)–(4.1.89), we obtain |r1k (x, t) − r2k (x, t)| ≤ C1 u(t) L 1 , |(r1k (x, t) ((r1k

− r2k (x, t))x |

− r2k )v2 )x x (t)

∀(x, t) ∈ [0, L] × [0, +∞),

(4.1.90)

≤ C1 (|u(x, t)| + u(t) L 1 ), ∀(x, t) ∈ [0, L] × [0, +∞),

(4.1.91)

≤ C1 (1 + v2x x (t))u(t) H 1 ,

(4.1.92)

∀t > 0.

Multiplying (4.1.81), (4.1.82) and (4.1.83) by u, v and θ respectively, adding them up and integrating the result over [0, L], recalling vx ≤ C1 (r1n−1 v)x and using (4.1.87)– (4.1.92), we deduce that for any small > 0, L β(r1n−1 v)2x + κr12n−2 θx2 1 d (u(t)2 + v(t)2 + C V θ (t)2 ) + dx 2 dt u1 0 ≤ ((r1n−1 v)x (t)2 + vx (t)2 + θθx (t)2 ) + C1 H1 (t)(u(t)2 + v(t)2 + θ (t)2 ) which, together with Lemma 4.1.1, leads to d (u(t)2 + v(t)2 + C V θ (t)2 ) + C1−1 (vx (t)2 + (r1n−1 v)x (t)2 + θθx (t)2 ) dt (4.1.93) ≤ C1 H1 (t)(u(t)2 + v(t)2 + θ (t)2 ) where, by (4.1.86), H1(t) = v1x x (t)2 + v2x x (t)2 + θ2x x (t)2 + 1 satisfies that for any t > 0, t H1(τ )dτ ≤ C1 (1 + t). (4.1.94) 0

By Lemma 4.1.1, the embedding theorem, (4.1.82) and (4.1.87)–(4.1.92), we get (r1n−1 v)x x (t)2 ≤ C1 vt (t)2 + (r1n−1 v)x (t)2L ∞ + θ (t)2H 1 + ((r1n−1 v2 )x x (t)2 + v2x x (t)2 )u(t)2H 1 1 ≤ (r1n−1 v)x x (t)2 + C1 (vt (t)2 + θ (t)2H 1 ) 2 + C1 ((r1n−1 v)x (t)2 + v2x x (t)2 u(t)2H 1 ) which gives vx x (t)2 + (r1n−1 v)x x (t)2 ≤ C1 vt (t)2 + C1 H1(t)((r1n−1 v)x (t)2 + u(t)2H 1 + θ (t)2H 1 ).

(4.1.95)

Using (4.1.81) and (4.1.13), we have n−1 u 1x (r2n−1 v2 )x − u 2x (r1n−1 v1 )x ((r1 − r2n−1 )v2 )x (r1n−1 v)x ux = + − . u1 u1 t u1 u 21 x x

(4.1.96)


185

Inserting (4.1.96) into (4.1.82) we obtain n−1 (r1 v2 )x u ((r1n−1 − r2n−1 )v2 )x ux 1−n = r1 vt + β − − βr11−n (r1n−1 − r2n−1 ) β u1 t u 1u 2 u2 x (r2n−1 v2 )x (θ2 u − θ u 2 θ2 −R + Rr11−n (r1n−1 − r2n−1 ) × u2 u1u2 u 2 x x x u 1x (r2n−1 v2 )x − u 2x (r1n−1 v1 )x ((r1n−1 − r2n−1 )v2 )x −β + β . (4.1.97) u1 u 21 x

ux u1 ,

Multiplying (4.1.97) by integrating it over [0, L], and using Lemma 4.1.1, (4.1.87)– (4.1.92) and (4.1.96), we conclude L d θ2 u 2x u x 2 β (t) + R dx 2 dt u 1 0 u 1u 2 ≤ C1 vt (t)2 + C1 H1(t)((r1n−1 v)x (t)2 + u(t)2H 1 + θ (t)2H 1 ). That is, d u x 2 (t) + C1−1 u x (t)2 dt u 1 ≤ C1 vt (t)2 + C1 H1(t)((r1n−1 v)x (t)2 + u(t)2H 1 + θ (t)2H 1 ).

(4.1.98)

Multiplying (4.1.82) by vt , integrating it over [0, L], and using Lemma 4.1.1, (4.1.87)– (4.1.92) and the identity (r1n−1 vt )x = (r1n−1 v)t x − (n − 1)(r1n−2 v1 v)x , we obtain n−1 d (r1 v)x 2 (t) + C1−1 vt (t)2 ≤ C1 H1(t)((r1n−1 v)x (t)2 + u(t)2H 1 + θ (t)2H 1 ). √ dt u1 (4.1.99) Similarly, multiplying (4.1.83) by θt , we get n−1 d r 1 θ x 2 (t) + C1−1 θt (t)2 ≤ C1 H1(t)((r1n−1 v)x (t)2 + u(t)2H 1 + θ (t)2H 1 ). √ dt u1 (4.1.100) Multiplying (4.1.99) by a large number N1 , then adding up the result, (4.1.93), (4.1.97) and (4.1.99), we finally conclude

d G 1 (t) ≤ C1 H1(t)((r1n−1 v)x (t)2 + u(t)2H 1 + θ (t)2H 1 ) dt ≤ C1 H1(t)G 1 (t) where

(4.1.101)

u x 2 G 1 (t) = u(t)2 + (t) + v(t)2 u1 (r n−1 v) 2 r n−1 θ 2 x x + N1 1√ (t) + C V θ (t)2 + 1√ (t) , u1 u1


186

which, combined with Gronwall’s inequality and (4.1.94), implies u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 t ≤ C1 G 1 (0) exp(C1 H1 (τ )dτ ) 0

≤ C1 exp(C1 t)(u 0 2H 1 + v0 2H 1 + θ0 2H 1 ), ∀t > 0. That is, S(t)(u 01 , v01 , θ01 ) − S(t)(u 02 , v02 , θ02 ) H 1

+

≤ C1 exp(C1 t)(u 01 , v01 , θ01 ) − (u 02 , v02 , θ02 ) H 1

+

(4.1.102)

which leads to the continuity of S(t) with respect to the initial data in H+1 . By (4.1.79)– (4.1.80), in order to derive (4.1.77), it suffices to show that for any (u 0 , v0 , θ0 ) ∈ H+1 , S(t)(u 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 1 → 0, as t → 0+ , +

(4.1.103)

which also yields S(0) = I

(4.1.104)

with I being the unit operator on H+1 . To derive (4.1.103), we choose a function sequence m m (u m 0 , v0 , θ0 ) which is smooth enough, for example, m m 1+α (u m ( n ) × C 2+α ( n ) × C 2+α ( n )) ∩ H+1 0 , v0 , θ0 ) ∈ (C

for some α ∈ (0, 1), such that m m (u m 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 1 → 0, as m → +∞. +

(4.1.105)

By the regularity results (see Theorem 4.1.3), we conclude that for arbitrary but fixed T > 0, there exists a unique global smooth solution (u m (t), v m (t), θ m (t)) ∈ (C 1+α (Q T ) × C 2+α (Q T ) × C 2+α (Q T )) ∩ H+1 , Q T = n × (0, T ). This gives for m = 1, 2, 3, . . . , m m + (u m (t), v m (t), θ m (t)) − (u m 0 , v0 , θ0 ) H 1 → 0, as t → 0 . +

(4.1.106)

Fixing T = 1, by the continuity of the operator S(t), (4.1.102) and (4.1.105), for any t ∈ [0, 1], (u m (t), v m (t), θ m (t)) − (u(t), v(t), θ (t)) H 1

+

= ≤

m m S(t)(u m 0 , v0 , θ0 ) − S(t)(u 0 , v0 , θ0 ) H+1 m m C1 (u m 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H+1 → 0,


187

as m → +∞. This together with (4.1.105) and (4.1.106) implies S(t)(u 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 1 = (u(t), v(t), θ (t)) − (u 0 , v0 , θ0 ) H 1 +

+


+

m m + (u m (t), v m (t), θ m (t)) − (u m 0 , v0 , θ0 ) H 1

+

m m + + (u m 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 1 → 0, as m → +∞, t → 0 . +

which gives (4.1.103) and (4.1.104). Thus the proof is complete.

Using Lemmas 4.1.1–4.1.2, (4.1.47)–(4.1.50) and (4.1.52)–(4.1.54), we complete the proof of Theorem 4.1.1.

4.1.2 Uniform a priori estimates in H 2 In this subsection we shall complete the proof of Theorem 4.1.2. We begin with the following lemma. Lemma 4.1.7. If (u 0 , v0 , θ0 ) ∈ H+2 , then problem (4.1.13)–(4.1.17) admits a unique generalized global solution (u(t), v(t), θ (t)) in H+2 , which defines a nonlinear C0 -semigroup S(t) (also denoted by S(t) by the uniqueness of solution in H+1 ) on H+2 such that for any (u 0 , v0 , θ0 ) ∈ H+2 , S(t)(u 0 , v0 , θ0 ) H 2 = (u(t), v(t), θ (t)) H 2 ≤ C2 , ∀t > 0,

(4.1.107)

S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)) ∈ C([0, +∞), H+2 ),

(4.1.108)

+

u(t) ∈ C

1/2

+

2

([0, +∞), H ), v(t), θ (t) ∈ C

1/2

1

([0, +∞), H ).

(4.1.109)

The proof of Lemma 4.1.7 can be divided into six lemmas. In what follows, our attention will be paid to the uniform estimate of the specific volume u in H 2. To this end, similarly to that in [450] for n = 1, we need to give a representation of u, which has been obtained for n = 2 or n = 3 in [170]. For n = 1, a similar representation with more general constitutive relations of p = p(u, θ ) (the pressure) and σ = σ (u, θ, vx ) (the stress) has been given in Chapter 2 (see, e.g., Lemma 2.1.3). Lemma 4.1.8. For each t ≥ 0, there exists a point x 0 = x 0 (t) ∈ [0, L] such that the specific volume u(x, t) has the following representation: u(x, t) =

D(x, t) R t θ (x, s)B(x, s) 1+ ds , B(x, t) β 0 D(x, s)

∀ ∈ [0, L], ∀x

(4.1.110)


188

where L x 1 1 u 0 (x) r01−n (y)v0 (y)d yd x β u∗ 0 0

x 1−n 1−n r (y, t)v(y, t)d y − r0 (y)v0 (y)d y ,

D(x, t) = u 0 (x) exp +

x x 0 (t )

(4.1.111)

0

t L 2 1 1 v + Rθ (x, s)d x ds β u∗ 0 0 n t L n (n − 1)a + r −n (x, s)v2 (x, s)d x ds nu ∗ 0 0 t L −n 2 r (y, s)v (y, s)d yds , +(n − 1)

B(x, t) = exp

0

u∗ =

L 0

(4.1.112)

x

u 0 (x)d x.

(4.1.113)

Proof. We adapt and modify the idea of the proof of Lemma 2.1.3 (see also [170]). Let θ β(r n−1 v)x −R , u u t x φ(x, t) = σ (x, s)ds + r01−n (y)v0 (y)d y σ (x, t) =

0

+(n − 1)

0

t 0

(4.1.114)

L

r −n (y, s)v2 (y, s)d yds.

(4.1.115)

x

Then by (4.1.14), a partial integration in the variable t, and (4.1. 6) and (4.1.10) (i.e.,vt = v, r x = r 1−n u), φx (x, t) = r 1−n (x, t)v(x, t). (4.1.116) Note that in view of (4.1.6) and (4.1.10), φ satisfies φt = β

θ (n − 1) (r n )x (r n−1 v)x −R + u u n u

L

r −n v 2 d y.

(4.1.117)

x

Multiplying (4.1.17) by u, using (4.1.13) and (4.1.16), we arrive at n−1 n (r )x (uφ)t − (r n−1 vφ)x = −v 2 − Rθ + β(r n−1 v)x + n n − 1 n L −n 2 v2 r r v dy . = − − Rθ + β(r n−1 v)x + n n x

L

r −n v 2 d y

x

(4.1.118)

Keeping in mind that v vanishes on the boundary and r (0, t) = a, we integrate (4.1.118)


over [0, L] × [0, t] to infer L (uφ)(x, t)d x = 0

L

u 0 (x)φ0 (x)d x −

0

−

n−1 n a n

0

t 0

L

L

189

1

0

v2 + Rθ d x ds n

r −n v 2 d x ds

(4.1.119)

0

where φ0 (x) = φ(x, 0). It follows from integration of (4.1.13) over [0, L] × [0, t] and use of (4.1.17) that L L u(x, t)d x = u 0 (x)d x ≡ u ∗ ∀t ≥ 0. (4.1.120) 0

0

Note that u > 0. If we apply the mean value theorem to (4.1.119) and use (4.1.120), then we conclude that for each t ≥ 0, there is an x 0 (t) ∈ [0, L] such that L 1 φ(x 0 (t), t) = φ(x, t)u(x, t)d x. (4.1.121) u∗ 0 Therefore from (4.1.115), (4.1.119) and (4.1.121), we get for any t ≥ 0, x0 (t ) t L t σ (x 0 (t), s)ds = φ(x 0 (t), t) − r01−n v0 d y − (n − 1) r −n v 2 d yds 0

t

0

0

x 0 (t )

(n − 1)a n t L −n 2 v2 ( + Rθ )d x ds − r v d x ds n nu ∗ 0 0 0 0 L x0 (t ) t L 1 r −n v 2 d x ds + u 0 φ0 d x − r01−n v0 d y. −(n − 1) u∗ 0 0 x 0 (t ) 0 1 =− u∗

L

(4.1.122)

Using (4.1.13), we may write (4.1.14) in the form r 1−n vt = β[log u]xt − R

θ = σx . u x

(4.1.123)

Integrate (4.1.123) over [0, t], then integrate over [x 0 (t), x] with respect to x. If we integrate by parts with respect to t, utilize (4.1.6), (4.1.10) and (4.1.122), then we infer t t x t θ β log u − R ds = β log u 0 + σ (x 0 (t), s)ds + r 1−n vt dsd y u 0 0 x 0 (t ) 0 t L 2 (n − 1)a n t L −n 2 1 v + Rθ d x ds − r v d x ds = β log u 0 − u∗ 0 0 n nu ∗ 0 0 t L L x 1 −(n − 1) r −n v 2 d yds + r 1−n vd y + u 0 φ0 d x u∗ 0 0 x x 0 (t ) x r01−n v0 d y, (4.1.124) − 0


190

which, when the exponentials are taken, turns into 1 R t θ (x, s) B(x, t) = exp{ ds}. D(x, t) u(x, t) β 0 u(x, s)

(4.1.125)

Multiplying (4.1.125) by Rθ/β and integrating over [0, t], we arrive at R t θ (x, s) R t θ (x, s)B(x, s exp{ ds} = 1 + ds. β 0 u(x, s) β 0 D(x, s)

Inserting (4.1.126) into (4.1.125), we obtain (4.1.110). Lemma 4.1.9. For any (u 0 , v0 , θ0 ) ∈

H+2 ,

(4.1.126)

the following estimate holds,

u(t) − u ¯ H 2 + u(t) − u ¯ W 1,∞ + (n − 1)r (t) − r¯ H 3 ≤ C2 ,

∀t > 0.

(4.1.127)

Proof. The proof is motivated by the idea in Chapter 2 (Lemma 2.3.9) for n = 1, but because B(x, t) depends on the variables x and t for n = 2 or n = 3, so the situation is more complicated than that in Chapter 2. Let B(x, t) = Z 1 (t)Z 2 (x, t) where t 1 2 v 1 1 + Rθ (x, s)d x ds Z 1 (t) = exp β u∗ 0 0 n t 1 n (n − 1)a + r −n (x, s)v2 (x, s)d x ds , nu ∗ 0 0

(n − 1) t L −n 2 r (y, s)v (y, s)d yds . Z 2 (x, t) = exp β 0 x

(4.1.128) (4.1.129)

Clearly, by Lemmas 4.1.1, (4.1.111)–(4.1.112) and (4.1.128)–(4.1.129), we easily deduce that for any t ≥ s ≥ 0, x ∈ [0, L], e−C1 (t −s) ≤ Z 2 (x, s)/Z 2 (x, t) ≤ 1,

(4.1.130) −C 1−1 (t −s)

−C1 (t −s)

e ≤ B(x, s)/B(x, t) ≤ Z 1 (s)/Z 1 (t) ≤ e , 1 = Z 2 (L, t) ≤ Z 2 (x, t) ≤ Z 2 (0, t), |Bx (x, t)|, |Bx x (x, t)| ≤ C1 B(x, t),

(4.1.131) (4.1.132)

0 < C1−1 ≤ D(x, t), D −1 (x, t) ≤ C1 , B −1 (x, t) ≤ Z 1−1 (t) ≤ 1.

(4.1.133)

Hence, by Lemma 4.1.1, (4.1.111) and the embedding theorem, we get Dx (x, t) ≤ C1 , Dx x (x, t) ≤ C2 ,

Dx (x, t) L ∞ ≤ C2 .

(4.1.134)

On the other hand, a straightforward calculation gives u x x (x, t) = I1 (x, t) + I2 (x, t) + I3 (x, t)

(4.1.135)


where

I1 = I11

R 1+ β

t 0

θ (x,s)B(x,s) ds , D(x,s)

191

(4.1.136)

Dx x (x,t) 2Dx (x,t)Bx (x,t) + D(x,t)Bx x (x,t) 2D(x,t)Bx2 (x,t) − + , (4.1.137) B(x,t) B 2 (x,t) B 3 (x,t) 2R(Dx (x,t)B(x,t) − D(x,t)Bx (x,t)) t θ x (x,s)B(x,s) + θ (x,s)Bx (x,s) I2 = D(x,s) β B 2(x,t) 0 θ (x,s)B(x,s)Dx (x,s) ds, (4.1.138) − D 2 (x,s) R D(x,t) t θx x (x,s)B(x,s) + 2θθx (x,s)Bx (x,s) + θ (x,s)Bx x (x,s) I3 = β B(x,t) 0 D(x,s) 2θθx (x,s)B(x,s)Dx (x,s) + 2θ (x,s)Bx (x,s)Dx (x,s) + θ (x,s)B(x,s)Dx x (x,s) − D 2 (x,s)

2 2θ (x,s)B(x,s)Dx (x,s) ds. (4.1.139) + D 3 (x,s)

I11 =

Thus by Lemma 4.1.1, Cauchy’s inequality and (4.1.130)–(4.1.139) imply 2 Bx (x, t) Bx x (x, t) Bx (x, t) I11 ≤ Dx x (x, t) + C2 2 + 2 + 3 B (x, t) B (x, t) B (x, t) −1

≤ C2 + C2 e−C1

t

≤ C2 ,

(4.1.140)

t 2 θ (x, s)B(x, s) ds I11 (x, t) d x (4.1.141) β D(x, s) 0 0 2 2 t L L t B(x, s) B(x, s) 2 ds d x + C1 ds d x ≤ C2 , Dx x (x, t) ≤ C1 0 0 B(x, t) 0 0 B(x, t)

L R

L

2

II2 ≤ C2

0 L

II3 2 ≤ C2

0

t 0 t 0

B(x, s) ds B(x, t) B(x, s) ds B(x, t)

2 d x ≤ C2 ,

(4.1.142)

2 dx

2 (|θθ x x (x, s)| + |θθx (x, s)|)B(x, s) ds d x B(x, t) 0 0 2 t L t B(x, s) ≤ C2 + C2 ds (θθx2x (x, s) + θx2 (x, s))dsd x ≤ C2 B(x, t) 0 0 0

+C2

L

t

which, combined with (4.1.135)–(4.1.142), Lemma 4.1.1 and the interpolation inequality, gives u(t) − u ¯ H 2 + u(t) − u ¯ W 1,∞ ≤ C2 , ∀t > 0. (4.1.143)


192

By (4.1.51) and (4.1.143), we derive that dˆx x x ≤ C2 which with (4.1.51)–(4.1.54) and the Poincaré´ inequality implies (r (t) − r¯ )x x x ≤ C1 (u x x (t) + u x (t) + u(t) − u ¯ L ∞ ) + C1 u(t) − u ¯ L 1 dˆx x x ¯ H2. ≤ C2 (u x x (t) + u x (t)) ≤ C2 u(t) − u

(4.1.144)

Thus the estimate (4.1.127) follows from (4.1.143)–(4.1.144) and (4.1.51)–(4.1.54). The proof is complete. Lemma 4.1.10. For any (u 0 , v0 , θ0 ) ∈ H+2 , the following estimates hold for any t > 0: θt (t)2 + vt (t)2 +

t 0

(vxt 2 + θθxt 2 )(τ )dτ ≤ C2 ,

(4.1.145)

vx x (t)2 + θθx x (t)2 + (n − 1)rt (t)2H 2 + u t (t)2H 1 + v(t)2W 1,∞ + θ (t) t

2 ¯ vx x x 2 + θθx x x 2 + (n − 1)rt (t)2H 3 + u t (t)2H 2 −θ W 1,∞ + 0 +v(t)2W 2,∞ + θ (t) − θ¯ 2W 2,∞ (τ )dτ ≤ C2 , (4.1.146) t [(n − 1)r − r¯ 2H 3 + u − u ¯ 2H 2 + u − u ¯ 2W 1,∞ ](τ )dτ ≤ C2 . (4.1.147) 0

Proof. By (4.1.14)–(4.1.15), Lemma 4.1.1 and the embedding theorem, we have vt (t) ≤ C1 (vx x (t) + θθx (t) + u x (t))

(4.1.148)

and θt (t) ≤ C1 (r x θx (t) + θθx x (t) + r x v(t)2L 4 + vx (t)2L 4 + r x v(t) + u x (t)) (4.1.149) ≤ C1 (θθx (t) H 1 + vx x (t) + u x (t)). Differentiating equation (4.1.14) with respect to t, then multiplying the resulting equation by vt in L 2 (0, L) and using the estimates rt = v, r x = r 1−n u, (r n−1 vt )x = (r n−1 v)t x − (n − 1)(r n−2 v 2 )x , v(t) ≤ C1 vx (t) ≤ C1 vx x (t),

(4.1.150) (4.1.151)

(r n−1 v)t x (t) ≥ C1−1 vt x (t) − C1 (vx (t) + vt (t)),

(4.1.152)

(r n−1 vt )x (t) + (r n−1 v)t x (t) ≤ C1 (vt x (t) + vx (t) + vt (t)),

(4.1.153)


193

we infer that for any > 0, 1 d vt (t)2 ≤ − 2 dt

L

0

β[(r n−1 v)t x ]2 d x + (r n−1 vt )x (t)2 + C1 ()[(r n−1 v)2x (t)2 u2

+(r n−2 v 2 )x (t)2 + θt (t)2 + (r n−1 v)x (t)2 ] + C1 (vx x (t) +θθx (t) + u x (t))vt (t) ≤ −(C1−1 − )vt x (t)2 + C1 ()(vx x (t)2 + vt (t)2 + θt (t)2 +u x (t)2 + θθx (t)2 ) which, by taking > 0 so small that 0 < ≤

1 2C 1 ,

gives

d vt (t)2 +C1−1 vxt (t)2 ≤ C1 (vx x (t)2 +u x (t)2 +θθx (t)2 +vt (t)2 +θt (t)2 ). dt (4.1.154) Thus it follows from Lemma 4.1.1 and (4.1.154) that

t

vt (t)2 +

vt x 2 (τ )dτ ≤ C2 , ∀t > 0.

(4.1.155)

0

On the other hand, by (4.1.14), Lemma 4.1.1, Lemma 4.1.9, the embedding theorem and Young’s inequality, we deduce vx x (t) ≤ C1 (vt (t) + vx (t) + u x (t) + θθx (t) + vx (t)1/2 vx x (t)1/2 ) 1 ≤ vx x (t) + C1 (vt (t) + vx (t) + u x (t) + θθx (t)) (4.1.156) 2 which with (4.1.13), (4.1.18) and Lemma 4.1.1 leads to vx x (t) ≤ C2 , v(t)W 1,∞ ≤ C1 vx x (t) ≤ C2 , (n − 1)rt (t) H 2 + u t (t) H 1 ≤ C1 v(t) H 2 ≤ C1 vx x (t) ≤ C2 .

(4.1.157) (4.1.158)

Differentiating equation (4.1.14) with respect to x, and using (4.1.13), we arrive at ∂ β ∂t

uxx u

+

θuxx u2

(4.1.159) n−1 v) n−1 v) u − Rθ u β(r − β(r Rθ θ x x x x x x + = r 1−n vt x + (n − 1)r −n u u u2 2β(r n−1 v)x x u x − 2Rθθ x u x 2Rθ u 2x − 2β(r n−1 v)x u 2x Rθθ x x . + + + u u2 u3

Multiplying (4.1.159) by u x x /u in L 2 (0, L), exploiting Lemmas 4.1.1–4.1.2, Lemma

194


4.1.5, Lemma 4.1.8 and the interpolation inequality, we deduce that u d u x x 2 x x 2 (t) + C1−1 (t) dt u u

1 u x x 2 (t) + C1 θθx (t)2 + u x (t)2 + (r n−1 v)x x (t)2 ≤ 4C1 u +(r n−1 v)x (t)2L ∞ u x (t)2 + θθx x (t)2 + vt x (t)2 + (r n−1 v)x x (t)2 u x (t)2L ∞ +θθx (t)2 u x (t)2L ∞ + u x (t)2 u x (t)2L ∞ + u x (t)2 (r n−1 v)x (t)2L ∞

1 u x x 2 (t) + C2 θθx (t)2 + u x (t)2 + vx x (t)2 + θθx x (t)2 ≤ 4C1 u +vt x (t)2 + u x x (t)u x (t)

1 u x x 2 (t) + C2 θθx (t)2 + u x (t)2 + vx x (t)2 + θθx x (t)2 + vt x (t)2 , ≤ 2C1 u i.e., 1 d u x x 2 u x x 2 (t) + (t) dt u 2C1 u 2 ≤ C2 (θθx (t) + u x (t)2 + vx x (t)2 + θθx x (t)2 + vt x (t)2 ) which with Lemma 4.1.1 and Lemma 4.1.5 gives t u x x (t)2 + u x x 2 (τ )dτ ≤ C2 ,

∀t > 0.

(4.1.160)

(4.1.161)

0

Similarly, we conclude from (4.1.15) that κa 2n−2 CV d θt (t)2 + θθxt (t)2 2 dt C1 κa 2n−2 ≤ θθxt (t)2 + C1 θθ x (t)2 + vx (t)2 + vx x (t)2 2C1 + vt (t)2 + θt (t)2 + vxt (t)2

(4.1.162)

which, together with Lemmas 4.1.1–4.1.2, Lemma 4.1.5 and Lemma 4.1.8, implies t θt (t)2 + θθx x (t)2 + θθxt 2 (τ )dτ ≤ C2 , ∀t > 0. (4.1.163) 0

Differentiating (4.1.14) and (4.1.15) with respect to x, using Lemma 4.1.1 and Lemma 4.1.5, we infer vx x x (t) ≤ C2 (vx (t) + u x (t) H 1 + θθx (t) H 1 + vt x (t)),

(4.1.164)

θθx x x (t) ≤ C2 (vx (t) H 1 + u x x (t) + θθx (t) H 1 + θt x (t)).

(4.1.165)


195

Therefore it follows from (4.1.154), (4.1.160) and (4.1.162)–(4.1.165) that

t

(vx x x 2 + θθx x x 2 )(τ )dτ ≤ C2 , ∀t > 0.

(4.1.166)

0

Moreover, appealing to (4.1.47)–(4.1.49), (4.1.13), (4.1.165)–(4.1.166), Lemma 4.1.1 and Lemma 4.1.5, we have rt (t) H 3 + v(t)W 2,∞ + u t (t) H 2 ≤ C2 vx (t) H 2 , ∀t > 0, θ (t) − θ¯ W 2,∞ ≤ C2 (θθx (t) H 2 + vx (t)), ∀t > 0.

(4.1.167) (4.1.168)

Hence the combination of (4.1.155), (4.1.157), (4.1.158), (4.1.161), (4.1.163)–(4.1.168), Lemma 4.1.1 and Lemma 4.1.5 yields the estimates (4.1.145)–(4.1.146). Now for n = 2, 3, similarly to (4.1.52)–(4.1.54), in view of (4.1.145)–(4.1.146), we derive r (t) − r¯ H 3 + u(t) − u ¯ W 1,∞ ≤ C2 u(t) − u ¯ H2 (4.1.169) which combined with (4.1.47)–(4.1.49), Lemma 4.1.1 and Lemma 4.1.5 gives the required estimate (4.1.146). The proof is complete. Combining Lemma 4.1.1 and Lemma 4.1.5 with Lemmas 4.1.8–4.1.10, we easily infer the following lemma. Lemma 4.1.11. For any (u 0 , v0 , θ0 ) ∈ H+2 , the estimate (4.1.28) holds. The estimate (4.1.107) and the global existence in H+2 of generalized solution (u(t), v(t), θ (t)) follows from Lemma 4.1.1, Lemma 4.1.5 and Lemmas 4.1.7–4.1.9. Similarly to (4.1.78), by Lemmas 4.1.8–4.1.10, we can prove that the relation (4.1.109) is valid. To complete the proof of Lemma 4.1.7, it suffices to prove the continuity of S(t) with respect to (u 0 , v0 , θ0 ) ∈ H+2 . This will be done in the next lemma. Lemma 4.1.12. The generalized global solution (u(t), v(t), θ (t)) in H+2 defines a nonlinear C0 -semi group S(t) on H+2 . Proof. The uniqueness of generalized global solutions in H+2 follows from that in H+1 . Thus S(t) satisfies (4.1.79) on H+2 and by Lemmas 4.1.8–4.1.10, S(t)L(H H+2 ,H H+2 ) ≤ C2 . In the same manner as in the proof of Lemma 2.3.10, we assume that (u 0 j , v0 j , θ0 j ) ∈ H+2 , ( j = 1, 2), (u j , v j , θ j ) = S(t)(u 0 j , v0 j , θ0 j ), and (u, v, θ ) = (u 1 , v1 , θ1 ) − (u 2 , v2 , θ2 ). Subtracting the corresponding equations (4.1.13)–(4.1.15) satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), we obtain equations (4.1.81)–(4.1.85). Similarly to (4.1.95), we have θθx x (t)2 + (r12n−2 θx )x 2 ≤ C1 (θt (t)2 + H1 (t)G 1 (t)).

(4.1.170)


196

Differentiating (4.1.82) with respect to x, we see that vt x =

βr1n−1 (r1n−1 v)x x x Rr n−1 θ2 u x x 1 + 1 + 2βr1n−1 (r1n−1 v)x x + R(x, t) u1 u1u2 u1 x (4.1.171)

where R(x, t) are the remaining terms. By Lemma 4.1.9, (4.1.87)–(4.1.92), we easily obtain R(t)2 ≤ C2 (1 + v2x x x (t)2 )(u(t)2H 2 + θ (t)2H 2 + (r1n−1 v)x x (t)2 ). (4.1.172) Here and hereafter in the proof of this lemma, C2 > 0 denotes the universal constant depending only on the H 2 norm of the initial data (u 0 j , v0 j , θ0 j ) and min u 0 j (x) and min θ0 j (x) ( j = 1, 2), but independent of t.

x∈[0,L]

x∈[0,L]

By (4.1.171)–(4.1.172) and the embedding theorem, we infer (r1n−1 v)x x x (t)2 ≤ C1 vt x (t)2 + C2 (u x x (t)2 + (r1n−1 v)x x (t)2L ∞ + R(t)2 ) 1 ≤ (r1n−1 v)x x x (t)2 + C1 vt x (t)2 + C2 (1 + v2x x x (t)2 )((r1n−1 v)x x (t)2 2 + u(t)2H 2 + θ (t)2H 2 ) which gives vx x x (t)2 + (r1n−1 v)x x x (t)2 ≤ C1 vt x (t)2 + C2 (1 + v2x x x (t)2 ) (r1n−1 v)x x (t)2 +u(t)2H 2 + θ (t)2H 2 . (4.1.173) By (4.1.81) and (4.1.171), we see that (r1n−1 v1 )x u x x ((r1n−1 − r2n−1 )v2 )x x x − u1 u 21 1 −2β(r1n−1 v)x x − r11−n R(x, t). (4.1.174) u1 x

θ2 u x x uxx )t + R = r11−n vt x + β β( u1 u1u2

Multiplying (4.1.174) by uux1x , integrating the resulting equation over [0, L] and using Lemmas 4.1.1–4.1.2, Lemmas 4.1.8–4.1.9 and (4.1.87)–(4.1.92), we conclude d uxx (t)2 + C1−1 u x x (t)2 ≤ C1 vt x (t)2 + C2 H2(t)(u(t)2H 2 + (r1n−1 v)x x (t)2 dt u 1 +θ (t)2H 2 ) ≤ C1 vt x (t)2 + C2 H2(t)[u(t)2H 2 + v(t)2H 2 +θ (t)2H 2 + vt (t)2 ]

(4.1.175)


197

where H2(t) = 1 + v2x x x (t)2 + v2t x (t)2 satisfies t H2(τ )dτ ≤ C2 (1 + t), ∀t > 0. 0

Similarly, differentiating (4.1.82) and (4.1.83) with respect to t, multiplying them by vt and θt respectively, integrating the resulting equations over [0, L] and using Lemmas 4.1.1–4.1.2, Lemmas 4.1.8–4.1.10 and (4.1.87)–(4.1.92), we finally deduce that d vt (t)2 + C1−1 vt x (t)2 ≤ C2 H2(t)(vt (t)2 + θt (t)2 + u(t)2H 1 dt +v(t)2H 1 + θ (t)2H 1 ), (4.1.176) d θt (t)2 + C1−1 θt x (t)2 ≤ C1 vt x (t)2 + C2 H2(t)(vt (t)2 + θt (t)2 + u(t)2H 1 dt +v(t)2H 1 + θ (t)2H 1 ). (4.1.177) Now multiplying (4.1.176) by a large number N2 > 0, then adding up the result, (4.1.175) and (4.1.177), we have d G 2 (t) ≤ C2 H2(t)(vt (t)2 + θt (t)2 + u(t)2H 2 + θ (t)2H 1 + v(t)2H 1 ) dt (4.1.178) ≤ C2 H2(t)(G 1 (t) + G 2 (t)) where G 2 (t) = uux1x (t)2 + N2 vt (t)2 + θt (t)2 . Thus adding (4.1.101) to (4.1.178) gives d G(t) ≤ C2 H2(t)G(t) dt where G(t) = G 1 (t) + G 2 (t), which, together with Gronwall’s inequality, (4.1.90) and (4.1.170), implies t 2 2 2 u(t) H 2 + v(t) H 2 + θ (t) H 2 ≤ C2 G(t) ≤ C2 G(0) exp(C2 H2(τ )dτ ) 0

≤ C2 exp(C2 t)(u 0 2H 2 + v0 2H 2 + θ0 2H 2 ), ∀t > 0.


+

≤ C2 exp(C2 t)(u 01 , v01 , θ01 ) − (u 02 , v02 , θ02 ) H 2

+

which leads to the continuity of semigroup S(t) with respect to the initial data in H+2 (and the uniqueness of generalized global solutions in H+2 ). Similarly to the proof of (4.1.78), by the continuity of the semigroup S(t) and the local regularity results, we can prove (4.1.108). Thus the proof is complete. The proof of Lemma 4.1.7 is now complete.


198

The next two lemmas concern the exponential stability of generalized global solution (u(t), v(t), θ (t)) in H 2 × H02 × H 2 (or equivalently, of semigroup S(t) on H+2 ) for the cases of n = 1, 2, 3. Lemma 4.1.13. For any (u 0 , v0 , θ0 ) ∈ H+2 , there exists a positive constant γ2 = γ2 (C2 ) ≤ γ1 such that for any fixed γ ∈ (0, γ2 ], the following estimates hold for any t > 0: t eγ t (θt (t)2 + vt (t)2 ) + eγ τ (vxt 2 + θθxt 2 )(τ )dτ ≤ C2 , (4.1.179) 0

eγ t u t (t)2H 1 + rt (t)2H 2 + vx x (t)2 + θθx x (t)2 + v(t)2W 1,∞ t

2 ¯ +θ (t) − θ W 1,∞ + eγ τ vx x x 2 + θθx x x 2 + rt 2H 3 + u t 2H 2 0 2 2 ¯ +vW 2,∞ + θ − θ W 2,∞ (τ )dτ ≤ C2 , (4.1.180) t

eγ τ r − r¯ 2H 3 + u − u ¯ 2H 2 + u − u ¯ 2W 1,∞ (τ )dτ ≤ C2 . (4.1.181) 0

Proof. Multiplying (4.1.154) by eγ t in L 2 ((0, L) × (0, t)), using Lemma 4.1.1 and Lemma 4.1.5, we deduce that for any fixed γ ∈ (0, γ1 ], t eγ t vt (t)2 + C1−1 eγ τ vxt 2 (τ )dτ (4.1.182) 0 t ≤ C2 + C1 (γ ) eγ τ (vt 2 + vx x 2 + θt 2 + θθx 2 + u x 2 )(τ )dτ ≤ C2 0

which, together with Lemmas 4.1.1, 4.1.5 and equations (4.1.157), (4.1.158), (4.1.164), (4.1.165), (4.1.167) and (4.1.168), yields

eγ t v(t)2W 1,∞ + vx x (t)2 + rt (t)2H 2 + u t (t)2H 1 (4.1.183) t

eγ τ vx x x 2 + v2W 2,∞ + u t 2H 2 + (n − 1)rt 2H 3 (τ )dτ ≤ C2 . + 0

Similarly, we can infer from (4.1.47)–(4.1.49), (4.1.52)–(4.1.56), Lemma 4.1.1 and Lemma 4.1.5 that for any fixed γ ∈ (0, γ1 ] and for any t > 0, ¯ 2 1,∞ ) + e (θt (t) + θθx x (t) + θ (t) − θ W γt

2

t

2

eγ τ (θθxt 2 + θθx x x 2 )(τ )dτ ≤ C2 .

0

Now multiplying (4.1.160) by eγ t and choosing γ so small that 1 , 0 < γ ≤ γ2 ≡ min γ1 , 4C1

(4.1.184)


199

we arrive at u d γt u x x 2 x x 2 e (t) ≤ C2 eγ t (θθx (t)2 + θθx x (t)2 (t) + (4C1 )−1 eγ t dt u u +vx x (t)2 + u x (t)2 + vt x (t)2 ) which, together with (4.1.182), Lemma 4.1.1 and Lemma 4.1.5, implies eγ t u x x (t)2 +

t 0

eγ τ u x x 2 (τ )dτ ≤ C2 , ∀t > 0.

(4.1.185)

By virtue of Lemmas 4.1.1 and 4.1.5, (4.1.182)–(4.1.185), (4.1.157), (4.1.158), (4.1.164), (4.1.165), (4.1.167)–(4.1.169), we can deduce the estimates (4.1.179)–(4.1.181). The proof is complete. Combining Lemmas 4.1.7–4.1.14 with Lemma 4.1.1 and Lemma 4.1.5, we easily derive the estimates (4.1.28)–(4.1.29) and hence the proof of Theorem 4.1.2 is now complete.

4.1.3 Results in Eulerian Coordinates In this subsection we will complete the proofs of Theorem 4.1.4–4.1.5. Now we return to problem (4.1.1)–(4.1.5) in Eulerian coordinates, and we have Lemma 4.1.14. For any (ρ0 , v0 , θ0 ) ∈ H Li ,G (i = 1, 2), there exists a unique generalized global solution (ρ, v, θ ) ∈ C([0, +∞), H Li ,G ) which defines a nonlinear C0 -semigroup S(t) on H Li ,G . Proof. For any given initial data (ρ0 , v0 , θ0 ) ∈ H Li ,G (i = 1, 2), it follows from the relationship (4.1.6), (4.1.7), (4.1.9) between the Eulerian coordinates and the Lagrangian coordinates that ( u 0 , v0 , θ0 ) = ( ρ10 , v0 , θ0 ) ∈ H+i . By Lemma 4.1.1, Lemma 4.1.6 and Lemma 4.1.7, there exists a unique generalized global solution ( u (ξ, t), v (ξ, t), θ (ξ, t)) ∈ i i C([0, +∞), H+) which defines a C0 -semigroup S(t) on H+ . It easy to see from Lemma 4.1.1, Lemma 4.1.6 and Lemma 4.1.7 and the relationship between the Lagrangian coordinates and the Eulerian coordinates (4.1.6), (4.1.7) that problem (4.1.1)–(4.1.5) admits a unique generalized global solution (ρ(r, t), v(r, t), θ (r, t)) ∈ C([0, +∞), H Li ,G ) which defines a C0 -semigroup S(t) on H Li ,G . The proof is complete. Proofs of Theorems 4.1.4–4.1.5 By Lemma 4.1.14, for any (ρ0 , v0 , θ0 ) ∈ H Li ,G , there exists a unique generalized global solution (ρ, v, θ ) ∈ C([0, +∞), H Li ,G ) which defines a C0 -semigroup S(t) on H Li ,G . Clearly, estimates (4.1.31)–(4.1.32) follow from (4.1.20)–(4.1.21). Exploiting the relationships (4.1.6), (4.1.7) and (4.1.9) between the Lagrangian coordinates and the


200

Eulerian coordinates, we infer that for a function f (x, t) ∈ L p (0, L) with any fixed 1 ≤ p < +∞ in Lagrange coordinates (x, t),

p

f (t) L p (0,L) =

L

b

f p (x, t)d x =

0

f p (r, t)r n−1 ρ(r, t)dr

a

which with (4.1.32) leads to −1/ p

C1

1/ p

f (t) L p (a,b) ≤ f (t) L p (0,L) ≤ C1 f (t) L p (a,b) .

(4.1.186)

Now letting p −→ +∞ in (4.1.186), we have f (t) L ∞ (a,b) = f (t) L ∞ (0,L) .

(4.1.187)

Thus using (4.1.186), (4.1.187), estimates (4.1.33), (4.1.34) in Theorem 4.1.4 and (4.1.37), (4.1.38) in Theorem 4.1.5 follow from (4.1.23), (4.1.24) and (4.1.28), (4.1.29), respectively. The proofs are complete.

4.2 Exponential Stability in H 4 In this section we shall establish the exponential stability in H 4 of solutions for the Navier-Stokes equations of problem (4.1.1)–(4.1.5) or (4.1.13)–(4.1.17).

4.2.1 Main Results In this subsection, we will present the main results. Now we first study problem (4.1.1)– (4.1.5) or (4.1.13)–(4.1.17). Similarly to the definitions of H+1 and H+2 in Section 4.1, we define

H+4 = (u, v, θ ) ∈ H 4[0, L] × H 4[0, L] × H 4[0, L] : u(x) > 0, θ (x) > 0, x ∈ [0, L], v|x=0 = v|x=L = 0, θx |x=0 = θ x |x=L = 0

which becomes a metric space when equipped with the metric induced from the usual norm. In the above, H 4 is the usual Sobolev space. We put · = · L 2 and use Ci (i = 1, 2, 3, 4) to stand for the generic constant depending only on the H+i norm of initial data, min u 0 (x) and min θ0 (x). x∈[0,L]

x∈[0,L]

Theorem 4.2.1. Assume that (4.1.19) holds. If (u 0 , v0 , θ0 ) ∈ H+4 , then there exists a unique global solution (u(t),v(t),θ (t)) ∈ C([0,+∞); H+4 ) to problem (4.1.13)–(4.1.17) which defines a nonlinear C0 -semigroup S(t) on H+4 with S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)).


201

For any (x, t) ∈ [0, L] × [0, +∞), besides (4.1.20)–(4.1.22), the following estimates hold: u(t) − u ¯ 2H 4 + u t (t)2H 3 + u t t (t)2H 1 + u x x (t)2W 1,∞ + v(t)2H 4 + vt (t)2H 2 + vt t (t)2 + vx x (t)2W 1,∞ + θ (t) − θ¯ 2H 4 + θt (t)2H 2 + θt t (t)2 + θθx x (t)2W 1,∞ ≤ C4 ,

(4.2.1)

+ v2H 5 + vt 2H 3 + vt t 2H 1 + vx x 2W 2,∞ + θ − θ¯ 2H 5 + θt 2H 3 + θt t 2H 1 + θθx x 2W 2,∞ (τ )dτ ≤ C4 .

(4.2.2)

t

u − u ¯ 2H 4 + u t 2H 4 + u t t 2H 2 + u t t t 2 + u x x 2W 1,∞ 0

Moreover, for any (u 0 , v0 , θ0 ) ∈ H+4 , there are constants C4 > 0 and γ4 = γ4 (C C4 ) > 0 such that for any fixed γ ∈ (0, γ4 ], there holds that for any t > 0, u(t) − u ¯ 2H 4 + u t (t)2H 3 + u t t (t)2H 1 + u x x (t)2W 1,∞ + v(t)2H 4 + vt (t)2H 2 + vt t (t)2 + vx x (t)2W 1,∞ + θ (t) − θ¯ 2H 4 0

+ θt (t)2H 2 + θt t (t)2 + θθx x (t)2W 1,∞ ≤ C4 e−γ t ,

t ¯ 2H 4 + u t 2H 4 + u t t 2H 2 + u t t t 2 + u x x 2W 1,∞ eγ τ u − u + v2H 5 + vt 2H 3 + vt t 2H 1 + vx x 2W 2,∞ + θ − θ¯ 2H 5 + θt 2H 3 + θt t 2H 1 + θθx x 2W 2,∞ (τ )dτ ≤ C4

(4.2.3)

(4.2.4)

and estimate (4.2.3) implies that the semigroup S(t) is exponentially stable on H+4 . Here u¯ = θ¯ =

1 L

L 0

1 CV L

u 0 (x)d x, r¯ (x) = (a n + n ux) ¯ 1/n ,

0

L

(C V θ0 + v02 /2)(x)d x.

(4.2.5) (4.2.6)

Corollary 4.2.1. Suppose that (u(t), v(t), θ (t)) is a solution obtained in Theorem 4.2.1 and satisfies the corresponding compatibility conditions. Then (u(t), v(t), θ (t)) is the classical solution verifying that for any fixed γ ∈ (0, γ4 ], (u(t) − u, ¯ v(t), θ (t) − θ¯ )2C 3+1/2 ×C 3+1/2 ×C 3+1/2 ≤ C4 e−γ t , ∀t > 0.

(4.2.7)

Remark 4.2.1. Theorem 4.2.1 and Corollary 4.2.1 are also valid for the boundary conditions v(0, t) = v(L, t) = 0, θ (0, t) = θ (L, t) = T0 with T0 > 0 being a constant, θ¯ = T0 .


202

4.2.2 Global Existence in H 4 In this subsection we prove the global existence of solutions in H+4 , while the existence of global solutions and the nonlinear C0 -semigroup S(t) on H+i (i = 1, 2) have been established in Sections 4.1.1–4.1.2. The next several lemmas concern the estimates in H+4 . Lemma 4.2.1. For any (u 0 , v0 , θ0 ) ∈ H+4 , there holds that for any t > 0, vt x (x, 0) + θt x (x, 0) ≤ C3 , vt t (x, 0) + θt t (x, 0) + vt x x (x, 0) + θt x x (x, 0) ≤ C4 , t t vt t x 2 (τ )dτ ≤ C4 + C4 (θt x x 2 + vt x x 2 )(τ )dτ, vt t (t)2 + 0 0 t t θt t x 2 (τ )dτ ≤ C4 () + C2 −1 θt x x 2 (τ )dτ θt t (t)2 + 0 0 t +C1 vt t x 2 (τ )dτ + C1 sup vt x (τ )2 0≤τ ≤t

0

(4.2.8) (4.2.9) (4.2.10)

(4.2.11)

with ∈ (0, 1) small enough. Proof. By Theorems 4.1.1–4.1.2 and (4.1.14), we easily infer that vt (t) ≤ C2 (vx (t) H 1 + u x (t) + θθx (t)).

(4.2.12)

Differentiating (4.1.14) with respect to x and using Theorems 4.1.1–4.1.2, we have

vt x (t) ≤ C2 (r n−1 v)x x (t) + θθx (t) + u x (t) + (r n−1 v)x x x (t) +θθx x (t) + u x x (t)

≤ C2 vx (t) + vx x x (t) + θθx (t) H 1 + u x (t) H 1 (4.2.13) or

vx x x (t) ≤ C2 vx (t) + u x (t) H 1 + θθx (t) H 1 + vt x (t) .

(4.2.14)

Differentiating (4.1.14) with respect to x twice, using Theorems 4.1.1–4.1.2 and a proper embedding theorem, we conclude

vt x x (t) ≤ C2 u x (t) H 2 + vx (t) H 3 + θθx (t) H 2 (4.2.15) or

vx x x x (t) ≤ C2 u x (t) H 2 + vx (t) H 2 + θθx (t) H 2 + vt x x (t) .

(4.2.16)

Similarly, it follows from (4.1.15) that θt (t) ≤ C2 (θθx (t) H 1 + vx (t) + u x (t)),

θt x (t) ≤ C2 θθx (t) H 2 + vx (t) H 1 + u x x (t)

(4.2.17) (4.2.18)


or and or

θθx x x (t) ≤ C2 θθx (t) H 1 + vx (t) H 1 + u x x (t) + θt x (t)

θt x x (t) ≤ C2 u x (t) H 2 + vx (t) H 2 + θθx (t) H 3

θθx x x x (t) ≤ C2 u x (t) H 2 + vx (t) H 2 + θθx (t) H 2 + θt x x (t) .

203

(4.2.19) (4.2.20)

(4.2.21)

By virtue of the boundary conditions (4.1.17) or (4.1.18) and the Poincaré´ inequality, we get vt (t) ≤ C1 vt x (t) ≤ C1 vt x x (t). (4.2.22) A simple calculation with Theorems 4.1.1–4.1.2 and (4.2.22) yields (r n−1 v)x x (t) ≤ C2 (vx (t) + vx x (t)) ≤ C2 vx x (t), u t t (t) ≤ C2 (vx (t) + vt x (t)), u t t x (t) ≤ C2 (u x (t) + vx x (t) + vt x x (t)).

(4.2.23) (4.2.24) (4.2.25)

Differentiating (4.1.14) with respect to t and using Theorems 4.1.1–4.1.2, Poincaré´ ’s inequality, (4.2.15) and (4.2.22)–(4.2.25), we easily deduce that

vt t (t) ≤ C2 θθx (t) + u x (t) + vx x (t) + vt x x (t) + θθxt (t) + θt (t) (4.2.26) which together with (4.2.15), (4.2.17) and (4.2.18) gives vt t (t) ≤ C2 (θθx (t) H 2 + vx (t) H 3 + u x (t) H 2 ).

(4.2.27)

In the same manner, we have

θt t (t) ≤ C2 θt (t) + θθx (t) + θt x (t) + θθx x (t) + θt x x (t) + vx (t) + vxt (t)

(4.2.28)

θt t (t) ≤ C2 (θθx (t) H 3 + vx (t) H 2 + u x (t) H 2 ).

(4.2.29)

and Thus estimates (4.2.8)–(4.2.9) follow from (4.2.13), (4.2.15), (4.2.18), (4.2.20), (4.2.27) and (4.2.29). Differentiating (4.1.14) with respect to t twice, multiplying the resulting equation by vt t in L 2 (0, L), performing an integration by parts and using Theorems 4.1.1–4.1.2 and Poincaré´ ’s inequality, we obtain that for any δ > 0,

d vt t (t)2 ≤ −(C1−1 − δ)vt t x (t)2 + C2 (δ) θθ x (t)2 + vx x (t)2 + u x (t)2 dt (4.2.30) +vt (t)2 + vxt (t)2 + vt t (t)2 + θt (t)2 + θt t (t)2 .


204

Choosing δ > 0 small enough in (4.2.30), using (4.2.27), (4.2.29) and Theorems 4.1.1– 4.1.2, and integrating with respect to t, we obtain estimate (4.2.10). Similarly, differentiating (4.1.15) with respect to t twice, multiplying the resulting equation by θt t in L 2 (0, L), integrating by parts and using Theorems 4.1.1–4.1.2, Poincaré´ ’s inequality and a proper embedding theorem, we deduce that for any ∈ (0, 1), d θt t (t)2 ≤ −(C1−1 − )θt t x (t)2 + vt t x (t)2 dt

(4.2.31)

+ C2 −1 θθx (t)2 + θt x (t)2 + vx (t)2 + vt x (t)2 + θt (t)2 + θt t (t)2 + θt x x (t)2 + C2 θt t (t)vt x (t)2 .

Thus choosing ∈ (0, 1) small enough in (4.2.31) and using (4.2.29) and Theorems 4.1.1–4.1.2, we can derive t θt t (t)2 + θt t x 2 (τ )dτ 0 t t vt t x 2 (τ )dτ + C4 () + C2 −1 θt t 2 (τ )dτ ≤ C1 0

t

+ C2

θt t 2 (τ )dτ

0

≤ C1

1/2

t 0

0

t

vt x 2 (τ )dτ

0

1/2 sup vt x (τ )

0≤τ ≤t

vt t x 2 (τ )dτ + C1 sup vt x (τ )2

+ C4 () + C2 −1

0≤τ ≤t

t

θt x x 2 (τ )dτ

0

which implies estimate (4.2.11). The proof is complete. Lemma 4.2.2. For any (u 0 , v0 , θ0 ) ∈ there holds that for any t > 0, t vt x (t)2 + vt x x 2 (τ )dτ 0 t ≤ C3 −6 + C2 2 (θt x x 2 + vt t x 2 )(τ )dτ, 0 t θt x x 2 (τ )dτ θt x (t)2 + 0 t ≤ C3 −6 + C2 2 (vt x x 2 + θt t x 2 )(τ )dτ

H+4 ,

(4.2.32)

(4.2.33)

0

with ∈ (0, 1) small enough. Proof. Differentiating (4.1.14) with respect to x and t, multiplying the resulting equation by vt x in L 2 (0, L), and integrating by parts, we arrive at 1 d vt x (t)2 = I0 (t) + I1 (t) 2 dt

(4.2.34)


205

with n−1 v) − Rθ β(r x I0 (t) = r n−1 vt x |x=L x=0 , u x t L n−1 v) − Rθ x n−1 β(r r I1 (t) = − vt x x d x. u 0

x

t

Using Theorems 4.1.1–4.1.2 and Sobolev’s interpolation inequality, we deduce that

I0 ≤ C2 vx x (t)1/2 vx x x (t)1/2 + vx x (t) + u x (t)1/2 u x x (t)1/2 + u x (t) + θθ x (t)1/2 θθx x (t)1/2 + θθx (t) + θt (t)1/2 θt x (t)1/2 + θt (t) + θt x (t) + θt x (t)1/2 θt x x (t)1/2 + vt x x (t)1/2 vt x x x (t)1/2 + vt x x (t) (vt x (t)1/2 vt x x (t)1/2 + vt x (t)) ≡ C2 (II01 + I02 )(vt x (t)1/2 vt x x (t)1/2 + vt x (t))

(4.2.35)

where I01 = vx x (t)1/2 vx x x (t)1/2 + vx x (t) + u x (t)1/2 u x x (t)1/2 + u x (t) + θθx (t)1/2 θθx x (t)1/2 + θθx (t) + θt (t)1/2 θt x (t)1/2 + θt (t) + θt x (t) and I02 = θt x (t)1/2 θt x x (t)1/2 + vt x x (t)1/2 vt x x x (t)1/2 + vt x x (t). Applying Young’s inequality several times, we have that for any ∈ (0, 1),

C2 I01 vt x (t)1/2 vt x x (t)1/2 + vt x (t) ≤

2 vt x x (t)2 + C2 −2 vx x (t)2H 1 + u x (t)2H 1 + θθx (t)2H 1 2

+ θt x (t)2 + θt (t)2 + vt x (t)2

(4.2.36)

and

C2 I02 vt x (t)1/2 vt x x (t)1/2 + vt x (t) ≤

2 vt x x (t)2 + 2 (θt x x (t)2 + vt x x x (t)2 ) 2 +C C2 −6 (θt x (t)2 + vt x (t)2 ).

(4.2.37)


206

Thus in view of (4.2.35)–(4.2.37) and Theorems 4.1.1–4.1.2, we conclude

I0 ≤ 2 vt x x (t)2 + vt x x x (t)2 + θt x x (t)2 + C2 −6 u x (t)2H 1 + vx x (t)2H 1 + θθx (t)2H 1 + θt x (t)2 + θt (t)2 + vt x (t)2

(4.2.38)

which leads to t t 2 I0 dτ ≤ (vt x x 2 + vt x x x 2 + θt x x 2 )(τ )dτ + C2 −6 , ∀t > 0. (4.2.39) 0

0

Similarly, by Theorems 4.1.1–4.1.2 and a proper embedding theorem, we get that for any ∈ (0, 1), L 2n−2 2 r vt x x I1 ≤ −β d x + 2 vt x x (t)2 u 0

+ C2 −2 u x (t)2 + vx x (t)2 + vxt (t)2 (4.2.40) + θθ x (t)2 + θt (t)2 + θt x (t)2 whence

t 0

I1 dτ ≤

−(C1−1

2

− )

0

t

vt x x 2 (τ )dτ + C2 −2 .

(4.2.41)

Taking ∈ (0, 1) small enough, we infer from (4.2.35), (4.2.39), (4.2.41) and Theorems 4.1.1–4.1.2 that for any t > 0, t t vt x (t)2 + vt x x 2 (τ )dτ ≤ C3 −6 + C2 2 (θt x x 2 + vt x x x 2 )(τ )dτ. (4.2.42) 0

0

Now we need to estimate vt x x x in (4.2.42) in term of vt t x . This observation is based on the fact that equations (4.1.14)–(4.1.15) are parabolic equations of second order for v and θ . In fact, differentiating (4.1.14) with respect to t and x, we can write n−1 v) − Rθ βr n−1 (r n−1 v)t x x x β(r x vt t x = r n−1 + D(t) (4.2.43) = u u x

tx

which, by Theorems 4.1.1–4.1.2 and a proper embedding theorem,

D ≤ C2 vx x (t) H 1 + θθx (t) H 1 + u x (t) H 1 + θt x (t) + θt (t) + θt x x (t) + vt x x (t) .

(4.2.44)

But a simple calculation with (4.2.43)–(4.2.44) yields vt x x x (t) ≤ C1 (r n−1 v)t x x x (t) + C2 (vx x (t) + vt x x (t) + vx x x (t))

≤ C1 vt t x (t) + C2 vx x (t) H 1 + θθx (t) H 1 + u x (t) H 1 + θt x (t) + θt (t) + vt x x (t) + θt x x (t) . (4.2.45)


207

Thus inserting (4.2.45) into (4.2.42), using Theorems 4.1.1–4.1.2 and taking ∈ (0, 1) small enough, we can derive the desired estimate (4.2.32). Analogously, using (4.1.15) and the estimate

θt x x x (t) ≤ C1 θt t x (t) + C2 u x (t) + vx x (t) H 1 + vt x x (t) + θθx (t) H 2 +θt x (t) + θt (t) + θt x x (t) , (4.2.46) we can derive estimate (4.2.33). The proof is now complete.

Lemma 4.2.3. For any (u 0 , v0 , θ0 ) ∈ H (4), there holds that for any t > 0, t vt x (t)2 + θt x (t)2 + (vt x x 2 + θt x x 2 )(τ )dτ 0 t ≤ C3 −6 + C2 2 (vt t x 2 + θt t x 2 )(τ )dτ

(4.2.47)

0

with ∈ (0, 1) small enough. Proof. Adding (4.2.32) to (4.2.33) and choosing ∈ (0, 1) small enough, we readily get the desired estimate (4.2.47). The proof is complete. Lemma 4.2.4. For any (u 0 , v0 , θ0 ) ∈ H+4 , there holds that for any t > 0, t

vt t x 2 + vt x x 2 vt t (t)2 + vt x (t)2 + θt t (t)2 + θt x (t)2 + 0 + θt t x 2 + θt x x 2 (τ )dτ ≤ C4 , t 2 2 (u x x x 2H 1 + u x x 2W 1,∞ )(τ )dτ ≤ C4 , u x x x (t) H 1 + u x x (t)W 1,∞ +

(4.2.48) (4.2.49)

0

vx x x (t)2H 1 + vx x (t)2W 1,∞ + θθx x x (t)2H 1 + θθx x (t)2W 1,∞ + u t x x x (t)2 + vt x x (t)2 + θt x x (t)2 t

vt t 2 + θt t 2 + vx x 2W 2,∞ + θθx x 2W 2,∞ + θt x x 2H 1 + 0 + vt x x 2H 1 + θt x 2W 1,∞ + vt x 2W 1,∞ + u t x x x 2H 1 (τ )dτ ≤ C4 ,

t 0

(vx x x x 2H 1 + θθx x x x 2H 1 )(τ )dτ ≤ C4 .

(4.2.50) (4.2.51)

Proof. Multiplying (4.2.10) and (4.2.11) by and 3/2 , respectively; adding the resultant to (4.2.47), and then taking ∈ (0, 1) small enough, we can obtain the estimate (4.2.48).


208

Differentiating (4.1.14) with respect to x, and using (4.1.13), we get ∂ u x x Rθ u x x Rθθ x − β(r n−1 v)x x 1−n −n β + = r v + (n − 1)r u t x ∂t u u u2 n−1 Rθθ x x β(r v)x u x − Rθ u x + + u u2 +

2Rθ u 2x − 2β(r n−1 v)x u 2x 2β(r n−1 v)x x u x − 2Rθθ x u x + u2 u3

≡ r 1−n vt x + E(x, t). Differentiating (4.2.52) with respect to x, we arrive at ∂ u x x x Rθ u x x x + = E 1 (x, t) β ∂t u u2 with

(4.2.52)

(4.2.53)

2u x u x x (r n−1 v)x (r n−1 v)x x x u x + u x x (r n−1 v)x x − E 1 (x, t) = β u2 u3 2Rθ u x u x x θx u x x + (1 − n)r 1−2n uvt x + r 1−n vt x x + E x (x, t). − 2 + u u3

Obviously, we can infer from Theorems 4.1.1–4.1.2 and (4.2.48) that

E 1 ≤ C2 u x (t) H 1 + vx x (t) H 1 + θθx (t) H 2 + vt x (t) H 1 which gives

Now

t

E 1 2 (τ )dτ ≤ C4 , ∀t > 0.

0 multiplying (4.2.53) by u xux x

(4.2.54)

(4.2.55)

in L 2 (0, L), we obtain

uxxx 2 d uxxx 2 + C1−1 ≤ C1 E 1 2 dt u u which combined with (4.2.55) and Lemma 4.2.1 leads to t u x x x 2 (τ )dτ ≤ C4 , ∀t > 0. u x x x (t)2 +

(4.2.56)

(4.2.57)

0

By (4.2.14), (4.2.16), (4.2.18), (4.2.21), (4.2.48), (4.2.57) and Theorems 4.1.1–4.1.2, we get t vx x x (t)2 + θθx x x (t)2 + (vx x x 2H 1 + θθx x x 2H 1 )(τ )dτ ≤ C4 , ∀t > 0 0

(4.2.58)


209

implying vx x (t)2L ∞ + θθx x (t)2L ∞ +

t 0

(vx x 2W 1,∞ + θθx x 2W 1,∞ )(τ )dτ ≤ C4 , ∀t > 0.

(4.2.59) Differentiating (4.1.14) with respect to t, using (4.2.47) and Theorems 4.1.1–4.1.2, we obtain

vt x x (t) ≤ C1 vt t (t) + C2 u x (t) + vx x (t) + vt x (t) + θθx (t) +θt (t) + θt x (t) ≤ C4 , ∀t > 0, (4.2.60) which with (4.2.16) implies t vx x x x (t)2 + (vt x x 2 + vx x x x 2 )(τ )dτ ≤ C4 ,

∀t > 0.

(4.2.61)

0

In the same manner, we can infer from (4.2.20)–(4.2.21) and (4.2.57)–(4.2.58) that t 2 2 θt x x (t) + θθx x x x (t) + (θt x x 2 + θθx x x x 2 )(τ )dτ ≤ C4 , ∀t > 0. (4.2.62) 0

Thus it follows from (4.2.58), (4.2.61)–(4.2.62) that, ∀t > 0, t vx x x (t)2L ∞ + θθx x x (t)2L ∞ + (vx x x 2L ∞ + θθx x x 2L ∞ )(τ )dτ ≤ C4 .

(4.2.63)

0

Now differentiating (4.2.53) with respect to x, we find ∂ uxxxx Rθ u x x x x β + = E 2 (x, t) ∂t u u2

(4.2.64)

with

2u x (r n−1 v)x u x x x (r n−1 v)x x u x x x + u x (r n−1 v)x x x x E 2 (x, t) = β − u2 u3 Rθθ x u x x x 2Rθ u x u x x x − + E 1x (x, t). + 3 u u2

Appealing to a proper embedding theorem, Theorems 4.1.1–4.1.2, (4.2.52) and (4.2.57)– (4.2.63), we can deduce that E x x (t) ≤ C4 (θθx (t) H 3 + u x (t) H 2 + vx (t) H 3 ) implying, by the expression of E 1 ,

E 1x (t) ≤ C4 vx (t) H 3 + u x (t) H 2 + vt x (t) H 2 + θθx (t) H 3

(4.2.65)


210

and further, by the expression of E 2 ,

E 2 (t) ≤ C4 vx (t) H 3 + u x (t) H 2 + vt x (t) H 2 + θθx (t) H 3 .

(4.2.66)

On the other hand, we can infer from (4.2.45), (4.2.46), (4.2.48) and Theorems 4.1.1– 4.1.2 that t (vt x x x 2 + θt x x x 2 )(τ )dτ ≤ C4 , ∀t > 0. (4.2.67) 0

Thus by virtue of (4.2.57), (4.2.61)–(4.2.62) and (4.2.66)–(4.2.67) and using Theorems 4.1.1–4.1.2, we get t

E 2 2 (τ )dτ ≤ C4 ,

∀t > 0.

(4.2.68)

0

Multiplying (4.2.64) by u xux x x in L 2 (0, L), we get u d u x x x x 2 x x x x 2 (t) + C1−1 (t) ≤ C1 E 2 (t)2 dt u u which combined with (4.2.68) implies t 2 u x x x x (t) + u x x x x 2 (τ )dτ ≤ C4 , ∀t > 0.

(4.2.69)

(4.2.70)

0

It is easy to verify from (4.2.26)–(4.2.29), (4.2.57)–(4.2.63) and Theorems 4.1.1–4.1.2 that t (vt t 2 + θt t 2 )(τ )dτ ≤ C4 , ∀t > 0. (4.2.71) 0

Differentiating (4.1.14) with respect to x three times, using Theorems 4.1.1–4.1.2 and Poincaré´ ’s inequality, we infer that vx x x x x (t) ≤ C1 vt x x x (t) + C2 (u x (t) H 3 + vx (t) H 3 + θθx (t) H 3 ).

(4.2.72)

Thus we conclude from (4.1.13), (4.2.61)–(4.2.62), (4.2.67), (4.2.70)–(4.2.72) and Theorems 4.1.1–4.1.2 that t (vx x x x x 2 + u t x x x 2H 1 )(τ )dτ ≤ C4 , ∀t > 0. (4.2.73) 0

Similarly, we can deduce that t

θθx x x x x 2 (τ )dτ ≤ C4 ,

∀t > 0

(4.2.74)

0

which with (4.2.59) and (4.2.73) gives t (vx x 2W 2,∞ + θθx x 2W 2,∞ )(τ )dτ ≤ C4 , ∀t > 0.

(4.2.75)

0

Finally, using (4.1.13), (4.2.57)–(4.2.63), (4.2.67), (4.2.70)–(4.2.71), (4.2.73)–(4.2.75) and Sobolev’s interpolation inequality, we can obtain the desired estimates (4.2.49)– (4.2.51). The proof is complete.


211

Lemma 4.2.5. For any (u 0 , v0 , θ0 ) ∈ H+4 , there holds that for any t > 0, u(t) − u ¯ 2H 4 + u t (t)2H 3 + u t t (t)2H 1 + v(t)2H 4 + vt (t)2H 2 + vt t (t)2 t

2 2 2 ¯ u − u ¯ 2H 4 + v2H 5 + vt 2H 3 +θ (t) − θ H 4 + θt (t) H 2 + θt t (t) + 0 ¯ 2 5 + θt 2 3 + θt t 2 1 (τ )dτ ≤ C4 , +vt t 2H 1 + θ − θ (4.2.76) H H H t (u t 2H 4 + u t t 2H 2 + u t t t 2 )(τ )dτ ≤ C4 . (4.2.77) 0

Proof. By equation (4.1.13) and (4.2.43)–(4.2.44), we have u t (t) H i ≤ C2 v H i+1 , i = 3, 4, u t t (t) ≤ C2 (v(t) H 1 + vt (t) H 1 ),

(4.2.78) (4.2.79)

u t t x (t) ≤ C2 (u x (t) + v(t) H 2 + vt (t) H 2 ), u t t x x (t) ≤ C1 (D(t) + vt t x (t))

≤ C2 u x (t) H 1 + v(t) H 3 + θθx (t) H 1 + vt (t) H 2 + θt (t) H 2 + vt t (t) H 1

(4.2.80)

(4.2.81)

and u t t t (t) ≤ C4 (vx (t) + vt x (t) + vt t x (t)).

(4.2.82)

Thus estimates (4.2.78)–(4.2.82), Theorems 4.1.1–4.1.2 and Lemma 4.2.4 imply (4.2.76)–(4.2.77). The proof is complete. By Lemmas 4.2.4–4.2.5, we can derive the global existence of solutions to (4.1.13)– (4.1.17) in H+4 with arbitrary initial datum (u 0 , v0 , θ0 ) ∈ H+4 and the uniqueness of a solution in H+4 follows from that of a solution in H+1 or H+2 .

4.2.3 A Nonlinear C0 -Semigroup S(t) on H 4 In this subsection we establish the existence of a nonlinear C0 -semigroup S(t) on H+4 . Lemma 4.2.6. The global solution (u(t), v(t), θ (t)) in H+4 to problem (4.1.13)–(4.1.17) defines a nonlinear C0 -semigroup S(t) on H+4 (also denoted by S(t) by the uniqueness of solution in H+1 and H+2 ) such that for any (u 0 , v0 , θ0 ) ∈ H+4 , we have S(t)(u 0 , v0 , θ0 ) H 4 = (u(t), v(t), θ (t)) H 4 ≤ C4 , ∀t > 0,

(4.2.83)

S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)) ∈ C([0, ∞); H+4 ), ∀t > 0.

(4.2.84)

+

+


212

Proof. The estimate (4.2.83) can be obtained from Lemmas 4.2.4–4.2.5. By Lemmas 4.2.4–4.2.5 and (4.2.83), we know that for any t > 0, the operator S(t) : (u 0 , v0 , θ0 ) ∈ H+4 −→ (u(t), v(t), θ (t)) ∈ H+4 exists, where (u(t), v(t), θ (t)) is the unique solution to problem (4.1.13)–(4.1.17) with the initial datum (u 0 , v0 , θ0 ) ∈ H+4 , and by the uniqueness of a global solution in H+4 , it satisfies on H+4 that for any t1 , t2 ∈ [0, +∞), S(t1 + t2 ) = S(t1 )S(tt2 ) = S(tt2 )S(t1 ).

(4.2.85)

Moreover, by Lemmas 4.2.4–4.2.5, S(t) is uniformly bounded on H+4 with respect to t > 0, i.e., S(t)L(H (4.2.86) H 4 ,H H 4 ) ≤ C4 , ∀t > 0. +

+

We here first verify the continuity of S(t) with respect to the initial data in H+4 for any t > 0. To this end, we assume that (u 0 j , v0 j , θ0 j ) ∈ H+4 , ( j = 1, 2), (u j , v j , θ j ) = S(t)(u 0 j , v0 j , θ0 j ), and (u, v, θ ) = (u 1 , v1 , θ1 )−(u 2 , v2 , θ2 ). Subtracting the corresponding equations (4.1.13)–(4.1.17) satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), we obtain u t = (r1n−1 v)x + [(r1n−1 − r2n−1 )v2 ]x , (4.2.87) n−1

n−1 n−1 n−1 (r1 v)x (r ((r v2 )x u − r2 )v2 )x vt = βr1n−1 − 1 + 1 u1 u 1u 2 u2 x n−1 v ) (r u − θ u θ 2 x 2 2 2 + β(r1n−1 − r2n−1 ) + Rr1n−1 u2 u1u2 x x θ 2 − R(r1n−1 − r2n−1 ) , (4.2.88) u2 x

2n−2 r1 r22n−2 θ2x u (r12n−2 − r22n−2 )θ2x θx 1 n−1 β(r1 v)x − + + C V θt = κ u1 u1u2 u1 u1 x

+ β((r1n−1 − r2n−1 )v2 )x − Rθ (r1n−1 v1 )x + β(r2n−1 v2 )x − Rθ2 u 2 (r1n−1 v)x − (r2n−1 v2 )x u + u 1 ((r1n−1 − r2n−1 )v2 )x u1u2

n−2 − 2μ(n − 1) r1 (v1 + v2 )v + (r1n−2 − r2n−2 )v22 ,

×

x

(4.2.89)

t = 0 : u = u 0 := u 01 − u 02 , v = v0 := v01 − v02 , θ = θ0 := θ01 − θ02 , x = 0, L : v = θx = 0 where

r j (x, t) = r0 j (x) +

t

(4.2.90)

x

n

u 0 j (y)d y)1/n

(4.2.91)

∀(x, t) ∈ [0, L] × [0, +∞).

(4.2.92)

v j (x, τ )dτ, r0 j (x) = (a + n

0

0

and r n−1 (x, t)rr j x (x, t) = u j (x, t), j

j = 1, 2,


213

By Lemmas 4.2.4–4.2.5, we know that for any t > 0, (u j (t) − u, ¯ v j (t), θ j (t) − θ¯ )2H 4 + u j t (t)2H 3 + u j t t (t)2H 1 + v j t (t)2H 2 t

u j − u +v j t t (t)2 + θθ j t (t)2H 2 + θθ j t t (t)2 + ¯ 2H (4) + v j 2H 5 0

¯ 2 5 + v j t 2 3 + v j t t 2 1 + θθ j t 2 3 + θθ j t t 2 1 + θθ j − θ H H H H H 2 2 2 +u j t H 4 + u j t t H 2 + u j t t t (τ )dτ ≤ C4 .

(4.2.93)

Here and hereafter in the proof of this lemma, C4 > 0 denotes the universal constant depending only on the H 4 norm of the initial data (u 0 j , v0 j , θ0 j ) and min u 0 j (x) and min θ0 j (x) ( j = 1, 2), but independent of t.

x∈[0,L]

x∈[0,L]

By (4.1.17)–(4.1.18) and (4.2.91)–(4.2.92), we have n n (0) − r20 (0) = 0 r1n (0, t) − r2n (0, t) = r10

which, together with (4.2.92), implies that for any (x, t) ∈ [0, L] × [0, +∞), x r1n (x, t) − r2n (x, t) = n u(y, t)d y, (r1n (x, t) − r2n (x, t))x = nu(x, t).

(4.2.94)

0

On the other hand, we can find that r1k (x, t) − r2k (x, t) = (r1n (x, t) − r2n (x, t))d(x, t)

(4.2.95)

where d(x, t) = 1/(r1 + r2 ) for n = 2, k = n − 1 = 1, or

d(x, t) = 1 for n = 2, k = 2n − 2 = 2,

or d(x, t) = (r1 + r2 )/(r12 + r1r2 + r22 ) for n = 3, k = n − 1 = 2, or d(x, t) = (r1 + r2 )(r12 + r22 )/(r12 + r1r2 + r22 ) for n = 3, k = 2n − 2 = 4, or d(x, t) = 0 for n = 2, k = n − 2 = 0, or

d(x, t) = 1/(r12 + r1r2 + r22 ) for n = 3, k = n − 2 = 1.

By (4.2.94)–(4.2.95) and Lemmas 4.2.4–4.2.5, we derive that for any (x, t) ∈ [0, L] × [0, +∞), |d(x, t)| + |d d x (x, t)| ≤ C1 , |d dx x (x, t)| + |d dt (x, t)| + |ddt x (x, t)| ≤ C2 , dx x x x (x, t)| + d dx x x x x (t) ≤ C4 , |d d x x x (x, t)| + |d

(4.2.96) (4.2.97)

dt x x x (x, t)| + |ddt t x (x, t)| ≤ C4 . |d dt t (x, t)| + |ddt x x (x, t)| ≤ C4 , |d

(4.2.98)


214

Thus it follows from (4.2.87)–(4.2.89), (4.2.94)–(4.2.98) and Lemmas 4.2.4–4.2.5 that j

j

j

j

(r1 − r2 )(t) H i+1 ≤ C4 u(t) H i , i = 0, 1, 2, 3, 4, j = n − 2, n − 1, 2n − 2, (4.2.99) (r1 − r2 )t (t) H 1 ≤ C2 (v(t) H 1 + u(t)), j = n − 2, n − 1, 2n − 2, j j (r1 − r2 )t t (t) ≤ C4 (v(t) H 1 + u(t) + vt x (t)), j = (r12n−2 − r22n−2 )θ2x (t) H 1 ≤ C2 u, (r1n−1 − r2n−1 )v2 (t) H i+1 ≤ C4 u(t) H i , i = 0, 1, 2, 3, 4, [(r1n−1 − r2n−1 )v2 ]t x (t) + [(r12n−2 − r22n−2 )θ2x ]t (t) H 1

n − 2, n − 1, (4.2.101)

≤ C4 (u(t) + v(t) H 1 ), j [(r1

j − r2 )v2 ]t x x (t)

(4.2.102) (4.2.103) (4.2.104)

≤ C4 (u(t) H 1 + v(t) H 1 + vt (t) + θ (t) H 1 ),

j = n − 1, 2n − 2, [(r1n−1 [(r1n−1

(4.2.100)

(4.2.105)

− r2n−1 )v2 ]t x x x (t) − r2n−1 )v2 ]t t x (t)

≤ C4 (1 + v2t x x x (t))(u(t) H 2 + v(t) H 3 ), (4.2.106)

≤ C4 (1 + v2t t x (t))(u(t) H 1 + v(t) H 1 + vt x (t)).

(4.2.107)

Multiplying (4.2.87), (4.2.88) and (4.2.89) by u, v and θ respectively in L 2 (0, L), adding up the resulting equations, recalling vx ≤ C1 (r1n−1 v)x and using (4.2.96)–(4.2.107), we deduce that for any small δ > 0, d

u(t)2 + v(t)2 + C V θ (t)2 dt

+ C1−1 vx (t)2 + (r1n−1 v)x (t)2 + θθx (t)2 ≤ C2 (u(t)2 + v(t)2 + θ (t)2 ).

(4.2.108)

By Theorems 4.1.1–4.1.2, a proper embedding theorem, (4.2.88) and (4.2.99)–(4.2.107), we infer (r1n−1 v)x x (t)2

≤ C1 vt (t)2 + (r1n−1 v)x (t)2L ∞ + θ (t)2H 1 + ((r1n−1 v2 )x x (t)2 +v2x x (t)2 )u(t)2H 1 ≤

1 n−1 (r v)x x (t)2 + C1 (vt (t)2 + θ (t)2H 1 ) 2 1 + C2 ((r1n−1 v)x (t)2 + u(t)2H 1 )

which gives vx x (t)2 ≤ C1 vt (t)2 + C2 (v(t)2H 1 + u(t)2H 1 + θ (t)2H 1 ).

(4.2.109)


215

Using (4.2.87) and noting that u j ( j = 1, 2) and v j ( j = 1, 2) satisfying (4.1.13), we have (r1n−1 v)x u 1x (r2n−1 v2 )x − u 2x (r1n−1 v1 )x ((r1n−1 − r2n−1 )v2 )x ux = + − . 2 u1 u1 t u1 u 1 x x (4.2.110) Inserting (4.2.110) into (4.2.88), we obtain

n−1 (r1 v2 )x u ((r n−1 − r2n−1 )v2 )x ux β = r11−n vt + β − 1 u1 t u1u2 u2 x n−1 (r2 v2 )x θ2 u − θ u 2 − βr11−n (r1n−1 − r2n−1 ) −R u2 u 1u 2 x x n−1 βu 1x (r2 v2 )x − u 2x (r1n−1 v1 )x θ2 + Rr11−n (r1n−1 − r2n−1 ) − u2 x u 21 ((r1n−1 − r2n−1 )v2 )x +β . (4.2.111) u1 x

ux u1

Multiplying (4.2.111) by in L 2 (0, L), and using Theorems 4.1.1–4.1.2, (4.2.96)– (4.2.107) and (4.2.109), we conclude L d θ2 u 2x u x 2 β (t) + R dx 2 dt u 1 0 u1u2 ≤ C1 vt (t)2 + C2 ((r1n−1 v)x (t)2 + u(t)2H 1 + θ (t)2H 1 ) whence d u x 2 (t) + C1−1 u x (t)2 ≤ C1 vt (t)2 + C2 (v(t)2H 1 + u(t)2H 1 + θ (t)2H 1 ). dt u 1 (4.2.112) Multiplying (11.2.89) by vt in L 2 (0, L), and using Theorems 4.1.1–4.1.2, (4.2.96)– (4.2.107) and the identity (r1n−1 vt )x = (r1n−1 v)t x − (n − 1)(r1n−2 v1 v)x , we get n−1 d (r1 v)x 2 (t) + C1−1 vt (t)2 ≤ C2 (v(t)2H 1 + u(t)2H 1 + θ (t)2H 1 ). (4.2.113) √ dt u1

Similarly, multiplying (4.2.89) by θt in L 2 (0, L), we have n−1 d r1 θx 2 (t) + C1−1 θt (t)2 ≤ C2 (v(t)2H 1 + u(t)2H 1 + θ (t)2H 1 ). (4.2.114) √ dt u1

Multiplying (4.2.113) by a large number N1 , then adding up the result, (4.2.108), (4.2.112) and (4.2.114), we conclude d G 1 (t) ≤ C2 G 1 (t) dt

(4.2.115)


216

where (r n−1 v) u 2 r n−1 θ 2 x x x 2 G 1 (t) = u(t)2 + (t) +v(t)2 + N1 1√ (t) +C V θ (t)2 + 1√ (t) . u1 u1 u1 Similarly to (4.2.109), we have θθx x (t)2 ≤ C2 (θt (t)2 + G 1 (t)).

(4.2.116)

Differentiating (4.2.88) with respect to x, we get that vt x =

βr1n−1 (r1n−1 v)x x x Rr1n−1 θ2 u x x 1 + +2βr1n−1 (r1n−1 v)x x +R(x, t) (4.2.117) u1 u1u2 u1 x

where R(x, t) are the remaining terms, which, by Lemmas 4.2.4–4.2.5, (4.2.96)– (4.2.107), satisfy ∂xi R(t)2 ≤ C4 (u(t)2H 2+i + θ (t)2H 2+i + v(t)2H 2+i ), i = 0, 1, 2.

(4.2.118)

By a proper embedding theorem, we infer from (4.2.117)–(4.2.118) that (r1n−1 v)x x x (t)2 ≤ C1 vt x (t)2 + C2 (u x x (t)2 + (r1n−1 v)x x (t)2L ∞ + R(t)2 )

1 ≤ (r1n−1 v)x x x (t)2 + C1 vt x (t)2 + C4 (r1n−1 v)x x (t)2 2 + u(t)2H 2 + θ (t)2H 2

implying vx x x (t)2 ≤ C1 vt x (t)2 + C4 (v(t)2H 2 + u(t)2H 2 + θ (t)2H 2 ).

(4.2.119)

By (4.2.87) and (4.2.117), we arrive at β

uxx u1

t

(r1n−1 v1 )x u x x ((r1n−1 − r2n−1 )v2 )x x x θ2 u x x 1−n +R = r1 vt x + β − u1u2 u1 u 21 1 −2β(r1n−1 v)x x − r11−n R(x, t) u1 x (4.2.120) ≡ R1 (x, t).

Multiplying (4.2.120) by uux1x in L 2 (0, L), using Lemmas 4.2.4–4.2.5, and (4.2.96)– (4.2.107), we conclude that d u x x 2 (t) + C1−1 u x x (t)2 (4.2.121) dt u 1 ≤ C1 vt x (t)2 + C4 u(t)2H 2 + v(t)2H 2 + θ (t)2H 2 + vt (t)2 .


217

Similarly, differentiating (4.2.88) and (4.2.89) with respect to t, multiplying the resulting equations by vt and θt in L 2 (0, L) respectively, using Lemmas 4.2.4–4.2.5, and (4.2.96)– (4.2.107), we deduce

d vt (t)2 + C1−1 vt x (t)2 ≤ C4 vt (t)2 + θt (t)2 + u(t)2H 1 + v(t)2H 1 dt (4.2.122) +θ (t)2H 1 ,

d θt (t)2 + C1−1 θt x (t)2 ≤ C1 vt x (t)2 + C4 vt (t)2 + θt (t)2 + u(t)2H 1 dt (4.2.123) +v(t)2H 1 + θ (t)2H 1 . Now multiplying (4.2.122) by a large number N2 > 0, then adding up the result, (4.2.121) and (4.2.123), we get d G 2 (t) ≤ C4 (G 1 (t) + G 2 (t)) (4.2.124) dt where G 2 (t) = uux1x (t)2 + N2 vt (t)2 + θt (t)2 . Differentiating (4.2.120) with respect to x, we arrive at θ2 u x x x uxxx β +R = R2 (x, t) (4.2.125) u1 t u1u2

with R2 (x, t) = R1x + β

u 1x u x x u 21

+ t

Rθ2 (u 1 u 2 )x u x x Rθ2x u x x − . u 1u 2 u 21 u 22

In view of (4.2.96)–(4.2.107) and (4.2.118), we get R1x (t)2 ≤ C1 vt x x (t)2 + C4 (u(t)2H 3 + θ (t)2H 3 + v(t)2H 3 ).

(4.2.126)

Thus

R2 (t)2 ≤ C1 R1x (t)2 + C4 u x x (t)2 + v(t)2H 3 + (r1n−1 − r2n−1 )v2 (t)2H 3

≤ C1 vt x x (t)2 + C4 u(t)2H 3 + v(t)2H 3 + θ (t)2H 3 + vt x (t)2 (4.2.127)

and

R2x (t)2 ≤ C4 R1x x (t)2 + u(t)2H 4 + v(t)2H 4 + (r1n−1 − r2n−1 )v2 (t)2H 4 ≤ C4 (vt x x (t)2 + vt x x x (t)2 ) + C4 (1 + r2n−1 v2 (t)2H 5 ) × (u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ).


ux x x u1

(4.2.128)

in L 2 (0, L) and using Lemmas 4.2.4–4.2.5, we have

u d u x x x 2 x x x 2 (t) + C1−1 (t) ≤ C1 R2 (t)2 . dt u 1 u1

(4.2.129)


218

Differentiating (4.2.125) with respect to x, we know that Rθ2 u x x x x uxxxx β + = R3 (x, t) u1 t u1u2 where

R3 (x, t) = β

u 1x u x x x u 21

− t

(4.2.130)

Rθ2x u x x x R(u 1 u 2 )x θ2 u x x x + + R2x (x, t) u1u2 u 21 u 22

which, along with Lemmas 4.2.4–4.2.5, (4.2.96)–(4.2.107) and (4.2.128), verifies R3 (t)2 ≤ C4 (vt x x (t)2 + vt x x x (t)2 ) + C4 (1 + v2 (t)2H 5 ) ×(u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ).

(4.2.131)

Now we hope that vt x x x (t) in (4.2.131) can be expressed by vt t x (t). To this end, we differentiate (4.2.88) with respect to t and x, and use the embedding theorem, (4.2.96)– (4.2.107) and Lemmas 4.2.4–4.2.5 to get (r1n−1 v)t x x x (t) ≤ C4 (vt t x (t) + vt x x (t)) + C4 (1 + (r1n−1 v2 )t (t) H 3 ) ×(u(t) H 2 + v(t) H 3 + θ (t) H 2 + θt (t)) which, with the expression of (r1n−1 v)t x x x , yields vt x x x (t) ≤ C4 (vt t x (t) + vt x x (t)) + C4 (1 + (r1n−1 v2 )t (t) H 3 )

× u(t) H 2 + v(t) H 2 + θ (t) H 2 + θt (t) + vt x (t) . (4.2.132) Thus, by (4.2.131), R3 (t)2 ≤ C4 (vt t x (t)2 + vt x x (t)2 ) + C4 (1 + v2 (t)2H 5 + v2t (t)2H 3 )

× u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 + θt (t)2 + vt x (t)2 . (4.2.133) Similarly to (4.2.129), we get u d u x x x x 2 x x x x 2 (t) + C1−1 (t) ≤ C1 R3 (t)2 . dt u1 u1

(4.2.134)

By (4.2.89) and (4.2.96)–(4.2.107), we conclude θt (t) ≤ C4 (θ (t) H 2 + u(t) H 1 + v(t) H 1 ),

(4.2.135)

or θθx x (t) ≤ C4 (θt (t) + θ (t) H 1 + u(t) H 1 + v(t) H 1 ),

(4.2.136)

θt x (t) ≤ C4 (θ (t) H 3 + u(t) H 2 + v(t) H 2 ),

(4.2.137)

and


219

or θθx x x (t) ≤ C4 θt x (t) + u(t) H 2 + v(t) H 1 + vt (t) (4.2.138) +θ (t) H 1 + θt (t) , θθx x x x (t) ≤ C4 u(t) H 3 + v(t) H 1 + θ (t) H 1 + vt (t) + θt (t) (4.2.139) +vt x (t) + θt x (t) + θt t (t) , (4.2.140) θt t (t) ≤ C4 u(t) H 3 + v(t) H 3 + θ (t) H 4 , (4.2.141) θt x x (t) ≤ C4 u(t) H 3 + v(t) H 3 + θ (t) H 4 . Similarly, from (4.2.88), vt (t) ≤ C4 u(t) H 1 + v(t) H 2 + θ (t) H 1 , (4.2.142) (4.2.143) vx x (t) ≤ C4 u(t) H 1 + v(t) H 1 + vt (t) + θ (t) H 1 , (4.2.144) vt x (t) ≤ C4 (θ (t) H 2 + u(t) H 2 + v(t) H 3 ), vx x x (t) ≤ C4 u(t) H 2 + v(t) H 1 + θ (t) H 1 + vt (t) (4.2.145) +θt (t) + vt x (t) , vx x x x (t) ≤ C4 u(t) H 3 + v(t) H 1 + θ (t) H 1 + vt (t) + vt x (t) (4.2.146) +vt t (t) + θt (t) + θt x (t) , (4.2.147) vt t (t) ≤ C4 u(t) H 3 + v(t) H 4 + θ (t) H 3 , (4.2.148) vt x x (t) ≤ C4 u(t) H 3 + v(t) H 4 + θ (t) H 3 . Differentiating (4.2.88) with respect to t twice, multiplying the resulting equation by vt t and using (4.2.96)–(4.2.107), (4.2.135)–(4.2.148) and Lemmas 4.2.4–4.2.5, we deduce that for any δ > 0, d vt t (t)2 + C1−1 vt t x (t)2 dt

(4.2.149)

≤ δvt t x (t)2 + C4 (δ)(1 + v2t t x (t)2 ) u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 +vt (t)2 + vt x (t)2 + vt t (t)2 + θt (t)2 + θt x (t)2 + θt t (t)2 .

Similarly to (4.2.149), we can deduce from (4.2.88)–(4.2.89) that for any δ > 0, d θt t (t)2 + C1−1 θt t x (t)2 dt ≤ δ(vt t x (t)2 + θt t x (t)2 ) + C4 (δ)(1 + v1t t x (t)2 + v2t t x (t)2

+ θ2t t x (t)2 ) u(t)2H 1 + v(t)2H 1 + θ (t)2H 1 + vt x (t)2 (4.2.150) + vt t (t)2 + θt (t)2 + θt x (t)2 + θt t (t)2 ,


220

d vt x (t)2 + C1−1 vt x x (t)2 dt ≤ C4 δ(vt x x (t)2 + vt t x (t)2 + θt x x (t)2 )

+ C4 (δ)(1 + v2t x x x (t)2 ) u(t)2H 2 + v(t)2H 1 + θ (t)2H 1 (4.2.151) + vt (t)2 + vt x (t)2 + θt (t)2 + θt x (t)2 , d θt x (t)2 + C1−1 θt x x (t)2 dt ≤ C1 δ(θt t x (t)2 + θt x x (t)2 ) + C4 (δ)(1 + θ2t x x x (t)2 )

× u(t)2H 2 + v(t)2H 1 + θ (t)2H 1 + vt (t)2 + θt (t)2 + vt x (t)2 + θt x (t)2 + vt t (t)2 + θt t (t)2 .

(4.2.152)

Put

u x x x 2 u x x x x 2 (t) + (t) . G 3 (t) = vt t (t) +vt x (t) +θt t (t) +θt x (t) +δ u1 u1 2

2

2

2

Then multiplying (4.2.129) and (4.2.134) by δ respectively, adding up the resulting equations, (4.2.148) and (4.2.149)–(4.2.151), and using (4.2.127), (4.2.133), and taking δ > 0 small enough, we get

d G 3 (t) + C4−1 vt t x (t)2 + vt x x (t)2 + θt t x (t)2 + θt x x (t)2 dt + u x x x (t)2 + u x x x x (t)2 ≤ C4 H3 (t)M(t) (4.2.153) where M(t) = u(t)2H 4 + v(t)2H 4 + θ (t)|2H 4 + vt (t)2 + θt (t)2 + vt x (t)2 + θt x (t)2 + vt t (t)2 + θt t (t)2 and by Lemmas 4.2.4–4.2.5, H3(t) = 1+v1t t x (t)2 +v2t t x (t)2 +θ2t t x (t)2 +θ2t (t)2H 3 +v2t (t)2H 3 +v2 (t)2H 5 verifies

t 0

H3(τ )dτ ≤ C4 (1 + t),

∀t > 0.

(4.2.154)

By (4.2.135), (4.2.137), (4.2.140), (4.2.142), (4.2.144), (4.2.147) and the definition of M(t), we know that u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ≤ M(t) ≤ C4 (u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ).

(4.2.155)


221

Let G(t) = G 1 (t) + G 2 (t) + G 3 (t). Then we easily infer from (4.2.136), (4.2.137)–(4.2.139), (4.2.143) and (4.2.145), (4.2.146) that

M(t) ≤ C4 u(t)2H 4 + v(t)2H 1 + θ (t)2H 1 + vt (t)2 + θt (t)2 +vt x (t)2 + θt x (t)2 + vt t (t)2 + θt t (t)2 ≤ C4 G(t).

(4.2.156)

On the other hand, we can find from the definition of G(t) that

G(t) ≤ C4 u(t)2H 4 + v(t)2H 1 + θ (t)2H 1 + vt (t)2 + θt (t)2 +vt x (t)2 + θt x (t)2 + vt t (t)2 + θt t (t)2 ≤ C4 M(t) which combined with (4.2.155)–(4.2.156) gives C4−1 (u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ) ≤ G(t) ≤ C4 (u(t)2H 4 + v(t)2H 4 + θ (t)2H 4 ).

(4.2.157)

Now we add up (4.2.105), (4.2.124) and (4.2.153) to arrive at d G(t) ≤ C4 H3(t)G(t) dt implying, with (4.2.157), ∀t > 0, u(t)2H 4

+ v(t)2H 4

+ θ (t)2H 4

≤ C4 G(t) ≤ C4 G(0) exp(C C4

t

H3(τ )dτ )

0 ≤ C4 exp(C C4 t)(u 0 2H 4 + v0 2H 4

+ θ0 2H 4 ).


+

≤ C4 exp(C C4 t)(u 10 , v10 , θ10 ) − (u 20 , v20 , θ20 ) H 4

+

which implies the continuity of semigroup S(t) with respect to the initial data in H+4 (and also the uniqueness of global solutions in H+4 ). In order to prove (4.2.84), by (4.2.85)–(4.2.86), it suffices to show that for any (u 0 , v0 , θ0 ) ∈ H+4 ,

S(t)(u 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 4 −→ 0 +

(4.2.158)


222

as t → 0+ , which also yields S(0) = I

(4.2.159)

with I being the unit operator (i.e., identity operator) on H+4 . To show (4.2.159), we m m 6 choose a function sequence which is smooth enough, for example, (u m 0 , v0 , θ0 ) ∈ H × 6 6 H × H such that m m (u m (4.2.160) 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 4 −→ 0 +

as m → +∞. By the local regularity result, we conclude that there is a small t0 > 0 such that there exists a unique smooth solution (u m (t), v m (t), θ m (t)) ∈ H 6 × H 6 × H 6 (∀t ∈ (0, t0 )). This implies that for m = 1, 2, . . . , m m (u m (t), v m (t), θ m (t)) − (u m 0 , v0 , θ0 ) H 4 −→ 0

(4.2.161)

+

as t → 0+ . By the continuity of the operator S(t), we conclude that for any t ∈ (0, t0 ), (u m (t), v m (t), θ m (t)) − (u(t), v(t), θ (t)) H 4

+

m m = S(t)(u m 0 , v0 , θ0 ) − S(t)(u 0 , v0 , θ0 ) H 4

+

m m ≤ C4 (t0 )(u m 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 4 −→ 0 +

as m → +∞, which along with (4.2.160)–(4.2.161) leads to S(t)(u 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 4 = (u(t), v(t), θ (t)) − (u 0 , v0 , θ0 ) H 4 +

+


+

m m +(u m (t), v m (t), θ m (t)) − (u m 0 , v0 , θ0 ) H 4

+

m m +(u m 0 , v0 , θ0 ) − (u 0 , v0 , θ0 ) H 4 −→ 0 +

as m → +∞ and t → 0+ , which gives (4.2.158) and (4.2.159). The proof is now complete.

4.2.4 Exponential Stability in H 4 In this subsection, based on the estimates established in Section 4.2.3, we shall show the exponential stability of solutions in H+4 or of the nonlinear C0 -semigroup S(t) on H+4 . (1)

(1)

Lemma 4.2.7. If (u 0 , v0 , θ0 ) ∈ H+4 , there exists a positive constant γ4 = γ4 (C C4 ) ≤ (1) γ2 (C2 ) such that for any fixed γ ∈ (0, γ4 ], there holds that for any t > 0, eγ t vt t (t)2 +

t 0

eγ τ vt t x 2 (τ )dτ ≤ C4 + C4

t 0

eγ τ (θt x x 2 + vt x x 2 )(τ )dτ, (4.2.162)


eγ t θt t (t)2 +

223

t

eγ τ θt t x 2 (τ )dτ 0 t t eγ τ θt x x 2 (τ )dτ C1 eγ τ vt t x 2 (τ )dτ ≤ C4 () + C2 −1 0

+ C1 e

γt

0

2

sup vt x (τ )

(4.2.163)

0≤τ ≤t

with ∈ (0, 1) small enough. Proof. The proofs of (4.2.162)–(4.2.163) are basically the same as those of (4.2.10)– (4.2.11). The difference here is to estimate (4.2.162)–(4.2.163) with weighted exponential function eγ t . Similarly to (4.2.30), multiplying (4.2.30) by eγ t and using Theorems 4.1.1– 4.1.2, we easily deduce that γt

2

e vt t (t) ≤

C4 − (C1−1

− δ − C1 γ )

t

0

eγ τ vt t x 2 (τ )dτ

+ C2 (δ) u x 2 + vt 2 + vx x 2 + vt x 2 e 0 2 (4.2.164) + vt t + θθx 2 + θt 2 + θt t 2 (τ )dτ. Thus taking γ and δ so small that δ ≤

t

γτ

1 4C 1

and 0 < γ ≤ min[

1 , γ2 (C2 )], using (4.2.26), 4C 12

(4.2.28) and Theorems 4.1.1–4.1.2, we can obtain estimate (4.2.162) from (4.2.164). Similarly to (4.2.31), using (4.2.28), we have eγ t θt t (t)2

t

≤ C4 + C2 γ eγ τ vx 2 + vt x 2 + θt 2 + θθx 2 + θt x 2 + θθx x 2 + θt x x 2 (τ )dτ 0 t

−1 eγ τ θθx 2 + θt x 2 + θt 2 + θt t 2 + vx 2 + vt x 2 (τ )dτ + C2 0 t t −1 − (C1 − ) eγ τ θt t x 2 (τ )dτ + C2 eγ τ vt t x 2 (τ )dτ 0 0 t t γτ 2 1/2 γ2 t + C3 ( e θt t (τ )dτ ) e sup vt x (τ )( vt x 2 (τ )dτ )1/2 0

0≤τ ≤t

t

0

≤ C4 () + C2 ( −1 + γ ) eγ τ θt x x 2 (τ )dτ − (C1−1 − ) 0 t + C2 eγ τ vt t x 2 (τ )dτ + eγ t sup vt x (τ )2 0

0

t

eγ τ θt t x 2 (τ )dτ

0≤τ ≤t

implying estimate (4.2.163) if we take 0 < γ ≤ min[1, min(

1 , γ2 (C2 ))] 4C 12

≡ γ4(1) and

> 0 small enough (for example, 0 < ≤ min[ 2C1 1 , 1]). The proof is complete.


224

(1)

Lemma 4.2.8. For any (u 0 , v0 , θ0 ) ∈ H+4 and for any fixed γ ∈ (0, γ4 ], there holds that for any t > 0, eγ t vt x (t)2 +

t

eγ τ vt x x 2 (τ )dτ 0 t eγ τ (θt x x 2 + vt x x x 2 )(τ )dτ, (4.2.165) ≤ C3 −6 + C2 2 0 t γt 2 γτ 2 e θt x (t) + e θt x x (τ )dτ 0 t ≤ C3 −6 + C2 2 eγ τ (vt x x 2 + θt x x x 2 )(τ )dτ (4.2.166) 0

with ∈ (0, 1) small enough. Proof. By (4.2.34), (4.2.38) and (4.2.40), we infer that t L r 2n−2 vt2x x 1 γt e vt x (t)2 + β d x dτ eγ τ 2 u 0 0 t t γ t γτ ≤ C3 + e vt x 2 (τ )dτ + |II0 |eγ τ dτ + 2 eγ τ vt x x 2 (τ )dτ 2 0 0 0 t

+ C2 −22 eγ τ u x 2 + vx x 2 + θθx 2 + vt x 2 + θt 2 + θt x 2 (τ )dτ 0 t ≤ C3 + C1 2 eγ τ (vt x x 2 + θt x x 2 + vt x x x 2 )(τ )dτ 0 t

eγ τ u x 2H 1 + vt x 2 + vx x 2H 1 + θt 2 + θθx 2H 1 + θt x 2 (τ )dτ + C2 −6 0 t ≤ C3 −6 + C1 2 eγ τ (vt x x 2 + θt x x 2 + vt x x x 2 )(τ )dτ 0 (1)

which gives estimate (4.2.165) for any fixed γ ∈ (0, γ4 ] and ∈ (0, 1) small enough. In the same manner, we can prove estimate (4.2.166). The proof is complete. (1)

Lemma 4.2.9. For any (u 0 , v0 , θ0 ) ∈ H+4 and for any fixed γ ∈ (0, γ4 ], there holds that for any t > 0, eγ t (vt x (t)2 + θt x (t)2 ) +

t

eγ τ (vt x x 2 + θt x x 2 )(τ )dτ

0

≤ C3 −6 + C2 2 with ∈ (0, 1) small enough.

t 0

eγ τ (θt t x 2 + vt t x 2 )(τ )dτ

(4.2.167)


225

Proof. Adding (4.2.165) to (4.2.166) and choosing ∈ (0, 1) small enough, we have eγ t (vt x (t)2 + θt x (t)2 ) +

t

eγ τ (vt x x 2 + θt x x 2 )(τ )dτ

0

≤ C3

−6

+ C2

t

2 0

eγ τ (θt x x x 2 + vt x x x 2 )(τ )dτ

which, combined with (4.2.45), (4.2.46), Theorems 4.1.1–4.1.2 and taking ∈ (0, 1) small enough, imply the estimate (4.2.167). The proof is complete. Lemma 4.2.10. For any (u 0 , v0 , θ0 ) ∈ H+4 , there is a positive constant γ4(2) ≤ γ4(1) such that for any fixed γ ∈ (0, γ4(2)], there holds that for any t > 0,

eγ t vt t (t)2 + vt x (t)2 + θt t (t)2 + θt x (t)2 t

eγ τ vt t x 2 + vt x x 2 + θt t x 2 + θt x x 2 (τ )dτ ≤ C4 , (4.2.168) + 0

t eγ t u x x x (t)2H 1 + u x x (t)2W 1,∞ + eγ τ (u x x x 2H 1 + u x x 2W 1,∞ )(τ )dτ ≤ C4 , 0

(4.2.169)

eγ t vx x x (t)2H 1 + vx x (t)2W 1,∞ + θθx x x (t)2H 1 + θθx x (t)2W 1,∞ + u t x x x (t)2 t

+ vt x x (t)2 + θt x x (t)2 + eγ τ vt t 2 + vx x x x 2H 1 + vt x x 2H 1 0

+ θt t 2 + θθx x x x 2H 1 + θt x x 2H 1 + vx x 2W 2,∞ + vt x 2W 1,∞ + θθx x 2W 2,∞ + θt x 2W 1,∞ + u t x x x 2H 1 (τ )dτ ≤ C4 . (4.2.170) Proof. Multiplying (4.2.162) and (4.2.163) by and 3/2 respectively, adding the resulting inequality to (4.2.167), and then taking > 0 small enough, we can obtain the desired estimate (4.2.168). Multiplying (4.2.56) by eγ t , using (4.2.54), (4.2.167) and Theorems (2) (1) 4.1.1–4.1.2 and choosing γ > 0 so small that 0 < γ ≤ γ4 ≡ min[ 2C1 1 , γ4 ], we conclude that for any t > 0, t t u u 1 x x x 2 x x x 2 eγ t (t) + eγ τ (τ )dτ ≤ C + C eγ τ E 1 (τ )2 dτ 3 1 u 2C1 0 u 0 t γτ 2 2 2 ≤ C3 + C2 e (u x H 1 + vx x H 1 + θθx H 2 + vt x 2H 1 )(τ )dτ ≤ C4 . 0

That is, γt

t

2

e u x x x (t) +

0

eγ τ u x x x 2 (τ )dτ ≤ C4 ,

∀t > 0.

(4.2.171)


226

In the same manner as the derivation of (4.2.58)–(4.2.63), and using (4.2.168), (4.2.171) and Theorems 4.1.1–4.1.2, we infer that

eγ t vx x x (t)2H 1 + vx x (t)2W 1,∞ + vt x x (t)2 + θθx x x (t)2H 1 + θθx x (t)2W 1,∞ t

2 +θt x x (t) + eγ τ vx x x 2H 1 + vx x 2W 1,∞ + θθx x x 2H 1 + θθx x 2W 1,∞ 0 2 +vt x x + θt x x 2 (τ )dτ ≤ C4 . (4.2.172) Similarly to (4.2.67), we have

t

eγ τ (vt x x x 2 + θt x x x 2 )(τ )dτ ≤ C4 ,

∀t > 0.

(4.2.173)

0

Similarly to (4.2.171), multiplying (4.2.169) by eγ t , using (4.2.66), (4.2.168), (4.12.171)– (4.2.173) and Theorems 4.1.1–4.1.2, we get that for any fixed γ ∈ (0, γ4(2) ], t u u 1 x x x x 2 x x x x 2 (t) + eγ τ eγ t (τ )dτ u 2C1 0 u t ≤ C4 + C1 eγ τ E 2 2 (τ )dτ 0 t ≤ C4 + C4 eγ τ (u x 2H 2 + vx 2H 3 + vt x 2H 2 + θθx 2H 3 )(τ )dτ ≤ C4 0

whence

γt

t

2

e u x x x x (t) +

eγ τ u x x x x 2 (τ )dτ ≤ C4 ,

∀t > 0.

(4.2.174)

0

Similarly to (4.2.71), we easily derive that for any fixed γ ∈ (0, γ4(2) ],

t 0

eγ τ (vt t 2 + θt t 2 )(τ )dτ ≤ C4 ,

∀t > 0.

(4.2.175)

Similarly to (4.2.73)–(4.2.74), using Theorems 4.1.1–4.1.2, (4.2.168) and (4.2.171)– (2) (4.2.175), we deduce that for any fixed γ ∈ (0, γ4 ],

t 0

eγ τ (vx x x x x 2 + θθx x x x x 2 + u t x x x 2H 1 )(τ )dτ ≤ C4 ,

∀t > 0.

(4.2.176)

Finally, the combination of estimates (4.2.171)–(4.2.176) above and using Sobolev’s interpolation inequality give the desired estimates (4.2.169)–(4.2.170). The proof is now complete.

4.3. Universal Attractors

227 (2)

Lemma 4.2.11. For any (u 0 , v0 , θ0 ) ∈ H+4 and for any fixed γ ∈ (0, γ4 ], there holds that for any t > 0,

eγ t u(t) − u ¯ 2H 4 + v(t)2H 4 + θ (t) − θ¯ 2H 4 + u t (t)2H 3 + u t t (t)2H 1 + vt (t)2H 2 t

+ vt t (t)2 + θt (t)2H 2 + θt t (t)2 + eγ τ u − u ¯ 2H 4 + v2H 5 + θ − θ¯ 2H 5 0 + vt 2H 3 + vt t 2H 1 + θt 2H 3 + |θt t 2H 1 (τ )dτ ≤ C4 , (4.2.177) t eγ τ (u t 2H 4 + u t t 2H 2 + u t t t 2 )(τ )dτ ≤ C4 . (4.2.178) 0

Proof. Using (4.2.78)–(4.2.82), Theorems 4.1.1–4.1.2 and Lemmas 4.2.4–4.2.5, we can derive estimates (4.2.177)–(4.2.178). The proof is complete. The proofs of Theorem 4.2.1 and Corollary 4.2.1 are now completed.

4.3 Universal Attractors In this section we are concerned with the infinite-dimensional dynamics of problems (4.1.1)–(4.1.5) for the Navier-Stokes equations for a polytropic viscous and heat-conductive ideal gas. Throughout this section we always assume that (4.1.19) holds. Thus, our first task now is to study the initial boundary value problem (4.1.9)–(4.1.14) where L is fixed. The aim of this section is to use the abstract framework established in Theorem 1.6.4 and the ideas in [450] to establish the existence of (maximal) universal attractors for this problem. Now let us first explain some mathematical difficulties in studying the dynamics of this problem. Firstly, from physical reasons, the special volume u and the absolute temperature θ should be positive for all time. These constraints give rise to some severe mathematical difficulties. For instance, we must work on incomplete metric spaces H+1 and H+2 , H+2 ⊂ H+1 which are usual Sobolev spaces with these constraints. Although in the literature, e.g., [151, 152] and the references cited there, some results on global existence of weak 1 solutions were established under the conditions that ρ0 ≥ 0, ρ0 ∈ L ∞ and ∈ L 1 , this ρ0 space is still incomplete and this framework of spaces seems too weak for the study of dynamics of compressible viscous and heat-conductive fluid. Secondly, the nonlinear semigroup S(t) defined by problem (4.1.13)–(4.1.17), where L is fixed, maps each H+1 and H+2 into itself, as proved in Sections 4.1–4.2. It is clear from equations (4.1.14) and (4.1.15) that we cannot continuously extend the semigroup S(t) to the closure of H+1 and H+2 .


228

Notice the following significant differences between the study of global existence and the study of existence of a (maximal) universal attractor: for the study of global existence, the initial datum is given while for the study of existence of a (maximal) universal attractor in certain metric space, the initial data are varying in that space. Since the (maximal) universal attractor is just the ω − li mi t set of an absorbing set in weak topology, the requirement on completeness of spaces is needed. To overcome this severe mathematical difficulty, we restrict ourselves to a sequence of closed subspaces of H+1 and H+2 (see the definition below). It turns out that it is crucial to prove that the orbit starting from any bounded set of this closed subspace will reenter this subspace and stay there after a finite time, which should be uniform with respect to all orbits starting from a bounded set; otherwise, there is no ground to talk about the existence of an absorbing set and a maximal universal attractor in this subspace. The proof of this fact becomes an essential part of this section and it will be done by delicate a priori estimates, using the spirit of paper [451] for the case of n = 1 (see also Chapter 2). Thirdly, two quantities, i.e., the total mass and energy are conserved. Indeed, if we integrate the equation (4.1.13) with respect to x and t and exploit the boundary conditions (4.1.17), we will end up with

L

L

u(x, t)d x =

0

u 0 (x)d x,

∀t > 0.

(4.3.1)

0

Next, if we multiply (4.1.14) by v, integrate the resultant and also integrate the equation (4.1.15) with respect to x and t, then add together, we finally get L

0

CV θ +

L v2 v2 dx = C V θ0 (x) + 0 (x) d x. 2 2 0

(4.3.2)

These two conservations indicate that there can be no absorbing set for initial data varying in the whole space. Instead, we should rather consider the dynamics in a sequence of closed subspaces defined by some parameters. In this regard, the situation is quite similar to those encountered for a single Cahn-Hilliard equation in the isothermal case (see, e.g., [407]), and for the coupled Cahn-Hilliard equations (see, e.g., [375]) and for a one-dimensional polytropic viscous ideal gas (see, e.g., [451]). Therefore, one of the key issues in the present section is how to choose these closed subspaces. Fourthly, (4.1.13)–(4.1.15) is a hyperbolic-parabolic coupled system. It turns out that in general the orbit is not compact. In order to prove the existence of a maximal attractor by the theory presented by Temam in [407], one has either to show uniform compactness of the orbit of semigroup S(t) for large time or to show that one can decompose S(t) into two parts, S1 (t) and S2 (t), with S1 being uniformly compact for large time and S2 going to zero uniformly. Since equations (4.1.13)–(4.1.15) represent a hyperbolicparabolic coupled system, the orbit is not compact. Moreover, since our system is quasilinear, the usual way of decomposition of S(t) into two parts for a semilinear system (see, e.g., [136]) does not seem feasible. To overcome this difficulty, we will adopt an approach motivated by an idea in [117] and [451] (see also, Theorem 1.6.4).


229

Finally, unlike the one-dimensional case, equations (4.1.13)–(4.1.15) look more complicated than the one-dimensional counterpart and they explicitly involve r , which, in turn, should satisfy (4.1.18). In other words, we are essentially considering a system of four equations with four dependent variables u, v, θ and r . It turns out that much more delicate estimates are needed. Let δi (i = 1, . . . , 5) be any given constants such that δ1 ∈ R, δ2 > 0, 0 < δ5 < eδ1 /L R δ2 , δ4 ≥ max 2(2δ /C , δ > 0 be arbitrarily given constants, and let L)CV /R 3 V

2

Hδ(i)

:= (u, v, θ ) ∈ H

L

:

(C V log(θ ) + R log(u))d x ≥ δ1 ,

0 L

δ5 ≤ δ3 ≤

(i)

(C V θ + v 2 /2)d x ≤ δ2 ,

0 L 0

ud x ≤ δ4 ,

δ3 2δ4 δ5 2δ2 , ≤u≤ , i = 1, 2, 4. ≤θ ≤ 2LC V C V L 2L L

Clearly, Hδ(i) is a sequence of closed subspaces of H+i (i = 1, 2, 4). We will see later on that the first three constraints are invariant. However, the last two constraints are not invariant. These two constraints are just introduced to overcome the difficulty that the original spaces H+i are incomplete. As mentioned before, it is crucial to prove that the (i) (i) orbit starting from any bounded set of Hδ will re-enter Hδ after a finite time. We use Ci , (i = 1, 2) to denote the universal constant depending only on the H+i norm of initial data, min u 0 (x) and min θ0 (x). Cδ denotes the universal conx∈[0,L]

x∈[0,L]

stant depending only on δi (i = 1, . . . , 5), but independent of initial data. Cδ(i) denotes the universal constant depending on both δ j ( j = 1, 2, 3, 4, 5), H+i norm of initial data, min θ0 (x) and min u 0 (x). x∈[0,L]

x∈[0,L]

Now our main theorems read as follows. Theorem 4.3.1. The nonlinear semigroup S(t) defined by the solution to problem (4.1.13)–(4.1.17) maps H+i (i = 1, 2) into itself. Moreover, for any δi (i = 1, . . . , 5) eδ1 /L R C /R , δ3 ] /C V L) V 2

with δ1 < 0, δ2 > 0, δ4 ≥ max[ 2(2δ Hδ(i)

> 0, 0 < δ5 < δ2 , it possesses in

a maximal universal attractor Ai,δ (i = 1, 2).

Remark 4.3.1. The set Ai =

δ1 ,δ2 ,δ3 ,δ4 ,δ5

Ai,δ (i = 1, 2) is a global non-compact attrac-

tor in the metric space H (i) in the sense that it attracts any bounded sets of H (i) with constraints u ≥ η1 , θ ≥ η2 with η1 , η2 being any given positive constants. Now we go back to the Eulerian coordinates and consider problem (4.1.1)–(4.1.5) with G n = (a, b)


230

being fixed. Let

and

(1) H L ,G = (ρ, v, θ ) ∈ H 1[a, b] × H 1[a, b] × H 1[a, b] : b s n−1 ρds = L, ρ(x) > 0, θ (x) > 0, a x ∈ [a, b], v|x=a = v|x=b = 0 ,

(2) H L ,G = (ρ, v, θ ) ∈ H 2[a, b] × H 2[a, b] × H 2[a, b] : b s n−1 ρds = L, ρ(x) > 0, θ (x) > 0, a

x ∈ [a, b], v|x=a = v|x=b = θx |x=a = θx |x=b = 0

where L > 0 is any given positive number. Let δi (i = 1, . . . , 5) be numbers as defined before, and let b

(i) (i) H L ,δ := (ρ, v, θ ) ∈ H L ,G : s n−1 ρds = L,

a

a

b

(C V log(θ ) − R log(ρ))r n−1 ρdr ≥ δ1 ,

b

δ5 ≤ a

δ3 ≤

bn

(C V θ + v 2 /2)r n−1 ρdr ≤ δ2 ,

L n δ5 2δ2 2L , , i = 1, 2, 4. ≤ δ4 , ≤θ ≤ ≤ρ≤ n −a 2LC V C V L 2δ4 δ3

(i)

(i)

Clearly, H L ,δ is a sequence of closed subspaces of H L ,G . Now we have Theorem 4.3.2. The nonlinear semigroup S(t) defined by the solution to problem (4.1.1)–(4.1.5) maps H L(i),G (i = 1, 2) into itself. Moreover, for any δi (i = 1, . . . , 5) with δ1 < 0, δ2 > 0, δ4 ≥ max[ (i) H L ,δ

eδ1 /L R ,δ ] 2(2δ2 /C V L)CV /R 3

> 0, 0 < δ5 < δ2 , it possesses in

a maximal universal attractor Ai,L ,δ (i = 1, 2, 4).

4.3.1 Nonlinear Semigroups on H 2 As mentioned in the previous section, for any initial data (u 0 , v0 , θ0 ) ∈ H+i (i = 1, 2), the results on global existence, uniqueness and asymptotic behavior of solutions to problem (4.1.13)–(4.1.17) have been established in Theorems 4.1.1–4.1.2, respectively. It has been proved in Theorems 4.1.1–4.1.2 that the operators S(t) defined by the solutions are C0 -semigroups on H+i , (i = 1, 2).


231

Now we go back to problem (4.1.1)–(4.1.5) in the Eulerian coordinates, and we have (i)

Lemma 4.3.1. For any (ρ0 , v0 , θ0 ) ∈ H L ,G (i = 1, 2) there exists a unique global (i) S(t) solution (ρ, v, θ ) ∈ C([0, +∞), H ) which defines a nonlinear C0 -semigroup L ,G

(i)

on H L ,G . (i)

Proof. For any given initial data (ρ0 , v0 , θ0 ) ∈ H L ,G , (i = 1, 2) it is clear from the relationship (4.1.6), (4.1.7), (4.1.9) between the Eulerian coordinates and the Lagrangian coordinates that ( u 0 , v0 , θ0 ) = ( ρ10 , v0 , θ0 ) ∈ H+i . By Lemmas 4.1.1–4.1.2, there is a unique global solution ( u (ξ, t), v (ξ, t), θ (ξ, t)) ∈ C([0, +∞); H+i ) which defines a C0 i semigroup S(t) on H+ . It easily follows from Lemma 4.1.2 and the relationship between the Lagrangian coordinates and the Eulerian coordinates (4.1.7), (4.1.8) that problem (4.1.1)–(4.1.5) admits a unique global solution (i)

(ρ(r, t), v(r, t), θ (r, t)) ∈ C([0, +∞); H L ,G ) (i)

which defines a C0 -semigroup on H L ,G . (1)

4.3.2 Existence of an Absorbing Set in Hδ

(1)

In this subsection we will show the existence of an absorbing set in Hδ . Throughout this subsection we always assume that the initial data belong to a bounded set of Hδ(1). First, (1) (1) we have to prove that the orbit starting from any bounded set in Hδ will re-enter Hδ after a finite time, which should be uniform with respect to all orbits starting from that bounded set. Lemma 4.3.2. If (u 0 , v0 , θ0 ) ∈ Hδ(1), then the following estimates hold:

L u(x,t)d x = u 0 (x)d x ≤ δ4 , ∀t > 0, (4.3.3) 0 0 L L v02 v2 δ5 ≤ CV θ + C V θ0 + (x,t)d x = (x)d x ≤ δ2 , ∀t > 0, (4.3.4) 2 2 0 0 t L 2n−2 2 L θx 2μ(2μ + nλ)r 2n−2 vx2 κr d x dτ (C V logθ + R logu)(x,t)d x + + − (2μ + (n − 1)λ)uθ uθ 2 0 0 0 L ≤− (C V logθ0 + R logu 0 )d x ≤ −δ1 , ∀t > 0. (4.3.5)

δ3 ≤

L

0


232

Proof. The estimates (4.3.3)–(4.3.4) have already been derived before (see (4.2.1) and (4.2.2)). A straightforward calculation, using (4.1.18)–(4.1.19) and (4.1.13), yields β 2μ(n − 1)(r n−2 v 2 )x ((r n−1 v)x )2 − uθ θ 2 λr n−1 vx 2μ(2μ + nλ) 2n−2 2 1 (n − 1)(2μ + (n − 1)λ) r −1 uv + r + vx = uθ 2μ + (n − 1)λ 2μ + (n − 1)λ ≥

2μ(2μ + nλ)r 2n−2 vx2 [2μ + (n − 1)λ]uθ

(4.3.6)

with β = 2μ + λ. Multiplying equation (4.1.15) by θ −1 and using (4.1.13), we easily get κ r 2n−2 θx β 2μ(n − 1)(r n−2 v 2 )x ((r n−1 v)x )2 − + (C V log θ + R log u)t = θ u uθ θ x

which, together with (4.3.6), yields the estimate (4.3.5). The proof is complete. Lemma 4.3.3. If (u 0 , v0 , θ0 ) ∈

then the following estimates hold:

vx2 θx2 (x, τ )d x dτ ≤ Cδ , ∀t > 0, + uθ 2 uθ 0 v(t) ≤ Cδ , ∀t > 0, L θ μ0 (x, t)d x ≤ Cδ , ∀μ0 ∈ [0, 1], ∀t > 0. 0 < Cδ−1 ≤

t 0

Hδ(1),

L

(4.3.7) (4.3.8) (4.3.9)

0

Proof. (4.3.8) is the direct result of (4.3.4). It follows from (4.3.4) that (4.3.8) and θ (t) L 1 ≤ Cδ hold. We can also deduce from (4.3.5) and a ≤ r ≤ b in (4.1.22) that L log θ d x ≤ Cδ − 0

holds. Applying the Jensen inequality to the convex function − log y yields (4.3.9). Combining (4.3.5) with (4.3.9), (4.3.3) and (4.1.19) yields (4.3.7). To estimate u, we need the following expression of u which is similar to that in the case n = 1 (see, Lemmas 2.1.3–2.1.4). Lemma 4.3.4. If (u 0 , v0 , θ0 ) ∈ Hδ(1), then 0 < Cδ−1 ≤ u(x, t) ≤ Cδ , 0 < Cδ−1 ≤ r x (x, t) ≤ Cδ ,

∀(x, t) ∈ [0, L] × [0, +∞). (4.3.10)


233

Proof. The proof is similar to that of (4.1.42), the difference here is that we shall note the dependence of constants on the parameters δi (i = 1, 2, . . . , 7). Because B(x, t) depends on the variables x and t for the case of n = 2 or 3, the situation now is more complicated than that for the case of n = 1. To this end, we let B(x, t) = Z 1 (t)Z 2 (x, t) where t 1 2 v 1 1 Z 1 (t) = exp ( + Rθ )(x, s)d x ds β u∗ 0 0 n (n − 1)a n t 1 −n 2 + r (x, s)v (x, s)d x ds , nu ∗ 0 0

(n − 1) t L −n 2 Z 2 (x, t) = exp r (y, s)v (y, s)d yds . β 0 x Thus, from Lemmas 4.3.1–4.3.2, Lemma 4.1.8 and the Cauchy inequality we easily deduce that for any t ≥ s ≥ 0, x ∈ [0, L], 0 < Cδ−1 ≤ D(x, t) ≤ Cδ , 1 2 (n − 1)a n −n 2 1 v + Rθ + r v (x, s)d x ≤ Cδ , 0 < Cδ−1 ≤ βu ∗ 0 n n 1 2 (n − 1)a n −n 2 1 v + Rθ + r v (x, s)d x 0 < Cδ−1 ≤ βu ∗ 0 n n L + (n − 1) r −n (y, s)v2 (y, s)d y] ≤ Cδ , 0

1 = Z 2 (L, t) ≤ Z 2 (x, t) ≤ Z 2 (0, t), eCδ t ≤ B(x, t) ≤ eCδ t , Z 1−1 (t)Z 1 (s)

−C δ−1 (t −s)

e

−Cδ (t −s)

≤

e

−Cδ (t −s)

≤ Z 2 (x, s)/Z 2 (x, t) ≤ 1,

≤e

,

e−Cδ (t −s) ≤ B(x, s)/B(x, t) ≤ Z 1 (s)/Z 1 (t) ≤ e

(4.3.11) (4.3.12)

(4.3.13) (4.3.14) (4.3.15) (4.3.16)

−C δ−1 (t −s)

.

(4.3.17)

Hence, similarly to that in Lemma 2.1.3, we can show that u(x, t) ≤ Cδ ,

∀(x, t) ∈ [0, L] × [0, +∞).

(4.3.18)

By Lemma 4.1.8 and (4.3.14), we have u(x, t) ≥ D(x, t)/B(x, t) ≥ Cδ−1 e−Cδ t , ∀(x, t) ∈ [0, L] × [0, +∞).

(4.3.19)

Then by the asymptotic behavior (4.1.24) in Theorem 4.1.1, (4.3.19) and a similar contradiction argument as in Theorem 4.1.1, we can prove that u(x, t) ≥ Cδ−1 , ∀(x, t) ∈ [0, L] × [0, +∞),

(4.3.20)

which, combined with (4.3.19), (4.1.18) and (4.1.22), gives the estimates (4.3.10). Thus the proof is complete.


234

(1)

Corollary 4.3.1. If (u 0 , v0 , θ0 ) ∈ Hδ , then

t

0

v(τ )2L ∞ dτ ≤ Cδ ,

∀t > 0.

(4.3.21)

Proof. The estimate (4.3.21) is the direct result of Lemmas 4.3.3–4.3.4, the Cauchy inequality, and the boundary condition (4.1.17). Thus the proof is complete. The next lemma concerns boundedness of θ from below. We need more delicate estimates to deal with the cases of n = 2 and n = 3. (1)

Lemma 4.3.5. If (u 0 , v0 , θ0 ) ∈ Hδ , then Cδ−1 ≤ θ (x, t),

∀(x, t) ∈ [0, L] × [0, +∞).

(4.3.22)

Proof. Let w = θ1 . By virtue of (4.1.19), we know that 0≤

2(n − 1)μ 2(n − 2)μ < 0 depending only on boundedness of this set such that for all t ≥ t0 , x ∈ [0, L], 2δ2 δ5 ≤ θ (x, t) ≤ , 2LC V LC V

2δ4 δ3 ≤ u(x, t) ≤ . 2L L

(4.3.28)

Proof. The proof is the same as in Lemma 2.5.5 for the case n = 1. So we can omit the detail here. Remark 4.3.2. It follows from Lemma 4.3.1 and Lemma 4.3.6 that for initial data be(1) (1) longing to a given bounded set of Hδ , the orbit will re-enter Hδ after a finite time. In what follows we prove that there is an absorbing set in Hδ(1). Since we assume that the initial data (u 0 , v0 , θ0 ) belong to an arbitrarily bounded set of Hδ(1), there is a positive constant B such that (u 0 , v0 , θ0 ) H 1 ≤ B. We use Cδ,B or , C to denote universal positive constants depending on B and δ , (i = 1, . . . , 5). Cδ,B i δ,B Then, similarly to the proof of Lemma 4.1.3, we have the following lemma. Lemma 4.3.7. The following inequalities hold, 2 1 v2 ¯ 2 ) ≤ E(u, v, S) ≤ v + Cδ,B (|u − u| ¯ 2 ). (4.3.29) + (|u − u| ¯ 2 + |S − S| ¯ 2 + |S − S| 2 Cδ 2

Lemma 4.3.8. There exists a positive constant γ1 = γ1 (Cδ,B ) > 0 such that for any fixed γ ∈ (0, γ1 ], the following estimates hold: t γt 2 2 2 2 ¯ ¯ e (v(t) H 1 + u(t) − u ¯ H 1 + θ (t) − θ H 1 + S(t) − S ) + eγ τ (u x 2 0

+ρx 2 + θθx 2 + vx 2 + vx x 2 + θθx x 2 )(τ )dτ ≤ Cδ,B , ∀t > 0

(4.3.30)

which implies that for any fixed γ ∈ (0, γ1 ], it holds that (u(t) − u, ¯ v(t), θ (t) − θ¯ )2H 1 ≤ Cδ,B e−γ t , Proof. The proof is similar to those of Lemmas 4.1.4–4.1.5.

∀t > 0.

(4.3.31)


236

Thus, the combination of Lemmas 4.3.2–4.3.8 yields the following result on exis(1) tence of an absorbing ball in Hδ . Theorem 4.3.3. Let R1 = R1 (δ) =

4(δ22 + C V2 δ42 ) C V2 L 2

and B1 = (u, v, θ ) ∈ Hδ(1), (u, v, θ )(1) H ≤ R1 . (1)

Then B1 is an absorbing ball in Hδ , i.e., there exists some t1 = t1 (Cδ,B ) ≥ t0 such that when t ≥ t1 , (u(t), v(t), θ (t))2H (1) ≤ R12 . (2)

4.3.3 Existence of an Absorbing Set in Hδ

In this subsection we are going to prove the existence of an absorbing set in Hδ(2). Throughout this section we always assume that the initial data belong to a bounded set in Hδ(2), i.e., (u 0 , v0 , θ0 ) H 2 ≤ B with B being any given positive constant. We first obtain +

the uniform estimates on H 2 norms of v and θ . Lemma 4.3.9. There exists a positive constant γ2 = γ2 (Cδ,B ) ≤ γ1 (Cδ,B ) such that for any fixed γ ∈ (0, γ2 ], and for all t > 0, 2δ22

+ Cδ,B e−γ t .

(4.3.32)

(r n−1 vt )x = (r n−1 v)t x − (n − 1)(r n−2 v 2 )x .

(4.3.33)

θ (t)2H 2 + v(t)2H 2 ≤

C V2 L 2

Proof. First, a straightforward calculation shows

Using the Poincaré´ inequality and (4.1.18), (4.1.19) and (4.1.22), (4.3.10), we can easily derive the estimates vt ≤ Cvt x ≤ Cδ,B ((r n−1 v)x + (r n−1 v)t x ) ≤ Cδ,B (vx + (r n−1 v)t x ).

(4.3.34)

Differentiating equation (4.1.14) with respect to t, then multiplying the resulting equation by vt eγ t and integrating the resultant over [0, L] × [0, t], by Theorems 4.1.1–4.1.2 and Young’s inequality and (4.3.33)–(4.3.34), we can easily get t 1 γt 1 e vt (t)2 + eγ τ (r n−1 v)xt 2 (τ )dτ 2 Cδ 0 t t ≤ Cδ,B + γ /2 eγ τ vt 2 (τ )dτ + Cδ,B eγ τ (vx 2 + θt 2 + vx x 2 )dτ 0 0 t γτ n−1 2 e (r v)xt (τ )dτ ≤ Cδ,B + Cδ,B γ 0 t + Cδ,B (γ ) eγ τ (θt 2 + vx 2 + vx x 2 )(τ )dτ 0


237

which implies, by Lemma 4.3.8, (4.1.14) and (4.3.33)–(4.3.34), that there exists a positive constant γ2 = γ2 (Cδ,B ) ≤ γ1 such that for any fixed γ ∈ (0, γ2 ], eγ t (vt (t)2 + vx x (t)2 ) +

t 0

eγ τ vxt 2 (τ )dτ ≤ Cδ,B .

(4.3.35)

In the same manner, by equation (4.1.15), Lemma 4.3.7, we can also get γt

2

t

2

e (θt (t) + θθx x (t) ) +

0

eγ τ θθxt 2 (τ )dτ ≤ Cδ,B

which, together with (4.3.35) and Lemma 4.3.8, gives (4.3.32). Thus the proof is com plete. Corollary 4.3.2. Let t2 = t2 (Cδ,B ) ≥ max(t1 (Cδ,B ), −γ γ2−1 ln(2δ22 /(C V2 L 2 Cδ,B ))). Then estimate (4.3.32) implies that for any t ≥ t2 (Cδ,B ), θ (t)2H 2 + v(t)2H 2 ≤

4δ22 C V2 L 2

.

(4.3.36)

The next lemma concerns the uniform estimate of u(t) on H 2. Lemma 4.3.10. There exists a positive constant γ3 = γ3 (Cδ,B ) ≤ γ2 such that for any fixed γ ∈ (0, γ3 ] and for all t > 0, there holds that u(t) − u ¯ 2H 2 ≤ Cδ,B e−γ t .

(4.3.37)

Proof. Differentiating equation (4.1.14) with respect to x, and using equation (4.1.13), we get ∂ uxx θ uxx + 2 ∂t u u β(r n−1 v)x u x − Rθ u x Rθθ x − β(r n−1 v)x x 1−n −n + =r vt x + (n − 1)r u u u2

n−1 2 n−1 2β(r Rθθ x x 2Rθ u x − 2β(r v)x x u x − 2Rθθ x u x v)x u 2x + + . (4.3.38) + u u2 u3

β

Multiplying (4.3.38) by u x x /u, then integrating the resultant over [0, L], by Young’s inequality and Lemmas 4.3.2–4.3.10, we can deduce that u 2 d u x x 2 xx (4.3.39) + Cδ−1 dt u u 1 u x x 2 ≤ + +Cδ,B (θθx 2 + u x 2 + (r n−1 v)x x 2 + θθx x 2 + vt x 2 ). 2Cδ u


238

Multiplying (4.3.39) by et /2Cδ and choosing γ so small that γ ≤ γ3 ≡ min(γ γ2 , 1/4Cδ ) and exploiting Lemmas 4.3.4–4.3.9, we obtain u (t) 2 u (0) 2 xx x x −t /2Cδ + Cδ,B e−γ t ≤ Cδ,B e−γ t ≤ e u(t) u(0) which, together with Lemmas 4.3.2–4.3.9, gives the estimate (4.3.37). The proof is com plete. Now letting R2 =

2(2δ22 +C V2 δ42 )1/2 CV L

and

2(2δ22 + C V2 δ42 ) , γ3−1 ln t3 = t3 (Cδ,B ) ≥ max t2 (Cδ,B ), −γ C V2 L 2 then we immediately infer the following theorem from Lemma 4.3.10 and Corollary 4.3.2. (2)

Theorem 4.3.4. The ball B2 = {(u, v, θ ) ∈ Hδ , (u(t), v(t), θ (t))2

H+2

(2)

≤ R22 } is an

absorbing ball in Hδ , i.e., when t ≥ t3 , (u(t), v(t), θ (t))2H 2 ≤ R22 . +

Having proved the existence of absorbing balls in Hδ(2) and Hδ(1), we can exactly follow the abstract framework established in Theorem 1.6.4 to conclude that Lemma 4.3.11. The set ω(B2 ) =

S(t)B2

(4.3.40)

s≥0t ≥s

where the closures are taken with respect to the weak topology of H (2), is included in B2 and nonempty. It is invariant by S(t), i.e., S(t)ω(B2 ) = ω(B2 ),

∀t > 0.

(4.3.41)

(2)

Remark 4.3.3. If we take B a bounded set in Hδ , we can also define ω(B) by (4.3.40) and when B is nonempty, ω(B) is also included in B2 , nonempty and invariant. Since B2 is an absorbing ball, it is clear that ω(B) ⊆ ω(B2 ). This shows that ω(B2 ) is maximal in the sense of inclusion. Lemma 4.3.12. The set satisfies

A2,δ = ω(B2 )

(4.3.42)

A2,δ is bounded and weakly closed in Hδ(2),

(4.3.43)


∀t ≥ 0,

(4.3.44)

lim d w (S(t)B, A2,δ ) = 0.

(4.3.45)

Hδ(2) , t −→+∞

Moreover, it is the maximal set in the sense of inclusion that satisfies (4.3.43), (4.3.44) and (4.3.45).


239

Proof. See, Theorem 2.5.3.

Following [117], we also call A2,δ the universal attractor of S(t) in Hδ(2). In order to discuss the existence of a universal attractor in Hδ(1), we need to prove the following lemma. Lemma 4.3.13. For every t ≥ 0, the mapping S(t) is continuous on bounded sets of Hδ(1) for the topology induced by the norm in L 2 × L 2 × L 2 . ¯ (i = 1, 2), Proof. Let (u 0i , v0i , θ0i ) ∈ Hδ(1), (i = 1, 2), (u 0i , v0i , θ0i ) H 1 ≤ R, (u i , vi , θi ) = S(t)(u 0i , v0i , θ0i ), and (u, v, θ ) = (u 1 , v1 , θ1 ) − (u 2 , v2 , θ2 ). Subtracting the corresponding equations (4.1.13)–(4.1.15) satisfied by (u 1 , v1 , θ1 ) and (u 2 , v2 , θ2 ), then multiplying the resulting equations by u, v, θ , respectively, adding together and integrating over [0, L], we get L β(r1n−1 v)2x + κr12n−2 θx2 1 d (u2 + v2 + C V θ 2 ) + dx 2 dt u1 0 L β(r1n−1 v2 )x − Rθ2 = + 1 u(r1n−1 v)x u u 1 2 0 R + β(r1n−1 v1 )x + β(r2n−1 v2 )x − Rθ2 n−1 + (r1 v)x θ d x u1

1 2n−2 κθ2x r2 [Rθ2 − β(r2n−1 v2 )x ](r2n−1 v2 )x R(r1n−1 v1 )x 2 + uθθx − uθ − θ dx u1u2 u 1u 2 u1 0 1 1 Rθ2 − β(r2n−1 v2 )x n−1 + 2μ(n − 1) r1n−2 (v1 + v2 )vθθ x d x + [(r1 −r2n−1 )v]x d x u 2 0 0 1 β(r1n−1 v1 )x β(r2n−1 v2 )x − Rθ2 β(r1n−1 v)x + θ− + + u [(r1n−1 −r2n−1 )v2 ]x d x u1 u2 u2 0 1 κθ2x (r12n−2 −r22n−2 )θθx − (4.3.46) − 2μ(n − 1)v22 (r1n−2 −r2n−2 )θθx d x u1 0 where

t

ri (x, t) = r0i (x) +

vi (x, τ )dτ, r0i (x) = a n + n

0

x

1/n u 0i (y)d y

(4.3.47)

0

and rin−1 (x, t)ri x (x, t) = u i (x, t), i = 1, 2,

∀(x, t) ∈ [0, L] × [0, +∞).

(4.3.48)

It follows from Theorem 4.1.1 that for any t > 0, t ¯ (vi 2H 2 + θi x 2 + θi x x 2 )(τ )dτ ≤ Cδ ( R), i = 1, 2 (u i (t), vi (t), θi (t))2H 1 + 0

(4.3.49)

240


¯ > 0 is a constant depending only on R¯ and δ. By (4.1.18) and (4.1.26), we where Cδ ( R) have x rin (x, t) = r¯ n (x) + n

(u i (y) − u)d ¯ y, i = 1, 2,

0

which implies that for any (x, t) ∈ [0, L] × [0, +∞), x u(y, t)d y, (r1n (x, t) − r2n (x, t))x = nu(x, t). r1n (x, t) − r2n (x, t) = n

(4.3.50)

0

Applying the mean value theorem to the function g(z) = z k/n over [r1n , r2n ] ⊆ [a n , bn ] with k > 0 being an arbitrarily given constant, by (4.3.50) and Theorem 4.1.1, we get |r1k (x, t) − r2k (x, t)| ≤ |g (z 0 )||r1n (x, t) − r2n (x, t)| ≤ Cδ u(t) L 1

(4.3.51)

where z 0 is a point between r1n and r2n (therefore, z 0 ∈ [a n , bn ]), and by Theorem 4.1.1, |g (z 0 )| ≤ Cδ . Similarly, applying the mean value theorem to the function h(z) = z n/ k over k k [r1 , r2 ] ⊆ [a k , bk ], we can deduce from Theorem 4.1.1 and (4.3.50)–(4.3.51) that nu(x, t) = [h(r1k ) − h(r2k )]x = h (z 1 )(r1k (x, t) − r2k (x, t))r2k (x, t)x + h (r1k )(r1k − r2k )x where z 1 is a point between r1k and r2k , hence, z 1 ∈ [a k , bk ]. By Theorem 4.1.1 and Lemma 4.3.4, 0 < Cδ−1 ≤ |h (z 1 )| ≤ Cδ , |[h (r1 )]x | ≤ Cδ . Therefore, nu(x, t) − h (z )(r k (x, t) − r k (x, t))r k (x, t) 1 x k k 1 2 2 |(r1 − r2 )x | = h (z 1 ) ≤ Cδ (|u(x, t)| + u(t) L 1 ).

(4.3.52)

Applying Young’s inequality and the imbedding theorem, we infer from (4.3.46)–(4.3.52) that d (u(t)2 + v(t)2 + C V θ (t)2 ) + Cδ−1 (vx (t)2 + (r1n−1 v(t))x 2 + θθx (t)2 ) dt 1 ≤ (vx (t)2 + (r1n−1 v(t))x 2 + θθx (t)2 ) 2Cδ ¯ (t)(u(t)2 + v(t)2 + θ (t)2 ) +Cδ ( R)H (4.3.53) where H (t) = (r1n−1 v2 )x 2L ∞ + (r2n−1 v2 )x 2L ∞ + (r1n−1 v1 )x 2 + v2x 2L ∞ +θ2 2L ∞ + v2 4L 4 + v1 2L ∞ + 1

(4.3.54)

t ¯ satisfying 0 H (τ )dτ ≤ Cδ ( R)(1 + t) for any t > 0. Then the assertion of this lemma follows from the Gronwall inequality and (4.3.54). The proof is complete.


241

Now we can again use Theorem 1.6.4 to obtain the following result on existence of (1) a universal attractor in Hδ . Theorem 4.3.5. The set A1,δ =

S(t)B1

(4.3.55)

s≥0 t ≥s

where the closures are taken with respect to the weak topology of H+1 , is the (maximal) (1) universal attractor in Hδ . Remark 4.3.4. Since A2,δ is bounded in H+2 , it is bounded in H+1 and by the invariance property (4.3.44), we have A2,δ ⊆ A1,δ . (4.3.56) On the contrary if we knew that A1,δ is bounded in H+2 then the opposite inclusion would hold.

4.3.4 Results of the Eulerian Coordinates (i)

By Lemma 4.3.1 for any (ρ0 , v0 , θ0 ) ∈ H L ,G , there is a unique global solution (ρ, v, θ ) ∈ (i)

(i)

C([0, +∞); H L ,G ) which defines a C0 -semigroup on H L ,G . It is easy to see from (i)

(4.1.11)–(4.1.12) and Lemma 4.3.4 that the existence of absorbing balls in Hδ presented (i) in Sections 4.3.2–4.3.3 implies the existence of absorbing balls in H L ,δ . Thus the general framework of Theorem 1.6.4 (see also, Ghidaglia [117]) also yields the existence of universal attractors Ai,L ,δ , (i = 1, 2).

4.3.5 Attractor in H 4 In this subsection we establish the existence of an attractor in H+4 . We define

H+4 = (u, v, θ ) ∈ H 4[0, L] × H 4[0, L] × H 4[0, L] : u(x) > 0, θ (x) > 0, x ∈ [0, L], v|x=0 = v| x=L = 0, θx |x=0 = θ x |x=L = 0

which becomes a metric space when equipped with the metrics induced from the usual norms. In the above, H 4 is the usual Sobolev space. Let = (u, θ ) = e − θ S = C V θ − θ (C V log θ + R log u), e = e(u, θ ) = e(u, S) = C V θ − internal energy, S = S(u, θ ) = C V log θ + R log u − entropy. The notation in this subsection is the same as that of Section 4.1.3. But we use Ci (i = 1, 2, 3, 4) to stand for the universal constant depending only on the H+i norm of


242

initial data, min u 0 (x) and min θ0 (x). Cδ denotes the universal constant depending x∈[0,L]

x∈[0,L]

only on δ j ( j = 1, . . . , 8), but independent of initial data. Cδ,Bi (i = 1, 2, 4) denotes the universal constant depending on both δ j ( j = 1, 2, . . . , 8), H (i) norm of initial data with (u 0 , v0 , θ0 ) H i ≤ Bi , min θ0 (x) and min u 0 (x). x∈[0,L]

x∈[0,L]

Our main results read as follows: Theorem 4.3.6. The nonlinear semigroup S(t) defined by the solution to (4.1.13)– (4.1.17) maps H+4 into itself. Moreover, for any δi (i = 1, . . . , 5) with δ1 < 0, δ2 > 0, δ /L R

(4)

e1 δ4 ≥ max[ 2(2δ /C C /R , δ3 ] > 0, 0 < δ5 < δ2 , it possesses in Hδ V L) V 2 (universal) attractor A4,δ .

a maximal

Proof. The proof is similar to those of Theorem 2.5.1 and Theorem 2.6.1. Remark 4.3.5. The set Ai = Ai,δ (i = 1, 2, 4) is a global non-compact attractor in the constraints u

δ1 ,δ2 ,δ3 ,δ4 ,δ5 (i) metric space Hδ in the sense that it attracts any bounded sets ≥ η1 , θ ≥ η2 with η1 , η2 being any given positive constants.

of Hδ(i) with

Now we go back to the Eulerian coordinates and consider problem (4.1.1)–(4.1.5) with G n = (a, b) being fixed. Let

= (ρ, v, θ ) ∈ H 4[a, b] × H 4[a, b] × H 4[a, b] : H L(4) ,G b s n−1 ρds = L, ρ(x) > 0, θ (x) > 0, a x ∈ [a, b], v| x=a = v|x=b = 0, θx |x=a = θ x |x=b = 0 where L > 0 is any given positive number. Let δi (i = 1, . . . , 5) be numbers as defined before, and let b

(4) (4) H L ,δ := (ρ, v, θ ) ∈ H L ,G : s n−1 ρds = L, a

b a

(C V log(θ ) − R log(ρ))r n−1 ρdr ≥ δ1 ,

δ5 ≤ δ3 ≤

b

a bn

(C V θ + v 2 /2)r n−1 ρdr ≤ δ2 ,

L 2L − an ≤ δ4 , . ≤ρ≤ n 2δ4 δ3

˜ Theorem 4.3.7. The nonlinear semigroup S(t) defined by the solution to problem (4) (4.1.1)–(4.1.5) maps H L ,G into itself. Moreover, for any δi (i = 1, . . . , 5) with δ1 < δ /L R

(4)

e1 0, δ2 > 0, δ4 ≥ max[ 2(2δ /C C /R , δ3 ] > 0, 0 < δ5 < δ2 , it possesses in H L ,δ a V L) V 2 maximal (universal) attractor A4,L ,δ .


243

4.4 Bibliographic Comments As far as the associated infinite-dimensional dynamics is concerned, we refer the readers to Section 2.7. Concerning compressible fluid, we mention the recent papers [451, 452] by Zheng and Qin for results on the existence of maximal universal attractors for a viscous and heat-conductive polytropic ideal gas (see also, Section 10.1.3). Three recent papers [97, 150, 151], came to our attention. In [150], the authors briefly described their recent investigation on the existence of a compact attractor for the one-dimensional isentropic compressible viscous flow in a finite interval. They work on the incomplete metric space

X = (ρ, u) ∈ H 1 × L 2 ;

1

ρd x = 1, ρ > 0, ρ −1 ∈ L ∞ .

0

In [151], the isentropic compressible viscous flow in a bounded domain in R3 is considered. Since it is based on the fundamental result on global existence of weak solution by P.L. Lions [235] and the uniqueness is not known, it is impossible to adopt the usual solution semigroup approach. As a result, the author adopted a quite different approach, i.e, he replaced the usual solution semigroup setting by simple time shifts; in other words, he worked on the space of ‘short’ trajectories, as mentioned. Therefore, the model under consideration in Section 4.3 is quite different from [97, 150, 151] in the following aspects: non-isentropic via isentropic; spherically symmetric motion via non-spherically symmetric motion; solution semigroup approach via simple time shift. For the well-posedness of the Navier-Stokes equations, we also consult Section 3.3.

Chapter 5

A Polytropic Viscous Gas with Cylinder Symmetry in R3 5.1 Main Results In this chapter we establish the exponential stability in H i (i = 1, 2, 4) of global weak solutions to the compressible Navier-Stokes equations with cylinder symmetry in R3 . The results of this chapter come from Qin [326], and Qin and Jiang [331]. Our attention in this chapter will be paid to the flows between two circular coaxial cylinders. We assume that the corresponding solutions depend only on the radial variable r ∈ G = {r ∈ R+ , 0 < a < r < b < +∞} and the time variable t ∈ R+ = [0, +∞). The reduced system of the three-dimensional equations now takes the following form (see e.g., Landau and Lifshitz [212], Frid and Shelukhin [106]): ρu ρt + (ρu)r + = 0, r v2 u + Pr − ν u r + ρ u t + uu r − = 0, r r r uv v − μ vr + = 0, ρ vt + uvr + r r r wr = 0, ρ(wt + uwr ) − μ wrr + r θr u + P ur + −Q=0 C V ρ(θt + uθr ) − κ θrr + r r where

u 2 u 2 vr 2 + μ vr − + wr2 + 2u r2 + 2 Q = λ ur + r r r

(5.1.1) (5.1.2) (5.1.3) (5.1.4) (5.1.5)

Chapter 5. A Polytropic Viscous Gas with Cylinder Symmetry in R3

246

and ρ, P, θ are mass density, pressure and absolute temperature respectively. The velocity → vector − v = (u, v, w) is given by the radial (u), angular (v), and axial (w) velocities, respectively. For simplicity, we consider a polytropic fluid P = γρθ , and the Duhem inequality, μ ≥ 0, and 3λ + 2μ ≥ 0 (ν = λ + 2μ). The parameters γ , C V , κ, λ and μ are physical constants. We consider the initial boundary value problem (5.1.1)–(5.1.5) subject to the following boundary and initial conditions → −v = − → 0 , θr = 0 at ∂ G, → − → t = 0 : (ρ, v , θ ) = (ρ0 (r ), − v 0 (r ), θ0 (r )), r ∈ G.

(5.1.6) (5.1.7)

First we find it convenient to transfer problems (5.1.1)–(5.1.7) into Lagrangian coordinates and draw the desired results. It is known that Eulerian coordinates (r, t) are connected to the Lagrangian coordinates (ξ, t) by the relation

t r (ξ, t) = r0 (ξ ) + u (ξ, τ )dτ (5.1.8) 0

where u (ξ, t) = u(r (ξ, t), t) and r0 (ξ ) = η−1 (ξ ),

η(r ) =

r a

sρ0 (s)ds, r ∈ G.

(5.1.9)

Here we note that if inf{ f ρ0 (s) : s ∈ (a, b)} > 0, then η is invertible. It follows from equation (5.1.1) and boundary condition (5.1.6) that

r(ξ,t ) ∂ sρ(s, t)ds = 0 ∂t a which, by (5.1.9), implies

r

a

r0

sρ(s, t)ds =

and G is transformed into = (0, L) with

b

L= sρ(s, t)ds = a

sρ0 (s)ds = ξ,

(5.1.10)

sρ0 (s)ds, ∀t ≥ 0.

(5.1.11)

a

a

b

Moreover, differentiating (5.1.10) with respect to ξ yields ∂ξ r (ξ, t) = r (ξ, t)−1 ρ −1 (r (ξ, t), t).

(5.1.12)

In general, for a function φ(r, t), if we denote φ(ξ, t) = φ(r (ξ, t), t), then we easily get ∂t φ(ξ, t) = ∂t φ(r, t) + u∂r φ(r, t), ∂r φ(r, t) −1 ∂ξ ρ (r, t). φ (ξ, t) = ∂r φ(r, t)∂ξ r (ξ, t) = r

(5.1.13)

5.1. Main Results

247

→ → In what follows, without danger of confusion, we denote ( ρ, − v , θ ) still by (ρ, − v , θ) and (ξ, t) by (x, t). We use τ = 1/ρ to denote the specific volume. Thus, by (5.1.12)– (5.1.13), equations (5.1.1)–(5.1.7) in Eulerian coordinates can be written in Lagrangian coordinates in the new variables (x, t), x ∈ , t ≥ 0 as follows : τt = (r u)x ,

v2 ν(r u)x − γ θ ut = r + , τ r x

uv (r v)x , vt = μr − τ r x

μτ w (r w)x wt = μr + 2 , τ r x 1 μ(r v)2x r 2 θx + [ν(r u)x − γ θ ](r u)x + C V θt = κ τ τ τ

(5.1.14) (5.1.15) (5.1.16) (5.1.17)

x

μr 2 w2x − 2μ(u 2 + v 2 )x + τ

(5.1.18)

subject to the initial and boundary conditions → → v (x, 0) = − v 0 (x), θ (x, 0) = θ0 (x), x ∈ , τ (x, 0) = τ0 (x), − → − − → → − v (0, t) = v (L, t) = 0 , θ x (0, t) = θ x (L, t) = 0, t ≥ 0 where Q=

2μ(u 2 + v 2 )x ν(r u)2x + μ(r v)2x + μr 2 w2x . − τ τ2

(5.1.19) (5.1.20)

(5.1.21)

In view of (5.1.8) and (5.1.12), r (x, t) is determined by rt (x, t) = u(x, t), r (x, t)r x (x, t) = τ (x, t),

1/2

x 2 r |t =0 = r0 (x) = a + 2 τ0 (y)d y .

(5.1.22)

0

In this section, we shall establish the exponential stability of global solutions in H 1 and H 2 to the compressible Navier-Stokes equations (5.1.1)–(5.1.7) or (5.1.14)–(5.1.20) with cylinder symmetry in R3 when the initial total energy is small enough. Now we study problem (5.1.14)–(5.1.20) where L > 0 is fixed. We introduce two spaces 3 → H+1 = (τ, − v , θ ) ∈ H 1[0, L] × H 1[0, L] × H 1[0, L] : → − → → τ (x) > 0, θ (x) > 0, x ∈ [0, L], − v |x=0 = − v |x=L = 0 ,


248

3 → H+2 = (τ, − v , θ ) ∈ H 2[0, L] × H 2[0, L] × H 2[0, L] : τ (x) > 0, θ (x) > 0, x ∈ [0, L], → − → − → v |x=0 = − v |x=L = 0 , θx |x=0 = θx |x=L = 0 which become two metric spaces when equipped with metrics induced from the usual norms. In the above, H 1, H 2 are the usual Sobolev spaces. In this chapter, we put · = · L 2 and use Ci (i = 1, 2, 4) to stand for the generic constant depending only on the H+i (i = 1, 2) norm of initial data, min τ0 (x) x∈[0,L]

and min θ0 (x). x∈[0,L]

Now our main results read as follows: Theorem 5.1.1. Let us set

L1

E0 = 0

2

− → 2 | v 0 (x)| + C V θ0 d x.

(5.1.23)

→ Then there exists a constant δ0 = δ0 (C1 ) > 0 such that as E 0 ≤ δ0 , for any (ττ0 , − v 0 , θ0 ) ∈ → − 1 1 H+, there exists a unique global weak solution (τ (t), v (t), θ (t)) ∈ H+ to problem (5.1.14)–(5.1.20) verifying → (τ (t), − v (t), θ (t)) ∈ C([0, +∞); H 1 ), +

→ → −v , τ , − → τ − τ¯ , − v , θ − θ¯ , τt , − v t , θt , θ x , → ¯ x, x x v x x , θ x x , r − r¯ , (r − r) (r − r¯ )x x , rt , rt x , rt x x ∈ L 2 ([0, +∞); L 2)

and the following estimates hold: 0 < C1−1 ≤ θ (x, t) ≤ C1 on [0, L] × [0, +∞), 0
0 and γ1 = γ1 (C1 ) > 0 such that for any fixed γ ∈ (0, γ1 ], we have that for any t > 0, ¯ 21 eγ t rt (t)2H 1 + r (t) − r¯ 2H 2 + τ (t) − τ¯ 2H 1 + τt (t)2 + θ (t) − θ H t → → +− v (t)2H 1 + eγ τ τ − τ¯ 2H 1 + τt 2H 1 + − v 2H 2 + θ − θ¯ 2H 2 0 → +− v t 2 + θt 2 + r − r¯ 2H 2 + rt 2H 2 (τ )dτ ≤ C1 (5.1.28)

5.2. Global Existence and Exponential Stability in H 1

249

where τ¯ =

1 L

L

τ0 (x)d x,

0 2

r¯ (x) = (a + 2τ¯ x)

1/2

.

θ¯ =

1 CV L

L

C V θ0 +

0

→ |− v 0 |2 (x)d x, (5.1.29) 2 (5.1.30)

→ Theorem 5.1.2. For any (ττ0 , − v 0 , θ0 ) ∈ H+2 , as E 0 ≤ δ0 , there exists a unique gener− → alized global solution (τ (t), v (t), θ (t)) ∈ H+2 to problem (5.1.14)–(5.1.20). In addi→ → v xxx, − v t x , θx x x , θt x , (r − r¯ )x x x , rt x x x ∈ tion to Theorem 5.1.1, we have τx x , τt x , τt x x , − 2 2 L ([0, +∞); L ) and the following estimates hold: → → rt (t)2H 2 + r (t) − r¯ 2H 3 + τ (t) − τ¯ 2H 2 + τt (t)2H 1 + − v (t)2H 2 + − v t (t)

t → → τ − τ¯ 2H 2 + τt 2H 2 + − v 2H 3 + − v t 2H 1 +θ (t) − θ¯ 2H 2 + θt (t)2 + 0 +θ − θ¯ 2H 3 + θt 2H 1 + r − r¯ 2H 3 + rt 2H 3 (τ )dτ ≤ C2 , ∀t > 0. (5.1.31) Further, there exist constants C2 > 0 and γ2 = γ2 (C2 )(≤ γ1 ) > 0 such that for any fixed γ ∈ (0, γ2 ], we have that for any t > 0, → → eγ t rt (t)2H 2 + r (t) − r¯ 2H 3 + τ (t) − τ¯ 2H 2 + τt (t)2H 1 + − v (t)2H 2 + − v t (t)2 t → γτ ¯ 2 2 + θt (t)2 + τ − τ¯ 2H 2 + τt 2H 2 + − +θ (t) − θ e v 2H 3 H 0 → +θ − θ¯ 2H 3 + − v t 2H 1 + θt 2H 1 + r − r¯ 2H 3 + rt 2H 3 (τ )dτ ≤ C2 . (5.1.32) Remark 5.1.1. Theorems 5.1.1–5.1.2 are also valid when the boundary conditions − → → → (5.1.20) are replaced by − v (0, t) = − v (L, t) = 0 , θ (0, t) = θ (L, t) = θ˜ where θ˜ > 0 is a constant and θ¯ is replaced by θ˜ . Remark 5.1.2. Similar results in H 4 to those in Chapter 4 also hold (see also Theorems 5.4.1–5.4.2).

5.2 Global Existence and Exponential Stability in H 1 In this section we shall complete the proof of Theorem 5.1.1. To this end, we assume in this section that all assumptions in Theorem 5.1.1 hold. The global existence of solutions in H+1 to problem (5.1.14)–(5.1.20) (or (5.1.1)–(5.1.7)) has been obtained in Frid and Shelukhin [106]. Theorem 5.1.1 is only concerned with the large-time behavior of solution in H+1 . The proof of Theorem 5.1.1 can be divided into the following eight lemmas.


250

→ Lemma 5.2.1. The global weak solution (τ (t), − v (t), θ (t)) ∈ H+1 to problem (5.1.14)– (5.1.20) satisfies the estimates 0 < a ≤ r (x, t) ≤ b, on [0, L] × [0, +∞), (5.2.1)

L

L e(x, t)d x = e(x, 0)d x ≡ E 0 , ∀t > 0, (5.2.2) 0 0

L

L

t L τQ κr 2 θx2 (x, s)d x ds = U (x, t)d x + + U0 (x)d x, ∀t > 0, θ τθ2 0 0 0 0 (5.2.3)

t L 2 2 2 u (r u)x τu + x + (x, s)d x ds ≤ C1 , ∀t > 0, (5.2.4) θ τθ τθ 0 0

t L 2 (τr −1 v − r vx )2 wx + (x, s)d x ds ≤ C1 , ∀t > 0, (5.2.5) τθ τθ 0 0

L

L 1 τ (x, t)d x = τ0 (x)d x = (b 2 − a 2 ), (5.2.6) 2 0 0

L −1 ≤ θ α (x, t)d x ≤ C2α , ∀α ∈ [0, 1], (5.2.7) 0 < C1α 0

where e(x, t) = U (x, t) =

1 − |→ v |2 + C V θ, 2

1 − |→ v |2 + C V (θ ) + γ (τ ), (θ ) = θ − log θ + 1. 2

Proof. First, we know from (5.1.22) that r x (0, t) = r −1 (0, t)τ (0, t) = a −1 τ (0, t) > 0, ∀t ≥ 0.

(5.2.8)

If r x (x, t) > 0 is violated on [0, L] × [0, +∞), by (5.2.8), there exists a y ∈ [0, L] and tˆ ∈ [0, ∞) such that r x (x, t) > 0 for 0 ≤ x < y, 0 ≤ t ≤ tˆ, but r x (y, tˆ) = 0. But by continuity, we have r x (x, t) ≥ 0, ∀x ∀ ∈ [0, y], ∀t ∈ [0, tˆ]. Hence, r (y, tˆ) ≥ r (0, tˆ) = a > −1 ˆ ˆ 0 and 0 = r x (y, t ) = r (y, t )τ (y, tˆ) > 0. This is a contradiction. Thus r x (x, t) > 0 and a = r (0, t) ≤ r (x, t) ≤ r (L, t) = b, ∀x ∀ ∈ [0, L], t ≥ 0, and estimate (5.2.1) is proved. Second, multiplying (5.1.15), (5.1.16), (5.1.17) and (5.1.18) by u, v, w and θ , respectively, adding up the results, and integrating the resulting equations in [0, L], using (5.1.20), we derive

L d e(x, t)d x = 0 dt 0 which gives (5.2.2). Similarly, by virtue of (5.1.15)–(5.1.18), we deduce that U (x, t)


251

satisfies κr 2 θx2 τQ r u[ν(r u)x − γ θ ] μr v(r v)x + μr 2 wwx = + Ut + + τθ2 θ τ τ 2 κ(θ − 1)r θx − 2μ(u 2 + v 2 ) + γ r u + τθ x which along with (5.1.20) yields (5.2.3). Third, by (5.1.21) and (5.2.1), we can deduce that 2μ(τ 2r −2 u 2 + r 2 u 2x ) + λ(r u)2x ν(r u)2x − 2μ(u 2 )x = τ τ 2 + (r u)2 u x , ≥ C1−1 τ u 2 + x τ μ(τr −1 v − r vx )2 μ(r v)2x − 2μ(v 2 )x = ≥ 0. τ τ Therefore, w2 (τr −1 v − r vx )2 u 2 + (r u)2x + x + ≥0 τ Q ≥ C1−1 τ u 2 + x τ τ τ

(5.2.9)

which gives (5.2.4) and (5.2.5). Fourth, with the help of (5.1.14), (5.1.20), (5.1.22) and (5.2.1), we get

L 0

L

τ0 (x)d x = 0

1 2 x=L 1 τ (x, t)d x = r = (b 2 − a 2 ) 2 x=0 2

which is (5.2.6). By (5.2.3), the Jensen inequality and the Young inequality, we get that for all α ∈ [0, 1], (L −1

L 0

θ α d x) ≤ L −1

L

(θ α )d x ≤ L −1

L

(α(θ ) + 2 − 2α)d x

L U0 (x)d x + (2 − 2α)L ≡ C1 . (5.2.10) ≤ L −1 αC V−1 0

0

0

−1 Thus (5.2.7) follows from (5.2.10) where L −1 C1α , L −1 C2α are two positive roots of the

function equation (y) = C1 . The proof is now complete.

Lemma 5.2.2. There exists a positive constant δ0 = δ0 (C1 ) > 0 such that, as E 0 ≤ δ0 , the following estimates hold, 0 < C1−1 ≤ τ (x, t) ≤ C1 , 0 < C1−1 ≤ r x (x, t) ≤ C1 , (x, t) ∈ [0, L] × [0, +∞). (5.2.11)


252

Proof. With the help of (5.1.14), we can rewrite (5.1.15) as u = σx r t with σ =

ν(r u)x − γ θ + τ

Let

φ(x, t) =

Then φ(x, t) satisfies

x 0

u0 dy + r0

φx = ur −1 ,

(5.2.12)

0

v2 − u 2 d y. r2

t

x

0

σ (x, s)ds.

φt = σ

whence

(5.2.13)

(5.2.14) (5.2.15)

(φτ )t = (r uφ)x − r uφx + ν(r u)x − γ θ + τ 0

x

v2 − u 2 d y. r2

(5.2.16)

Integrating (5.2.16) over [0, L], exploiting (5.1.20), (5.1.22) and (5.2.15), we infer

L

t L

L φτ d x = φ 0 τ0 d x − (u 2 + γ θ )d x ds 0

0

t

0

0 x

v2 − u 2 d yd x ds r2 0 0 0

t L

L 1 2 (u + v 2 ) + γ θ d x ds φ 0 τ0 d x − = 2 0 0 0

t L 2 2 2 b v −u d x ds. + 2 0 0 r2 +

L

(r 2 /2)x

Note that there exists a point a(t) ∈ [0, L] such that

L ∗ φ(x, t)τ (x, t)d x τ φ(a(t), t) =

(5.2.17)

(5.2.18)

0

where, by (5.2.6), ∗

τ =

L 0

τdx =

L 0

τ0 d x =

b2 − a 2 . 2

Thus by (5.2.14) and (5.2.17)–(5.2.18),

t

a(t ) σ (a(t), s)ds = φ(a(t), t) − u 0r0−1 d y 0 0 L

t L 1 1 2 2 (u + v ) + γ θ d x ds τ0 φ 0 d x − = ∗ τ 2 0 0 0 a(t )

b2 t L v2 − u 2 + d x ds − u 0r0−1 d y. (5.2.19) 2 0 0 r2 0


253

Now by virtue of (5.1.14), we can rewrite (5.2.12) as u v2 − u 2 θ = σx = ν(log τ )t x − γ + . r t τ x r2

(5.2.20)

Integrating (5.2.20) over [a(t), x] × [0, t], and using (5.2.19), we derive

t θ (x, s) ν log τ (x, t) − γ ds 0 τ (x, s)

t

t x 2

x u0 v − u2 u − dy = ν log τ0 (x) + σ (a(t), s)ds − d y + r2 r0 0 0 0 a (t ) r L

t L 1 1 2 2 = ν log τ0 (x) + ∗ (u + v ) + γ θ d x ds τ0 φ 0 d x − τ 2 0 0 0 a(t )

t x 2 b2 t L v2 − u 2 u0 v − u2 + d yds − d y − d yds 2 2 0 0 r0 r r2 0 0 0

x u0 u − d y. (5.2.21) + r0 a (t ) r That is,

τ −1 (x, t) exp γ ν −1

t 0

θ (x, s) ds = D −1 (x, t)B(x, t) τ (x, s)

(5.2.22)

where

L

x

x u u0 1 1 d y − τ φ d x + d y , (5.2.23) 0 0 ν τ∗ 0 a (t ) r 0 r0

t L

t L 2 b2 v − u2 1 1 2 2 B(x, t) = exp (u + v ) + γ θ d x ds − d x ds ντ ∗ 0 0 2 2ντ ∗ 0 0 r2

1 t x v2 − u 2 + d x ds . (5.2.24) ν 0 0 r2

D(x, t) = τ0 (x) exp

Multiplying (5.2.22) by γ ν −1 θ , we arrive at

t d θ (x, s) −1 exp γ ν ds = γ ν −1 θ D −1 (x, t)B(x, t) dt 0 τ (x, s) which with (5.2.22) gives τ (x, t) =

t θ (x, s)B(x, s) D(x, t) 1 + γ ν −1 ds . B(x, t) D(x, s) 0

Now for any 0 ≤ s ≤ t, we get from (5.2.24)

B(x, s)/B(x, t) = exp −

t s

(5.2.25)

A(x, ξ )dξ

(5.2.26)


254

where 1 A(x, t) = ∗ ντ

L

L 2 b2 v − u2 1 2 2 (u + v ) + γ θ d x − dx 2 2ντ ∗ 0 r2

x

v2 − u 2 d x. r2

1 + ν

0

0

(5.2.27)

By virtue of Lemma 5.2.1, we easily infer from (5.2.23) and (5.2.27) 0 < C1−1 ≤ D(x, t) ≤ C1 ,

A(x, t) ≤ C1 , ∀(x, t) ∈ [0, L] × [0, +∞).

(5.2.28)

In the sequel, we shall prove that there exists a positive constant δ0 > 0 such that, as E 0 ≤ δ0 , A(x, t) ≥ C1−1 > 0, ∀(x, t) ∈ [0, L] × [0, +∞). (5.2.29) In fact, we can deduce from Lemma 5.2.1 and (5.2.27)

x 2

L

L 2 b2 b 1 1 1 u 1 + 2 u2d x − 1 − 2 v2 d x A(x, t) = dy + 2ντ ∗ 0 r ν 0 r2 2ντ ∗ 0 r

x 2

L 1 γ v + dy + ∗ θdx 2 ν 0 r ντ 0

L

L

L 2 γ (r + b2 − 2τ ∗ )u 2 1 d x + θ d x − [b 2 − r 2 ]v 2 d x ≥ ∗ ∗ 2 ∗ 2 2ντ 0 2ντ r 0 0 2ντ r

L

L γ −1 b 2 − a 2 γ 1 θdx − [b 2 − r 2 ]v 2 d x ≥ ∗ C11 − E0 ≥ ∗ ∗ 2 ντ 0 τ ν ντ ∗ a 2 0 2ντ r which implies (5.2.29); here in (5.2.7) we take α = 1. Combining (5.2.26)–(5.2.29), we derive that as E 0 ≤ δ0 , for any 0 ≤ s ≤ t, −1

e−C1 (t −s) ≤ B(x, s)/B(x, t) ≤ e−C1

(t −s)

.

(5.2.30)

On the other hand, there exists a point b(t) ∈ [0, L] such that θ (b(t), t) = L

−1

L

θ (x, t)d x.

0

This gives

|θ 1/2 (x, t) − θ 1/2 (b(t), t)| ≤

x b(t )

≤ C1

(θ 1/2 ) y (y, t)d y

L 0

1/2 θx2 d x τθ2

L

τθdx

0

≤ C1 V 1/2 (t)[ max τ (y, t)]1/2 y∈[0,L]

1/2 (5.2.31)


with V (t) =

L

θx2 0 τ θ 2 d x.

255

Thus we derive from (5.2.31) and Lemma 5.2.1 that θ (x, t) ≤ C1 + C1 V (t) max τ (y, t),

(5.2.32)

θ (x, t) ≥ C1−1 − C1 V (t) max τ (y, t).

(5.2.33)

y∈[0,L]

y∈[0,L]

Now by virtue of (5.2.25) and (5.2.28)–(5.2.30) and (5.2.32), we derive

t −1 −1 θ (x, s)e−C1 (t −s)ds τ (x, t) ≤ C1 e−C1 t + C1 0

t ≤ C1 + C1 V (s) max τ (y, s)ds y∈[0,L]

0

which, by the Gronwall inequality, implies

t V (s)ds ≤ C1 . max τ (y, t) ≤ C1 exp C1

y∈[0,L]

(5.2.34)

0

Similarly, from (5.2.25), (5.2.28)–(5.2.29) and (5.2.33)–(5.2.34) it follows that there exists some time t0 > 0 such that as t ≥ t0 ,

t −1 θ (x, s)e−C1 (t −s)ds τ (x, t) ≥ C1 0

t −1 −C 1−1 t − C1 V (s)e−C1 (t −s)ds ≥ C1 /2 ≥ C1 − C1 e 0

where we have observed that

t

−C1 (t −s) −C1 t /2 0≤ V (s)e ds ≤ C1 e +

t

t /2

0

V (s)ds → 0

as t → +∞. Also from (5.2.25)–(5.2.26) and (5.2.28), we get for 0 < t ≤ t0 , τ (x, t) ≥ D(x, t)B −1 (x, t) ≥ C1−1 e−C1 t ≥ C1−1 e−C1 t0 ≥ C1−1 which along with (5.2.34) yields the first estimate of (5.2.11). The second estimate is easily derived from the first one and (5.1.22), (5.2.1). The proof is now complete. +∞ 2 Remark 5.2.1. Before proving (5.2.11), if we can prove that 0 v(s) ds ≤ C1 , then we can remove the smallness of E 0 in Lemma 5.2.2. Lemma 5.2.3. For any α ∈ (0, 1], we have

t L 2 u 2 + (r u)2x + w2x θx (x, s)d x ds ≤ C1 , ∀t > 0, + (5.2.35) θα θ 1+α 0 0

L

t

t θ α − L −1 θ α d x2L ∞ ds ≤ C1 , θθx (s)2 ds ≤ C1 sup θ (s)1+α L∞ , 0

0

t 0

0

(5.2.36)

u(s)2L ∞ ds ≤ C1 ,

0≤s≤t

t 0

(u x (s)2 + wx (s)2 )ds ≤ C1 sup θ (s)αL ∞ . (5.2.37) 0≤s≤t


256

Proof. We only prove the case for α ∈ (0, 1); the case of α = 1 is the direct result of L Lemmas 5.2.1–5.2.2. Multiplying (5.1.18) by θ −α ( 0 θ 1−α d x)−1 , we find

−1

τQ dx θα 0 0

L

L CV d d log = γ L −1 log τ d x + θ 1−α d x dt 0 1 − α dt 0

−1 L

L

L (r u)x 1−α θ +γ θ 1−α d x − L −1 θ 1−α d x d x. τ 0 0 0 L

L

θ 1−α d x

ακr 2 θ 2 x τ θ 1+α

+

(5.2.38)

Thus the integration of (5.2.38) with respect to t, and using Lemmas 5.2.1–5.2.2, we have

t L 2 τQ θx + α (x, s)d x ds θ 1+α θ 0 0 1/2

1/2

L

t L

L (r u)2x dx ≤ C1 + C1 θdx θ 1−α − L −1 θ 1−α d x L ∞ ds θ 0 0 0 0

t L 2

t L

L (r u)2x θx d x ds + ε ≤ C1 + C1 d x θ 1−α d x ds 1+α θ 0 0 0 0 θ 0

t L 2 θx d x ds ≤ C1 (ε) + C1 ε 1+α θ 0 0 which, by choosing ε > 0 small enough, and exploiting (5.2.9) and Lemmas 5.2.1–5.2.2, yields (5.2.35). By (5.2.35) and Lemmas 5.2.1–5.2.2, we have

t

L

t L α −1 α 2 θ − L θ d x L ∞ ds ≤ C1 θ α−1 θx d x ds 0

0

0

≤ C1

0

Similarly, we have

t

t u(s)2L ∞ ds ≤

L

t

L

0

t 0

0

(u x 2 + wx 2 )(s)ds ≤

0

0

0

0 L

t

0

θx2 θ

dx 2−α

L

θ α d x ds ≤ C1 .

0

L u 2x dx θ d x ds ≤ C1 , θ 0 u 2x + w2x d x ds sup θ (s)αL ∞ θα 0≤s≤t

≤ C1 sup θ (s)αL ∞ ,

t

0

L

θθx (s)2 ds ≤

0


0≤s≤t L

t

0

0

θx2

θ 1+α

1+α d x ds sup θ (s)1+α L ∞ ≤ C1 sup θ (s) L ∞ . 0≤s≤t

0≤s≤t


257

Lemma 5.2.4. For all t > 0, the following estimates hold:

v(t)2L ∞ + v(t)2H 1 + 2

ττx (t) +

t

0

L 0

t 0

(vt 2 + v2H 1 + v2L ∞ )(s)ds ≤ C1 ,

(ττx2 + θ τx2 )d x ds ≤ C1 + C1 sup θ (s)αL ∞ ,

(5.2.39) (5.2.40)

0≤s≤t

w(t)2L ∞ + w(t)2H 1 +

t (r w) 2 x wt 2 + (s)ds ≤ C1 . τ 0 x

(5.2.41)

Proof. Multiplying (5.1.16) by v in L 2 (0, L), and using the Poincaré´ inequality and Lemmas 5.2.1–5.2.3, we deduce 1 d v(t)2 + μ 2 dt

L 0

(r v)2x d x ≤ C1 τ

L 0

|uv 2 |d x

≤ C1 εv(t)2 + C1 u(t)2L ∞ v(t)2

L (r v)2x d x + C1 u(t)2L ∞ v(t)2 ≤ C1 ε τ 0 which, by taking ε > 0 so small that C1 ε ≤ μ/2, applying the Gronwall inequality and the embedding theorem, implies

L 0

((r v)x 2 + v2L ∞ + v2H 1 )(s)ds ≤ C1 .

(5.2.42)

Analogously, multiplying (5.1.16) by vt in L 2 (0, L), and exploiting Lemmas 5.2.1–5.2.3, we deduce μ d 2 dt

(r v)2x d x + vt (t)2 τ 0

L

L μ L uvvt (r v)x (r v)2x dx = ru dx − μ uvd x − 2 2 0 τ r τ 0 0 x x

L 1 uv vt 1 L (r u)x uv ≤ vt (t)2 + C1 v(t)2L ∞ − vt + dx − + 2 uvd x u 4 2 0 τ r r r 0

L 2 1 (r v)x dx ≤ vt (t)2 + C1 v(t)2L ∞ + C1 u(t)2L ∞ 2 τ 0 L

which, along with (5.2.37), (5.2.42) and the Gronwall inequality, yields (5.2.39). By means of (5.1.14), we rewrite (5.1.15) as u r

−

νττx γ (θ τx − τ θx ) v 2 − u 2 = + . τ t τ2 r2

(5.2.43)


258

Multiplying (5.2.43) by ur − ντττx in L 2 (0, L), and using Lemmas 5.2.1–5.2.3, we infer that for any α ∈ (0, 1],

L 2 νττx 1 d θ τx u 2 dx − + νγ 3 2 dt r τ 0 τ r

L v2 − u 2 u νττx γ (θ τx − τ θx ) γ νττx θx + − dx = + r τ rτ2 τ 2r r2 0

L

L 2 θx 2 2 ≤ε θ τx d x + C1 u(t) L ∞ + C1 d x + C1 θ (t)αL ∞ u(t)2L ∞ 1+α 0 0 θ νττx u (5.2.44) + C1 u(t)2L ∞ + v(t)2L ∞ − . r τ Picking ε > 0 small enough and exploiting Lemmas 5.2.1–5.2.3 and (5.2.39), we infer from (5.2.44)

t L u νττx 2 θ τx2 d x ds − + r τ 0 0 ≤ C1 + C1 sup θ (s)αL ∞ + C1

t 0

0≤s≤t

u νττx (u2L ∞ + v2L ∞ ) − ds. r τ

(5.2.45)

This implies

t u νττx sup − (u2L ∞ + v2L ∞ )(s)ds ≤ 2(C1 + C1 sup θ (s)αL ∞ )1/2 + C1 τ 0 s∈[0,t ] r 0≤s≤t α/2

≤ C1 + C1 sup θ (s) L ∞ 0≤s≤t

which, combined with Lemma 5.2.1 and (5.2.45), gives 2

ττx (t) +

t

0

L 0

θ τx2 d x ds ≤ C1 + C1 sup θ (s)αL ∞ .

(5.2.46)

0≤s≤t

By (5.2.33), we infer from (5.2.46) that

t

0

L 0

τx2 d x ds ≤ C1

t

0

L 0

θ τx2 d x ds + C1

≤ C1 + C1 sup θ (s)αL ∞

t

V (s)dsττx 2

0

0≤s≤t

which, together with (5.2.46), yields (5.2.40). The proof of (5.2.41) is similar to that of (5.2.39). The proof is now complete.


259

Lemma 5.2.5. The following estimates hold: u(t)2H 1 +

t (r u)x )x 2 (s)ds u t 2 + u2L ∞ + u2H 1 + ( τ 0 ≤ C1 + C1 sup θ (s)1+α L ∞ , ∀α ∈ (0, 1],

(5.2.47)

0≤s≤t

(1+α)/4

u(t) L ∞ ≤ C1 + C1 sup θ (s) L ∞

,

α ∈ (0, 1].

(5.2.48)

0≤s≤t

Proof. Similarly to (5.2.39), we derive

(r u)2x d x + u t 2 τ 0

L ν L (r u)2 1 τ θx − θ τx ≤ (r u)[ 2 x ]x d x + u t d x + u t (t)2 + C1 v(t)2L ∞ 2 2 0 4 τ τ 0

L 1 γθ 1 v2 (r u)(r u)x u t + d x + u t 2 ≤ − τ r r τ x r2 4 0

L 2

L θx 1+α 2 ∞ +C1 v(t) L ∞ + C1 θ (t) L ∞ d x + C1 θ (t) L θ τx2 d x 1+α 0 θ 0

L 2

L 1 θx 1+α 2 2 d x + C1 θ (t) L ∞ θ τx2 d x ≤ u t (t) + C1 v(t) L ∞ + C1 θ (t) L ∞ 1+α 2 0 θ 0

L (r u)2x d x + C1 u(t) L ∞ (r u)x (θθx + θ τx ) +C1 u(t)2L ∞ τ 0

L (r u)2x d x) +C1 v(t)2L ∞ (u(t)2 + τ 0

L 2

L 1 θx ∞ ≤ u t (t)2 + C1 v(t)2L ∞ + C1 θ (t)1+α d x + C θ (t) θ τx2 d x 1 L L∞ 1+α 2 θ 0 0

L 2 (r u) x dx +C1 (v(t)2L ∞ + u(t)2L ∞ ) τ 0

ν d 2 dt

L

which, by the Gronwall inequality and the embedding theorem, yields (5.2.47). By the interpolation inequality and Lemma 5.2.1, we obtain u(t) L ∞ ≤ C1 u(t)1/2 u x (t)1/2 + C1 u(t) ≤ C1 + C1 u x (t)1/2 (1+α)/4

≤ C1 + C1 sup θ (s) L ∞ 0≤s≤t

which is (5.2.48). The proof is complete.


260

Lemma 5.2.6. The following estimates hold for any t > 0:

t (θt 2 + θθx 2H 1 )(s)ds ≤ C1 + C1 sup θ (s)2(1+α) , ∀α ∈ (0, 1], θθ x (t)2 + L∞ 0

0≤s≤t

(5.2.49) 0
0. Similarly, we infer from Lemmas 5.2.1–5.2.5,

t

t

L 1 t (r u)2x (r u)x 2L 4 + θ 2−α |II2 |ds ≤ θt 2 ds + C1 d x + (r v)x 2L 4 ∞ L 4 0 θα 0 0 0 + r wx 2L 4 + u2L ∞ u x 2 + v2L ∞ vx 2 ds


261

1/4

3/4

t

t (r u)x 2 2 θt ds + C1 (r u)x ds ds τ 0 0 0 x 1/4

3/4

t

t (r v)x 2 2 + C1 sup θ 2−α + C ds (r v) ds 1 x L∞ τ 0 0 0≤s≤t x 1/4

3/4

t

t (r w)x 2 + C1 r wx 2 ds ds τ 0 0 x

t

t 2 2 + C1 sup u(s) L ∞ u x (s) ds + vx (s)2 ds

1 ≤ 4

1 ≤ 4

t

2

0

0≤s≤t

t 0

0

3(1+α)/2

θt 2 ds + C1 + C1 sup θ (s) L ∞ 0≤s≤t

+ C1 sup θ (s)2−α L∞ . 0≤s≤t

(5.2.54) Choosing ε > 0 small enough in (5.2.53), we deduce from (5.2.53)–(5.2.54)

t 2(1+α) θt (s)2 ds ≤ C1 + C1 sup θ (s) L ∞ sup θθx (s)2 + 0

0≤s≤t

(5.2.55)

0≤s≤t

which with (5.1.18) gives (5.2.49). By the interpolation inequality, we derive from Lemma 5.2.1, 1/3

θ (t) L ∞ ≤ C1 θθx (t)2/3 θ (t) L 1 + C1 θ (t) L 1 ≤ C1 + C1 θθx (t)2/3 .

(5.2.56)

Inserting (5.2.56) into (5.2.55) and picking α > 0 so small that 0 < α < 1/2, we have sup θθx (s) ≤ C1 0≤s≤t

which, along with (5.2.56) and the embedding theorem, gives θ (x, t) ≤ C1 ,

∀(x, t) ∈ [0, L] × [0, +∞).

(5.2.57)

Estimate (5.2.51) follows from Lemmas 5.2.1–5.2.5 and (5.1.14)–(5.1.18), and (5.2.57). In what follows, we shall prove the first inequality of (5.2.50). Note that from (5.2.52)– (5.2.54) and (5.2.57),

L 2 2 (r u) 2 d r θx x 2 d x ≤ C1 + C1 θθx (t) + + (r u)x 2 dt 0 τ τ x (r v) 2 x 2 2 (5.2.58) + + (r v)x + wx τ x with

(r u) 2 r 2 θx2 x d x + θθx (t)2 + + (r u)x 2 τ τ 0 0 x (r v) 2 x 2 2 + + (r v)x + wx dt < +∞. τ x

L ∞

(5.2.59)

262


Using Theorem 1.2.4, (5.2.58)–(5.2.59), and the Poincaré´ inequality, we get that as t → +∞, θθx (t) −→ 0, θ (t) − θ¯ L ∞ −→ 0. (5.2.60) By (5.2.60), we know that there exists some time t1 > 0 such that as t ≥ t1 > 0, θ (x, t) ≥

θ¯ > 0, f or a.e. x ∈ [0, L]. 2

(5.2.61)

Let m = 1/θ. Choosing δ > 0 to satisfy 0
0 such that for any fixed δ ∈ (0, δ1 ], the following estimate holds: → ¯ 21 eδt − v (t)2H 1 + τ (t) − τ¯ 2H 1 + θ (t) − θ (5.2.66) H

t → eδs (τ − τ¯ 2H 1 + ρ − ρ ¯ 2H 1 + θ − θ¯ 2H 2 + − v 2H 2 )(s)ds ≤ C1 , ∀t > 0 + 0

with ρ¯ = 1/τ¯ . Proof. By using (5.1.14)–(5.1.18) and noting that τ¯t = 0, θ¯t = 0, we find that → E(ρ −1 , − v , S) satisfies

ρ θ¯ κ(r θ x )2 → ν(r u)2x + μ(r v)2x + μr 2 w2x + E t (ρ −1 , − v , S) + θ θ 2 = κ(1 − θ¯ /θ )ρr θx + νρ(r u)(r u)x + μρ(r v)(r v)x + μr w(r w)x − μw2 ¯ u − 2μ(1 − θ¯ /θ )(u 2 + v 2 )x , −γ (ρθ − ρ¯ θ)r (5.2.67) x ν(ρx /ρ)2 /2 + ρx r −1 u/ρ + r −2 v 2 ρx /ρ + γ θρx2 /ρ = ρ(r −1 u)x (r u)x t

−γρx θx − [ρr −1 u(r u)x ]x

(5.2.68)

with ρ¯ = 1/τ¯ . Multiplying (5.2.67), (5.2.68) by eδt , δeδt respectively, and then adding


264

the results up, we get ∂ ¯ ν(r u)2x + μ(r v)2x + μr 2 w2x + κ(r 2 θx )2 /θ /θ M(t) + eδt θρ ∂t + δeδt γ θρx2 /ρ + γρx θx − ρ(r −1 u)x (r u)x + r −2 u 2 ρx /ρ = δeδt E + δν(ρx /ρ)2 /2 + δρx r −1 u/ρ + eδt κ(1 − θ¯ /θ )ρr 2 θx ¯ + ρ νr u − δ(r −1 u)x (r u)x + μρr v(r v)x + μr w(r w)x − μw2 − γ r u(ρθ − ρ¯ θ) x − 2μ(1 − θ¯ /θ )(u 2 + v 2 )x eδt

(5.2.69)

where M(t) = eδt [E + δν(ρx /ρ)2 /2 + δr −1 uρx /ρ]. Integrating (5.2.69) over [0, L] × [0, t], by Lemmas 5.2.1–5.2.3, Young’s inequality, and Poincaré´ ’s inequality, integrating by parts in the last term, we deduce that

L M1 (t) ≡ M(t)d x +

t

0

+δ

0 L

0

t

0

¯ eδs θρ[ν(r u)2x + μ(r v)2x + μr 2 w2x + κ(r θx )2 /θ )]/θ (x, s)d x ds L

0

eδs γ θρx2 /ρ + γρx θx − ρ(r −1 u)x (r u)x + r −2 u 2 ρx /ρ (x, s)d x ds

t

−v |2 /2 + δr −2 u 2 /(2ν) + δν(ρ /ρ)2 eδs |→ x 0 0

t ¯ 2 ) d x ds + C1 δ + C1 (|τ − τ¯ |2 + |S − S| eδs (vx 2 + u x 2 )(s)ds 0

t −1 δs 2 2 2 + C1 δ e (u L ∞ + v L ∞ )θθx (s)ds.

≤ C1 + δ

L

(5.2.70)

0

On the other hand, by Lemmas 5.2.1–5.2.5, and the Poincaré´ inequality, we deduce (r −1 u)x ≤ C1 (r u)x , γρx θx ≤

γ r 2 θx2 ρ γ θρx2 + , 4ρ a 2θ

u(t)4L 4 ≤ C1 (r u)x 2 , r −2 u 2 ρx γ θρx2 u4 ≤ + 4 . ρ 4ρ γ r ρθ

With the help of (5.2.71)–(5.2.72) and Lemmas 5.2.1–5.2.5, we conclude 1 M1 (t) ≥ eδt (1/2 − δa −2 /ν)u2 + v2 + w2 2 1 −1 2 ¯ 2) + δνρx /ρ + C1 (τ − τ¯ 2 + S − S 4

t L + eδs C1−1 (θ¯ /θ − C1 δ)(r u)2x + C1−1 (r v)2x + w2x 0 0 + (κ θ¯ /θ − γ δ/a 2 )ρ(r θx )2 /θ + δθρx2 /(2ρ) d x ds

(5.2.71) (5.2.72)


265

→ ¯ 2 + ρx (t)2 + ττx (t)2 ) ≥ C1−1 eδt (− v (t)2 + τ (t) − τ¯ 2 + S(t) − S

t → + C1−1 v x 2 + θθx 2 + ρx 2 + ττx 2 )(s)ds eδs (− (5.2.73) 0

where there exists a δ0 > 0 such that as δ ∈ (0, δ0 ], we have 0 < 1/2 − δa −2 /ν > 0, θ¯ /θ − C1 δ > 0,

κ θ¯ − γ δ/a 2 > 0. θ

(5.2.74)

By the mean value theorem, Poincaré´ ’s inequality and Lemmas 5.2.1–5.2.6, we have ¯ ≤ C1 (τ − τ¯ + θ − θ¯ ) S − S → ≤ C (ττ + θθ + − v ),

(5.2.75)

¯ + τ − τ¯ ). θ − θ¯ ≤ C1 (S − S

(5.2.76)

1

x

x

x

Thus, using (5.2.75)–(5.2.76), Lemmas 5.2.1–5.2.5 and the Poincaré´ inequality, we infer that for any δ ∈ (0, δ0 ], → ¯ 2 + ττx (t)2 ) eδt (− v (t)2 + τ (t) − τ¯ 2 + S(t) − S

t → eδs (− v x 2 + θθx 2 + ττx 2 + ρx 2 )(s)ds + 0

t → eδs (− v x 2 + θθx 2 + ττx 2 + ρx 2 )(s)ds ≤ C1 + C1 δ 0

t +C1 (v2L ∞ + u2L ∞ )eδs θθx 2 (s)ds 0

which, after taking δ1 = min[δ0 , 2C1 1 ], implies that for any fixed δ ∈ (0, δ1 ], → ¯ 2 + ττx (t)2 + ρx (t)2 ) eδt (− v (t)2 + τ (t) − τ¯ 2 + S(t) − S

t → eδs (− v x 2 + θθx 2 + ττx 2 + ρx 2 )(s)ds + 0

t ≤ C1 + C1 (v2L ∞ + u2L ∞ )eδs θθx 2 (s)ds. (5.2.77) 0

Multiplying (5.1.15), (5.1.16), (5.1.17) and (5.1.18) by −eδt u x x , −eδt vx x , −eδt wx x and −eγ t θx x , respectively, then integrating them over [0, L] × [0, t], and adding the results up, using the Young’s inequality, the imbedding theorem and the estimates νρu x x (r u)x x ≥ C1−1 u 2x x − C1 (u 2 + u 2x + τx2 ), ρ(r 2 θx )x θx x ≥ C1−1 θx2x − C1 θx2 ,


266

we finally deduce that

t 1 δt − → e (→ v x (t)2 + C V θθx (t)2 ) + C1−1 eδs (− v x x 2 + θθx x 2 )(s)ds 2 0

δ t δs − 1 t e (→ v x 2 + C V θθx 2 )(s)ds − θθx x 2 eδs ds ≤ C1 + 2 0 4 0

t → + C1 eδs (− v x 2 + ττx 2 + (r u)x 2L ∞ + θθx 2 + (r v)x 2L ∞ )(s)ds 0

t

t → → eδs (− v x 2 + θθx 2 )(s)ds + C1 eδs (− v x 2 + θθx 2 ττx 2 )ds ≤ C1 + C1 δ 0 0

t 1 → + eδs (− v x x 2 + θθx x 2 )ds. (5.2.78) 2C1 0

Summing up (5.2.77) and (5.2.78), there exists a positive constant δ1 ≤ δ1 such that as δ ∈ (0, δ1 ], and applying the Gronwall inequality, we can obtain the required estimate (5.2.66). The proof is now complete.

5.3 Global Existence and Exponential Stability in H 2 In this section we shall complete the proof of Theorem 5.1.2. We begin with the following lemma. → v 0 , θ0 ) ∈ H+2 , then if E 0 ≤ δ0 problem (5.1.14)–(5.1.20) admits Lemma 5.3.1. If (ττ0 , − → a unique global weak solution (τ (t), − v (t), θ (t)) in H 2 . Moreover, there exists a positive +

constant δ2 = δ2 (C2 )(≤ δ1 ) such that for any δ ∈ [0, δ2 ], the following estimate holds: t → eδt τ (t) − τ¯ 2H 2 + − v (t)2H 2 + θ (t) − θ¯ 2H 2 + eδs τ − τ¯ 2H 2 0 → → + − v 2H 3 + θ − θ¯ 2H 3 + − v t (t)2H 1 + θt 2H 1 ds ≤ C2 , ∀t > 0. (5.3.1)

Proof. Differentiating (5.1.15) with respect to t, multiplying the resulting equation by u t eδt , integrating the resulting equation over [0, L] × [0, t], using Lemmas 5.2.1–5.2.5, Lemma 5.2.8 and the estimates u t |t =0 ≤ C2 , u t x ≤ C1 ((r u)t x + u H 1 ), we deduce that for δ > 0 small enough,

t

t eδs (r u)t x 2 ds ≤ C2 + C2 δ eδs (r u)t x 2 ds u t (t)2 eδt + C1−1 0 0

t δs 2 2 2 e (u H 2 + u t + vt + v2 + θt 2 )(s)ds + C1 0

t ≤ C2 + C2 δ eδs (r u)t x 2 ds. 0


That is, u t (t)2 eδt +

t 0

267

eδs (r u)t x 2 ds ≤ C2 , ∀t > 0, δ ∈ [0, δ2 ]

(5.3.2)

where δ2 = δ2 (C2 )(≤ δ1 ) > 0 is a constant. By (5.1.15), Lemmas 5.2.1–5.2.5 and Lemma 5.2.8, we easily derive → ¯ H1 ) u x x (t) ≤ C1 (u t (t) + τ (t) − τ¯ H 1 + − v (t) H 1 + θ (t) − θ which, together with Lemmas 5.2.1–5.2.6, and (5.3.2), yields

t 2 2 δt (u t (t) + u(t) H 2 )e + eδs (u2H 3 + u t 2H 1 )ds ≤ C2 , ∀t > 0, δ ∈ [0, δ2 ]. 0

(5.3.3)

Similarly to (5.3.3),

t 2 2 2 2 δt vt (t) + v(t) H 2 + wt (t) + w(t) H 2 e + eδs (v2H 3 + vt 2H 1 0

+w2H 3 + wt 2H 1 )ds ≤ C2 , ∀t > 0, δ ∈ [0, δ2 ].

(5.3.4)

Differentiating equation (5.1.15) with respect to x, and using (5.1.14), we arrive at

θ τx x ν(r u)x τx − γ θ τx ∂ τx x γ θ x − ν(r u)x x −1 −2 + 2 = r ut x + r τ + ν ∂t τ τ τ τ2

2ν(r u)x x τx − 2γ θx τx 2γ θ τx2 − 2ν(r u)x τx2 γ θx x + + + τ τ2 τ3 +

τ v2 2vvx − 3 . r r

(5.3.5)

Multiplying (5.3.5) by τx x /τ in L 2 (0, L), exploiting Lemmas 5.2.1–5.2.6, Lemma 5.2.8 and the interpolation inequality, we deduce that τ d 1 τx x 2 x x 2 τ x x 2 (t) + C1−1 (t) ≤ (t) + C1 (θθx (t)2 + ττx (t)2 dt τ τ 2C1 τ 2 2 +(r u)x x (t) + θθx x (t) + u t x (t)2 + vx (t)2 ). (5.3.6) Multiplying (5.3.6) by eδt , using Lemmas 5.2.1–5.2.6, Lemma 5.2.8 and (5.3.1)–(5.3.5), there exists a constant δ2 = δ2 (C2 ) ≤ δ2 ≤ δ1 such that when δ ∈ [0, δ2 ], ττx x (t)2 eδt + The proof is now complete.

t 0

eδs ττx x 2 ds ≤ C2 , ∀t > 0.

(5.3.7)

Proof of Theorem 5.1.2. Combining Lemmas 5.2.1–5.2.6, Lemma 5.2.8 and Lemma 5.3.1, we can complete the proof of Theorem 5.1.2.


268

5.4 Global Existence and Exponential Stability in H 4 As before we define the space for fixed L > 0, → v , θ ) ∈ H 4[0, L] × (H 4[0, L])3 H+4 = (τ, − × H 4[0, L] : τ (x) > 0, θ (x) > 0, x ∈ [0, L], − → → − → v |x=0 = − v |x=L = 0 , θx |x=0 = θx |x=L = 0 . Our main results read as follows. Theorem 5.4.1. Let E 0 be as in (5.1.23) in Theorem 5.1.1. Then there exists a constant → δ0 > 0 such that as E 0 ≤ δ0 , for any (ττ0 , − v 0 , θ0 ) ∈ H+4 , problem (5.1.14)–(5.1.20) − → has a unique global solution (τ (t), v (t), θ (t)) ∈ H+4 such that for any (x, t) ∈ [0, L] × [0, +∞), the following estimates hold: → ¯ 24 τ (t) − τ¯ 2H 4 + − v (t)2H 4 + θ (t) − θ H

t → ¯ 2 5 }(s)ds ≤ C4 . + {τ − τ¯ 2H 4 + − v 2H 5 + θ − θ H

(5.4.1)

0

→ Theorem 5.4.2. Under the conditions of Theorem 5.4.1, for any (ττ0 , − v 0 , θ0 ) ∈ H+4 , there exists constants C4 > 0 and γ4 = γ4 (C C4 ) > 0 such that for any fixed γ ∈ (0, γ4 ] and for any t > 0, the following estimates hold: → eγ t (τ (t) − τ¯ 2H 4 + − v (t)2H 4 + θ (t) − θ¯ 2H 4 )

t → eγ s {τ − τ¯ 2H 4 + − v 2H 5 + θ − θ¯ 2H 5 }(s)ds ≤ C4 . +

(5.4.2)

0

→ Corollary 5.4.1. Assume that (τ (t), − v (t), θ (t)) ∈ H+4 is a global solution obtained in Theorems 5.4.1–5.4.2 and satisfies the corresponding compatibility conditions; then it is also the classical global solution verifying that for any fixed γ ∈ (0, γ4 ], → (τ (t) − τ¯ , − v (t), θ (t) − θ¯ )2 3+ 1 C

2 ×(C

3+ 1 3 3+ 1 2 ) ×C 2

≤ C4 e−γ t .

(5.4.3)

5.4.1 Global Existence of Solutions in H 4 In this subsection, we shall establish the global existence in H 4 and complete the proof of Theorem 5.4.1. We begin with the following lemma.


269

→ Lemma 5.4.1. For any (ττ0 , − v 0 , θ0 ) ∈ H+4 and any t > 0, we have u t x (x, 0) + vt x (x, 0) + wt x (x, 0) + θt x (x, 0) ≤ C4 ,

(5.4.4)

(5.4.5) u t t (x, 0) + vt t (x, 0) + wt t (x, 0) + θt t (x, 0) ≤ C4 , (5.4.6) u t x x (x, 0) + vt x x (x, 0) + wt x x (x, 0) + θt x x (x, 0) ≤ C4 ,

t

t u t t (t)2 + u t t x 2 (s)ds ≤ C4 + C4 (u t x x 2 + vt x x 2 + θt x x 2 )(s)ds, 0

0 t

vt t (t)2 +

(5.4.7)

t

vt t x 2 (s)ds ≤ C4 + C4

(u t x x 2 + vt x x 2 )(s)ds,

0

0

0

0≤s≤t

(5.4.8)

t

t wt t x 2 (s)ds ≤ C4 + C4 (u t x x 2 + wt x x 2 )(s)ds, (5.4.9) wt t (t)2 + 0 0

t

t θt t (t)2 + θt t x 2 (s)ds ≤ C4 + C1 ε sup u t x 2 + C2 ε−1 θt x x 2 (s)ds

+ C1 ε

t 0

0

(u t t x 2 + vt t x 2 + wt t x 2 )(s)ds.

(5.4.10)

Proof. Differentiating (5.1.15) with respect to x, we have ν(r u) − γ θ ν(r u) − γ θ v2 x x +r + x xx τ τ r x ν(r u) − γ θ (ν(r u)x − γ θ )ττx xx x − = rx τ τ2 ν(r u) (ν(r u)x x − γ θx )ττx x x x − γ θx x − +r τ τ2 2ν(r u)x τx 2 − 2γ θ τx2 ν(r u)x x τx + ν(r u)x τx x − γ θx τx − γ θ τx x − + 2 τ τ3 v2 τ 2vvx − 3 . + (5.4.11) r r

ut x = rx

Using the Gagliardo-Nirenberg inequality and the Young inequality, we have 1

1

3 4

1 4

3

1

ττx L ∞ ≤ C(ττx 2 ττx x 2 + ττx ) ≤ C(ττx + ττx x ), ττx 2L 4

≤ C(ττx ττx x + ττx )

(5.4.12)

2

≤ C(ττx 2 ττx x 2 + ττx 2 ) ≤ C(ττx + ττx x ).

(5.4.13)

Using Theorems 5.1.1–5.1.2 and a simple calculation, we obtain (r u)x x ≤ C2 (u x + u x x ) ≤ C2 u x x , (r u)x x x ≤ C2 (u x x + u x x x ) ≤ C2 u x H 2 .

(5.4.14) (5.4.15)


270

Thus, by (5.4.11)–(5.4.15), we have u t x ≤ C2 (u x H 2 + θθx H 1 + ττx L ∞ u x + ττx L ∞ u x x +ττx H 1 + ττx L ∞ θθx + ττx 2L 4 + vx ) which along with Theorem 5.1.1–5.1.2 gives, u t x ≤ C2 (u x H 2 + ττx H 1 + θθx H 1 + vx )

(5.4.16)

or u x x x ≤ C2 (u x H 1 + ττx H 1 + θθx H 1 + vx + u t x ).

(5.4.17)

Differentiating (5.1.15) with respect to x twice, using Theorems 5.1.1–5.1.2 and the Gagliardo-Nirenberg inequality and the Young inequality, we have u t x x ≤ C2 (u x H 3 + ττx H 2 + θθx H 2 + vx H 1 )

(5.4.18)

or u x x x x ≤ C2 (u x H 2 + ττx H 2 + θθx H 2 + vx H 1 + u t x x ).

(5.4.19)

Differentiating (5.1.16) with respect to x, we arrive at vt x = μr x

(r v)

xx

(r v) (r v)x τx (r v)x x τx xxx −2 + μr 2 τ τ τ2 2 uvr x (uv)x 2(r v)x τx + 2 . − + 3 τ r r

−

τ (r v)x τx x − τ2

(5.4.20)

Using Theorems 5.1.1–1.1.2 and the Gagliardo-Nirenberg inequality and the Young inequality, we derive from (5.4.20) vt x ≤ C2 (u x + ττx H 1 + vx H 2 )

(5.4.21)

vx x x ≤ C2 (u x + ττx H 1 + vx H 1 + vt x ).

(5.4.22)

or Differentiating (5.1.15) with respect to x twice, using Theorems 5.1.1–5.1.2 and the Gagliardo-Nirenberg inequality and the Young inequality, we have vt x x ≤ C2 (u x H 1 + ττx H 2 + vx H 3 )

(5.4.23)

vx x x x ≤ C2 (u x H 1 + ττx H 2 + vx H 2 + vt x x ).

(5.4.24)

or Similarly, by (5.1.17) and using Theorems 5.1.1–5.1.2 and the Gagliardo-Nirenberg inequality and the Young inequality, we deduce that wt x ≤ C2 (ττx H 1 + wx H 2 ),

(5.4.25)


or

271

wx x x ≤ C2 (ττx H 1 + wx H 1 + wt x ),

(5.4.26)

wt x x ≤ C2 (ττx H 2 + wx H 3 ),

(5.4.27)

wx x x x ≤ C2 (ττx H 2 + wx H 2 + wt x x ).

(5.4.28)

and or

Differentiating (5.1.18) with respect to x, we arrive at (r 2 θ ) 1 r 2 θx τx x x − [ν(r u)x − γ θ ] (r u)x + 2 x x τ τ τ μ(r v)2x τx 1 2μ(r v)x (r v)x x − + ν(r u)x − γ θ (r u)x x + τ τ τ2 2 2 2 2 μr wx τx μ(r wx )x − + − 2μ(u 2 + v 2 )x x τ τ2 2ττ θ + 2τ θ + r 2 θ 4τ τx θx + 2r 2 τx θx x r 2 θx τx x 2r 2 θx τx2 x x xx xxx − =κ − + τ τ2 τ2 τ3 2 2ν(r u)x (r u)x x − γ (r u)x θx ν(r u)x τx − γ θ τx (r u)x γ θ (r u)x x + − − τ τ τ2 μττx (r v)2x μττx ω2x r 2 2μτ ω2x + 2μωx ωx x r 2 2μ(r v)x (r v)x x − − + + τ τ τ2 τ2 2 2 − 4μ(u x + vx + uu x x + vvx x ). (5.4.29)

C V θt x = κ

Using Theorems 5.1.1–5.1.2, we can infer that θt x ≤ C2 ττx L ∞ θθx + θθx x x + ττx L ∞ θθx x + θθ x L ∞ ττx x + θθ x L ∞ ττx 2L 4 + θθx L ∞ (r u)x + ττx L ∞ (r u)x 2L 4 + ττx L ∞ (r u)x + (r u)x x + ττx L ∞ (r v)x 2L 4 + ωx 2L 4 + ωx L ∞ ωx x + ττx L ∞ ωx 2L 4 + u x 2L 4 + u x x + vx x + vx 2L 4 + (r u)x L ∞ (r u)x x + (r v)x L ∞ (r v)x x which, along with Theorems 5.1.1–5.1.2 and the Gagliardo-Nirenberg inequality and the Young inequality, implies θt x ≤ C2 (θθx H 2 + u x H 1 + ττx H 1 + vx H 1 + wx H 1 )

(5.4.30)

θθx x x ≤ C2 (θθx H 1 + u x H 1 + ττx H 1 + vx H 1 + wx H 1 + θt x ).

(5.4.31)

or

Differentiating (5.1.18) with respect to x twice, using Theorems 5.1.1–5.1.2 and the Gagliardo-Nirenberg inequality and the Young inequality, we have θt x x ≤ C2 (θθx H 3 + u x H 2 + ττx H 2 + vx H 2 + wx H 2 )

(5.4.32)


272

or θθx x x x ≤ C2 (θθx H 2 + u x H 2 + ττx H 2 + vx H 2 + wx H 2 + θt x x ). (5.4.33) By virtue of the boundary condition (5.1.20) and the Poincaré´ inequality, we get u t ≤ C1 u t x ≤ C1 u t x x , vt ≤ C1 vt x ≤ C1 vt x x , wt ≤ C1 wt x ≤ C1 wt x x .

(5.4.34)

Differentiating (5.1.15) with respect to t, we arrive at ν(r u) − γ θ ν(r u) − γ θ ν(r u)x τx − γ θ τx xx x txx tx u t t = rt − + r τ τ τ2 [nu(r u)x − γ θ ]t τx + [nu(r u)x − γ θ ]τt x ν(r u)x x τt − γ θx τt − − 2 τ τ2 2 v rt 2vvt 2[nu(r u)x − γ θ ]ττx τt − 2 . + + (5.4.35) r τ3 r Using Theorems 5.1.1–5.1.2 and the Gagliardo-Nirenberg inequality and the Young inequality, we infer from (5.4.35) u t t ≤ C2 {θθx + θt x + θt + u x x + u t x x + ττx + vt }.

(5.4.36)

Similarly, we deduce from (5.1.16)–(5.1.17), vt t ≤ C2 {vx x + vt x x + ττx + u t + u x x }, wt t ≤ C2 {wx x + wt x x + ττx + u x x }.

(5.4.37) (5.4.38)

By (5.1.18), we have θt ≤ C2 (θθx H 1 + u x H 1 + ττx H 1 + vx H 1 + wx H 1 ).

(5.4.39)

Thus inserting (5.4.18), (5.4.23), (5.4.34), (5.4.39) into (5.4.36), we get u t t ≤ C2 (θθx H 2 + u x H 3 + ττx H 2 + vx H 3 + wx H 1 ).

(5.4.40)

Similarly, inserting (5.4.18), (5.4.23) into (5.4.37), we have vt t ≤ C2 (θθx H 2 + u x H 3 + ττx H 2 + vx H 3 ).

(5.4.41)

Inserting (5.4.27) into (5.4.38), we get wt t ≤ C2 (u x H 1 + ττx H 2 + wx H 3 ).

(5.4.42)

Differentiating (5.1.18) with respect to t and using (5.4.30), (5.4.32), (5.4.39) and Theorems 5.1.1–5.1.2 and the Poincaré´ inequality, we have θt t ≤ C2 θθx x + θt x + θt x x + u x H 1 + ττx + u t x + vt x + wt x ≤ C2 θθx H 3 + u x H 2 + ττx H 2 + vx H 2 + wx H 2 . (5.4.43)


273

Thus estimates (5.4.4)–(5.4.6) follow from (5.4.16), (5.4.18), (5.4.21), (5.4.23), (5.4.25), (5.4.27), (5.4.29), (5.4.32) and (5.4.40)–(5.4.43). Differentiating (5.1.15) with respect to t twice, multiplying the resulting equation by u t t in L 2 (0, L), using Theorems 5.1.1–5.1.2 and the Poincaré´ inequality, we obtain for any δ > 0, d u t t 2 ≤ −(C1−1 − δ)u t t x 2 + C2 (δ) θθ x 2 + u x x 2 + ττx 2 + u t x 2 dt (5.4.44) +θt 2 + u t t 2 + θt t 2 + vt t 2 . Choosing δ > 0 small enough, integrating with respect to t, using Theorems 5.1.1–5.1.2 and (5.4.36)–(5.4.39), we can derive from (5.4.44)

t

t 2 2 u t t + u t t x ds ≤ C4 + C4 (u t x x 2 + vt x x 2 + θt x x 2 )(s)ds. (5.4.45) 0

0

In the same manner, by (5.1.16) and (5.1.17), we have

t

t 2 2 vt t x (s)ds ≤ C4 + C4 (u t x x 2 + vt x x 2 )(s)ds, vt t (t) + 0 0

t

t 2 2 wt t x (s)ds ≤ C4 + C4 (u t x x 2 + wt x x 2 )(s)ds. wt t (t) + 0

(5.4.46) (5.4.47)

0

Differentiating (5.1.18) with respect to t twice, multiplying the resulting equation by θt t in L 2 (0, L), using Theorems 5.1.1–5.1.2 and the Poincaré´ inequality, we obtain d θt t 2 ≤ −(C1−1 − ε)θt t x 2 + ε u t t x 2 + vt t x 2 + wt t x 2 + C2 ε−1 θθx 2 dt +θt x 2 + u x 2 + u t x 2 + θt 2 + θt t 2 + θt x x 2 + C2 θt u t x 2 . (5.4.48) Choosing ε > 0 small enough, integrating with respect to t, using (5.4.43), we have

t

t θt t x 2 (s)ds ≤ C1 ε (u t t x 2 + vt t x 2 + wt t x 2 )(s)ds θt t (t)2 + 0 0

t

t 1 1 +C C4 (ε) + C2 sup u t x ( θt t 2 ds) 2 ( u t x 2 ds) 2 +C2 ε−1

0≤s≤t

0

0

t

(θt x x 2 + θt t 2 )(s)ds

t 2 −1 ≤ C4 (ε) + C1 ε sup u t x + C2 ε θt x x 2 (s)ds 0≤s≤t

+C1 ε

0

t 0

0

(u t t x 2 + vt t x 2 + wt t x 2 )(s)ds.


(5.4.49)


274

→ Lemma 5.4.2. For any (ττ0 , − v 0 , θ0 ) ∈ H+4 , there holds that for any t > 0,

t

u t x 2 +

vt x 2 +

u t x x 2 ds ≤ C3 ε−6 + C2 ε2

0 t 0

vt x x 2 ds ≤ C3 ε−6 + C2 ε2

t

0

t 0

(u t t x 2 + θt x x 2 )(s)ds,

(5.4.50)

vt t x 2 (s)ds,

(5.4.51)

t

t wt x 2 + wt x x 2 ds ≤ C3 ε−6 + C2 ε2 wt t x 2 (s)ds, 0 0

t θt x x 2 ds θt x 2 + 0

t ≤ C3 ε−6 + C2 ε2 (u t x x 2 + θt t x 2 + vt x x 2 + wt x x 2 )(s)ds

(5.4.52)

(5.4.53)

0

with any ε ∈ (0, 1) small enough. Proof. Differentiating (5.1.15) with respect to x and t, multiplying the resulting equation by u t x in L 2 (0, L),

ν(r u) − γ θ v2 x r + ut x d x x τ r tx 0 x=L ν(r u) − γ θ v2 x + ut x = r x x=0 τ r t

L 2 v ν(r u)x − γ θ r − + ut x x d x x τ r t 0 = I0 + I1

1 d u t x 2 = 2 dt

L

where x=L ν(r u) − γ θ v2 x + ut x I0 = r x x=0 τ r t ν(r u) − γ θ ν(r u) − γ θ x=L x x ut x = rt +r x tx x=0 τ τ ν(r u) − γ θ x=L x ut x = r tx x=0 τ ν(r u) − γ θ [ν(r u)x − γ θ ]ττx x=L xx x − ut x = r t x=0 τ τ2 ν(r u) ν(r u)x x (r u)x [ν(r u)t x − γ θt ]ττx txx − = r − τ τ2 τ2 [ν(r u)x − γ θ ](r u)x x 2[ν(r u)x − γ θ ]ττx (r u)x x=L ut x − + x=0 τ2 τ3

(5.4.54)


and

275

ν(r u) − γ θ v2 x r + ut x x d x x τ r t 0

L [ν(r u)x − γ θ ]ττx v 2 ν(r u)x x − γ θx − r + u t x x d x. =− τ r t τ2 0

I1 = −

L

Using Sobolev’s interpolation inequality and Theorems 5.1.1–5.1.2, we deduce that 1

1

1

1

1

1

I0 ≤ C2 {u x x 2 u x x x 2 + u x x + ττx 2 ττx x 2 + ττx + θt 2 θt x 2 1

1

1

1

+θt + u t x x 2 u t x x x 2 + u t x x }(u t x 2 u t x x 2 + u t x ) ≡ I01 + I02 where 1

1

1

1

I01 = C2 (u x x 2 u x x x 2 + u x x + ττx 2 ττx x 2 + ττx 1

1

1

1

+θt 2 θt x 2 + θt )(u t x 2 u t x x 2 + u t x ) and

1

1

1

1

I02 = C2 (u t x x 2 u t x x x 2 + u t x x )(u t x 2 u t x x 2 + u t x ). Applying Young’s inequality several times, we have that for any ε ∈ (0, 1), ε2 u t x x 2 + C2 ε−2 {u x x 2H 1 + ττx 2H 1 + θt 2H 1 + u t x 2 }, 2 ε2 ≤ u t x x 2 + ε2 u t x x x 2 + C2 ε−6 u t x 2 2

I01 ≤ I02 whence

I0 ≤ ε2 (u t x x 2 + u t x x x 2 ) + C2 ε−6 (u x x 2H 1 + ττx 2H 1 + θt 2H 1 + u t x 2 ). (5.4.55) Using Theorems 5.1.1–5.1.2 again, we infer from (5.4.55)

t

t 2 I0 ds ≤ ε (u t x x 2 + u t x x x 2 )ds + C2 ε−6 . 0

(5.4.56)

0

Similarly to (5.4.56), using Theorems 5.1.1–5.1.2 and a proper embedding theorem, we get

L 2 2 r ut x x I1 ≤ −ν d x + ε 2 u t x x 2 + C2 ε−2 {u x x 2 + ττx 2 τ 0 +θt 2 + u t x 2 + θθx 2 + θt x 2 + vt 2 + v2 } whence

t 0

I1 ds ≤

−(C1−1

t

2

−ε )

0

u t x x 2 ds + C2 ε−2 .

(5.4.57)

(5.4.58)


276

Inserting (5.4.56) and (5.4.58) into (5.4.54), taking ε ∈ (0, 1) small enough, we conclude

t

t u t x 2 + u t x x 2 ds ≤ C3 ε−6 + C2 ε2 u t x x x 2 ds. (5.4.59) 0

0

Differentiating (5.1.15) with respect to x and t, we arrive at ν(r u) − γ θ v2 ν(r u)t x x x x + D(t). ut t x = r + ( )t x = r x tx τ r τ

(5.4.60)

Using Theorems 5.1.1–5.1.2 and a proper embedding theorem, we get D ≤ C2 (u x H 2 + θθ x H 1 + θt x H 1 + ττx H 1 + vx + vt x + u t x x ). (5.4.61) Using (5.4.60) and (5.4.61), we have u t x x x ≤ C1 u t t x +C C2 (u x H 2 +θθx H 1 +θt x H 1 +ττx H 1 +vx +vt x +u t x x ). (5.4.62) Inserting (5.4.62) into (5.4.59) and using Theorems 5.1.1–5.1.2, taking ε ∈ (0, 1) small enough, we can derive the desired estimate (5.4.50). Differentiating (5.1.16) with respect to x and t, multiplying the resulting equation by vt x in L 2 (0, L), we arrive at

L 0

vt x vt t x d x =

L 0

μr

(r v) x

τ

x

−

uv vt x d x, r tx

i.e., x=L L (r v) (r v) uv uv 1 d x x vt x 2 = μr μr − vt x − − vt x x d x x x x=0 2 dt τ r t τ r t 0 (5.4.63) = A0 + A1 where x=L (r v) (r v) (r v) x=L uv x x x A0 = μr vt x − vt x = μ rt +r , x t x t x x=0 x=0 τ r τ τ

L (r v)x uv A1 = − μr − vt x x d x x τ r t 0

L (r v) uv (r v)x x μrt + μr − ( )t vt x x d x. =− x tx τ τ r 0 Using Sobolev’s interpolation inequality and Theorems 5.1.1–5.1.2, we deduce that 1

1

1

1

1

1

A0 ≤ C2 {vx x 2 vx x x 2 + vx x + ττx 2 ττx x 2 + ττx + u x x 2 u x x x 2 1

1

1

1

+u x x + vt x x 2 vt x x x 2 + vt x x }(vt x 2 vt x x 2 + vt x ) ≡ A01 + A02

(5.4.64)


277

where 1

1

1

1

A01 = C2 (vx x 2 vx x x 2 + vx x + ττx 2 ττx x 2 + ττx 1

1

1 2

1 2

1

1

+u x x 2 u x x x 2 + u x x )(vt x 2 vt x x 2 + vt x ), 1 2

1 2

A02 = C2 (vt x x vt x x x + vt x x )(vt x vt x x + vt x ).

(5.4.65) (5.4.66)

Applying Young’s inequality several times, we have ε2 vt x x 2 + C2 ε−2 {vx x 2H 1 + ττx 2H 1 + u x x 2H 1 + vt x 2 }, (5.4.67) 2 ε2 A02 ≤ vt x x 2 + ε2 vt x x x 2 + C2 ε−6 vt x 2 . (5.4.68) 2 A01 ≤

Therefore it follows from (5.4.64)–(5.4.68) A0 ≤ ε2 (vt x x 2 + vt x x x 2 ) + C2 ε−6 (vx x 2H 1 + ττx 2H 1 + u x x 2H 1 + vt x 2 ) which together with Theorems 5.1.1–5.1.2, implies

t

t 2 A0 ds ≤ ε (vt x x 2 + vt x x x 2 )ds + C2 ε−6 . 0

(5.4.69)

0

Similarly, using Theorems 5.1.1–5.1.2 and a proper embedding theorem, we get

L 2 2 r vt x x A1 ≤ −μ d x +ε 2 vt x x 2 +C2 ε−2 vx x 2 +ττx 2 +vt x 2 +u t 2 +u2 τ 0 (5.4.70) whence

t

t A1 ds ≤ −(C1−1 − ε2 ) vt x x 2 ds + C2 ε−2 . (5.4.71) 0

0

Integrating (5.4.63) with respect to t, using (5.4.69) and (5.4.71), taking ε ∈ (0, 1) small enough, we can obtain

t

t 2 2 −6 2 vt x + vt x x ds ≤ C3 ε + C2 ε vt x x x 2 ds. (5.4.72) 0

0

Differentiating (5.1.16) with respect to x and t, we have (r v) uv (r v)t x x x x vt t x = μr + D1 (t). − ( )t x = μr x tx τ r τ

(5.4.73)

Using Theorems 5.1.1–5.1.2 and a proper embedding theorem, we derive D1 ≤ C2 (vx H 2 + ττx H 1 + τt H 1 + rt H 1 + u t H 1 + vt x H 1 ).

(5.4.74)

Using (5.4.72) and (5.4.73), we get vt x x x ≤ C1 vt t x + C2 (vx H 2 + ττx H 1 + τt H 1 + rt H 1 + u t H 1 + vt x H 1 ). (5.4.75)


278

Inserting (5.4.75) into (5.4.72), using Theorems 5.1.1–5.1.2, taking ε ∈ (0, 1) small enough, we can derive the desired estimate (5.4.51). Differentiating (5.1.17) with respect to x and t, multiplying the resulting equation by wt x in L 2 (0, L), we have x=L (r w) μτ w 1 d x wt x 2 = μr + 2 wt x x t x=0 2 dt τ r

L μτ w (r w)x μr − + 2 wt x x d x x t τ r 0 = B0 + B1 (5.4.76) where

x=L x=L (r w) (r w) (r w) μτ w x x x − 2 wt x = μ rt +r }wt x B0 = μr x t x tx x=0 x=0 τ τ τ r

and

(r w) μτ w x μr + 2 wt x x d x x t τ r 0

L (r w) μτ w (r w)x x μrt wt x x d x. + μr − =− x tx τ τ r2 t 0

B1 = −

L

Using Sobolev’s interpolation inequality and Theorems 5.1.1–5.1.2, we deduce that 1

1

1

1

1

1

B0 ≤ C2 {wx x 2 wx x x 2 + wx x + ττx 2 ττx x 2 + ττx + u x x 2 u x x x 2 1

1

1

1

+u x x + wt x x 2 wt x x x 2 + wt x x }(wt x 2 wt x x 2 + wt x ) (5.4.77) ≡ B01 + B02 where 1

1

1

1

B01 = C2 (wx x 2 wx x x 2 + wx x + ττx 2 ττx x 2 + ττx 1

1

1

1

+u x x 2 u x x x 2 + u x x )(wt x 2 wt x x 2 + wt x ), 1

1

1

1

B02 = C2 (wt x x 2 wt x x x 2 + wt x x )(wt x 2 wt x x 2 + wt x ). Applying Young’s inequality several times, we have ε2 wt x x 2 + C2 ε−2 {wx x 2H 1 + ττx 2H 1 + u x x 2H 1 + wt x 2 }, 2 ε2 ≤ wt x x 2 + ε2 wt x x x 2 + C2 ε−6 wt x 2 . 2

B01 ≤

(5.4.78)

B02

(5.4.79)

Hence it follows from (5.4.77)–(5.4.79) B0 ≤ ε2 (wt x x 2 + wt x x x 2 ) + C2 ε−6 (wx x 2H 1 + ττx 2H 1 + u x x 2H 1 + wt x 2 ). (5.4.80)


279

Using Lemmas 5.1.1–5.1.2 again, we have

t

t 2 B0 ds ≤ ε (wt x x 2 + wt x x x 2 )ds + C2 ε−6 , 0

r 2 wt2x x d x + ε2 wt x x 2 + C2 ε−2 wx x 2 + ττx 2 τ 0 2 (5.4.82) +wt x + τt 2H 1 + rt 2H 2

B1 ≤ −μ

whence

(5.4.81)

0

t 0

L

B1 ds ≤ −(C1−1 − ε2 )

t

wt x x 2 ds + C2 ε−2 .

(5.4.83)

0

Integrating (5.4.76) with respect to t, using (5.4.83) and (5.4.82), taking ε ∈ (0, 1) small enough, we can derive

t

t wt x 2 + wt x x 2 ds ≤ C3 ε−6 + C2 ε2 wt x x x 2 ds. (5.4.84) 0

0

Differentiating (5.1.17) with respect to x and t, we arrive at (r w) μτ w (r w)t x x x x + D2 (t). − = μr wt t x = μr 2 x tx tx τ τ r

(5.4.85)

Using Theorems 5.1.1–5.1.2 and a proper embedding theorem, we get D2 ≤ C2 (wx H 2 + ττx H 1 + τt H 1 + rt H 1 + wt x H 1 ).

(5.4.86)

Using (5.4.85) and (5.4.86), we have wt x x x ≤ C1 wt t x + C2 (wx H 2 + ττx H 1 + τt H 1 + rt H 1 + wt x H 1 ). (5.4.87) Inserting (5.4.87) into (5.4.84), using Theorems 5.1.1–5.1.2, taking ε ∈ (0, 1) small enough, we can derive the desired estimate (5.4.52). Differentiating (5.1.18) with respect to x and t, multiplying the resulting equation by wt x in L 2 (0, L), we have x=L r 2θ γ θ (r u)x 1 d x θt x 2 = κ + τ Q θt x − t x=0 2 dt τ x τ

L 2 r θx γ θ (r u)x + τ Q θt x x d x = M0 + M1 κ − − t τ x τ 0 where x=L r 2θ γ θ (r u)x x + τ Q θt x − , M0 = κ t x=0 τ x τ

L 2 γ θ (r u)x r θx + τ Q θt x x d x. M1 = − κ − t τ x τ 0

(5.4.88)


280

Using Sobolev’s interpolation inequality and Theorems 5.1.1–5.1.2, we deduce that 1 1 1 1 M0 ≤ C2 θθx x 2 θθx x x 2 + θθx x + θt x x 2 θt x x x 2 + θt x x + ττx 1

1

1

1

1

1

+ττx 2 ττx x 2 + u t x 2 u t x x 2 + u t x + vt x 2 vt x x 2 + vt x 1

1

1

1

1

1

+wt x 2 wt x x 2 + wt x + u x 2 u x x 2 + u x + vx 2 vx x 2 1 1 1 1 +vx + wx 2 wx x 2 + wx (θt x 2 θt x x 2 + θt x ) = M01 + M02

(5.4.89)

where

1 1 1 1 M01 = C2 θθx x 2 θθx x x 2 + θθx x + ττx 2 ττx x 2 + ττx 1

1

1

1

1

1

+u x 2 u x x 2 + u x + vx 2 vx x 2 + vx + wx 2 wx x 2 1 1 +wx + u t x + vt x + wt x (θt x 2 θt x x 2 + θt x ), (5.4.90) 1 1 1 1 1 1 M02 = C2 u t x 2 u t x x 2 + vt x 2 vt x x 2 + wt x 2 wt x x 2 + wt x 1 1 1 1 +θt x x 2 θt x x x 2 + θt x x (θt x 2 θt x x 2 + θt x ). (5.4.91) Applying Young’s inequality several times, we have from (5.4.90)–(5.4.91), M01 ≤

M02 ≤

ε2 θt x x 2 + C2 ε−2 {θθ x x H 1 + u x x H 1 + vx H 1 + wx H 1 2 +ττx H 1 + u t x 2 + vt x 2 + wt x 2 + θt x 2 }, (5.4.92) ε2 θt x x 2 + ε2 (u t x x 2 + vt x x 2 + wt x x 2 + θt x x x 2 ) 2 +C C2 ε−6 (u t x 2 + vt x 2 + wt x 2 + θt x 2 ).

(5.4.93)

Thus we infer from (5.4.92)–(5.4.93) M0 ≤ ε2 (θt x x 2 + u t x x 2 + vt x x 2 + wt x x 2 + θt x x x 2 ) + C2 ε−6 θθx x H 1 + u x x H 1 + vx H 1 + wx H 1 + ττx H 1 + u t x 2 + vt x 2 + wt x 2 + θt x 2 which with Theorems 5.1.1–5.1.2 yields

t

t 2 M0 ds ≤ ε (θt x x 2 + u t x x 2 + vt x x 2 + wt x x 2 + θt x x x 2 )ds + C2 ε−6 . 0

0

(5.4.94) Using Theorems 5.1.1–5.1.2 and a proper embedding theorem, we deduce

L 2 2 r θt x x d x + ε2 θt x x 2 + C2 ε−2 θθx x 2 + ττx 2 + θt x 2 M1 ≤ −κ τ 0 2 +τt H 1 + u t x 2 + vt x 2 + wt x 2 + u x 2 + vx 2 + wx 2


whence

t 0

M1 ds ≤ −(C1−1 − ε2 )

t 0

281

θt x x 2 ds + C2 ε−2 .

(5.4.95)

Integrating (5.4.88) with respect to t, using (5.4.94) and (5.4.95), taking ε ∈ (0, 1) small enough, we can derive

t

t θt x 2 + θt x x 2 ds ≤ C3 ε−6 +C C2 ε 2 (θt x x x 2 +u t x x 2 +vt x x 2 +wt x x 2 )ds. 0

0

(5.4.96)

Differentiating (5.1.18) with respect to x and t, we arrive at r 2 θt x x x + D3 (t) τ which, by Theorems 5.1.1–5.1.2 and a proper embedding theorem, yields D3 ≤ C2 θt x H 1 + θθx H 2 + ττx H 1 + τt H 2 + u x H 1 + vx H 1 + wx H 1 + u t H 1 + vt H 1 C V θt t x = κ

+ wt H 1 + u t x x + vt x x + wt x x + r x H 1 + rt H 2 .

(5.4.97)

(5.4.98)

By virtue of (5.4.97) and (5.4.98), we get θt x x x ≤ C1 θt t x + C2 θt x H 1 + θθx H 2 + ττx H 1 + τt H 2 + u x H 1 + vx H 1 + wx H 1 + u t H 1 + vt H 1 + wt H 1 + u t x x + vt x x + wt x x + r x H 1 + rt H 2 . (5.4.99) Inserting (5.4.99) into (5.4.96), we can derive the desired estimate (5.4.53). The proof is now complete. − → 4 Lemma 5.4.3. For any (ττ , v , θ ) ∈ H , there holds that for any t > 0, 0

0

0

+ t

u t x 2 + vt x 2 + wt x 2 + θt x 2 + (u t x x 2 + vt x x 2 + wt x x 2 + θt x x 2 )ds 0

t −6 2 2 ≤ C3 ε + C2 ε (θt t x + u t t x 2 + vt t x 2 + wt t x 2 )ds (5.4.100) 0

with any ε ∈ (0, 1) small enough. Proof. Adding (5.4.50)–(5.4.52) to (5.4.53), taking ε ∈ (0, 1) small enough, we can derive the desired estimate (5.4.100). → v , θ ) ∈ H 4 , there holds that for any t > 0, Lemma 5.4.4. For any (ττ , − 0

2

2

0

2

0

+

2

u t t + vt t + wt t + θt t + u t x 2 + vt x 2 + wt x 2 + θt x 2

t + (u t t x 2 + vt t x 2 + wt t x 2 + θt t x 2 + u t x x 2 0

+ vt x x 2 + wt x x 2 + θt x x 2 )(s)ds ≤ C4 with any ε ∈ (0, 1) small enough.

(5.4.101)


282

3

Proof. Multiplying (5.4.7)–(5.4.9) by ε respectively, multiplying (5.4.10) by ε 2 , adding the resultant to (5.4.100), taking ε ∈ (0, 1) small enough, we can derive the desired estimate (5.4.101). The proof is complete. → Lemma 5.4.5. For any (ττ , − v , θ ) ∈ H 4 , there holds that for any t > 0, 0

ττx x x 2H 1 +

t 0

0

+

0

ττx x x 2H 1 ds ≤ C4 ,

u x x x 2H 1 + vx x x 2H 1 + wx x x 2H 1 + θθx x x 2H 1

t + (u x x x x 2H 1 + vx x x x 2H 1 + wx x x x 2H 1 + θθx x x x 2H 1 )ds ≤ C4 .

(5.4.102)

(5.4.103)

0

Proof. Differentiating (5.1.15) with respect to x and using (5.1.14), we get γ θ − ν(r u) γ θ τx x ν(r u)x τx − γ θ τx γ θx x ∂ τx x x xx −1 −2 )+ + + = r u + r τ ν ( t x ∂t τ τ τ τ2 τ2 2γ θ τx2 − 2ν(r u)x τx2 2ν(r u)x x τx − 2γ θx τx + + τ2 τ3 2 2vvx v τ − 2 + 4 = r −1 u t x + E(x, t), (5.4.104) r r where γ θ − ν(r u) ν(r u)x τx − γ θ τx γ θx x x xx + (5.4.105) + τ τ τ2 2ν(r u)x x τx − 2γ θx τx 2γ θ τx2 − 2ν(r u)x τx2 2vvx v2 τ + + − + . τ2 τ3 r2 r4

E(x, t) = r −2 τ

Differentiating (5.4.104) with respect to x, we have ν

γ θ τx x x ∂ τx x x ( )+ = E 1 (x, t) ∂t τ τ2

(5.4.106)

where E 1 (x, t) = ν

2ττx τx x (r u)x γ θx τx x + τx x (r u)x x − − 2 τ τ3 τ2 2γ θ τx x τx + − r −3 τ u t x + r −1 u t x x + E x (x, t). τ3

(r u)

x x x τx

(5.4.107)

By a proper calculation, we can derive E 1 ≤ C2 (u x H 2 + ττx H 1 + θθx H 2 + vx H 1 + u t x H 1 ). Using Theorems 5.1.1–5.1.2 and (5.4.108), we have

t E 1 2 ds ≤ C4 . 0

(5.4.108)

(5.4.109)



τx x x τ

283

in L 2 (0, L), using the Poincaré´ inequality, we obtain

τ 2 d τx x x 2 xxx + C1−1 ≤ C1 E 1 2 . dt τ τ

(5.4.110)

Integrating (5.4.110) with respect to t and using Theorems 5.1.1–5.1.2 and (5.4.109), we conclude

t ττx x x (t)2 + ττx x x (s)2 ds ≤ C4 . (5.4.111) 0

By (5.4.17), (5.4.22), (5.4.26), (5.4.31) and Theorems 5.1.1–5.1.2 and (5.4.101), we infer (5.4.112) u x x x (t)2 + vx x x (t)2 + wx x x (t)2 + θθx x x (t)2

t + (u x x x 2H 1 + vx x x 2H 1 + wx x x 2H 1 + θθx x x 2H 1 )(s)ds ≤ C4 . 0

Differentiating (5.1.15) with respect to t, using Theorems 5.1.1–5.1.2, we can deduce u t x x (t) ≤ C1 u t t + C2 θθx + θt x + u x x + u t x + ττx + vt . (5.4.113) Using Theorems 5.1.1–5.1.2 again and (5.4.101), we have u t x x (t) ≤ C4 which combined with (5.4.19) leads to

t 2 (u t x x 2 + u x x x x 2 )(s)ds ≤ C4 . u x x x x (t) +

(5.4.114)

(5.4.115)

0

In the same manner, we get vt x x (t) ≤ C4 , wt x x (t)

t

2

≤ C4 , vx x x x (t) +

(vt x x 2 + vx x x x 2 )(s)ds

0

≤ C4 ,

t 2 wx x x x (t) + (wt x x 2 + wx x x x 2 )(s)ds ≤ C4

(5.4.116) (5.4.117)

0

which, combined with (5.4.115)–(5.4.117), give u x x x x (t)2 + vx x x x (t)2 + wx x x x (t)2

t + (u x x x x 2 + vx x x x 2 + wx x x x 2 )(s)ds ≤ C4 .

(5.4.118)

0

In the same manner, we get

t

2

θθx x x x (t) +

0

θθx x x x (s)2 ds ≤ C4 .

(5.4.119)


284

Differentiating (5.4.106) with respect to x, we arrive at γ θ τx x x x ∂ τx x x x + ν = E 2 (x, t) ∂t τ τ2

(5.4.120)

where E 2 (x, t) =

ν(r u)x x x x τx + ν(r u)x x τx x x − γ θx τx x x τ2 2γ θ τx x x τx − 2ν(r u)x τx x x τx + + E 1x (x, t). τ3

(5.4.121)

By (5.4.105) and using Theorems 5.1.1–5.1.2, we can derive E x x (t) ≤ C4 (θθx H 3 + u x H 3 + ττx H 2 + vx H 2 )

(5.4.122)

which, combined with (5.4.107) and using Theorems 5.1.1–5.1.2, implies E 1x (t) ≤ C4 (θθx H 3 + u x H 3 + ττx H 2 + vx H 2 + u t x H 2 ).

(5.4.123)

Hence we infer from (5.4.121) and (5.4.123), E 2 (t) ≤ C4 (u x H 3 + ττx H 2 + θθx H 3 + vx H 2 + u t x H 2 ). Using (5.4.62), (5.4.101) and Theorems 5.1.1–5.1.2, we have

t (u t x x 2 + u t x x x 2 )(s)ds ≤ C4 .

(5.4.124)

(5.4.125)

0

Using (5.4.109), (5.4.118), (5.4.123), (5.4.1124) and Theorems 5.1.1–5.1.2, we obtain

t E 2 (s)2 ds ≤ C4 . (5.4.126) 0


τx x x x τ in

L 2 (0, L) and using the Poincaré´ inequality, we get

τ d τx x x x 2 x x x x 2 + C1−1 ≤ C1 E 2 2 . dt τ τ

(5.4.127)

Integrating (5.4.127) with respect to t and using Theorems 5.1.1–5.1.2 and (5.4.126), we can derive

t

ττx x x x (t)2 +

ττx x x x (s)2 ds ≤ C4 , ∀t > 0.

(5.4.128)

0

Differentiating (5.1.15) with respect to x three times and using Theorems 5.1.1–5.1.2, we get u x x x x x (t) ≤ C2 (u x (t) H 3 + ττx (t) H 3 + θθx (t) H 3 + vx (t) H 2 + u t x x x (t)). (5.4.129)


285

Thus we conclude from (5.4.109), (5.4.112), (5.4.117), (5.4.118), (5.4.124)–(5.4.129) and Theorems 5.1.1–5.1.2 that

t u x x x x x 2 (s)ds ≤ C4 . (5.4.130) 0

Similarly, we can deduce that

t (vx x x x x 2 + wx x x x x 2 + θθx x x x x 2 )(s)ds ≤ C4 .

(5.4.131)

0

Finally, using (5.4.108), (5.4.109), (5.4.117), (5.4.118), (5.4.125), (5.4.130) and (5.4.131), we can obtain the desired estimates (5.4.102) and (5.4.103). The proof is complete. Proof of Theorem 5.4.1. Using Theorems 5.1.1–5.1.2 and Lemma 5.4.5, we can easily prove Theorem 5.4.1.

5.4.2 Exponential Stability in H+4 In this subsection, based on the estimates established in Sections 5.1.1–5.4.1, we will show the exponential stability of a global solution in H+4 . → v 0 , θ0 ) ∈ H+4 , there exists a constant 0 < γ4(1) = γ4(1) (C C4 ) ≤ Lemma 5.4.6. For any (ττ0 , − (1) γ2 (C2 ) > 0 such that for any fixed γ ∈ (0, γ4 ],

t γt 2 e u t t (t) + eγ s u t t x (s)2 ds 0

t eγ s (u t x x 2 + vt x x 2 + wt x x 2 + θt x x 2 )(s)ds, (5.4.132) ≤ C4 + C4 0

t eγ t vt t (t)2 + eγ s vt t x (s)2 ds 0

t eγ s (u t x x 2 + vt x x 2 )(s)ds, (5.4.133) ≤ C4 + C4 0

t eγ t wt t (t)2 + eγ s wt t x (s)2 ds 0

t eγ s (u t x x 2 + wt x x 2 )(s)ds. (5.4.134) ≤ C4 + C4 0

Proof. Multiplying (5.4.44) by eγ t and integrating the resulting inequality with respect to t, integrating by parts and using the Poincaré´ inequality, we can derive

t

t eγ t u t t (t)2 ≤ C4 − (C1−1 − δ − C1 γ ) eγ s u t t x (s)2 ds + C2 (δ) eγ s θθx 2 0 0 2 2 2 2 2 2 2 (5.4.135) +u x x + ττx + u t x + θt + u t t + θt t + vt t (s)ds.


286

Taking γ and δ so small that 0 < δ ≤

1 4C 1

and 0 < γ ≤ mi n[1, mi n(

1 , γ2 (C2 ))] 4C 12

≡

γ4(1) , using (5.4.36)–(5.4.38), (5.4.43) and Theorems 5.1.1–5.1.2, we can obtain estimate (5.4.132) from (5.4.135). Similarly, we can obtain estimate (5.4.133) and (5.4.134). The proof is complete. (1) − → Lemma 5.4.7. For any (ττ , v , θ ) ∈ H 4 , and for any fixed γ ∈ (0, γ ], there holds 0

0

+

0

4

that for any t > 0,

t

t eγ t θt t (t)2 + eγ s θt t x (s)2 ds ≤ C2 ε−1 eγ s θt x x (s)2 ds + C2 eγ t sup u t x 2 0

0

t

+ C4 (ε) + C2 ε

0≤s≤t

eγ s (u t t x 2 + vt t x 2 + wt t x 2 )(s)ds.

(5.4.136)

0

Proof. Multiplying (5.4.48) by eγ t and integrating the resulting inequality with respect to t, using (5.4.43), we have

t

t eγ t θt t (t)2 ≤ C4 (ε) + γ eγ s θt t (s)2 ds − (C1−1 − ε) eγ s θt t x (s)2 ds 0 0

t

t γs 2 2 2 −1 +ε e (u t t x + vt t x + wt t x )(s)ds + C2 ε eγ s θθx 2 0 0 +θt x 2 + θt 2 + θt t 2 + θt x x 2 + u x 2 + u t x 2 (s)ds +C C2 e

γ 2t

t

sup 0

0≤s≤t

1

γs

2

e θt t ds

t

2

1 2

2

u t x ds

0

t ≤ C4 (ε) + C2 (ε−1 + γ ) eγ s θt x x (s)2 ds − (C1−1 − ε) eγ s θt t x (s)2 ds 0 0

t eγ s (u t t x 2 + vt t x 2 + wt t x 2 )(s)ds + C2 eγ t sup u t x (s)2 . +C C2 ε t

0

0≤s≤t

Taking ε ∈ (0, 1) small enough, we can derive (5.4.136). (1) → Lemma 5.4.8. For any (ττ0 , − v 0 , θ0 ) ∈ H+4 , and for any fixed γ ∈ (0, γ4 ], there holds that for any t > 0,

t

t eγ t u t x (t)2 + eγ s u t x x (s)2 ds ≤ C3 ε−6 + C2 ε2 eγ s u t x x x (s)2 ds, 0

eγ t vt x (t)2 +

0

t

eγ s vt x x (s)2 ds ≤ C3 ε−6 + C2 ε2

0

γt

e wt x (t) +

0

t

eγ s vt x x x (s)2 ds, (5.4.138)

0

t

2

(5.4.137)

γs

2

e wt x x (s) ds ≤ C3 ε

−6

+ C2 ε

2 0

t

eγ s wt x x x (s)2 ds, (5.4.139)


eγ t θt x (t)2 +

t

eγ s θt x x (s)2 ds ≤ C3 ε−6 + C2 ε2 0 + vt x x x 2 + wt x x x 2 + θt x x x 2 (s)ds.

287

t

eγ s u t x x x (s)2

0

(5.4.140)

Proof. Multiplying (5.4.54) by eγ t and integrating the resulting inequality with respect to t, using (5.4.55) and (5.4.57), we have

t L 1 γt r 2 u 2t x x e u t x (t)2 + ν d xds (5.4.141) eγ s 2 τ 0 0

t γ t γs e u t x 2 ds + ε2 eγ s (u t x x 2 + u t x x x 2 )(s)ds ≤ C3 + 2 0 0

t +C C2 ε−6 (u x x 2H 1 + ττx 2H 1 + θt 2H 1 + θθx 2 + u t x 2 + vt 2 )(s)ds. 0

Taking ε ∈ (0, 1) small enough, using Theorems 5.1.1–5.1.2 and for any γ ∈ (0, γ4(1) ], we have

t

t eγ t u t x (t)2 + eγ s u t x x (s)2 ds ≤ C3 ε−6 + C2 ε2 eγ s u t x x x (s)2 ds. 0

0

(5.4.142) In the same manner, we can derive (5.4.138)–(5.4.140). (1) → Lemma 5.4.9. For any (ττ0 , − v 0 , θ0 ) ∈ H+4 and for any fixed γ ∈ (0, γ4 ], there holds that for any t > 0, eγ t (u t x (t)2 + vt x (t)2 + wt x (t)2 + θt x (t)2 ) (5.4.143)

t + eγ s (u t x x 2 + vt x x 2 + wt x x 2 + θt x x 2 )(s)ds 0

t eγ s u t t x 2 + vt t x 2 + wt t x 2 + θt t x 2 (s)ds. ≤ C3 ε−6 + C2 ε2 0

Proof. Adding (5.4.137)–(5.4.139) to (5.4.140) and choosing ε ∈ (0, 1) small enough, we have eγ t (u t x (t)2 + vt x (t)2 + wt x (t)2 + θt x (t)2 )

t + eγ s (u t x x 2 + vt x x 2 + wt x x 2 + θt x x 2 )(s)ds 0

t −6 2 ≤ C3 ε + C2 ε eγ s u t x x x 2 + vt x x x 2 + wt x x x 2 + θt x x x 2 ds. 0

Using (5.4.62), (5.4.76), (5.4.87), (5.4.99) and Theorems 5.1.1–5.1.2, taking ε ∈ (0, 1) small enough, we can obtain (5.4.143). The proof is complete.


288

(2) (1) → Lemma 5.4.10. For any (ττ0 , − v 0 , θ0 ) ∈ H+4 , there exists a constant 0 < γ4 ≤ γ4 such (1) that for any fixed γ ∈ (0, γ4 ], there holds that for any t > 0,

eγ t u t t (t)2 + vt t (t)2 + wt t (t)2 + θt t (t)2

+ u t x (t)2 + vt x (t)2 + wt x (t)2 + θt x (t)2

t eγ s u t t x 2 + vt t x 2 + wt t x 2 + θt t x 2 + u t x x 2 + 0 + vt x x 2 + wt x x 2 + θt x x 2 (s)ds ≤ C4 .

(5.4.144) 3

Proof. Multiplying (5.4.132)–(5.4.133) by ε respectively, multiplying (5.4.136) by ε 2 , adding the resulting inequality to (5.4.143), taking ε ∈ (0, 1) small enough, we can obtain (5.4.144). → Lemma 5.4.11. For any (τ0 , − v 0 , θ0 ) ∈ H+4 , there exists a constant 0 < γ4(2) ≤ γ4(1) such that for any fixed γ ∈ (0, γ4(1)], there holds that for any t > 0, e

γt

ττx x x (t)2H 1

+ 0

t

eγ s ττx x x (s)2H 1 ds ≤ C4 ,

(5.4.145)

eγ t u x x x (t)2H 1 + vx x x (t)2H 1 + wx x x (t)2H 1 + θθx x x (t)2H 1 (5.4.146)

t eγ s u x x x x 2H 1 + vx x x x 2H 1 + wx x x x 2H 1 + θθx x x x 2H 1 (s)ds ≤ C4 . + 0

Proof. Multiplying (5.4.110) by eγ t and integrating the resulting inequality with respect to t, using (5.4.108), we get

t τ (t) 2 τ 2 xxx xxx eγ t eγ s (5.4.147) ≤ C3 + (γ − C1−1 ) (s)ds τ (t) τ 0

t + C2 eγ s u x 2H 2 + ττx 2H 1 + vx 2H 1 + θθx 2H 2 + u t x 2H 1 (s)ds. 0

(2)

Taking γ > 0 small enough such that 0 < γ ≤ γ4 and Theorems 5.1.1–5.1.2, we get 2 γ t τx x x

1 e + τ 2C1 whence γt

t

2

e ττx x x (t) +

0

0

t

(1)

≡ mi n[ 2C1 1 , γ4 ], using (5.4.144)

τ 2 xxx eγ s ds ≤ C4 τ

eγ s ττx x x (s)2 ds ≤ C4 , ∀t > 0.

(5.4.148)


289

By (5.4.17), (5.4.22), (5.2.26) and (5.4.31), we obtain u x x x (t)2 + vx x x (t)2 + wx x x (t)2 + θθx x x (t)2 ≤ C2 u x (t)2H 1 + vx (t)2H 1 +wx (t)2H 1 + θθx (t)2H 1 + ττx (t)2H 1 + u t x (t)2 +vt x (t)2 + wt x (t)2 + θt x (t)2 .

(5.4.149)

Using Theorems 5.1.1–5.1.2 and Lemma 5.4.10, we can derive (5.4.150) eγ t u x x x (t)2 + vx x x (t)2 + wx x x (t)2 + θθx x x (t)2

t eγ s u x x x 2 + vx x x 2 + wx x x 2 + θθx x x 2 (s)ds ≤ C4 . + 0

By (5.4.19), (5.4.24), (5.4.28) and (5.4.33), we infer u x x x x (t)2 + vx x x x (t)2 + wx x x x (t)2 + θθx x x x (t)2 ≤ C2 u x (t)2H 2 + vx (t)2H 2 + wx (t)2H 2 + θθx (t)2H 2 + ττx (t)2H 2 (5.4.151) + u t x x (t)2 + vt x x (t)2 + wt x x (t)2 + θt x x (t)2 . Using (5.4.148), (5.4.150) and Theorems 5.1.1–5.1.2 and Lemma 5.4.10, we can derive (5.4.152) eγ t u x x x x (t)2 + vx x x x (t)2 + wx x x x (t)2 + θθx x x x (t)2

t eγ s u x x x x 2 + vx x x x 2 + wx x x x 2 + θθx x x x 2 (s)ds ≤ C4 . + 0

Multiplying (5.4.127) by eγ t and integrating the resulting inequality with respect to t, using (5.4.124), we get

t τ τ 1 x x x x 2 x x x x 2 eγ t eγ s (5.4.153) + ds τ 2C1 0 τ

t eγ s u x 2H 3 + ττx 2H 2 + vx 2H 2 + θθx 2H 3 (s)ds. ≤ C3 + C4 0

Using Theorems 5.1.1–5.1.2 and (5.4.19), (5.4.33), (5.4.148), we have

t

t τ τ 1 x x x x 2 x x x x 2 eγ t eγ s eγ s {u x x x x 2 + θθx x x x 2 }ds + ds ≤ C4 + C4 τ 2C1 0 τ 0

t eγ s u x 2H 2 + ττx 2H 2 + vx 2H 2 ≤ C4 + C4 0 2 (5.4.154) +θθx H 2 + u t x x 2 + wx 2H 2 + θt x x 2 (s)ds.


290

Using Theorems 5.1.1–5.1.2 and Lemma 5.4.10 and (5.4.148), we can derive

t τ τ 1 x x x x 2 x x x x 2 eγ t eγ s + ds ≤ C4 τ 2C1 0 τ whence

γt

t

2

e ττx x x x (t) +

eγ s ττx x x x (s)2 ds ≤ C4 , ∀ t > 0.

(5.4.155)

0

By (5.4.129), we have

t eγ s u x x x x x (s)2 ds 0

t eγ s u x 2H 3 + ττx 3H 2 + vx 2H 2 + θθx 2H 3 + u t x x x 2 (s)ds. ≤ C2 0

Using (5.4.148)–(5.4.155) and Theorems 5.1.1–5.1.2 and Lemma 5.4.10, we obtain

t eγ s u x x x x x (s)2 ds ≤ C4 . (5.4.156) 0

In the same manner, we have

t eγ s vx x x x x 2 + wx x x x x 2 + θθx x x x x 2 (s)ds ≤ C4 .

(5.4.157)

0

Combining with (5.4.148) and (5.4.155), we can derive (5.4.145). Combining with (5.4.150), (5.4.152), (5.4.156) and (5.4.157), we obtain (5.4.146). The proof is complete. Proof of Theorem 5.4.2. Using Theorems 5.1.1–5.1.2 and Lemma 5.4.11, we can prove Theorem 5.4.2.

5.5 Bibliographic Comments In the one-dimensional case for the compressible Navier-Stokes equations, we refer to Section 3.3. In two or three dimensions, we consult Section 3.3, and for this case, we also know that the global existence and large-time behavior of smooth solutions to the equations of a viscous and heat-conductive polytropic ideal gas in general domains have been investigated only for sufficiently small smooth initial data (see, e.g., [82, 412]). Particularly, the exponential decay of global smooth solutions with small initial data has been established in the general domains by Matsumura and Nishida (see, e.g., [255–257]), while in this chapter, we do not need the smallness of the initial density ρ0 (we only need the smallness of the initial total energy which does not include the initial density ρ0 ). This is a new ingredient of this chapter.


291

Note that the circular coaxial cylinder symmetric domain in R3 is an unbounded domain. However, under our assumptions that our solutions depend only on one spatial variable r ∈ G = {r ∈ R+ : 0 < a ≤ r ≤ b}, the related domain G to equations is a bounded domain. Moreover, there are some essential differences between our results and those results of Matsumura and Nishida [255–257] in the following aspects: the circular coaxial cylinder symmetric unbounded domain via the general bounded domain; the small total initial energy via the small smooth initial data; the weak solutions via the smooth solutions. For the spherically symmetric motion of a viscous and heat-conductive polytropic ideal gas in an annular bounded domain or in an exterior domain, the global existence and uniqueness of generalized solutions for arbitrary large initial data have been proved in [110, 111, 167, 170, 174, 325, 335, 452] for various boundary conditions. In [106], Frid and Shelukhin discussed the vanishing shear viscosity and established the global existence in H 1 of solutions to the compressible fluids for flows with cylinder symmetry. Qin [326], and Qin and Jiang [331] established the global existence of solutions in H i (i = 1, 2) and H 4, respectively. In this chapter, these results in [326, 331] have been introduced. It is worth pointing out some difficulties encountered in this chapter. Since we are about to study the large-time behavior of global solutions, all the estimates should be uniform, that is, they should be independent of any length of time. This will result in some severe mathematical difficulties. The first difficulty encountered here is to establish uniform point-wise positive lower and upper bounds of the specific volume τ = 1/ρ (cf. (5.1.25)). To derive this, we need the smallness of the initial total energy. The second difficulty arising here is to prove the point-wise positive lower bound (cf. (5.1.24)) of the absolute temperature which should be positive for all time from the physical point of view. To do this, we have carried out the following two steps: (1) we prove the result of large-time behavior of the absolute temperature, with which we can prove the absolute temperature, has a uniform point-wise positive lower bound for a sufficiently large time (see, (5.2.61)); (2) with the help of the delicate estimates, we make a transform m = 1/θ (see, the formula above (5.2.62)) to prove the absolute temperature has a uniform pointwise positive lower bound on any bounded time interval (see, the formula below (5.2.63)). Combining these two aspects we finally derive the positive lower bound of the absolute temperature (see, (5.2.64)). The third difficulty is that we need some estimates of the absolute temperature with the fractional order (see (5.2.35) in Lemma 5.2.3), since from the proofs of our main results, we easily find that the case of α = 1 is not applicable for our proofs. This is why we have to establish Lemma 5.2.3. The forth difficulty here is that equations under consideration and the constitutive relations for the cylinder symmetric case (cf. (5.1.1)–(5.1.5) or (5.1.14)–(5.1.22)) seem more difficult than those (cf. (4.1.1)– (4.1.5) or (4.1.13)–(4.1.18)) of the spherically symmetric model in Chapter 4 (see, e.g., [110, 111, 167, 170, 174, 325, 335, 452]), so we need more delicate estimates to prove our results in this chapter.

Chapter 6

One-dimensional Nonlinear Thermoviscoelasticity In this chapter we shall study the global existence and asymptotic behavior of solutions to a 1D nonlinear thermoviscoelasticity system. The more general constitutive relation will be studied and our assumptions on the growth exponents of the temperature include cases not studied ever before. The results of this chapter are adopted from Qin [315, 317, 320, 324]. We shall use in this chapter the same notation as that in Chapter 2.

6.1 Global Existence and Asymptotic Behavior of Solutions This section is concerned with global existence and asymptotic behavior, as time tends to infinity, of solutions to system in one-dimensional nonlinear thermoviscoelasticity. The referential (Lagrangian) form of the conservation laws of mass, momentum, and energy for a one-dimensional material with the reference density ρ0 = 1 is u t − vx = 0, vt − σx = 0,

v2 e+ − (σ v)x + Q x = 0, 2 t

(6.1.1) (6.1.2) (6.1.3)

and the second law of thermodynamics is expressed by the Clausius-Duhem inequality Q ≥ 0. (6.1.4) ηt + θ x

294

Chapter 6. One-dimensional Nonlinear Thermoviscoelasticity

Here subscripts indicate partial differentiations, u, v, σ, e, Q, η and θ denote the deformation gradient, velocity, stress, internal energy, heat flux, specific entropy and temperature, respectively. We consider the problem (6.1.1)–(6.1.3) in the region {0 ≤ x ≤ 1, t ≥ 0} under initial conditions u(x, 0) = u 0 (x), v(x, 0) = v0 (x), θ (x, 0) = θ0 (x) on [0, 1],

(6.1.5)

and boundary conditions of the form σ (0, t) = γ v(0, t), σ (1, t) = −γ v(1, t), θ (0, t) = θ (1, t) = T0 ,

(6.1.6)

where γ = 0 or γ = 1 , and T0 > 0 is the reference temperature. The boundary condition (6.1.6) with γ = 1 , boundary damping, indicates that the endpoints of the interval [0,1] are connected to some sort of dash pot. For one-dimensional homogeneous, thermoviscoelastic materials, e, σ, η and q are given by the constitutive relations (see Dafermos [74], Dafermos and Hsiao [77]) e = e(u, θ ), σ = σ (u, θ, vx ), η = η(u, θ ), Q = Q(u, θ, θ x )

(6.1.7)

which in order to be consistent with (6.1.4), must satisfy σ (u, θ, 0) = u (u, θ ), η(u, θ ) = −θ (u, θ ), (σ (u, θ, w) − σ (u, θ, 0))w ≥ 0, Q(u, θ, g)g ≤ 0

(6.1.8) (6.1.9)

where = e − θ η is the Helmholtz free energy function. We assume that e(u, θ ), p(u, θ ), μ(u) and k(u, θ ) are twice continuously differentiable on 0 < u < +∞ and 0 ≤ θ < +∞, and interrelated by eu (u, θ ) = − p(u, θ ) + θ pθ (u, θ ), σ (u, θ, vx ) = − p(u, θ ) + μ(u)vx , (6.1.10) Q(u, θ, θ x ) = −k(u, θ )θθ x so as to be consistent with (6.1.4) or (6.1.8)–(6.1.9). We assume that μ(u) satisfies μ(u)u ≥ μ0 > 0

(6.1.11)

with some constant μ0 > 0. Furthermore, we will be concerned with solid-like materials, so we require that p(u, θ ) be compressive for small u and tensile for large u, at any < ∞ such that temperature, i.e., there are 0 < u≤U p(u, θ ) ≥ 0, 0 < u < u , 0 ≤ θ < ∞, < u < ∞, 0 ≤ θ < ∞. p(u, θ ) ≤ 0, U

(6.1.12) (6.1.13)

Therefore the assumptions (6.1.12)–(6.1.13) imply that there exists a constant η0 with such that u ≤ η0 ≤ U u (η0 , T0 ) = − p(η0 , T0 ) = 0. (6.1.14)

6.1. Global Existence and Asymptotic Behavior of Solutions

295

We also require that p(u, θ ) possess the following monotone condition (see, e.g., Jiang [164]): − pu (u, T0 ) > 0, for any u ≤ u ≤ U if γ = 0 in (6.1.6), − pu (u, T0 ) ≥ p0 > 0, for any 0 < u < ∞ if γ = 1 in (6.1.6).

(6.1.15) (6.1.16)

Here p0 is a constant and u , (1 − λ)η0 + λ min u 0 (x)}) − 2E 0 (λ)] − 1), (6.1.17) u := M −1 ( min [M(min{ 1/2

λ∈[0,1]

x∈[0,1]

, (1 − λ)η0 + λ max u 0 (x)}) + 2E (λ)] + 1), (6.1.18) U := M −1 ( max [M(max{U 0 1/2

λ∈[0,1]

x∈[0,1]

1

E 0 (λ) = (1 + 2γ 2 / p0)

[E((1 − λ)η0 + λu 0 (x), (1 − λ)T T0 + λθ0 (x))

0

+λ2 v02 (x)/2]d x + γ 2 η02 ,

M(u) =

u 1

(6.1.19)

μ(w)dw, E(u, θ ) = (u, θ ) − (η0 , T0 ) − (θ − T0 )θ (u, θ ), (6.1.20)

and (u, θ ) is the Helmholtz free energy function. We can show that u is a priori bounded, u < u(x, t) < U (see Lemma 6.2.3). We assume that there are exponents q, r satisfying one of the conditions 0 ≤ r ≤ 2/3, (3r + 2)/6 < q,

(6.1.21)

2/3 < r < 3, (4r + 2)/7 < q,

(6.1.22)

3 ≤ r, (5r + 1)/8 < q

(6.1.23)

and concerning the growth of the temperature, we require that there be positive constants ν, N1 , N possibly depending on u and/or U such that for any u ≤ u ≤ U , 0 ≤ θ < ∞, 0 ≤ e(u, 0), ν(1 + θ r ) ≤ eθ (u, θ ) ≤ N(1 + θ r ), −N N1 (1 + θ

r+1

r+1

(6.1.24)

) ≤ pu (u, θ ) ≤ −N(1 + θ ), | pθ (u, θ )| ≤ N(1 + θ r ),

(6.1.25) (6.1.26)

ν(1 + θ q ) ≤ k(u, θ ) ≤ N(1 + θ q ), |ku (u, θ )| + |kuu (u, θ )| ≤ N(1 + θ q ).

(6.1.27) (6.1.28)

Without loss of generality, we assume that the initial velocity satisfies 1 v0 (x)d x = 0 if γ = 0.

(6.1.29)

0

For the initial data, we assume that for some α4 ∈ (0, 1), (u 0 (x), v0 (x), θ0 (x)) ∈ H 1+α4 × H 2+α4 × H 2+α4 and u 0 (x) > 0, θ0 (x) > 0 for all x ∈ [0, 1], and the initial data are compatible with the boundary condition (6.1.6). We are now in a position to state our main theorem.

296


Theorem 6.1.1. Under the assumptions (6.1.11)–(6.1.13), (6.1.15)–(6.1.16), (6.1.21)– (6.1.29) and above assumptions on the initial data, the problem (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6) admits a unique global solution (u(t), v(t), θ (t)) ∈ BT1+α4 × HT2+α4 × 2+α HT 4 for any T > 0 with u < u(x, t) 0 on [0, 1]×[0, ∞). Moreover, there exist positive constants t0 , C1 , C1 such that for all t ≥ t0 , there holds v(t) H 1 + θ (t) − T0 H 1 + u(t) − η0 H 1 ≤ C1 exp(−C1 t)

(6.1.30)

and as t → +∞, we have

2

u(t) − η0 H 1 → 0, v(t) H 1 → 0, v(t) L ∞ → 0,

(6.1.31)

θθx (t) → 0, θ (t) − T0 H 1 → 0, θ (t) − T0 L ∞ → 0,

(6.1.32)

2

γ (v (0, t) + v (1, t)) → 0, p(u, θ ) H 1 → 0, σ (u, θ, vx ) → 0

(6.1.33)

where (η0 , 0, T0 ) is the unique solution to the corresponding stationary problem to (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6). Remark 6.1.1. With different assumptions on the exponents q and r , similar conclusions to those in Theorem 6.1.1 hold for the boundary conditions σ (0, t) = v(0, t),

σ (1, t) = −v(1, t),

Q(0, t) = Q(1, t) = 0

and for the boundary conditions Q(0, t) = Q(1, t) = 0 or

θ (0, t) = θ (1, t) = T0

σ (0, t) = 0, σ (1, t) = −v(1, t) or σ (0, t) = v(0, t), σ (1, t) = 0. Remark 6.1.2. Similar global existence results in Theorem 6.1.1 can be established for the boundary conditions σ (0, t) = σ (1, t) = 0, Q(0, t) = Q(1, t) = 0 and σ (0, t) = σ (1, t) = 0, θ (0, t) = θ (1, t) = T0 . With the exponents q = 0 and 0 ≤ r < 7/18, Dafermos [74] established the global existence of smooth solutions for the first case of boundary conditions above. The global existence of smooth solutions in Theorem 6.1.1 is based on a priori estimates that can be used to continue a local solution globally in time. Existence and uniqueness of local solutions (with positive u and θ ) can be obtained by linearization of the problem (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6), and by use of the Banach contraction mapping theorem. After uniform a priori estimates have been established in Section 6.2, the global existence of smooth large solutions can be obtained by the same approaches as in Kawashima and Nishida [191] and Nagasawa [283–286]. So the most important step is to derive uniform a priori estimates.

6.2. Uniform A Priori Estimates

297

6.2 Uniform A Priori Estimates Theorem 6.2.1. Let (u, v, θ ) be a smooth solution as described in Theorem 6.1.1, then we have for any T > 0, 4) 4) 4) |||u|||(1+α + ||v||(2+α + |||θ |||(2+α ≤ C. T T T

The proofs of Theorem 6.1.1 and Theorem 6.2.1 are divided into a series of lemmas. Lemma 6.2.1. There holds that θ (x, t) > 0,

on

[0, 1] × [0, ∞).

Proof. The proof of (6.2.1) is similar to that of (2.1.32).

(6.2.1)

Lemma 6.2.2. If u ≤ u(x, t) ≤ U for all x ∈ [0, 1] and t ∈ [0, τ ], τ > 0, then ωγ 1 1 1 ν 1 (θ (x, t) − T0 )2 dx + (u(x, t) − η0 )2 d x + v(x, t)d x 2 0 θ (x, t) + T0 2 0 2 0 1 v 2 (x) d x ≡ e0 , ∀0 ≤ t ≤ τ, E(u 0 (x), θ0 (x)) + 0 ≤ (6.2.2) 2 0 where ωγ = 0 for γ = 0 and ωγ = p0 for γ = 1, E(u, θ ) and p0 are the same as in (6.1.20) and (6.1.16), respectively. Proof. Recalling the definition of E(u, θ ), and noting that eθ (u, θ ) = −θ θθ (u, θ ) and (u, θ ) = e(u, θ ) − θ η(u, θ ) satisfies −θ (u, θ ) = η(u, θ ), u (u, θ ) = σ (u, θ, 0) = − p(u, θ ),

(6.2.3)

using (6.1.1)–(6.1.3), (6.1.11) and (6.2.3), we deduce after a direct calculation that 2 k(u, θ )θθ x2 v2 (θ − T0 )k(u, θ )θθ x μvx + T0 + = σ v + ∂t E(u, θ ) + . (6.2.4) 2 θ θ2 θ x Integrating (6.2.4) over Q t ≡ [0, 1] × [0, t] and using (6.1.6) leads to t 1 2 1

v2 k(u, θ )θθ x2 μvx E(u, θ ) + d x ds (x, t)d x + T0 + 2 θ θ2 0 0 0 t 1

v02 2 2 E(u 0 , θ0 ) + d x. (6.2.5) +γ [v (0, s) + v (1, s)]ds = 2 0 0 Recalling that uu = − pu , if we use the mean value theorem (or the Taylor theorem), (6.1.24), and (6.1.14)–(6.1.15), we see that E(u, θ ) − (u, T0 ) + (η0 , T0 ) ≥

ν (θ − T0 )2 2 (θ + T0 )

(6.2.6)

298


and (u, T0 ) − (η0 , T0 ) ≥

ωγ (u − η0 )2 2

for u ≤ u ≤ U , which adding to (6.2.6) gives E(u, θ ) ≥

ωγ ν (θ − T0 )2 (u − η0 )2 f or u ≤ u ≤ U . + 2 θ + T0 2

(6.2.7)

Inserting (6.2.7) into (6.2.5) yields the lemma.

Next we want to bound the deformation gradient u(x, t). To this end, we rewrite (6.1.2), using (6.1.20), as follows: vt + p(u, θ )x = M(u)t x .

(6.2.8)

Lemma 6.2.3. There holds that u < u(x, t) < U , on [0, 1] × [0, ∞)

(6.2.9)

where u and U are defined by (6.1.17) and (6.1.18) respectively. Proof. We integrate (6.2.8) over [0, y] × [s, τ ] and [y, 1] × [s, τ ], 0 ≤ y ≤ 1, 0 ≤ s < τ , respectively, and apply the boundary condition (6.1.6) to obtain

τ

M(u(y, τ )) − M(u(y, s)) =

p(y, t)dt + γ

s

τ

y

v(0, t)dt +

s

0

(v(x, τ ) − v(x, s))d x (6.2.10)

and

τ

M(u(y, τ )) − M(u(y, s)) = s

γ

p(y, t)dt − γ s

1

v(1, t)dt −

(v(x, τ ) − v(x, s))d x

y

(6.2.11) where p(y, t) = p(u(y, t), θ (y, t)). We add (6.2.11) to (6.2.10) and take u t = vx into account to deduce M(u(y, τ )) − M(u(y, s))

τ 1 γ γ τ 1 1 (v(x, τ ) − v(x, s))d x = p(y, t)dt − vx (x, t)d x dt + − 2 s 0 2 s 0 y τ γ 1 = p(y, t)dt − (u(x, τ ) − u(x, s))d x 2 0 s

1 γ 1 + (v(x, τ ) − v(x, s))d x. (6.2.12) − 2 0 y


299

By Lemma 6.2.2 and the Schwarz inequality, recalling the definition (6.1.19), we see that if u ≤ u(x, t) ≤ U for 0 ≤ x ≤ 1, 0 ≤ t ≤ τ , then we have γ 1 1 γ 1 (v(τ, x) − v(s, x))d x (u(x, τ ) − u(x, s))d x + − 2 0 2 0 y 1/2 1/2 1 1 2 2 ≤ γ max u (x, ·)d x + max v (x, ·)d x (6.2.13) [0,τ ]

≤

γ (2η02

[0,τ ]

0

+ 4e0 / p0 )

1/2

+ (2e0 )

1/2

0

1/2

≤ 2((1 + 2γ 2 / p0 )e0 + γ 2 η02 )1/2 ≡ 2E 0 (1).

In particular, (6.1.17) and (6.1.18) yield u 0 and y ∈ [0, 1] such that u < u(x, t) < U f or x ∈ [0, 1], 0 ≤ t < τ, but u(y, τ ) = u or u(y, τ ) = U . (6.2.14) Note that u < u . If u(y, τ ) = u, then either u(y, t) < u for 0 ≤ t ≤ τ , or u(y, t) < u for 0 ≤ s < t ≤ τ , but u(y, s) = u . Recalling that, on account of (6.2.14), u ≤ u(x, t) ≤ U for 0 ≤ x ≤ 1 and 0 ≤ t ≤ τ, in the former case we apply (6.2.12) with s = 0 and utilize (6.2.12) and (6.2.13) to deduce 1/2

M(u(y, τ )) > M(u 0 (y)) − 2E 0 (1) − 1

(6.2.15)

while in the latter case (6.2.12) combined with (6.2.12) and (6.2.13) implies 1/2

M(u(y, τ )) > M( u ) − 2E 0 (1) − 1.

(6.2.16)

In either case, by (6.1.17), M(u(y, τ )) > M(u) which contradicts u(y, τ ) = u. Hence u < u(x, t), 0 ≤ x ≤ 1, 0 ≤ t < ∞. Similarly, we can show that u(y, τ ) = U is a contradiction. This shows u(x, t) 0, 1 t 1+r 2 [(θ/T T0 − log(θ/T T0 ) − 1) + θ + v ](x, t)d x + γ [v 2 (0, s) + v 2 (1, s)](x, s)ds 0

+

0

t 0

1 0

(1 + θ q )θθx2 v2 [ x + ]d x ds ≤ C. θ θ2

(6.2.17)

Proof. In view of (6.1.24)–(6.1.25), we have uu (u, T0 ) = − pu (u, T0 ) > 0 for u ∈ [u, U ]. Therefore it follows from the Taylor theorem and (6.1.29) that E(u, θ ) − (u, θ ) + (u, T0 ) + (θ − T0 )θ (u, θ ) = (u, T0 ) − (η0 , T0 ) − u (η0 , T0 )(u − η0 ) 1 (1 − ξ )uu (η0 + ξ(u − η0 ), T0 )dξ ≥ 0. = (u − η0 )2 0

300


Thus, E(u, θ ) ≥ (u, θ ) − (u, T0 ) − (θ − T0 )θ (u, θ ) 1 2 = −(T T0 − θ ) (1 − τ )θθ (u, θ + τ (T T0 − θ ))dτ

0 1

(1 − τ ){1 + [θ + τ (T T0 − θ )]r } dτ ≥ ν(T T0 − θ )2 θ + τ (T T0 − θ ) 0

νT T (T T r −θ r ) ν(T T 1+r −θ 1+r ) T0 (θ/T T0 − log(θ/T T0 ) − 1) + 0 r0 − 0 r+1 , = νT T0 − log(θ/T T0 ) − 1), for r = 0. 2νT T0 (θ/T

for r > 0,

≥ νT T0 (θ/T T0 − log(θ/T T0 ) − 1) + C5 θ r+1 − C6

which, combined with (6.2.5), yields (6.2.17).

Remark 6.2.1. It follows from the convexity of the function − ln y that there exist two positive constants r1 , r2 only depending on the initial data such that 1 1 0 < r1 ≤ θ d x ≤ r2 . T0 0 Remark 6.2.2. By the mean value theorem and (6.1.14), we have | p(u, θ )| ≤ C(1 + θ r+1 )

(6.2.18)

for any u ∈ [u, U ] and θ > 0. Lemma 6.2.5. The following estimates hold for any t > 0, t v(s)2L ∞ ds ≤ C, 0 t vx (s)2 ds ≤ C(1 + sup θ (s) L ∞ )β 0

(6.2.19) (6.2.20)

0≤s≤t

with β = max(r + 1 − q, 0). Proof. If γ = 0, we infer from (6.1.29) and (6.1.2) that 1 1 vd x = v0 d x = 0 0

0

which implies that for any t > 0, there is a point x 0 (t) ∈ [0, 1] such that v(x 0 (t), t) = 0. We infer from (6.2.21) that

v(x, t) =

x x 0 (t )

v y (y, t)d y.

(6.2.21)

(6.2.22)


301

Thus by Lemma 6.2.4 and (6.2.22), we have t

0

v(s)2L ∞ ds

t

≤ 0

≤C

1 0

t 0

0

1

vx2 dx θ

1

θ d x ds

0

vx2 d x ds ≤ C θ

which gives (6.2.19). If γ = 1, we have x v 2 (x, s) = (v(0, s) + v y (y, s)d y)2 0 1 1 |v y (y, s)|2 2 dy ≤ 2 v (0, s) + θ (y, s)d y θ (y, s) 0 0 1 |v y (y, s)|2 dy . ≤ 2 v2 (0, s) + θ (y, s) 0

(6.2.23)

Thus it follows from (6.2.23) and Lemma 6.2.4 that

t 0

v(s)2L ∞ ds

≤2

t 0

2

v (0, s)ds +

t 0

1 0

v 2y (y, s) θ (y, s)

d yds ≤ C

which also gives (6.2.19). Multiplying (6.1.2) by v, integrating the resultant over Q t and using (6.1.1), (6.1.6), (6.1.11), (6.1.14), Lemma 6.2.4, the mean value theorem and Remark 6.2.2, yields t t 1 1 2 2 2 v + γ (v (0, s) + v (1, s))ds + μ(u)vx2 d x ds 2 0 0 0 t 1 1 p(u, θ )vx d x ds + v0 2 = 2 0 0 t 1 t 1 1 = ( p(u, θ ) − p(u, T0 ))vx d x ds + p(u, T0 )vx d x ds + v0 2 2 0 0 0 0 t 1 t 1 ≤C pθ (u, θ2 )(θ − T0 )2 ds + μ(u)vx2 d x ds 2 0 0 0 1 u0 1 u p(ξ, T0 )dξ d x + p(ξ, T0 )dξ d x + C + 0

u

t

0

1

u t

1 μ(u)vx2 d x ds + C (1 + θ r )(θ − T0 )2 ds 2 0 0 0 1 t 1 t 1 2 (r−1)/2 2 μ(u)vx d x ds + C (1 + θ ) (θ − T0 ) L ∞ (1 + θ )r+1 d x ds ≤C+ 2 0 0 0 0 ≤C+

302


t 1 t 1 μ(u)vx2 d x ds + C [(1 + θ )(r−1)/2(θ − T0 )]x 2L 1 ds 2 0 0 0 t 1 1 t 1 (1 + θ q )θθx2 2 β μ(u)vx d x ds + C(1 + sup θ (s) L ∞ ) d x ds ≤C+ 2 0 0 θ2 0 0 0≤s≤t 1 t 1 μ(u)vx2 d x ds (6.2.24) ≤ C(1 + sup θ (s) L ∞ )β + 2 0 0 0≤s≤t ≤C+

which gives (6.2.20); here min(θ, T0 ) ≤ θ2 ≤ max(θ, T0 ) ≤ C(1 + θ ).

Lemma 6.2.6. There holds that for any (x, t) ∈ [0, 1] × [0, +∞), (6.2.25) C − C V (t) ≤ θ 2m 1 (x, t) ≤ C + C V (t) 1 (1+θ q )θθx2 ∞ d x satisfying 0 V (t)dt < ∞. with 0 ≤ m 1 ≤ m = (q +r +1)/2 and V (t) = 0 θ2 Proof. The Cauchy inequality, Lemma 6.2.4 and (6.1.6) imply 1 1 m1 m 1 −1 1/2 θ ≤C +C |θ θx |d x ≤ C + C V (t)[ (1 + θ )2m 1 −q d x]1/2 0

0

1

≤ C + C V 1/2 (t)[

(1 + θ )r+1 d x]1/2 ≤ C + C V 1/2 (t)

0

which implies (6.2.25). Lemma 6.2.7. The following estimates hold for any t > 0: t 1 (1 + θ )2m v 2 d x ds ≤ C, 0 0 t 1

u x (t)2 + 0

0

t 0

(1 + θ r+1 )u 2x d x ds ≤ C(1 + sup θ (s) L ∞ )β , (1 + θ )2m u 2x d x ds ≤ C(1 + sup θ (s) L ∞ )β .

0

(6.2.28)

0≤s≤t

Proof. It follows from Lemma 6.2.4 that t 1 t (1 + θ )2m v 2 d x ds ≤ C 0

(6.2.27)

0≤s≤t

1 0

(6.2.26)

0

1 0

v 2 d x ds + C

t 0

V (s)v2 ds ≤ C.

The equation (6.1.2) can be rewritten as (v − μ(u)u x )t + pu (u, θ )u x = − pθ (u, θ )θθ x .

(6.2.29)

Multiplying (6.2.29) by v − μ(u)u x , then integrating the resultant over Q t leads to t 1 1 v − μ(u)u x 2 − μ(u) pu (u, θ )u 2x d x ds 2 0 0 t 1 1 [ pu u x v + pθ θx (v − μ(u)u x )]d x ds. = v0 − μ(u 0 )u 0x 2 − 2 0 0


303

Using Lemmas 6.2.1–6.2.6 and noting the facts t 0

1 0

θ 2 (1 + θ r )2 v 2 d x ds ≤ C(1 + sup θ (s) L ∞ )δ 1 + θq 0≤s≤t

t 0

1

(1 + θ )2m v 2 d x ds

0

≤ C(1 + sup θ (s) L ∞ )δ , t 0

1 0

(1 + θ r )2 θx2 d x ds 1 + θ r+1

0≤s≤t 1

t

(1 + θ )r−1 θx2 d x ds, t ≤ C(1 + sup θ (s) L ∞ )β V (s)ds

≤C

0

0

0

0≤s≤t β

≤ C(1 + sup θ (s) L ∞ ) , 0≤s≤t

we arrive at 2

u x +

t

≤C +C

0

1 0

t 0

≤C +C

[1 + θ r+1 ]u 2x d x ds

t 0

[(1 + θ r+1 )|u x v| + (1 + θ r )|θθ x (v − μ(u)u x )|]d x ds

0 1 0

(1 + θ 1+r )(u 2x + Cv 2 )d x ds + C

t

V (s)ds

0

t 1 θ 2 (1 + θ r )2 2 v d x ds + C (1 + θ 1+r )u 2x d x ds q 1 + θ 0 0 0 0 t 1 (1 + θ r )2 θx2 d x ds +C 1 + θ 1+r 0 0 t 1 t β 1+r 2 (1 + θ )u x d x ds + C v2L ∞ ds ≤ C(1 + sup θ (s) L ∞ ) + C +C

t

1

1

0

0≤s≤t

0

0

with β = max(r + 1 − q, 0) ≥ δ = max(r + 1 − 2q, 0). Thus for small > 0 we have (6.2.27). The proof of (6.2.28) is similar to that of (6.2.26) if we exploit (6.2.27). Lemma 6.2.8. The following estimates hold for any t > 0:

t

2

vx (t) + 0

vt (s)2 ds + γ (v 2 (0, t) + v 2 (1, t)) ≤ C(1 + sup θ (s) L ∞ )β4 ,

(6.2.30)

0≤s≤t

t 0

vx x (s)2 ds ≤ C(1 + sup θ (s) L ∞ )β5 0≤s≤t

(6.2.31)

304


with β1 = max(2r + 2 − q, 0), β2 = max(3β, β1 ), β3 = (β2 + 3β)/2, β4 = max(β1 , β3 , 3β), β5 = max(β2 , β4 ). Proof. By (6.1.2), we have vx x = μ−1 (u)(vt + pu u x + pθ θx − μ (u)u x vx ). It follows from Lemmas 6.2.1–6.2.7 that t vx x (s)2 ds 0 t (vt 2 + (1 + θ r+1 )u x 2 + (1 + θ r )θθx 2 + vx 2L ∞ u x 2 )ds ≤C 0

≤ C(1 + sup θ (s) L ∞ )β

t 0

0≤s≤t

+ C(1 + sup θ (s) L ∞ )

β1

0≤s≤t

0 t

0 t

+ C(1 + sup θ (s) L ∞ )β

1

(1 + θ )2m u 2x d x ds

V (s)ds + C

0

t

vt (s)2 ds

(vx vx x + vx 2 )(s)ds

0

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )

2β

+ C(1 + sup θ (s) L ∞ )

0≤s≤t

+ C(1 + sup θ (s) L ∞ )

β

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )

t 0

0≤s≤t

t

1/2

2

vx ds

0

+ C(1 + sup θ (s) L ∞ )β β1

0≤s≤t

t

+C

vt (s)2 ds

0 t

1/2

2

vx x ds

vx (s)2 ds

t

vt (s)2 ds

0

+ C(1 + sup θ (s) L ∞ )

0

+C

0≤s≤t

β1

t

3β/2

1/2 2

vx x ds

0

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )β1 + C(1 + sup θ (s) L ∞ )3β 0≤s≤t t

vt (s)2 ds +

+C 0

1 2

0≤s≤t

t

vx x (s)2 ds

0

which implies t t vx x 2 ds ≤ C(1 + sup θ (s) L ∞ )β2 + C vt 2 ds. 0

0≤s≤t

0

(6.2.32)


305

By virtue of (6.1.1) and (6.1.6), we easily get μ(u)vx vt |x=1 (6.2.33) γ d 2 d v (1, t) + ( p(u(1, t), T0 )v(1, t)) − pu (u(1, t), T0 )vx (1, t)v(1, t), =− 2 dt dt μ(u)vx vt |x=0 (6.2.34) d γ d 2 v (0, t) + ( p(u(0, 1), T0 )v(0, t)) − pu (u(0, t), T0 )vx (0, t)v(0, t). = 2 dt dt On the other hand, multiplying (6.1.2) by vt , then integrating the resultant over Q t , using the Nirenberg inequality, (6.2.32) and Lemmas 6.2.1–6.2.7, yields t vx (t)2 + vt (s)2 ds + γ (v 2 (0, t) + v 2 (1, t)) 0 t t ≤C + vt (s)2 ds + C (vx 3L 3 + pu u x + pθ θx 2 )(s)ds 0

0

+ C| p(u(1, t), T0 )v(1, t) − p(u(0, t), T0 )v(0, t)| t +C (| pu (u(1, t), T0 )vx (1, t)v(1, t)| + | pu (u(0, t), T0 )vx (0, t)v(0, t)|)(s)ds 0 t t vt (s)2 ds + C (vx 5/2 vx x 1/2 + vx 3 )(s)ds + Cv(t) L ∞ ≤ 0

0

+ C(1 + sup θ (s) L ∞ )2β + C(1 + sup θ (s) L ∞ )β1 0≤s≤t

+C

t 0

v(s)2L ∞ ds

0≤s≤t

t 0

≤ C(1 + sup θ (s) L ∞ )β1 +

0≤s≤t

1/2

0

1/4

1/2

vx (s)2L ∞ ds t

vt (s)2 ds + Cvx (t)1/2

3/4 vx (s)2 ds 0 0 t × sup vx (s) + C sup vx (s) vx (s)2 ds t

+C

0≤s≤t

+C

t 0

t

vx x (s)2 ds

0≤s≤t

1/2

vx (s)2 ds

≤ C(1 + sup θ (s) L ∞ ) 0≤s≤t

+C β1

t 0

1/4

vx (s)2 ds 2

+ sup vx (s) + 0≤s≤t t

+ C[(1 + sup θ (s) L ∞ )β2 + 0≤s≤t

0

0

vt (s)2 ds]1/4

× (1 + sup θ (s) L ∞ )3β/4 sup vx (s) 0≤s≤t

0≤s≤t

0

t 0

t

1/4 vx x (s)2 ds

vt (s)2 ds

306


+ C sup vx (s)(1 + sup θ (s) L ∞ )β + C(1 + sup θ (s) L ∞ )β/2 0≤s≤t

0≤s≤t

+ C(1 + sup θ (s)

L∞

)

β/4

0≤s≤t

[(1 + sup θ (s)

0≤s≤t

L∞

)

β2

t

+

vt (s)2 ds]1/4

0

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )β1 + C(1 + sup θ (s) L ∞ )β3 /2 sup vx (t) 0≤s≤t

0≤s≤t

t

+ sup vx (t)2 +

vt (s)2 ds

0

0≤s≤t

1/4

t

+C

0≤s≤t

vt (s)2 ds

(1 + sup θ (s) L ∞ )3β/4 sup vx (s)

0

0≤s≤t

0≤s≤t

+ C sup vx (s)(1 + sup θ (s) L ∞ )β + C(1 + sup θ (s) L ∞ )(β+β2 )/4 0≤s≤t

0≤s≤t

+ C(1 + sup θ (s) L ∞ )β/4 0≤s≤t

t 0

1/4 vt (s)2 ds

0≤s≤t

+ C(1 + sup θ (s) L ∞ )β/2

≤ C(1 + sup θ (s) L ∞ )β1 + 2 sup vx (s)2 + 2 0≤s≤t

0≤s≤t t

vt (s)2 ds

0

0≤s≤t

+ C(1 + sup θ (s) L ∞ )β3 + C(1 + sup θ (s) L ∞ )3β 0≤s≤t

0≤s≤t

≤ C(1 + sup θ (s)

L∞

)

β4

2

+ 2 sup vx (s) + 2

0≤s≤t

t

vt (s)2 ds

0

0≤s≤t

which gives (6.2.30) for small > 0 and (6.2.31) follows from (6.2.30) and (6.2.32). Corollary 6.2.1. The following estimates are valid for any t > 0: t p x (s)2 ds ≤ C(1 + sup θ (s) L ∞ )β1 , 0 t

0 t 0

(6.2.35)

0≤s≤t

vx (s)3L 3 ds ≤ C(1 + sup θ (s) L ∞ )β6 ,

(6.2.36)

vx (s)2L ∞ ds ≤ C(1 + sup θ (s) L ∞ )β7 ,

(6.2.37)

0≤s≤t

0≤s≤t

with β6 = (β5 + 3β + 2β4 )/4 and β7 = (β + β5 )/2. Proof. By Lemmas 6.2.1–6.2.7, we have t t 1 p x (s)2 ds ≤ C [(1 + θ )2r+2 u 2x + (1 + θ )2r θx2 ](x, s)d x ds 0

0

0

≤ C(1 + sup θ (s) L ∞ )2β + C(1 + sup θ (s) L ∞ )β1 0≤s≤t

≤ C(1 + sup θ (s) L ∞ ) 0≤s≤t

0≤s≤t β1


t 0

vx (s)3L 3 ds ≤ C

t

≤C

t

307

[vx 5/2 vx x 1/2 + vx 3 ](s)ds

0

1/4

t

2

vx x (s) ds

0

3/4 2

vx (s) ds

t

+

0

2

vx (s) ds

0

≤ C(1 + sup θ (s) L ∞ )(β5 +3β+2β4 )/4 + C(1 + sup θ (s) L ∞ ) 0≤s≤t

sup vx (s)

0≤s≤t β4 /2+β

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )

β6

0≤s≤t

and

t 0

vx (s)2L ∞ ds ≤ C

[vx vx x + vx 2 ](s)ds

0

1/2

t

≤C

t

2

vx ds

0

≤ C(1 + sup θ (s) L ∞ )

t

1/2

2

vx x ds

0 (β+β5 )/2

t

+C

vx 2 ds

0

+ C(1 + sup θ (s) L ∞ )β

0≤s≤t

0≤s≤t

≤ C(1 + sup θ (s)

L∞

)

β7

0≤s≤t

with β5 ≥ 3β.

Lemma 6.2.9. The following estimates are valid for any t > 0: v(t) H 1 ≤ C(1 + sup θ (s) L ∞ )β4 /2 ,

(6.2.38)

v(t) L ∞ ≤ C(1 + sup θ (s) L ∞ )β4 /4 ,

(6.2.39)

0≤s≤t 0≤s≤t

t


(6.2.40)

(1 + θ )2m v 2 vx2 d x ds ≤ C(1 + sup θ (s) L ∞ )3β4 /2 ,

(6.2.41)

0

0

t 0

v(t)4L 4

1

1 0

0≤s≤t

t

+γ

0≤s≤t t

4

4

vvx 2 (s)ds

(v (0, s) + v (1, s))ds +

0

0

≤ C(1 + sup θ (s) L ∞ )β .

(6.2.42)

0≤s≤t

Proof. From the Nirenberg inequality, Lemmas 6.2.7–6.2.8, we easily deduce (6.2.38)– (6.2.40). Multiplying (6.1.2) by v 3 , integrating the resultant over Q t , and using (6.1.6) and Lemmas 6.2.7–6.2.8, gives t t 1 v(t)4L 4 + γ (v 4 (0, s) + v 4 (1, s))ds + v 2 vx2 d x ds 0

≤C +C

0

t 0

1 0

| p(u, θ )v 2 vx |d x ds

0

308


≤C +C

t 0

1 ≤C+ 2

t 0

1

(1 + θ )r+1 v 2 |vx |d x ds

0 1 0

v 2 vx2 d x ds

+C

≤ C(1 + sup θ (s) L ∞ )β + 0≤s≤t

1 2

t 0

t 0

1 0 1 0

(1 + θ )2(r+1) v 2 d x ds v 2 vx2 d x ds

which implies (6.2.42), and (6.2.41) is the direct result of Lemma 6.2.4, (6.2.38)–(6.2.39) and (6.2.42). Lemma 6.2.10. There holds that for any t > 0, t 1 (T T0 − θ )2 (1 + θ )q+r θx2 1+r 2 θ (t) + [ + (1 + θ )q+r−1 θx2 ](x, s)d x ds θ2 0 0 ≤ C(1 + sup θ (s) L ∞ )β8

(6.2.43)

0≤s≤t

with β9 = min[max(2r + 1 − 2q, 0), max(3r + 3 − 2q, 0)/2], β10 = min[max(r − q, 0) + β, δ/2 + 3β4 /4, (β + β1 )/2], β11 = min[max(q − r, 0) + β, (q + 2 + β)/2, (2 max(1 − r, 0) + 3β4 )/4], β8 = max(3β/2, β9 , β10 , β11 , β12 ). β12 = β + 1, Proof. Let E 1 (u, θ ) = E(u, θ ) + C6 , A(t) =

t 0

1 0

(T T0 − θ )2 (1 + θ )q+r θx2 d x ds. θ2

Thus we know from the proof of Lemma 6.2.4 that E 1 (u, θ ) ≥ C5 θ r+1 > 0 and (6.2.4) can be rewritten as 2 k(u, θ )θθ x2 (θ − T0 )k(u, θ )θθ x v2 μvx + + T0 = σ v + . (6.2.44) ∂t E 1 (u, θ ) + 2 θ θ θ2 x Multiplying (6.2.44) by E 1 + v2 , integrating the resultant over Q t yields t 1 2 2 2 k(u, θ )θθ x2 μvx E 1 (u, θ ) + v + 2T + (E 1 (u, θ ) + v 2 /2)d x ds T 0 2 θ θ2 0 0 t v 2 (0, s) v 2 (1, s) + v 2 (1, s) E 1 (1, s) + v 2 (0, s) E 1 (0, s) + ds + 2γ 2 2 0 t 1 v02 2 (θ − T0 )k(u, θ )θθ x = E 1 (u 0 , θ0 ) + − 2 σv + (E x + vvx )d x ds. 2 θ 2

0

0

(6.2.45)


309

Noting (6.2.3) and (6.1.20), we can deduce (θ − T0 )θθx . (6.2.46) θ Inserting (6.2.46) and (6.1.10)–(6.1.11) into (6.2.45), and using Lemma 6.2.6 and Corollary 6.2.1, leads to t 1 2 (1 + θ q )θθx2 vx r+1 2 θ + + (θ r+1 + v 2 )d x ds θ θ2 0 0 t 1 (θ − T0 )2 eθ (u, θ )k(u, θ )θθ x2 μv 2 vx2 + d x ds + θ2 0 0 t 1 |(θ − T0 )k(u, θ )vvx θx | ≤ C +C | pv(E x + vvx )| + μ|vvx E x | + θ 0 0 |(T T0 − θ )k(u, θ )θθ x | − p(u, θ )u x + pθ (u, θ )(θ − T0 )u x d x ds, + θ E x (u, θ ) = − p(u, θ )u x + pθ (u, θ )(θ − T0 )u x + eθ (u, θ )

i.e., θ ≤

t

r+1 2

+

0

0

1

[(θ r−1 + θ q+r−1 )θθx2 + (T T0 − θ )2 (1 + θ )q+r θx2 θ −2 ]d x ds

C + I1 + I2 + I3 + I4 .

(6.2.47)

By Lemmas 6.2.1–6.2.9, we have t 1 pv(E x + vvx )d x ds I1 ≤ C 0

≤C

0 1

t 0

0

[(1 + θ )2r+2 |vu x | + (1 + θ )2r+1 |(T T0 − θ )θθx v|θ −1

+ (1 + θ )r+1 v 2 |vx |]d x ds (1)

(2)

(3)

= I1 + I1 + I1

(6.2.48)

where (1) I1

≤C

t 0

1 0

(1 + θ )2r+2 u 2x d x ds

≤ C(1 + sup θ (s) L ∞ )

t 0

β

0

0

t

0≤s≤t 1

1/2 t

0

1/2

1 0

1

1/2 (1 + θ )

0

2r+2 2

v d x ds

1/2

(1 + θ )2m u 2x d x ds

(1 + θ )2m v 2 d x ds

≤ C(1 + sup θ (s) L ∞ )3β/2 , 0≤s≤t

(6.2.49)

310


t

(1 + θ )2r+1 |(T T0 − θ )vθθx | d x ds θ 0 0 t 1 1 ≤ A(t) + C (1 + θ )3r+2−q v 2 d x ds 8 0 0 1 ≤ A(t) + C(1 + sup θ (s) L ∞ )max(2r+1−2q,0) 8 0≤s≤t

I1(2) ≤ C

1

(6.2.50)

or (2) I1

≤C

t 0

V (s)ds

1/2 t 0

≤ C(1 + sup θ (s) L ∞ )

1 0

1/2 (1 + θ )

4r+4−q 2

v d x ds

max(3r+3−2q,0)/2

t 0

0≤s≤t

1 0

1/2 (1 + θ )

2m 2

v d x ds

≤ C(1 + sup θ (s) L ∞ )max(3r+3−2q,0)/2 0≤s≤t

which, combined with (6.2.50), yields 1 A(t) + C(1 + sup θ (s) L ∞ )β9 , 8 0≤s≤t t 1 ≤C (1 + θ )r+1 v 2 |vx |d x ds

I1(2) ≤ (3)

I1

0

≤C

0

t 0

1

(1 + θ )

2r+2 2

v d x ds

(6.2.51)

1/2 t

0

0

1 0

1/2 v 2 vx2 d x ds

β

≤ C(1 + sup θ (s) L ∞ ) .

(6.2.52)

0≤s≤t

Similarly, t I2 ≤ C

1

|vvx E x |d x ds t 1 (1 + θ )r |(T T0 − θ )θθx vvx | (1) (2) r+1 (1 + θ ) |u x vvx | + d x ds ≤ I2 + I2 ≤C θ 0 0 0

0

where (1)

I2 ≤ C ≤C

t 0

1 0

(1 + θ )r+1 |u x vvx |d x ds

t 0

1 0

(1 + θ )2r+2 u 2x d x ds

≤ C(1 + sup θ (s) L ∞ )3β/2 0≤s≤t

1/2 t 0

1 0

1/2 v 2 vx2 d x ds (6.2.53)


311

and t

(1 + θ )r |(T T0 − θ )θθx vvx | d x ds θ 0 0 t 1 1 ≤ A(t) + C (1 + θ )r−q v 2 vx2 d x ds 8 0 0 1 ≤ A(t) + C(1 + sup θ (s) L ∞ )max (r−q,0)+β 8 0≤s≤t

I2(2) ≤ C

1

(6.2.54)

or t

(1 + θ )r+1 |θθx vvx | d x ds θ 0 0 1/2 1/2 t 1 t 2r+2−q 2 2 ≤C V (s)ds (1 + θ ) v vx d x ds

I2(2) ≤ C

1

0

0

≤ C(1 + sup θ (s) L ∞ )

0

(β+β1 )/2

(6.2.55)

0≤s≤t

or (2) I2

≤C

t 0

1 0

1/2 (1 + θ )2r+2−q v 2 vx2 d x ds

≤ C(1 + sup θ (s) L ∞ )

δ/2

t 0

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )

1 0

1/2 (1 + θ )2m v 2 vx2 d x ds

(2δ+3β4 )/4

0≤s≤t

which, combined with (6.2.54) and (6.2.55), implies (2)

I2

≤

1 A(t) + C(1 + sup θ (s) L ∞ )β10 . 8 0≤s≤t

(6.2.56)

By Lemmas 6.2.1–6.2.9, we get t

(1 + θ q )|(T T0 − θ )vvx θx | d x ds θ 0 0 t 1 1 ≤ A(t) + C (1 + θ )q−r v 2 vx2 d x ds 8 0 0 1 ≤ A(t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+β 8 0≤s≤t

I3 ≤ C

1

(6.2.57)

312


or

t

(1 + θ )q+1 |vvx θx | d x ds θ 0 0 1/2 1/2 t 1 t q+2 2 2 ≤C V (s)ds (1 + θ ) v vx d x ds

I3 ≤ C

1

0

0

≤ C(1 + sup θ (s) L ∞ )

0

(q+2+β)/2

(6.2.58)

0≤s≤t

or I3 ≤ C

t 0

1 0

1/2 (1 + θ )q+2 v 2 vx2 d x ds

≤ C(1 + sup θ (s) L ∞ )

max(1−r,0)/2

t 0

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )

0

1

1/2 (1 + θ )2m v 2 vx2 d x ds

[2 max(1−r,0)+3β4 ]/4

0≤s≤t

which, together with (6.2.57) and (6.2.58), gives I3 ≤

1 A(t) + C(1 + sup θ (s) L ∞ )β11 . 8 0≤s≤t

(6.2.59)

Similarly, t 1 (T T0 − θ )k(u, θ )θθ x [− p(u, θ )u x + pθ (θ − T0 )u x ]d x ds I4 ≤ C θ 0 0 t 1 (1 + θ )q+r+1 |(T T0 − θ )θθ x u x | + (T T0 − θ )2 (1 + θ )q+r |u x θx | d x ds ≤C θ 0 0 t 1 1 A(t) + C ≤ (1 + θ )2m+1 u 2x d x ds 16 0 0 t 1 (T T0 − θ )2 (1 + θ )q+r θx2 + Cθ u 2x d x ds + θ 16θ 0 0 1 ≤ A(t) + C(1 + sup θ (s) L ∞ )β12 . (6.2.60) 8 0≤s≤t

Therefore, (6.2.43) follows from (6.2.47)–(6.2.60). Corollary 6.2.2. There holds that for any t > 0, t 1 (1 + θ )q+r θx2 d x ds ≤ C(1 + sup θ (s) L ∞ )β13 0

0

with β13 = max(β8 , r ).

0≤s≤t

(6.2.61)


313

Proof. We write

θ = θ − T0 ln θ.

Then

(θ − T0 )θθx θx , θx = (6.2.62) θx + T0 . θx = θ θ Thus it follows from Lemma 6.2.4, Lemma 6.2.10, (6.2.62) and Young’s inequality that t 0

1 0


≤C

≤ C(1 + sup θ (s) L ∞ )β8

t

(1 + θ )q+r θx2 ]d x ds θ2 0 0 t + C(1 + sup θ (s) L ∞ )r V (s)ds

0≤s≤t

1

[(1 + θ )q+r θx2 +

0

0≤s≤t

≤ C(1 + sup θ (s)

L∞

)

β13

0≤s≤t

which implies (6.2.61). Lemma 6.2.11. The following estimates hold for any t > 0: t 0

(1 + θ )2m+1 vx2 d x ds ≤ C(1 + sup θ (s) L ∞ )β14 ,

(6.2.63)


(6.2.64)

0

t 0

1

1

0≤s≤t

0

t 0

0≤s≤t 1

0

(1 + θ )q−r vx4 d x ds ≤ C(1 + sup θ (s) L ∞ )β16 0≤s≤t

where β14 = max(β4 + 1, 2m + 1 + β), β15 = min(β17 , β6 + q + 1), β16 = min[q2 + (3β4 + β5 )/2, max(q − r, 0) + β4 + (β + β5 )/2], β17 = q1 + (5β4 + β5 )/4, q1 = max[(q + 1 − 3r )/4, 0], q2 = max[(q − 3r − 1)/2, 0]. Proof. It is obvious from Lemma 6.2.5 and Lemma 6.2.9 that t 0

and

0

t 0

1

1 0

(1 + θ )2m+1 vx2 d x ds ≤ C(1 + sup θ (s) L ∞ )β4 +1 0≤s≤t

(1 + θ )2m+1 vx2 d x ds ≤ C(1 + sup θ (s) L ∞ )2m+1+β

which result in (6.2.63).

0≤s≤t

(6.2.65)

314


The interpolation inequality and Lemmas 6.2.1–6.2.10 give t 1 (1 + θ )q+1 |vx |3 d x ds 0

0

t

≤ C(1 + sup θ (s) L ∞ )q1

0

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )

q1

0≤s≤t

+ sup vx (s) 0≤s≤t

0

1 0

3

|vx | d x ds +

sup vx (s)

t 0

t 0

0≤s≤t

+ sup vx (s)5/2 0≤s≤t

(1 + θ )3m/2 |vx |3 d x ds

0

t

≤ C(1 + sup θ (s) L ∞ )q1 0≤s≤t

1

t

V 0

0

t

3/4

0

vx (s)2 ds

vx (s)2 ds + sup vx (s)5/2

1

t 0

0≤s≤t 1/4

3

(s)|vx | d x ds

3/44

t 0

vx x (s)2 ds

1/4

1/4

vx x (s)2 ds

vx (s)2 ds

≤ C(1 + sup θ (s) L ∞ ) (1 + sup θ (s) L ∞ )(2β4 +3β+β5 )/4 q1

0≤s≤t

0≤s≤t

+ (1 + sup θ (s) L ∞ )(5β4 +β5 )/4

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )β17

(6.2.66)

0≤s≤t

and t 0

1 0

(1 + θ )q+1 |vx |3 d x ds ≤ C(1 + sup θ (s) L ∞ )q+1 0≤s≤t

t 0

vx (s)3L 3 ds

≤ C(1 + sup θ (s) L ∞ )q+1+β6 0≤s≤t

which, combined with (6.2.66), yields (6.2.64). Similarly, t 1 t 1 (1 + θ )q−r vx4 d x ds ≤ C(1 + sup θ (s) L ∞ )q2 (1 + θ )m vx4 d x ds 0

0

≤ C(1 + sup θ (s) L ∞ ) ≤ C(1 + sup θ (s) 0≤s≤t

t 0

L∞

)

t

q2

0≤s≤t

+

0≤s≤t

0

t

q2

0

vx (s)4L 4 ds

t

+

V 0

1/2

0

(s)vx (s)4L 4 ds

(vx (s)3 vx x(s) + vx (s)4 )ds

0

V 1/2 (s)vx (s)3 vx x (s)ds +

t 0

V 1/2 (s)vx (s)4 ds


315

≤ C(1 + sup θ (s) L ∞ ) 0≤s≤t

t

×

1/2

+ sup vx (s)

0

sup vx (s)

1/2 2

vx (s) ds

0

0≤s≤t

t

2

t

3

vx (s) ds + sup vx (s)

0

0≤s≤t

×

t

vx x (s) ds 2

t

2

2

0

q2

0≤s≤t

1/2

0

1/2 V (s)ds

2

vx x (s) ds

t

+ sup vx (s)3

1/2

0

0≤s≤t

t

V (s)ds

1/2 vx (s)2 ds

0

q 2 ≤ C(1 + sup θ (s) L ∞ ) (1 + sup θ (s) L ∞ )(2β4+β+β5 )/2 0≤s≤t

0≤s≤t

+ (1 + sup θ (s)

L∞

)

(3β4 +β5 )/2

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )q2 +(3β4 +β5 )/2

(6.2.67)

0≤s≤t

or t 0

1 0

(1 + θ )q−r vx4 d x ds ≤ C(1 + sup θ (s) L ∞ )max(q−r,0) 0≤s≤t

0

t

vx (s)4L 4 ds

≤ C(1 + sup θ (s) L ∞ )max(q−r,0)+(2β4+β+β5 )/2 0≤s≤t

with β5 ≥ β4 ≥ 3β, which with (6.2.67) leads to (6.2.65).

Lemma 6.2.12. There holds that for any t > 0, 0

1

(1 + θ )2q θx2 d x +

t 0

1 0

(1 + θ )q+r θt2 d x ds ≤ C(1 + sup θ (s) L ∞ )β18 (6.2.68) 0≤s≤t

where β19 = [max(3q + 2 − r, 0) + β13 + β5 ]/2, β20 = min[β19 , (3q + 4 + β5 )/2], β21 = max[2 max(q − r, 0) + 2β + β13 , max(q − r, 0) + β + (β13 + β14 )/2, max(q − r, 0) + β + (β13 + β16 )/2], β22 = max[max(q − r, 0) + q + 2 + β, 2 max(q − r, 0) + r + 2 + 2β, max(q − r, 0) + β + (r + 2 + β14 )/2, max(q − r, 0) + β + (r + 2 + β16 )/2], β23 = min(β21 , β22 ),

316


β24 = max[(max(q − r, 0) + β7 + β13 )/2, (2β7 + β13 )/3, (2β7 + β13 + β14 )/4, (2β7 + β13 + β16 )/4], β25 = max[(q + 2 + β7 )/2, (2β7 + r + 2)/3, (2β7 + r + 2 + β14 )/4, (2β7 + r + 2 + β16 )/4], β26 = min(β24 , β25 ), β18 = max[β14, β15 , β16 , β20 , β23 , β26 ]. Proof. The proof is similar to that of Lemma 2.1.10 or of Lemma 2.2.10. Let θ H (x, t) = H (u, θ ) = k(u, ξ )dξ, 0

X (t) =

t 0

1 0

(1 + θ )q+r θt2 d x ds, Y (t) =

1 0

(1 + θ )2q θx2 d x.

Then it is easy to verify that Ht = Hu vx + kθt , Hxt = [kθθx ]t + Hu vx x + Huu vx u x + ku u x θt . Multiplying (6.1.3) by Ht and integrating the resultant over Q t results in t 1 t 1 (eθ θt + θ pθ vx − μvx2 )H Ht d x ds + kθθx Ht x d x ds 0 0 0 0 t t (kθθ x Ht )(0, s)ds = 0. − (kθθ x Ht )(1, s)ds + 0

(6.2.69)

0

But we know from (6.1.32)–(6.1.36) that Huu | ≤ C(1 + θ )q+1 . |H Hu | + |H By Lemmas 6.2.1–6.2.10 and Corollary 6.2.2, we can see that t 1 eθ θt Ht d x ds ≥ C0 X (t) − C(1 + sup θ (s) L ∞ )β14 , 0

0

(6.2.70)

(6.2.71)

0≤s≤t

t 1 t 1 2 ≤C (θ p v − μv )H H d x ds [(1 + θ )q+r+2 vx2 + (1 + θ )q+1 |vx |3 θ x t x 0

0

q+r+1

q

0 0 )vx2 |θt |]d x ds

+(1 + θ ) |vx θt | + (1 + θ C0 X (t) + C(1 + sup θ (s) L ∞ )β14 ≤ 8 0≤s≤t

+C(1 + sup θ (s) L ∞ )β15 + C(1 + sup θ (s) L ∞ )β16 , 0≤s≤t

0≤s≤t

(6.2.72)


317

t 0

1

kθθx (kθθ x )t d x ds ≥ CY (t) − C,

(6.2.73)

0

t 1 kθθx (H Hu vx x + Huu vx u x )d x ds 0 0 t 1 [(1 + θ )2q+1 |θθx |(|vx x | + |vx u x |)]d x ds ≤C 0

≤C

0

t 0

1 0

1/2 (1 + θ )4q+2 θx2 d x ds

≤ C(1 + sup θ (s) L ∞ )

1/2 vx x 2 ds

max(3q+2−r,0)/2

0

× (1 + sup θ (s) L ∞ )

β5 /2

1/2 vx u x 2 ds

0 1 0

1/2


t

+ sup u x

0≤s≤t

t

+

0

t

0≤s≤t

≤ C(1 + sup θ (s) L ∞ )

t

0

0≤s≤t

1/2 vx 2L ∞ ds

(max(3q+2−r,0)+β13 +β5 )/2

0≤s≤t

+ C(1 + sup θ (s) L ∞ )(max(3q+2−r,0)+β13 +β7 +β)/2 0≤s≤t

≤ C(1 + sup θ (s) L ∞ )β19

(6.2.74)

0≤s≤t

with β5 ≥ β + β7 and β5 ≥ β4 ≥ 3β. Similarly, t 1 kθθx (H Hu vx x + Huu vx u x )d x ds 0

≤C

0

t 0

1 0

1/2

(1 + θ )4q+2 θx2 d x ds

≤ C(1 + sup θ (s) L ∞ )(3q+4)/2

0≤s≤t

t

1/2 vx x ds

0

t

2

t

+

1/2

1/2 2

vx u x ds

0

V (s)ds

0

× (1 + sup θ (s) L ∞ )β5 /2 + (1 + sup θ (s) L ∞ )(β+β7 )/2 0≤s≤t

≤ C(1 + sup θ (s) L ∞ )

0≤s≤t (3q+4+β5 )/2

0≤s≤t

which with (6.2.74) gives t 1 kθθx (H Hu vx x + Huu vx u x )d x ds ≤ C(1 + sup θ (s) L ∞ )β20 . 0 0 0≤s≤t

(6.2.75)

318


On the other hand, using Lemmas 6.2.1–6.2.11 and the embedding theorem W 1,1 → L∞, we can easily see that t

t 1 kθθx ku u x θt d x ds ≤ C (1 + θ )q |kθθx u x θt |d x ds 0 0 0 0 t 1 C0 X (t) + C ≤ (kθθ x )2 (1 + θ )q−r u 2x d x ds 8 0 0 t C0 max(q−r,0)+β ≤ X (t) + C(1 + sup θ (s) L ∞ ) kθθx 2L ∞ ds 8 0 0≤s≤t t 1 C0 max(q−r,0)+β 2 ∞ X (t) + C(1 + sup θ (s) L ) [kθθx + |kθθx (kθθ x )x |d x]ds ≤ 8 0 0 0≤s≤t C0 X (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+β ≤ 8 0≤s≤t t 1 max(q−r,0) ∞ × (1 + sup θ (s) L ) (1 + θ )q+r θx2 d x ds +

≤

1

0

0

0≤s≤t

t

1 0


0

1/2 t 0

1 0

1/2 (1 + θ )

q−r

2

|(kθθ x )x | d x ds

C0 X (t) + C(1 + sup θ (s) L ∞ )2 max(q−r,0)+β+β13 8 0≤s≤t + C(1 + sup θ (s) L ∞ )max(q−r,0)+β+β13 /2

0≤s≤t

× X (t) + ≤

t 0

1 0

1/2 [(1 + θ )q+r+2 vx2

+ (1 + θ )q−r vx4 ]d x ds

C0 X (t) + C(1 + sup θ (s) L ∞ )2 max(q−r,0)+2β+β13 4 0≤s≤t + C(1 + sup θ (s) L ∞ )max(q−r,0)+β+(β13+β14 )/2 0≤s≤t

+ C(1 + sup θ (s) L ∞ )max(q−r,0)+β+(β13+β16 )/2 0≤s≤t

≤

C0 X (t) + C(1 + sup θ (s) L ∞ )β21 . 4 0≤s≤t

But we also know that t 1 t 1 C0 X (t) + C kθ θ k u θ d x ds ≤ (kθθ x )2 (1 + θ )q−r u 2x d x ds x u x t 8 0 0 0 0 C0 X (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+β ≤ 8 0≤s≤t

(6.2.76)


(1 + θ )q |θθx ||(kθθx )x |d x ds 0 0 0 0 t C0 max(q−r,0)+β q+2 X (t) + C(1 + sup θ (s) L ∞ ) (1 + sup θ (s) L ∞ ) V (s)ds ≤ 8 0 0≤s≤t 0≤s≤t 1/2 1/2 t 1 t 2 q 2 + V (s)ds θ (1 + θ ) |(kθθ x )x | d x ds ×

t

319

1

(1 + θ )2q θx2 d x ds +

0

0

t

1

0

C0 X (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+q+2+β ≤ 8 0≤s≤t

1/2 t 1 + C(1 + sup θ (s) L ∞ )max(q−r,0)+β+(r+2)/2 (1 + θ )q−r |(kθθ x )x |2 d x ds 0

0≤s≤t

0

C0 X (t) + C(1 + sup θ (s) L ∞ )max(q−r,0)+q+2+β ≤ 4 0≤s≤t + C(1 + sup θ (s) L ∞ )2 max(q−r,0)+2β+r+2 0≤s≤t



C0 X (t) + C(1 + sup θ (s) L ∞ )β22 . ≤ 4 0≤s≤t

(6.2.77)

Hence (6.2.76) and (6.2.77) imply t 1 C0 X (t) + C(1 + sup θ (s) L ∞ )β23 kθθx ku u x θt d x ds ≤ 4 0 0 0≤s≤t

(6.2.78)

where β23 = min(β21 , β22 ). For η = 0 or 1, we have from (6.1.1), (6.1.6), (6.1.28) and Lemmas 6.2.1–6.2.11 that T 0 |H Ht ( η, t)| = |(H Hu vx )( η, t)| = (k(u( η, t), ξ ))u dξ vx ( η, t) 0

≤ C|vx ( η, t)| ≤ Cvx L ∞ , t 1/2 1/2 t t 2 2 (kθθ x Ht )( ≤C η , s)ds v (s) ds kθ θ ds ∞ ∞ x x L L 0 0 0 1/2 t

≤ C(1 + sup θ (s) L ∞ )β7 /2 0≤s≤t

1

kθθx 2 +

0

|kθθx (kθθ x )x |d x ds

0

≤ C(1 + sup θ (s) L ∞ )β7 /2 0≤s≤t

t 1 × (1 + sup θ (s) L ∞ )max(q−r,0) (1 + θ )q+r θx2 d x ds 0≤s≤t

0

0

320


+

t 0

1 0


1/2 t 0

1 0

1/2 1/2 (1 + θ )

q−r

2

|(kθθ x )x | d x ds

≤ C(1 + sup θ (s) L ∞ )(max(q−r,0)+β7 +β13 )/2 + C(1 + sup θ (s) L ∞ )(2β7 +β13 )/4

0≤s≤t

0≤s≤t

1/4

× X (t) + (1 + sup θ (s) L ∞ )β14 + (1 + sup θ (s) L ∞ )β16 0≤s≤t

0≤s≤t

C0 X (t) + C(1 + sup θ (s) L ∞ )[max(q−r,0)+β7 +β13 ]/2 ≤ 8 0≤s≤t + C(1 + sup θ (s) L ∞ )(2β7 +β13 )/3 + C(1 + sup θ (s) L ∞ )(2β7 +β13 +β14 )/4 0≤s≤t

0≤s≤t

+ C(1 + sup θ (s) L ∞ )(2β7 +β13 +β16 )/4 0≤s≤t

C0 X (t) + C(1 + sup θ (s) L ∞ )β24 . ≤ 8 0≤s≤t Similarly, t (kθθ x Ht )( η, s)ds 0

≤ C(1 + sup θ (s) L ∞ ) 0≤s≤t

t

+

V (s)ds

0

β7 /2

t q+2 (1 + sup θ (s) L ∞ ) V (s)ds 0

0≤s≤t

1/2 t 0

(6.2.79)

1

1/2 1/2

θ 2 (1 + θ )q |(kθθ x )x |2 d x ds

0

≤ C(1 + sup θ (s) L ∞ )(β7 +q+2)/2 + C(1 + sup θ (s) L ∞ )(2β7 +r+2)/4 0≤s≤t

×

t 0

1

1/4

0≤s≤t

(1 + θ )q−r |(kθθ x )x |2 d x ds

0

≤ C(1 + sup θ (s) L ∞ )(β7 +q+2)/2 + C(1 + sup θ (s) L ∞ )(2β7 +r+2)/4

0≤s≤t

0≤s≤t

× X (t) + (1 + sup θ (s) L ∞ )

β14

0≤s≤t

≤

1/4

+ (1 + sup θ (s) L ∞ )

β16

0≤s≤t

C0 X (t) + C(1 + sup θ (s) L ∞ )(β7 +q+2)/2 + C(1 + sup θ (s) L ∞ )(2β7 +r+2)/3 8 0≤s≤t 0≤s≤t + C(1 + sup θ (s) L ∞ )(2β7+r+2+β14 )/4 + C(1 + sup θ (s) L ∞ )(2β7 +r+2+β16 )/4 0≤s≤t

C0 X (t) + C(1 + sup θ (s) L ∞ )β25 . ≤ 8 0≤s≤t

0≤s≤t

(6.2.80)


321

Thus (6.2.79) and (6.2.80) give t C0 (kθθ x Ht ( X (t) + C(1 + sup θ (s) L ∞ )β26 . η, s)ds ≤ 8 0 0≤s≤t

(6.2.81)

Therefore it follows from (6.2.69), (6.2.71)–(6.2.73), (6.2.75), (6.2.78) and (6.2.81) that X (t) + Y (t) ≤ C(1 + sup θ (s) L ∞ )β14 + C(1 + sup θ (s) L ∞ )β15 0≤s≤t

0≤s≤t

+ C(1 + sup θ (s) L ∞ )

β16

+ C(1 + sup θ (s) L ∞ )β20

0≤s≤t

0≤s≤t

+ C(1 + sup θ (s) L ∞ )

β23

+ C(1 + sup θ (s) L ∞ )β26

0≤s≤t

0≤s≤t

which gives (6.2.68). Lemma 6.2.13. The following estimates hold for any t > 0:

1 0

(θθx2 + u 2x + vx2 )(x, t)d x +

t 0

θ (t)| L ∞ ≤ C, 1 0

(6.2.82)

[u 2x + θx2 + θt2 + vt2 + vx2 + vx2 x ](x, s)d x ds ≤ C. (6.2.83)

Proof. Note that after a lengthy calculation, (6.1.21)–(6.1.23) imply β18 < 2q + r + 3. Similarly to (2.1.103), we deduce 2q+r+3

θ (t) L ∞

≤ C + CY (t) ≤ C(1 + sup θ (s) L ∞ )β18 0≤s≤t

1 2q+r+3 ≤ sup θ (s) L ∞ +C 2 0≤s≤t which gives (6.2.82). (6.2.83) is the direct result of Lemmas 6.2.4–6.2.12.

Remark 6.2.3. Similar to the proof of (2.1.104), we can deduce that 2q+2r+4

θ (t) L ∞

≤ C(1 + sup θ (s) L ∞ )β8 +β18 .

(6.2.84)

0≤s≤t

However, it is easy to check that the ranges of q obtained from (6.2.84) are smaller than those from (6.1.21)–(6.1.23). In what follows, we show the results on asymptotic behavior. Lemma 6.2.14. The following estimates hold for any t > 0, t ( p2 + σ 2 )(s)ds ≤ C,

(6.2.85)

0

d p2 ≤ C( p2 + θt 2 + 1), dt d v2 ≤ C(vt 2 + 1). dt

(6.2.86) (6.2.87)

322


Proof. Integrating (6.1.2) on [0, x] and noting (6.1.6) gives x vt d y − γ v(0, t). p(u, θ ) = μ(u)vx − 0

Hence from Lemmas 6.2.1–6.2.13, we have t t 2 p ds ≤ C (vx 2 + vt 2 + γ v 2 (0, s))ds ≤ C 0

which implies

0

t

t

2

σ ds ≤ C

0

( p2 + vx 2 )ds ≤ C.

0

Clearly, we have d p2 ≤ C( p2 + pt 2 ) ≤ C( p2 + vx 2 + θt 2 ) dt ≤ C(1 + p2 + θt 2 ). The proof of (6.2.87) is the same as that of (6.2.86).

Lemma 6.2.15. The following estimates are valid for any t > 0:

d u x 2 ≤ vx x 2 + u x 2 , dt

1 d θθx 2 + C9 (1 + θ )q−r θx2x d x ≤ C(vx x 2 + 1), dt 0 t 1 (1 + θ )q−r θx2x d x ds ≤ C. θθx 2 + 0

(6.2.88) (6.2.89) (6.2.90)

0

Proof. See, e.g., Lemma 2.1.13.

It is the most difficult to prove vx → 0 as t → ∞, thus the following lemma plays a very important role in proving it. Lemma 6.2.16. The following estimates hold, t Z (s)ds ≤ C, ∀t > 0,

(6.2.91)

0

d Z(t) ≤ C(vx x 2 + θt 2 + vt 2 + 1) dt

(6.2.92)

where 1 Z (t) : = 2

1 0

μ(u)vx2 d x +

γ 2 (v (0, t) + v 2 (1, t)) + p(u(0, t), T0 )v(0, t) 2

− p(u(1, t), T0 )v(1, t)) + C10 ( p2 + θθx 2 + u x 2 + v2 )

(6.2.93)


323

with a sufficiently large positive constant C10 such that γ Z(t) ≥ C(vx 2 + v2 + θθx 2 + u x 2 ) + (v 2 (0, t) + v 2 (1, t)). 2

(6.2.94)

Proof. Denote w(t) =

1 2

γ 2 (v (0, t) + v 2 (1, t)) 2 0 + p(u(0, t), T0 )v(0, t) − p(u(1, t), T0 )v(1, t). 1

μ(u)vx2 d x +

(6.2.95)

Thus multiplying (6.1.2) by vt , integrating the resultant over [0, 1], integrating by parts and noting (6.2.33)–(6.2.34), gives vt (t)2 + w (t) 1 1 1 3 μ (u)vx d x − px vt d x − pt (u(0, t), T0 )v(0, t) + pt (u(1, t), T0 )v(1, t) = 2 0 0 1 ≤ C(vx 3L 3 + u x 2 + θθx 2 ) + vt 2 2 + | pu (u(0, t), T0 )vx (0, t)v(0, t)| + | pu (u(1, t), T0 )vx (1, t)v(1, t)| 1 ≤ C(vx (t)3L 3 + u x (t)2 + θθx (t)2 + vx (t)2L ∞ + v(t)2L ∞ ) + vt (t)2 , 2 i.e., 1 w (t)+ vt 2 ≤ C(vx (t)3L 3 +u x (t)2 +θθx (t)2 +vx (t)2L ∞ +v(t)2L ∞ ). (6.2.96) 2 But due to (6.1.14) and using the mean value theorem, we have p(u, θ ) = pu (u 1 , θ1 )(u − η0 ) + pθ (u 1 , θ1 )(θ − T0 )

(6.2.97)

where min(u, η0 ) ≤ u 1 ≤ max(u, η0 ) and min(θ, T0 ) ≤ θ1 ≤ max(θ, T0 ). Hence by the mean value theorem and (6.1.25), we get |u − η0 | ≤ C(| p| + |θ − T0 |), u − η0 ≤ C( p + θ − T0 ) ≤ C( p + θθx ).

(6.2.98) (6.2.99)

By the Nirenberg inequality and (6.2.98)–(6.2.99), we have p L ∞ ≤ C(u − η0 L ∞ + θ − T0 L ∞ ) ≤ C(u − η0 1/2 u x 1/2 + u − η0 + θθ x ) ≤ C{( p1/2 + θθx 1/2 )u x 1/2 + p + θθ x } ≤ C( p + θθx + u x ) and

v L ∞ ≤ C(v1/2 vx 1/2 + v) ≤ C(v + vx ).

(6.2.100) (6.2.101)

324


Thus, p L ∞ v L ∞ ≤ C( p + θθx + u x )(v + vx ) ≤ vx 2 + C(v2 + p2 + u x 2 + θθx 2 ).

(6.2.102)

Clearly, we have γ 2 (v (0, t) + v 2 (1, t)) − 2 p L ∞ v L ∞ 2 γ ≥ (C11 − 2)vx 2 + (v 2 (0, t) + v 2 (1, t)) 2 2 −C12 ( p + v2 + u x 2 + θθx 2 ).

w(t) ≥ C11 vx 2 +

(6.2.103)

Taking > 0 sufficiently small in (6.2.103) (2 < C11 ) and taking C10 > 0 sufficiently large in (6.2.93) (C10 > C12 ), gives (6.2.94). On the other hand, it is easy to get from (6.2.93) Z (t) ≤ C(vx 2 + p2 + θθx 2 + u x 2 + p2L∞ + v2L∞ ) γ (6.2.104) + (v 2 (0, t) + v 2 (1, t)) 2 γ ≤ C(vx 2 + p2 + θθx 2 + u x 2 + v2L ∞ ) + (v 2 (0, t) + v 2 (1, t)) 2 which implies (6.2.91) by Lemmas 6.2.1–6.2.13. From (6.2.93), (6.2.96) and Lemmas 6.2.14–6.2.15, we easily deduce d d Z (t) = w (t) + C10 ( p2 + θθx 2 + u x 2 + v2 ) dt dt ≤ C(vx 3L 3 + u x 2 + θθx 2 + vx 2L ∞ + v2L ∞ + p2 + θt 2 + vx x 2 + vt 2 + 1) ≤ C(vx 5/2 vx x 1/2 + vx 3 + u x 2 + θθx 2 + vx vx x + vx 2 + vvx + v2 + p2 + θt 2 + vx x 2 + vt 2 + 1) ≤ C(vx x 2 + θt 2 + vt 2 + 1).

Lemma 6.2.17. As t → +∞, we have Z (t) → 0,

(6.2.105)

u − η0 H 1 → 0, u x → 0, u − η0 L ∞ → 0, v H 1 → 0, vx → 0,

(6.2.106) (6.2.107)

θ − T0 H 1 → 0, θθx → 0, θ − T0 L ∞ → 0,

(6.2.108)

2

2

p → 0, γ (v (0, t) + v (1, t)) → 0, p(u, θ ) H 1 → 0, σ (u, θ ) → 0

(6.2.109) (6.2.110)

6.3. Exponential Stability and Maximal Attractors

325

where (η0 , 0, T0 ) is the unique solution to the corresponding stationary problem to (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6). Moreover, there exist positive constants t0 , C1 , C1 such that for all t ≥ t0 , there holds v(t) H 1 + θ (t) − T0 H 1 + u(t) − η0 H 1 ≤ C1 exp(−C1 t).

(6.2.111)

Proof. Estimate (6.2.105) is the direct consequence by applying Theorem 1.2.4 and Lemma 6.2.16. It is obvious that (6.2.106)–(6.2.110) are the consequence of (6.2.105). It is easy to verify that (η0 , 0, T0 ) is the unique solution to the corresponding stationary problem to (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6). Now since (u(t) − η0 , v(t), θ (t) − T0 ) can be small in the H 1 norm for sufficiently large t, we can deduce the desired estimate (6.2.111) by the same method as that in Section 2.3. Proofs of Theorem 6.1.1 and Theorem 6.2.1 By the standard argument (see, e.g., Theorems 2.1.1–2.1.2) and using Lemmas 6.2.1–6.2.17, we complete the proofs of Theorem 6.1.1 and Theorem 6.2.1. Remark 6.2.4. It follows from the proofs of Lemmas 6.2.1–6.2.17 and Theorem 6.1.1 that all the constants in Lemmas 6.2.1–6.2.17 depend only on the H 1 norm of the initial data (u 0 , v0 , θ0 ). Thus the following results of global existence, uniqueness and the same results of the asymptotic behavior as Theorem 6.1.1 hold. Corollary 6.2.3. If (u 0 , v0 , θ0 ) ∈ H 1 × H 1 × H 1, then problem (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6) admits a unique generalized solution (u(t), v(t), θ (t)) in the sense that u ∈ L ∞ (0, +∞; H 1), u t ∈ L ∞ (0, +∞; L 2 ), (v, θ ) ∈ L ∞ (0, +∞; H 1) ∩ L 2 (0, +∞; H 2) ∩ H 1(0, +∞; L 2 ). Moreover, (6.2.105)–(6.2.111) hold.

6.3 Exponential Stability and Maximal Attractors In this section we prove global existence, exponential stability of solutions and existence of maximal attractors in H i (i = 1, 2, 4) for problem (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6). We assume that e(u, θ ), p(u, θ ), μ(u), σ (u, θ, vx ) and k(u, θ ) are sufficiently smooth functions on 0 < u < +∞ and 0 ≤ θ < +∞ with the constitutive relations (6.1.11)–(6.1.13). Furthermore, we consider a kind of solid-like materials, so we require that p(u, θ ) be compressive for small u and tensile for large u, at any temperature, i.e., there are < ∞ such that (6.1.12)–(6.1.13) hold. 0 0, θ (x) > 0, x ∈ [0, 1],

1 0

v(x)d x = 0 if γ = 0 i n (6.1.6), θ |x=0 = θ |x=1

= T0 > 0 ,

H+2 = (u, v, θ ) ∈ H 2[0, 1] × H 2[0, 1] × H 2[0, 1] : u(x) > 0, θ (x) > 0, x ∈ [0, 1],

1 0

v(x)d x = 0 if γ = 0 i n (6.1.6), θ |x=0 = θ |x=1 = T0 ,

(− p(u, θ ) + μ(u)vx )|x=0 = γ v|x=0 , (− p(u, θ ) + μ(u)vx )|x=1 = −γ v|x=1 and H+4 = (u, v, θ ) ∈ H 4[0, 1] × H 4[0, 1] × H 4[0, 1] : u(x) > 0, θ (x) > 0, x ∈ [0, 1],

1

v(x)d x = 0 if γ = 0 i n (6.1.6), θ |x=0 = θ |x=1 = T0 ,

0

(− p(u, θ ) + μ(u)vx )|x=0 = γ v|x=0 , (− p(u, θ ) + μ(u)vx )|x=1 = −γ v|x=1


327

which become three metric spaces when equipped with the metrics induced from the usual norms. In the above, H 1, H 2, H 4 are the usual Sobolev spaces. Let 1 Hδi := (u, v, θ ) ∈ H+i : (E(u, θ ) + v 2 /2)d x ≤ δ1 , δ2 ≤ u ≤ δ3 , δ4 ≤ θ ≤ δ5 , , 0

i = 1, 2, 4 where E(u, θ ) =: (u, θ ) − (η0 , T0 ) − θ (u, θ )(θ − T0 )

(6.3.1)

where δi (i = 1, . . . , 5) are any given constants satisfying 0 < δ1 , 0 < δ2 < η0 < δ3 , 0 < δ4 < T0 < δ5 .

(6.3.2)

Obviously, Hδi (i = 1, 2, 4) is a sequence of closed subspaces of H+i (i = 1, 2, 4). We shall see later on that the first constraint is invariant, while the last two constraints are not invariant. These two constraints are just introduced to overcome the difficulty that the original spaces H+i (i = 1, 2, 4) are incomplete. It should be pointed out that it is very crucial to prove that the orbit starting from any bounded set of Hδi will re-enter Hδi and stay there after a finite time. We use Ci (i = 1, 2, 4) to denote the universal positive constant depending only on the H i norm of initial data. Cδ (sometimes Cδ ) stands for the universal positive constant depending only on δi (i = 1, . . . , 5), but independent of initial data. C B˜ i ,δ denotes the universal positive constant depending on δ j ( j = 1, . . . , 5), H i norm of initial data (u 0 , v0 , θ0 ) with (u 0 , v0 , θ0 ) H i ≤ B˜ i (i = 1, 2, 4). Without danger of confusion we use the same symbol to denote the state functions as well as their values along a thermodynamic process, e.g., p(u, θ ), and p(u(x, t), θ (x, t)) and p(x, t). Similarly to the proofs of those of Theorems 2.3.1–2.3.2 and Theorem 2.4.1, we can prove the following Theorems 6.3.1–6.3.3. Theorem 6.3.1. Assume that e, p, σ and k are C 2 functions on 0 < u < +∞ and 0 ≤ θ < +∞, and the assumptions (6.1.11)–(6.1.13), (6.1.15)–(6.1.16), (6.1.21)–(6.1.29) hold. If (u 0 , v0 , θ0 ) ∈ H+1 , then the unique global weak solution (u(t), v(t), θ (t)) ∈ H+1 to the problem (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6) defines a nonlinear C0 -semigroup S(t) on H+1 mapping H+1 into itself and satisfying that for any (u 0 , v0 , θ0 ) ∈ H+1 , S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)) ∈ C([0, +∞); H+1 )

(6.3.3)

and being continuous with respect to initial data, i.e., S(t)(u 01 , v01 , θ01 ) − S(t)(u 02 , v02 , θ02 ) H 1 ≤ C1 (u 01 , v01 , θ01 ) − (u 02 , v02 , θ02 ) H 1 +

+

(6.3.4)

328


where (u j (t), v j (t), θ j (t)) ( j = 1, 2) is the unique global solution with initial datum (u 0 j , v0 j , θ0 j ) ∈ H+1 ( j = 1, 2). Moreover, for any (u 0 , v0 , θ0 ) ∈ H+1 , there exist constant C1 > 0 and γ1 = γ1 (C1 ) > 0 such that for any fixed γ ∈ [0, γ1 ] and for any t > 0, the following inequality holds: eγ t (u(t) − η0 2H 1 + v(t)2H 1 + θ (t) − T0 2H 1 + u t (t)2 ) +

t 0

eγ τ (u − η0 2H 1

+v2H 2 + θ − T0 2H 2 + u t 2H 1 + vt 2 + θt 2 )(τ )dτ ≤ C1

(6.3.5)

which implies that the semigroup S(t) is exponentially stable on H+1 for any fixed γ ∈ (0, γ1 ]. Remark 6.3.1. The estimate (6.3.5) implies the corresponding estimate of exponential decay of solution in Theorem 6.1.1 (see also (6.1.30)) which holds only for a large time. Theorem 6.3.2. Assume that e, p, σ and k are C 3 functions on 0 < u < +∞ and 0 ≤ θ < +∞, and the assumptions (6.1.11)–(6.1.13), (6.1.15)–(6.1.16) and (6.1.21)– (6.1.29) hold. If (u 0 , v0 , θ0 ) ∈ H+2 , then there exists a unique global weak solution (u(t), v(t), θ (t)) ∈ H+2 to the problem (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6) which defines a nonlinear C0 -semigroup S(t) (also denoted by S(t) by the uniqueness of solution in H+1 ) on H+2 mapping H+2 into itself and satisfying that for any (u 0 , v0 , θ0 ) ∈ H+2 , S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)) ∈ C([0, +∞); H+2 )

(6.3.6)

and being continuous with respect to initial data, i.e., S(t)(u 01 , v01 , θ01 ) − S(t)(u 02 , v02 , θ02 ) H 2 ≤ C2 (u 01 , v01 , θ01 ) − (u 02 , v02 , θ02 ) H 2 + + (6.3.7) where (u j (t), v j (t), θ j (t)) ( j = 1, 2) is the unique global solution with initial datum (u 0 j , v0 j , θ0 j ) ∈ H+2 ( j = 1, 2). Moreover, for any (u 0 , v0 , θ0 ) ∈ H+2 , there exist constants C2 > 0 and 0 < γ2 = γ2 (C2 ) ≤ γ1 (C1 ) such that for any fixed γ ∈ [0, γ2 ] and for any t > 0, the following inequality holds: eγ t (u(t) − η0 2H 2 + v(t)2H 2 + θ (t) − T0 2H 2 + u t (t)2H 1 ) +

t 0

+v2H 3 + θ − T0 2H 3 + u t 2H 2 + vt 2H 1 + θt 2H 1 )(τ )dτ ≤ C2

eγ τ (u − η0 2H 2 (6.3.8)

which implies that the semigroup S(t) is exponentially stable on H+2 for any fixed γ ∈ (0, γ2 ]. Theorem 6.3.3. Assume that e, p, σ and k are C 5 functions on 0 < u < +∞ and 0 ≤ θ < +∞, and assumptions (6.1.11)–(6.1.13), (6.1.15)–(6.1.16) and (6.1.21)–(6.1.29) hold. If (u 0 , v0 , θ0 ) ∈ H+4 , then there exists a unique global solution (u(t), v(t), θ (t)) ∈ C([0, +∞); H+4 ) to problem (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6) which defines a non-


329

linear C0 -semigroup S(t) on H+4 mapping H+4 into itself and satisfying that, for any (u 0 , v0 , θ0 ) ∈ H+4 , S(t)(u 0 , v0 , θ0 ) = (u(t), v(t), θ (t)) ∈ C([0, +∞); H+4 )

(6.3.9)

and being continuous with respect to initial data, i.e., S(t)(u 01 , v01 , θ01 ) − S(t)(u 02 , v02 , θ02 ) H 4 ≤ C4 (u 01 , v01 , θ01 ) − (u 02 , v02 , θ02 ) H 4 + + (6.3.10) where (u j (t), v j (t), θ j (t)) ( j = 1, 2) is the unique global solution with initial datum (u 0 j , v0 j , θ0 j ) ∈ H+4 ( j = 1, 2). Moreover, for any (u 0 , v0 , θ0 ) ∈ H+4 , there are constants C4 > 0 and 0 < γ4 = γ4 (C C4 ) ≤ γ2 such that for any fixed γ ∈ [0, γ4 ], the following estimates hold for any t > 0: eγ t (u(t) − η0 2H 4 + u t (t)2H 3 + u t t (t)2H 1 + v(t)2H 4 + vt (t)2H 2 + vt t (t)2 + θ (t) − T0 2H 4 + θt (t)2H 2 + θt t (t)2 ) t eγ τ (u − η0 2H 4 + u t 2H 4 + u t t 2H 2 + u t t t 2 + v2H 5 + 0

+ vt 2H 3 + vt t 2H 1 + θ − T0 2H 5 + θt 2H 3 + θt t 2H 1 )(τ )dτ ≤ C4

(6.3.11)

which implies that the semigroup S(t) is exponentially stable on H+4 for any fixed γ ∈ (0, γ4 ]. Corollary 6.3.1. Under assumptions of Theorem 6.3.3 and if corresponding compatibility conditions hold, the global solution (u(t), v(t), θ (t)) obtained in Theorem 6.3.3 is the classical solution verifying that for any fixed γ ∈ (0, γ4 ] and for any t > 0, (u(t) − η0 , v(t), θ (t) − T0 )2C 3+1/2 ×C 3+1/2 ×C 3+1/2 ≤ C4 e−γ t .

(6.3.12)

Theorem 6.3.4. Assume that e, p, σ and k are C i+1 (i = 1, 2, 4) functions on 0 < u < +∞ and 0 ≤ θ < +∞, and assumptions (6.1.11)–(6.1.13), (6.1.15)–(6.1.16), and (6.1.21)–(6.1.29) hold. Then semigroup S(t) defined on H+i (i = 1, 2, 4) by the solution to problem (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6) maps H+i (i = 1, 2, 4) into itself. Moreover, for any δi (i = 1, . . . , 5) satisfying (6.3.2), it possesses in Hδi (i = 1, 2, 4) a universal (maximal) attractor Ai,δ (i = 1, 2, 4). Corollary 6.3.2. Under assumptions in Hsiao and Luo [159], i.e., (6.7.3)–(6.7.7) (in Section 6.7) and f (u) is strictly monotone decreasing in u, namely, f (u) < 0, ∀u ∈ [u, U ]

(6.3.13)

where u and U are positive constants depending only on the initial data, but not on any length of time, and 0 < u ≤ u˜ ≤ U˜ ≤ U , the same conclusions as in Theorems 6.1.1–6.1.3 and Corollary 6.3.1 also hold for the model in Hsiao and Luo [159] with the boundary conditions of stress-free and thermally insulated endpoints, where η0 is the 1 unique root of f (u) = 0 and T0 should be replaced by θ¯ ≡ C1V 0 (C V θ0 + 12 v02 )(x)d x.

330


Corollary 6.3.3. Under assumptions in Hsiao and Jian [157], i.e., (6.7.3) and (6.7.6)– (6.7.9), the same conclusions as in Theorems 6.3.1–6.3.3 and Corollary 6.3.1 also hold where η0 is the unique root of f (u) = 0 for the model in Hsiao and Jian [157] with the boundary condition (6.1.6). Remark 6.3.2. The corresponding estimate (6.3.12) obtained in Corollaries 6.3.2–6.3.3 for the models in Hsiao and Luo [159], Hsiao and Jian [157] (where T0 is replaced by ¯ was obtained for sufficiently large time (see, e.g., Theorem 1.5 in Hsiao and Luo [159] θ) and Theorem 1.4 in Hsiao and Jian [157]). Thus this estimate has improved those in [157, 159]. Remark 6.3.3. The corresponding estimate (6.3.12) in Corollary 6.3.1 for the models in [157, 159] was obtained in Greenberg and MacCamy [129] for the case of isothermal viscoelasticity (i.e., θ ≡ constant). Thus our results have extended the case of isothermal viscoelasticity to the non-isothermal case – thermoviscoelastic materials. Remark 6.3.4. It is easy to verify that our approaches in the proofs of Theorems 6.3.1– 6.3.3 also apply to all boundary conditions involving pinned or stress-free endpoints which are either held at constant temperature or insulated. For these boundary conditions, if we could establish uniform estimates (i.e., these estimates depend only on the initial and boundary data, but independent of any length of time) in H 1 similar to those in Theorem 6.3.1 with γ = 0, then we readily obtain similar results to those in Theorems 6.3.1–6.3.3 and Corollary 6.3.1. Otherwise, if we only derive estimates in H 1 depending on the length of time and similar to those in Theorems 6.3.1 with γ = 0, we only obtain the corresponding global existence results in Theorems 6.3.2–6.3.3 where all constants depend on the length of any given time, but no large time behavior solutions can be obtained. But it is noteworthy that the strict monotonicity of the pressure p(u, θ ) in u (the deformation gradient) should be assumed, which is very helpful for deriving estimates of u in H i (i = 1, 2, 4). Remark 6.3.5. Similar conclusions to those in Theorem 6.3.4 also hold for the results for the models in Hsiao and Jian [157] and Hsiao and Luo [159], but for the boundary conditions of the model in Hsiao and Luo [159], we have to modify appropriately the definitions of H+i and Hδi (i = 1, 2, 4) as follows: 1 H+ = {(u, v, θ ) ∈ H 1[0, 1] × H 1[0, 1] × H 1[0, 1] : u(x) > 0, θ (x) > 0, x ∈ [0, 1],

H+2

1

v(x)d x = 0 ,

0 = (u, v, θ ) ∈ H 2[0, 1] × H 2[0, 1] × H 2[0, 1] : u(x) > 0, θ (x) > 0, x ∈ [0, 1],

1

v(x)d x = 0 θx |x=0 = θx |x=1 = 0,

0

(− f (u)θ + μ(u)vx )|x=0 = (− f (u)θ + μ(u)vx )|x=1 = 0


331

and 4 H+ = (u, v, θ ) ∈ H 4[0, 1] × H 4[0, 1] × H 4[0, 1] : u(x) > 0, θ (x) > 0, x ∈ [0, 1],

1

v(x)d x = 0, θx |x=0 = θx |x=1 = 0,

0

(− f (u)θ + μ(u)vx )|x=0 = (− f (u)θ + μ(u)vx )|x=1 = 0 which become three metric spaces when equipped i with the metrics induced from the usual norms. In the above, H 1, H 2, H 4 are the usual Sobolev spaces. Let 1 Hδi := (u, v, θ ) ∈ H+i : δ0 ≤ (C V θ + v 2 /2)d x ≤ δ1 , δ2 ≤ u ≤ δ3 , δ4 ≤ θ ≤ δ5 , , 0

i = 1, 2, 4 where δ0 , δ1 , . . . , δ5 > 0 are some parameters satisfying 0 < δ2 < η0 < δ3 , 0 < δ4 < δ0 /C V < δ1 /C V < δ5 . Remark 6.3.6. The set Ai = Ai,δ is a global noncompact attractor in the metδ1 ,...,δ5

ric space H+i (i = 1, 2, 4) in the sense that it attracts any bounded sets of H+i with constraints u ≥ u ∗ , θ ≥ θ ∗ with u ∗ , θ ∗ being any given positive constants.

6.4 Exponential Stability in H 1 and H 2 In this section, we complete the proofs of Theorems 6.3.1–6.3.2. Lemma 6.4.1. The unique global weak solution (u(t), v(t), θ (t)) in H+1 defines a nonlinear C0 -semigroup S(t) on H+1 . Moreover, for any (u 0 , v0 , θ0 ) ∈ H+1 , the global weak solution (u(t), v(t), θ (t)) to the problem (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6) satisfies (u(t), v(t), θ (t)) = S(t)(u 0 , v0 , θ0 ) ∈ C([0, +∞), H+1 ),

(6.4.1)

u(t) ∈ C 1/2 ([0, +∞), H 1), v(t), θ (t) ∈ C 1/2 ([0, +∞), L 2 ).

(6.4.2)

Proof. See, e.g., Lemma 2.3.2.

Lemma 6.4.2. The following estimate holds for the global weak solution (u(t),v(t),θ (t)) in H+1 obtained in Lemma 6.4.1, 0 < C1−1 ≤ θ (x, t), Proof. See, e.g., Lemma 2.3.3.

∀(x, t) ∈ [0, 1] × [0, +∞).

(6.4.3)

332


Proof of Theorem 6.3.1. We begin with Lemmas 6.4.1–6.4.2 and repeat the same process as the proofs of Theorem 2.3.1 to be able to complete the proof of Theorem 6.3.1. Proof of Theorem 6.3.2. Employing Theorem 6.3.1 and repeating the same argument as the proof of Theorem 6.3.2, we can prove the conclusions of Theorem 6.3.2.

6.5 Exponential Stability in H 4 In this section, we complete the proof of Theorem 6.3.3. Proof of Theorem 6.3.3. Based on Theorems 6.3.1–6.3.2, we follow the proof of Theorem 2.4.1 to finish the proof of Theorem 6.3.3.

6.6 Universal Attractors in H i (i = 1, 2, 4) In this section, we shall complete the proof of Theorem 6.3.4.

6.6.1 Existence of An Absorbing Set in Hδ1 In this subsection we establish the existence of an absorbing set in Hδ1. The situation is different from those encountered in the treatment of a viscous polytropic ideal gas (see, e.g., [445, 446]) and a viscous heat conductive real gas (see, e.g., Qin and Muñ˜ oz Rivera [337]; see also Chapter 2). Throughout this subsection we always suppose that the initial datum belongs to a bounded set B1 of Hδ1, i.e., (u 0 , v0 , θ0 ) H 1 ≤ B˜ 1 , B˜ 1 being some +

positive constant. First we have to show the orbit starting from B1 will re-enter Hδ1 and stay there after a finite time, which should be uniform with respect to all orbits starting from B1 . Lemma 6.6.1. If (u 0 , v0 , θ0 ) ∈ Hδ1, then the following estimates hold for any t > 0: t 1 1 μvx2 k(u, θ )θθ x2 E(u, θ ) + v 2 (x, t)d x + T0 (x, τ )d x dτ + 2 θ θ2 0 0 0 t 1 1 2 2 2 +γ E(u 0 , θ0 ) + v0 (x)d x ≤ δ1 , (6.6.1) [v (0, τ ) + v (1, τ )]dτ = 2 0 0 1 1 r ωγ ν [1 + min (θ, T0 )] (θ − T0 )2 (x, t)d x + (u − η0 )2 (x, t)d x 2 0 θ + T0 2 0 1 1 E(u 0 , θ0 ) + v02 (x)d x ≤ δ1 , (6.6.2) ≤ 2 0 u δ ≤ u(x, t) ≤ Uδ , (x, t) ∈ [0, 1] × [0, +∞), (6.6.3)

1

6.6. Universal Attractors in H i (i = 1, 2, 4)

1 0

333

[θ/T T0 − log(θ/T T0 ) − 1 + θ r+1 ](x, t)d x t 1

vx2 (1 + θ q )θθx2 + (x, τ )d x dτ ≤ Cδ + θ θ2 0 0

(6.6.4)

where u δ : ≡ M −1 ( min M(min{ ˜, (1 − λ)η0 } + λδ2 ) − 2E δ

− 1),

(6.6.5)

Uδ : ≡ M −1 ( max M(max{ ˜, (1 − λ)η0 } + λδ3 ) + 2E δ

+ 1),

(6.6.6)

1/2

λ∈[0,1]

1/2

λ∈[0,1]

E δ : ≡ (1 + 2γ 2 / p0 )δ1 + γ 2 η02

(6.6.7)

and ωγ = 0 if γ = 0 in (6.1.6) or ωγ = 1 if γ = 1 in (6.1.6). Proof. Integrating (6.2.4) over [0, 1] × [0, +∞) and using (6.1.6) lead to (6.6.1). Recalling that (u 0 , v0 , θ0 ) ∈ Hδ1 and uu (u, θ ) = − pu (u, θ ) > 0, similarly to (6.2.6), we have E(u, θ ) − (u, T0 ) + (η0 , T0 ) ≥

ν[1 + minr (θ, T0 )] (θ − T0 )2 2(θ + T0 )

and (u, T0 ) − (η0 , T0 ) ≥

(6.6.8)

ωγ (u − η0 )2 2

which with (6.6.8) gives E(u, θ ) ≥

ωγ ν[1 + minr (θ, T0 )] (θ − T0 )2 + (u − η0 )2 . 2(θ + T0 ) 2

(6.6.9)

Therefore (6.6.2) follows from (6.6.1) and (6.6.9). Similarly to the proof of Lemma 6.2.3, we have (6.2.12). Similarly to (6.2.13), we can derive from (6.6.1)–(6.6.2), y 1 1 γ 1 (v(x, τ ) − v(x, s))d x (u(x, τ ) − u(x, s))d x + − 2 0 2 0 y

1 1 1

≤ γ max

t ∈[0,τ ]

≤

γ (2η02

2

u 2 (x, t)d x

0

+ 4δ1 / p0 )

1/2

1

+ max

+ (2δ1 )

t ∈[0,τ ]

2

v 2 (x, t)d x

0

1/2 1/2

≤ 2[(1 + 2γ 2 / p0 )δ1 + γ 2 η02 ]1/2 ≡ 2E δ .

(6.6.10)

Note that (u 0 , v0 , θ0 ) ∈ Hδ1 and (6.6.5)–(6.6.6) imply u δ < δ2 ≤ u 0 (x) ≤ δ3 < Uδ , x ∈ [0, 1].

(6.6.11)

334


Therefore, if u δ < u(x, t) < Uδ is violated on [0, 1] × [0, +∞), then there exist τ > 0 and y ∈ [0, 1] such that u δ < u(x, t) < Uδ f or x ∈ [0, 1], 0 ≤ t < τ, but u(y, τ ) = u δ or u(y, τ ) = Uδ . (6.6.12) Note that (6.6.5)–(6.6.6) imply u δ < u. ˜ If u(y, τ ) = u δ , then either u(y, t) < u˜ for 0 ≤ t ≤ τ , or u(y, t) < u˜ for 0 ≤ s < t ≤ τ , but u(y, s) = u. ˜ By (6.6.12), for the former case we apply (6.2.12) with s = 0 and employ (6.6.10)–(6.6.11) to derive 1/2

M(u(y, τ )) > M(u 0 (y)) − 2E δ

1/2

− 1 ≥ M(δ2 ) − 2E δ

−1

(6.6.13)

while for the latter case (6.2.12) with (6.6.10)–(6.6.11) leads to 1/2

M(u(y, τ )) > M(u) ˜ − 2E δ

− 1.

(6.6.14)

Thus in either case, by (6.6.12), we have M(u(y, τ )) > M(u δ ) which contradicts u(y, τ ) = u δ in (6.6.12). Hence u δ < u(x, t),

0 ≤ x ≤ 1, 0 < t < +∞.

(6.6.15)

Similarly, we can show that u(y, τ ) = Uδ is a contradiction. This shows that u(x, t) < Uδ f or 0 ≤ x ≤ 1, 0 ≤ t < +∞. Thus the combination of (6.6.15) and (6.6.16) implies (6.6.3). Similarly to (6.2.17), (6.6.4) follows from (6.6.1)–(6.6.3).

(6.6.16)

Corollary 6.6.1. Under assumptions of Lemma 6.6.1, there exist two positive constants Cδ , Cδ such that 1 1 θ d x ≤ Cδ , ∀t > 0. (6.6.17) 0 < Cδ ≤ T0 0

Proof. See, e.g., Remark 6.2.1. Lemma 6.6.2. If (u 0 , v0 , θ0 ) ∈ Hδ1, then the following estimate holds, 0 < Cδ−1 ≤ θ (x, t), Proof. See, e.g., Lemma 2.8.5.

∀(x, t) ∈ [0, 1] × [0, +∞).

(6.6.18)

Lemma 6.6.3. For initial data belonging to an arbitrary fixed bounded set B of Hδ1 there is t0 > 0 depending only on boundedness of this bounded set B1 such that, for all t ≥ t0 , x ∈ [0, 1], δ2 ≤ u(x, t) ≤ δ3 , δ4 ≤ θ (x, t) ≤ δ5 . (6.6.19) Proof. See, e.g., Lemma 2.5.5.

6.6. Universal Attractors in H i (i = 1, 2, 4)

335

Remark 6.6.1. It follows from Lemma 6.6.1 and Lemma 6.6.3 that for initial data belonging to a given bounded set B1 of Hδ1, the orbit will re-enter Hδ1 and stay there after a finite time. Now we use Lemmas 6.6.1–6.6.3, and follow the proofs of Lemmas 2.5.8–2.5.9 to obtain the following lemma. Lemma 6.6.4. There exists a positive constant γ1 = γ1 (C B˜ 1 ,δ ) such that for any fixed γ ∈ (0, γ1 (C B˜ 1 ,δ )], the following estimate holds, (u(t) − η0 , v(t), θ (t) − T0 )2H 1 ≤ C B1 ,δ e−γ t , ∀t > 0.

(6.6.20)

+

Thus the following result on the existence of an absorbing set in Hδ1 follows from Lemma 6.6.4. Theorem 6.6.1. Let

R1 = R1 (δ) = 2 δ32 + T02

and

Bˆ 1 = {(u, v, θ ) ∈ Hδ1, (u, v, θ ) H 1 ≤ R1 }. +

Then Bˆ 1 is an absorbing ball in Hδ1, i.e., there exists some t1 = t1 (C B˜ 1 ,δ ) = max{−γ1−1 log[2(δ32 + T02 )/C B˜ 1 ,δ ], t0 } ≥ t0 such that when t ≥ t1 ,

(u(t), v(t), θ (t))2H 1 ≤ R12 . +

6.6.2 Existence of An Absorbing Set in Hδ2 Throughout this subsection we always assume that the initial datum belongs to an arbitrarily fixed bounded set B2 in Hδ2, i.e., (u 0 , v0 , θ0 ) H 2 ≤ B˜ 2 , with B˜ 2 > 0 being some + constant. Following the proofs of Lemmas 2.5.10–2.5.11, we readily obtain the following lemma. Lemma 6.6.5. There exists a positive constant γ2 = γ2 (C B˜ 2 ,δ ) ≤ γ1 such that for any fixed γ ∈ (0, γ2 (C B˜ 2 ,δ )], the following estimate holds, (u(t) − η0 , v(t), θ (t) − T0 )2H 2 ≤ C B˜ 2 ,δ e−γ t , +

∀t > 0.

(6.6.21)

Now if we set

γ2−1 log[2(δ32 + T02 )/C B˜ 2 ,δ ] t2 = t2 (C B˜ 2 ,δ ) ≥ max t1 (C B˜ 1 ,δ ), −γ R2 = R2 (δ) = R1 (δ) = 2 δ32 + T02 ,

then the following result on the existence of an absorbing set in Hδ2 is readily obtained.

336


Theorem 6.6.2. The ball Bˆ2 = (u, v, θ ) ∈ Hδ2 : (u, v, θ )2H 2 ≤ R22 +

is an absorbing ball in

Hδ2,

i.e., when t ≥ t2 , (u(t), v(t), θ (t))2

H+2

≤ R22 .

6.6.3 Existence of An Absorbing Set in Hδ4 Throughout this subsection we always assume that the initial data belonging to an arbitrarily fixed bounded set B4 in Hδ4, i.e., (u 0 , v0 , θ0 ) H 4 ≤ B˜ 4 , with B˜ 4 > 0 being some + constant. Following the proofs of Lemmas 2.6.1–2.6.3, we easily obtain the following lemma. Lemma 6.6.6. There exists a positive constant γ4 = γ4 (C B˜ 4 ,δ ) ≤ γ2 (C B˜ 2 ,δ ) such that for any fixed γ ∈ (0, γ4 ], the following estimate holds, (u(t) − η0 , v(t), θ (t) − T0 )2H 4 ≤ C B˜ 4 ,δ e−γ t , ∀t > 0. (6.6.22) + γ4−1 log[2(δ32 + T02 )/C B˜ 4,δ ] , Now if we take t4 = t4 (C B˜ 4 ,δ ) = max t2 (C B˜ 2 ),δ , −γ

then we readily derive the following result on the existence of an absorbing set Bˆ4 in Hδ4. Lemma 6.6.7. The ball Bˆ 4 = {(u, v, θ ) ∈ Hδ4, (u, v, θ )2

H+4

in Hδ4, i.e., when t ≥ t4 (C B˜ 4 ,δ ), we have

≤ R42 } is an absorbing set

(u(t), v(t), θ (t))2H 4 ≤ R42 . +

Proof of Theorem 6.3.4. Since we have proved the existence of absorbing balls Bˆ 1 , Bˆ 2 and Bˆ 4 in Hδ1, Hδ2 and Hδ4 respectively, by Theorem 1.6.4, we can complete the proof.

6.7 Bibliographic Comments For solid-like materials, Dafermos [74], Dafermos and Hsiao [77] considered the following boundary conditions (stress free and thermally insulated): σ (0, t) = σ (1, t) = 0, Q(0, t) = Q(1, t) = 0, t ≥ 0,

(6.7.1)

and established existence of global smooth solutions to (6.1.1)–(6.1.3), (6.1.5) and (6.1.10) by applying the Leray-Schauder fixed point theorem. The techniques in Dafermos [74] work when only one end of the body is stress-free while the other is fixed. Jiang [164] established the global existence of a smooth solution to the problem (6.1.1)–(6.1.3) and (6.1.5)–(6.1.6) with constitutive relations e = e(u, θ ), σ = − p(u, θ ) + μ(u)vx , Q = −k(u, θ )θθx

(6.7.2)

where the viscosity μ(u) satisfies μ(u)u ≥ μ0 > 0, 0 < u < +∞,

(6.7.3)


337

for some constant μ0 . It is well known that the large-time behavior of the system (6.1.1)– (6.1.3) is of great interest since the pressure function p(u, θ ) is not necessary monotone in u. Unfortunately, the problem has been open till now. Hsiao and Luo [159] first considered a kind of solid-like material with the constitutive relations e = c0 θ, σ = − p(u, θ ) + μ(u)vx , p(u, θ ) = f (u)θ, Q = −k(u)θθ x ,

(6.7.4)

and (6.7.3) where c0 > 0 is a constant, f (u) and k(u) are twice continuously differentiable for u > 0 such that k(u) > 0, for u > 0;

(6.7.5)

f (u) ≥ 0

(6.7.6)

f (u) ≤ 0

0 0, (0, 0, 0) = 0, (0, 0, 0) = 0, (0, 0, 0) > 0, ∂θ ∂u x ∂θ ∂θθx and is a C 2 -function satisfying

(0) > 0.

(7.1.5) (7.1.6)

(7.1.7)

Concerning the kernel we assume that k(t) ∈ C 2 (R+ ) and that k(t) is a strongly positive definite kernel, i.e., for any T > 0, and for any y(t) ∈ L 1loc (R+ ), the following inequality holds: T

T

(k ∗ y)(t) y(t)dt ≥ C0

0

(k ∗ y)2 (t)dt

(7.1.8)

0

with C0 > 0 being a constant t independent of T , and the sign ∗ denotes the convolution product, i.e., k ∗ y(·, t) = 0 k(t − τ )y(·, τ )dτ ; additionally we assume that there exist positive constants c0 ≤ c1 , and c2 such that k(t) > 0, |k (t)| ≤ c2 k(t), k (t) + c0 k(t) ≤ 0 ≤ k (t) + c1 k(t), ∀t ≥ 0.

(7.1.9)

To simplify notation we will introduce ∂ Q/∂θθ x ∂S (0, 0) = α, (0, 0, 0) = μ > 0, ∂θ (θ + τ0 )∂ N/∂θ (0) ∂ N/∂u x (0, 0) = β, =γ >0 ∂ N/∂θ τ0 ∂ N/∂θ (0, 0) −

(7.1.10) (7.1.11)

with the product αβ > 0. For the initial data we assume that u 0 ∈ H 3(), u 1 ∈ H 2(), u 2 ∈ H 1(), ∀ ∈ . θ0 ∈ H 3(), θ1 ∈ H 2(), |θ0 (x)| < τ0 , ∀x

(7.1.12) (7.1.13)

By u 2 and θ1 we are denoting u 2 : = [S(u x , θ )]x |t =0 , ∂ N/∂u x Q(u x , θx , θ )x θ1 : = − u 1,x |t =0 + |t =0 ∂ N/∂θ (θ + τ0 )∂ N/∂θ

(7.1.14) (7.1.15)

satisfying the compatibility conditions u 0 = u 1 = u 2 = θ0 = θ1 = 0

at x = 0, x = 1.

(7.1.16)

7.1. Main Results

341

We put · = · L 2 (0,1). By Hη we denote the Hessian matrix of the function η. The matrix Aτ denotes the transposed matrix of matrix A. We use C, C0 , C1 , . . . , to denote the generic constants independent of time t > 0. We are now in a position to state our main result. Theorem 7.1.1. Under assumptions (7.1.5)–(7.1.9) and (7.1.12)–(7.1.15), there exists a small constant 0 < 0 < min(1, ρ0 ), ρ0 = min[1, τ0 /2] such that for any ∈ (0, 0 ) and for any initial data satisfying

u 0 2H 2 (0,1) + u 1 2H 2 (0,1) + u 2 2H 1 (0,1) + θ0 2H 2 (0,1) + θ1 2H 2 (0,1) < 2 , (7.1.17) problem (7.1.1)–(7.1.4) admits a unique global solution (u(t), θ (t)) satisfying 3

u(t) ∈

C j ([0, +∞), H 3− j ()),

j =0

(k ∗ θ )(t), θ (t) ∈

1

C j ([0, +∞), H 3− j ()),

(7.1.18)

j =0

(k ∗ θ )(t), θ (t) ∈ C 2 ([0, +∞), L 2 ()),

(7.1.19)

(k ∗ ∂ti θx )(t) ∈ L 2 ([0, +∞), L 2 ()), (i = 0, 1, 2), (k

j ∗ ∂t θx x )(t),

2

2

θx x x (t) ∈ L ([0, +∞), L ()), ( j = 0, 1).

(7.1.20) (7.1.21)

Moreover, there exist positive constants C1 , C2 such that for any t > 0,

u(t) 2H 3 + u t (t) 2H 2 + u t t (t) 2H 1 + u t t t 2 + θ (t) 2H 3 + θt (t) 2H 2 + θt t (t) 2 +

2 1

(k ∗ ∂ti θx )(t) 2 +

(k ∗ ∂ti θx x )(t) 2 + (k ∗ θx x x )(t) 2 i=0

≤ C1 e

−C2 t

.

i=0

(7.1.22)

Remark 7.1.1. The technique in this section also works for boundary conditions u(0, t) = u x (1, t) = θ x (0, t) = θ (1, t) = 0 and u x (0, t) = u(1, t) = θ (0, t) = θ x (1, t) = 0. Therefore the conclusions in Theorem 7.1.1 also hold for the above boundary conditions. The main purpose of this chapter is to show that the solution of the problem (7.1.1)– (7.1.4) exists globally in time and is exponentially stable. The main difficulties to prove our results arise from the complication of nonlinearities in the system, thermal memory effect and a point-wise term. To overcome these difficulties, we use multiplicative techniques, an inequality related to the point-wise term and some technical ideas involving positive kernels and delicate estimates.

342

Chapter 7. A Nonlinear One-dimensional Thermoelastic System

7.2 Global Existence and Exponential Stability In this section we will show global existence as well as exponential stability of the solution, that is, we prove Theorem 7.1.1. We introduce the definitions of a (strongly) positive definite kernel. Definition 7.2.1. A function b(t) ∈ L 1loc [0, +∞) is said to be positive definite (or of positive type) if for any w ∈ C[0, +∞) and every T > 0, there holds T w(t)(b ∗ w)(t)dt ≥ 0 (7.2.1) 0

where

(b ∗ w)(t) =

t 0

b(t − τ )w(τ )dτ.

(7.2.2)

Definition 7.2.2. A function b(t) ∈ L 1loc [0, +∞) is said to be strongly positive definite if there is an ε > 0 such that the function t → b(t) − εe−t is positive definite. Equivalently, for any φ ∈ L 1loc [0, +∞) and every T > 0, there holds where Q(φ, T, b) =

T 0

Q(φ, T, e) ≤ C Q(φ, T, b) φ(t)(b ∗ φ)(t)dt and e = e−t , C = ε−1 .

As the terminology suggests, strongly positive definite implies positive definite. These definitions are easy to check directly. Using transform techniques we can readily show that (see also, e.g., Renardy, Hrusa and Nohel [361], and Staffans [393–395]) Lemma 7.2.1. The kernel b(t) ∈ L 1 (0, +∞) is (strongly) positive definite if and only if (resp., there exists a constant C0 ) such that C0 ˆ ˆ , ∀ w ∈ (−∞, +∞) (7.2.3) Re b(i w) ≥ 0 resp., Re b(i w) ≥ 1 + w2 +∞ ˆ ˆ where b(w) is the Laplace transform of b, i.e., b(w) = 0 e−wt b(t)dt. From the viewpoint of applications, it is useful to know that certain types of sign conditions guarantee strong positive definiteness. More precisely, we have Corollary 7.2.1. If b ∈ C 2 [0, +∞) and (−1) j b( j )(t) ≥ 0, ∀t ≥ 0, j = 0, 1, 2; b ≡ 0,

(7.2.4)

then b is strongly positive definite. On the other hand, even if b is assumed to be very smooth, strong positive definiteness of b does not imply that (7.2.4) holds. It does not even imply that b ≥ 0 on [0, +∞). Indeed, it is easy to verify that the function b given by b(t) = e−t cos t satisfies (7.2.3) with C0 > 0 and hence is strongly positive definite.

(7.2.5)

7.2. Global Existence and Exponential Stability

343

As the above example shows, strong positive definiteness does not imply any global sign conditions. However, if a strongly positive definite function is sufficiently regular, then statements can be made regarding its point-wise behavior near zero. In particular, if b, b , b ∈ L 1 [0, +∞), b is strongly positive definite

(7.2.6)

b(0) > 0, b (0) < 0.

(7.2.7)

then

That (7.2.6) implies b(0) > 0 follows from (7.2.3) and the inversion formula 1 +∞ ˆ w)dw. Re b(i b(0) = π −∞

(7.2.8)

To see that (7.2.6) implies b (0) < 0, observe that ˆ w) = −b (0) lim w2 Re b(i

w→∞

as can be verified using two integrations by parts and the Riemann-Lebesgue lemma. This limit must be strictly positive by (7.2.3). ˆ ∈ L 1 (R+ ) is a strongly positive definite kernel satisfying Lemma 7.2.2. Assume that k(t) 1 + kˆ (t) ∈ L (R ); then for any y(t) ∈ L 1loc (R+ ), it follows that t t |kˆ ∗ y(τ )|2 dτ ≤ β0 k1 y(τ ) kˆ ∗ y(τ )dτ (7.2.9) 0

0

+∞ 2 + 4( +∞ |kˆ (t)|dt)2 and β > 0 is a constant such that the ˆ where k1 = ( 0 |k(t)|dt) 0 0 ˆ − β0 e−t is a positive definite kernel. function k(t) Proof. Define

y(τ ), 0 ≤ τ ≤ t, 0, otherwise. By the Plancherel identity and the fact that convolution is mapped into point-wise multiplication by the Fourier transform, 2 t ∞ τ 2 ˆ ˆ |k ∗ y(τ )| dτ ≤ k(τ − s)yt (s)ds dτ 0 0 0 +∞ 1 ˜ˆ 2 |k(w)| | y˜t (w)|2 dw (7.2.10) = 2π −∞ yt (τ ) =

where

∞ ∞ ˜ˆ ≤ ˆ ˆ e−iwt k(t)dt |k(t)|dt, |k(w)| = 0 0 ∞ ∞ ˜ˆ |wk(w)| = (e−iwt − 1)kˆ (t)dt ≤ 2 |kˆ (t)|dt 0

and f˜ denotes the Fourier transform of f .

0

(7.2.11) (7.2.12)

344


Square these two inequalities (7.2.11) and (7.2.12), and add (7.1.12) to (7.2.11) to get

˜ˆ 2 |k(w)| ≤

k1 1 + w2

which, combined with (7.2.10), yields 2 +∞ t τ | y˜t (w)|2 ˆk(τ − s)y(s)ds dτ ≤ k1 dw 2π −∞ 1 + w2 0 0 +∞ +∞ t τ k1 −|τ −s| yt (τ ) e yt (s)dsdτ = k1 y(τ ) e−(τ −s) y(s)dsdτ = 2 −∞ −∞ 0 0 t t y(τ )(e−τ ∗ y)(τ )dτ ≤ k1 β0 y(τ )(kˆ ∗ y)(τ )dτ = k1 0

0

which gives (7.2.9). We now introduce the definition of the resolvent of a kernel b(t). Definition 7.2.3. Let b(t) ∈ L 1 [0, +∞) satisfying f (t) − (b ∗ f )(t) = g

for some f, f g ∈ L 1loc [0, +∞). The function r (t) is said to be the resolvent of −b(t) if and only if it solves r (t) + (b ∗ r )(t) = −b(t), ∀t ≥ 0 and

f (t) = g(t) + (r ∗ g)(t),

∀t ≥ 0.

With the same assumptions as in Theorem 7.1.1, we can show that the problem (7.1.1)–(7.1.4) admits a unique local solution (u(t), θ (t)) such that u(t) ∈

3

C j ([0, Tm ), H 3− j ()),

j =0

(k ∗ θ )(t), θ (t) ∈

1

C j ([0, Tm ), H 3− j ()),

(7.2.13)

j =0

(k ∗ θ )(t), θ (t) ∈ C 2 ([0, Tm ), L 2 ()), (k ∗ ∂ti θx )(t) ∈ L 2 ([0, Tm ), L 2 ()), (i = 0, 1, 2), (7.2.14) j

(k ∗ ∂t θx x )(t), θ x x x (t) ∈ L 2 ([0, Tm ), L 2 ()), ( j = 0, 1), |θ (x, t)| ≤ ρ0 , ∀(x, t) ∈ × [0, Tm )

(7.2.15) (7.2.16)

where [0, Tm ) is the maximal existence interval of solution (u(t), θ (t)) (see, Renardy, Hrusa and Nohel [361]). Therefore, in order to obtain a global smooth solution we need to show that

u(t) H 3 (0,1) + θ (t) H 3 (0,1) ≤ C, ∀t ≥ 0 (7.2.17)


345

where C > 0 is a constant independent of t. To this end we reduce system (7.1.1)– (7.1.2) to

where

u t t − u x x + αθθx = f

in (0, 1) × [0, Tm ),

(7.2.18)

θt − μθθx x − γ k ∗ θx x + βu t x = g

in (0, 1) × [0, Tm ),

(7.2.19)

∂S ∂S (u x , θ ) + α θx , (u x , θ ) − 1 u x x + ∂u x ∂θ

∂ N/∂u x ∂ Q/∂θθx g= − β ut x − μ θx x − < biggl[ (θ + τ0 )∂ N/∂θ ∂ N/∂θ ∂ Q/∂u x ∂ Q/∂θ uxx + θx + (θ + τ0 )(∂ N/∂θ ) (θ + τ0 )(∂ N/∂θ )

t (θθx ) − γ θx x (τ )dτ + k(t − τ ) (θ + τ0 )∂ N/∂θ 0 f =

(7.2.20)

(7.2.21)

and

t

(k ∗ θ x x )(·, t) =

k(t − τ )θθx x (·, τ )dτ.

(7.2.22)

0

For simplicity, we put ∂S ∂S (u x , θ ) + α, (u x , θ ) − 1, η2 = ∂u x ∂θ ∂ Q/∂θθ x ∂ N/∂u x − μ, W2 = − β, W1 = (θ + τ0 )∂ N/∂θ ∂ N/∂θ ∂ Q/∂u x ∂ Q/∂θ W3 = , W4 = , (θ + τ0 )(∂ N/∂θ ) (θ + τ0 )(∂ N/∂θ ) (θθx ) W5 = − γ. (θ + τ0 )∂ N/∂θ η1 =

It follows from (7.1.9) that the kernel k(t) decays exponentially as time goes to infinity. Thus we can choose δ ∈ δ0 ≡ min(1, c0 /2) so small that for any t ≥ 0, c

˜ := eδt k(t) ≤ c3 e− 20 t k(t)

(7.2.23)

and ˜ ˜ > 0, |k˜ (t)| ≤ 2c2 k(t), k(t) Let us write

c0 ˜ ˜ k˜ (t) + k(t) ∀t ≥ 0. (7.2.24) ≤ 0 ≤ k˜ (t) + c1 k(t), 2

v(x, t) = eδt u(x, t),

φ(x, t) = eδt θ (x, t).

(7.2.25)

346


Then equations (7.2.18)–(7.2.19) can be rewritten as vt t − vx x + αφx = F

φt − μφx x − γ k˜ ∗ φx x + βvt x = G where

in (0, 1) × [0, Tm ),

(7.2.26)

in (0, 1) × [0, Tm ),

(7.2.27)

F = f eδt + 2δvt − δ 2 v, G = geδt + δφ + δβvx .

To prove (7.2.17), it suffices to show that the solution (v(t), φ(t)) is bounded in H 3 × H 3. ˜ ) Let l(t) be the resolvent kernel of − γ k(t μ , i.e., ˜ − γ (l ∗ k)(t). ˜ μl(t) = −γ k(t) Then by (7.2.27), we get φx x (x, t) = g1 (x, t) + (l ∗ g1)(x, t) with g1 =

μ−1 (φ

t

(7.2.28)

+ βvt x − G).

˜ Lemma 7.2.3. Assume that (7.1.9) is valid . Then the resolvent kernel l(t) of −γ k(t)/μ satisfies that for any t > 0, +∞

2 (|l (τ )| + |l(τ )|)(τ )dτ ≤ C3 (7.2.29) 0

provided that δ is small enough. ˜ ˜ k˜ (t) decay exponenProof. Note that (7.1.9) implies k(t), k˜ (t) ∈ L 1 [0, +∞) and k(t), tially. This implies by the standard theory for Volterra equations (see, e.g., Dafermos [68], Renardy, Hrusa and Nohel [361], Staffans [393–395]) that l(t), l (t) ∈ L 1 [0, +∞) and l(t), l (t) decay exponentially. Thus (7.2.29) follows. To facilitate our analysis let us introduce the linear problem t − μx x

in (0, 1) × [0, Tm ), (7.2.30) Vt t − Vx x + αx = F − γ k˜ ∗ x x + βV Vt x = G in (0, 1) × [0, Tm ), (7.2.31) V (x, 0) = V0 , Vt (x, 0) = V1 , (x, 0) = 0 , V (0, t) = V (1, t) = (0, t) = (1, t) = 0.

Without loss of generality we assume that α > 0, β > 0. In what follows we will study the asymptotic behavior of the linearized system (7.2.30)–(7.2.31). To this end we define the functions 1 1 2 E 1 (t; V, ) = (V V + Vx2 + αβ −1 2 )d x, (7.2.32) 2 0 t 1 1 2 (V V + Vt2x + αβ −1 2t )d x, (7.2.33) E 2 (t; V, ) = 2 0 tt 1 1 2 (V V + Vx2x + αβ −1 2x )d x. (7.2.34) E 3 (t; V, ) = 2 0 tx


347

Multiplying (7.2.30) and (7.2.31) by Vt and αβ −1 respectively, and summing up the product result, we have d E 1 (t;V,) = −μαβ −1 dt

1 0

2x d x −γ αβ −1

1 0

k˜ ∗x x d x +

1 0

(F Vt +αβ −1 G)d x.

(7.2.35) Assuming regular initial data and noting that Vt and t satisfy the same boundary conditions, we have 1 1 d E 2 (t; V, ) = −μαβ −1 2t x d x − γ αβ −1 (7.2.36) k˜ ∗ t x t x d x dt 0 0 1 1 ˜ (F Ft Vt t + αβ −1 Gt t )d x − γ αβ −1 k(t) 0x t x d x. + 0

0

Multiplying (7.2.30) and (7.2.31) by −V Vx xt and −αβ −1 x x respectively, and summing up the product result, we obtain 1 1 d E 3 (t; V, ) = −μαβ −1 2x x d x − γ αβ −1 k˜ ∗ x x x x d x + αx Vt x |x=1 x=0 dt 0 0 1 − (F Vx xt + αβ −1 Gx x )d x. (7.2.37) 0

A point-wise term involving the second-order derivatives appears in (7.2.37), which is not possible to be estimated by the usual Sobolev’s inequalities. To overcome this difficulty we will use the following lemma which has been proved in Muñ˜ oz Rivera and Barreto [277]. Lemma 7.2.4. Let us take (w0 , w1 , f1 ) ∈ H01(0, L) ∩ H 2(0, L) × H01(0, L) × H 1(0, T ; L 2 (0, L)) and let w be the solution of the problem wt t − w x x = f 1

in (0, L) × [0, T ),

w(x, 0) = w0 , wt (x, 0) = w1 , in (0, L), w(0, t) = w(1, t) = 0 on (0, T );

(7.2.38) (7.2.39) (7.2.40)

then the following identity holds: L L d 1 L 2 L 2 2 [wx (0, t) + wx (L, t)] = x− wt wx d x + (wx + wt2 )d x 4 dt 0 2 2 0 L L x− f 1 wx d x. − (7.2.41) 2 0

348


Proof. Multiplying (7.2.38) by (x − L/2)wx and integrating it over [0, L], we have L L L (x − L/2)wt t wx d x − (x − L/2)wx x wx d x = (x − L/2) f 1 vx d x. (7.2.42) 0

0

0

Since wt (0, t) = wt (L, t) = 0, a straightforward calculation yields L L L d (x − L/2)wt t wx d x = (x − L/2)wt wx d x − (x − L/2)wt wxt d x dt 0 0 0 L 1 L 2 d (x − L/2)wt wx d x + wt d x. (7.2.43) = dt 0 2 0 On the other hand, we derive L 1 L (x − L/2)wx x wx d x = (x − L/2)d(w2x ) 2 0 0 1 L 2 L 2 2 wx d x. = [wx (L, t) + wx (0, t)] − 4 2 0

(7.2.44)

Thus (7.2.41) follows from (7.2.42)–(7.2.44). Motivated by Lemma 7.2.4 we define the functional 1 1 x− Vt x Vt t d x. E 4 (t; V ) = − 2 0

(7.2.45)

By equation (7.2.18) and Lemma 7.2.4, we easily get 1 1 2 1 d E 4 (t; V ) = − Vt2x (0, t) + Vt2x (1, t) + (V V + Vt2t )d x dt 4 2 0 tx 1 1 1 1 t x Vt x d x − Ft Vt x d x. (7.2.46) x− x− +α 2 2 0 0 Now we introduce the functions 1 Vt x d x, E 5 (t; V, ) = 0

1

E 6 (t; V ) =

Vt x Vx d x,

(7.2.47)

0

1

n(t; V , ) = m(t; V , ) =

0 1 0

[V Vt2x + Vx2x + 2t + 2x ](t)d x,

(7.2.48)

[V Vt2x + Vx2x + 2t x + 2x x ](t)d x.

(7.2.49)

Thus by Poincaré´ ’s inequality, we have n(t; V, ) ≤ Cm(t; V, )

(7.2.50)


349

with C > 0 being a constant independent of t. Let us introduce the functional K (t; V, ) = N1 E 1 (t; V , ) + N2 E 2 (t; V, ) + N3 E 3 (t; V, ) + + E 5 (t; V, ) +

β E 6 (t; V ). 2

β E 4 (t; V ) 6 (7.2.51)

Under the above notation, we have Lemma 7.2.5. There exist positive constants Ni (i = 1, 2, 3) and γ0 , γ1 such that K (t; V, ) satisfies the inequality d μα 1 β 1 2 K (t; V, ) ≤ − [N N1 2x + N2 2t x + N3 2x x ]d x − (V Vx x + Vt2x )d x dt 4β 0 8 0 αγ N1 1 ˜ β 2 Vt x (0, t) + Vt2x (0, t)] − − [V k ∗ x x d x 48 β 0 αγ N3 1 ˜ αγ N2 1 ˜ − k ∗ t x t x d x − k ∗ x x x x d x β β 0 0 6γ 2 1 ˜ (k ∗ x x )2 d x + R(t; V , ), (7.2.52) + β 0 where

1 α α R(t; V, ) = N1 F Vt + G d x + N2 Ft Vt t + Gt t d x β β 0 0 1 1 α − F Vx xt + Gx x d x (Fx − GV Vt x )d x − N3 β 0 0 1 1 β 1 β Ft Vt x d x + − x− F 2d x 6 0 2 4 0 1 β 1 αγ 2 N2 ˜ 2 − F Vx x d x + 20x d x k (t) 2 0 2μβ 0

1

and γ0 n(t; V, ) ≤ K (t; V , ) ≤ γ1 n(t; V, ).

(7.2.53)

Proof. By (7.2.30) and (7.2.31), we get 1 1 1 1 d E 5 (t;V,) = −β Vt2x d x + α 2x d x + μ x x Vt x d x + γ Vt x k˜ ∗ x x d x dt 0 0 0 0 1 1 Vx x x d x − (Fx − GV Vt x )d x (7.2.54) − 0

and d E 6 (t; V ) = dt

1 0

0

Vt2x d x

−

1 0

Vx2x d x

+α

1 0

x Vx x d x −

1 0

F Vx x d x.

(7.2.55)

350


Thus by (7.2.54) and (7.2.55), we deduce that d β (E 5 (t; V, ) + E 6 (t; V )) dt 2 1 β 1 2 αβ 1 2 =− (V V + Vx x )d x + x Vx x d x + α 2x d x 2 0 tx 2 0 0 1 1 1 Vt x x x d x + γ Vt x k˜ ∗ x x d x − Vx x x d x +μ 0

1

− 0

0

β (Fx − GV Vt x )d x − 2

0

1

F Vx x d x.

By (7.2.30) we easily know 1 1 2 Vt t d x ≤ 3 (V Vx2x + α 2 2x + F 2 )d x 0

(7.2.56)

0

(7.2.57)

0

which, together with (7.2.46), implies 1 2 3 1 2 3α 2 1 2 d E 4 (t; V ) ≤ − [V Vt x (0, t) + Vt2x (1, t)] + (V Vx x + Vt2x )d x + dx dt 4 2 0 2 0 x 1 1 1 1 3 1 2 x− t x Vt x d x + x− Ft Vt x d x. F dx − +α 2 2 0 2 0 0 (7.2.58) Using Young’s inequality, we conclude 1 1 1 αβ 1 t x Vt x d x + μ x− x x Vt x d x + γ Vt x k˜ ∗ x x d x 6 0 2 0 0 β 1 2 α2 β 1 2 6μ2 1 2 6γ 2 1 ˜ ≤ Vt x d x + t x d x + x x d x + ( k ∗ x x )2 d x 8 0 24 0 β 0 β 0 and

1 2 1 β 1 2 2 αβ αβ −1 −1 x Vx x d x ≤ V dx + 2x d x 2 8 0 xx β 2 0 0 which, together with (7.2.56) and (7.2.58), imply β d β E 4 (t;V ) + E 5 (t;V,) + E 6 (t;V ) dt 6 2 1 β 2 β 1 2 2 2 V (0,t) + Vt x (1,t)] − ≤ − [V (V Vx x + Vt x )d x + C4 2x d x 24 t x 8 0 0 1 α2 β 1 2 6γ 2 1 ˜ 6μ2 1 2 x x d x + t x d x − (Fx − GV Vt x )d x + ( k ∗ x x )2 d x + β 0 24 0 β 0 0 1 β 1 2 β 1 β 1 x− Ft Vt x d x + F dx − F Vx x d x (7.2.59) − 6 0 2 4 0 2 0


351

α β 2 with C4 = β2 ( αβ 2 − 1) + 4 + α. By Nirenberg’s inequality and Young’s inequality, we easily derive 1/4 1/4 1 1 x=1 2 2 αx Vt x |x=0 ≤ C5 x d x x x d x (|V Vt x (0, t)| + |V Vt x (1, t)|) 2

μα ≤ 2β

0 1

0

0

β 2x x d x + (V V 2 (0, t) + Vt2x (1, t)) + C6 48N N3 t x

1 0

2x d x. (7.2.60)

˜ is a strongly positive definite kernel if δ is small enough, that is, Noting that k(t) ˜ there is δ1 < δ0 , when δ ∈ (0, δ1 ), k(t) is a strongly positive definite kernel, and by (7.2.23)–(7.2.24) and Lemma 7.2.2, there is a positive constant k1∗ , independent of δ, such ˆ ˜ that when k(t) = k(t) in Lemma 7.2.2, k1 ≡ k1 (δ) ≤ k1∗ and β0 ≡ β0 (δ) ≤ β0∗ ≡ max β0 (δ). Now we choose N1 , N2 , N3 so large that

δ∈(0,δ1 )

N1 >

6γβ0∗ k1∗ + 24μ 2β(C C4 + C6 N3 ) αβ 2 + 1, N2 > + 1, N3 > + 1. μα 6μ α

(7.2.61)

Thus from (7.2.35)–(7.2.37), (7.2.51), (7.2.59)–(7.2.60) and Cauchy’s inequality, relation (7.2.52) follows. By equation (7.2.40), Cauchy’s inequality and choosing N1 , N2 large enough, we know that there exist positive constants γ0 and γ1 such that (7.2.53) holds. The proof is complete. Now define

1

N (t; v, φ) =

0 1

N1 (t; v, φ) = 0

(vx2 x + vt2x + vt2x x + vt2t x + φx2x + φt2x + φt2x x )d x,

(7.2.62)

(vx2 x + vt2x + vt2x x + vt2t x + φt2x )d x.

(7.2.63)

By the smallness condition of initial data, we have N (0; u, θ ) < 2 .

(7.2.64)

Using equations (7.2.7)–(7.2.8) and (7.2.64), there exists a constant β3 > 1 such that N (0; v, φ) ≤ β3 N (0; u, θ ) < β3 2 from which it follows that there exists a constant β4 > 0 such that n(0; v, φ) + n(0; vt , φt ) < β4 2 .

(7.2.65)

From (7.2.23)–(7.2.24) it follows that there exists a constant α1 > 0, independent of δ, such that

4αγ 2 +∞ ˜ 2 N1 k (t) + N2 (2k˜ 2 (t) + k˜ 2 (t)) + N3 k˜ 2 (t) dt α1 > μβ 0 4γ 2 +∞ ˜ 2 + k (t)dt. β 0

352


Using the continuity of the solution, it follows that N (t; v, φ) ≤ α0 2 , ∀t ∈ [0, t0 )

(7.2.66)

for some t0 ∈ (0, Tm ), where 2(2γ γ2 + 1) 3γ1 β4 24 max(1, β 2 ) γ 2 k 2 (0) + α1 + β3 , γ2 = 1 + C3 + . α0 = γ0 2 μ2 μ2

t1 = sup τ1 > 0|N (t; v, φ) ≤ α0 2 in [0, τ1 ) .

Define

(7.2.67)

Then we have either t1 = Tm or t1 < Tm . We will show that the latter case will not happen. To this end, we assume that t1 < Tm . By Sobolev’s embedding theorem and (7.2.66)–(7.2.67), we obtain |vx (x, t)| + |φ(x, t)| + |φx (x, t)| + |φt (x, t)| ≤ C7 , ∀(x, t) ∈ [0, 1] × [0, t1) (7.2.68) which implies |u x (x, t)| + |θ (x, t)| + |θθx (x, t)| + |θt (x, t)| ≤ C8 , ∀(x, t) ∈ [0, 1] × [0, t1 ). (7.2.69) Thus if is small enough, we have |u x (x, t)| + |θ (x, t)| + |θθ x (x, t)| < ρ0 . Define ν=

sup

|x|+|y|≤ρ0

(7.2.70)

|∂ ρ ηi (x)|, |∂ ρ W j (y)|; i = 1, 2; j = 1, . . . , 5; 0 ≤ ρ ≤ 3

where ∂ ρ denotes the partial derivatives of order |ρ|. Recalling the definitions of ηi and Wi and using the inequalities above, we deduce W j | ≤ C9 , j = 1, . . . , 5, |ηi | ≤ C9 , i = 1, 2; |W t k(t − τ )W W5 θx x dτ ≤ C9 |k˜ ∗ φx x |e−δt , ∀(x, t) ∈ [0, 1] × [0, t1 )

(7.2.71) (7.2.72)

0

with C9 = C9 (ν) > 0 being a constant. By Nirenberg’s inequality and (7.2.67), we easily derive that |vt x (x, t)| + |vt t (x, t)| + |φt x (x, t)| ≤ C10 , ∀(x, t) ∈ [0, 1] × [0, t1 ) which, together with (7.2.58), implies |u t x (x, t)| + |u t t (x, t)| + |θt x (x, t)| ≤ C11 , ∀(x , t) ∈ [0, 1] × [0, t1 ).

(7.2.73)


353

By equation (7.2.36), (7.2.68)–(7.2.72) and (7.2.35), we get |vx x (x, t)| ≤ C + C(|u x x (x, t)| + |θθx (x, t)|)eδt ≤ C + C|vx x (x, t)| which implies |vx x (x, t)| + |u x x (x, t)|eδt ≤ C12 , ∀(x, t) ∈ [0, 1] × [0, t1 ).

(7.2.74)

Similarly, it follows from (7.2.37), (7.2.38) and Lemma 7.2.3 that

φx x (t) L ∞ ≤ C sup g1 (τ ) L ∞ ≤ C + C sup g(τ ) L ∞ τ ∈[0,t ]

τ ∈[0,t ]

≤ C + C sup φx x (τ ) L ∞ τ ∈[0,t ]

which yields

θθx x (t) L ∞ + sup φx x (τ ) L ∞ ≤ C13 , ∀t ∈ [0, t1 ) τ ∈[0,t ]

(7.2.75)

provided that is small enough. Noting that

vx x x = vt t x + αφx x + eδt ∇η1 · (u x x , θx )u x x + ∇η2 · (u x x , θx )θθx + η1 u x x x + η2 θx x + 2δvt x − δ 2 vx and differentiating (7.2.36) with respect to x, we obtain from equation (7.2.69), (7.2.71)– (7.2.72) and Nirenberg’s inequality that

vx x x (t) 2 ≤ C[N N1 (t; v, φ) + φx x (t) 2 ] + C 2 vx x x (t) 2 which gives N1 (t; v, φ) + φx x (t) 2 ) ≤ 2 , ∀t ∈ [0, t1 ),

vx x x (t) 2 ≤ C(N

(7.2.76)

provided that is small enough. Similarly, differentiating (7.2.37) with respect to x and using (7.2.69), (7.2.71)–(7.2.76), we get

G x (t) 2 ≤ C( 2 + δ 2 ) N1 (t; v, φ) + φx x (t) 2 + φx x x (t) 2

(7.2.77) + (k˜ ∗ φx x )(t) 2 + (k˜ ∗ φx x x )(t) 2 ,

φx x x (t) 2 ≤ C N1 (t; v, φ) + G x (t) 2 + (k˜ ∗ φx x x )(t) 2 , which imply

φx x x (t) 2 ≤ C N1 (t; v, φ)+ φx x (t) 2 + (k˜ ∗φx x )(t) 2 + (k˜ ∗φx x x )(t) 2 (7.2.78)

if + δ is small enough. In the next lemmas we will estimate R(t; V , ).

354


Lemma 7.2.6. Under the same assumptions as in Theorem 7.1.1, the following inequalities hold for any t ∈ [0, t1 ),

1 1 d Ft Vt t d x ≤ C(δ + )m(t; V , ) − η1 Vt2x d x, 2 dt 0 0 1 1 1 d Ft Vx x d x ≤ C(δ + )m(t; V , ) + η1 Vx2x d x, 2 dt 0 0 1 1 2 x− Ft Vt x d x ≤ C(δ + )m(t; V , ) + [V V (0, t) + Vt2x (1, t)], 2 4 tx 0 1 F Vt d x ≤ C(δ + )m(t; V , ), 1

0 1

0

F Vx x d x ≤ C(δ + )m(t; V , ),

1 0

(7.2.79) (7.2.80) (7.2.81) (7.2.82) (7.2.83)

F 2 d x ≤ C(δ + )m(t; V , ).

(7.2.84)

Proof. We only consider the case of (V, ) = (vt , φt ) and F = Ft to prove (7.2.79). The case of (V, ) = (v, φ) and F = F is simple. By (7.2.9) and noting that Ft t = ft t eδt + 2δδ f t eδt + δ 2 f eδt + 2δvt t t − δ 2 vt , f t t = η1t t u x x + 2η1t u x xt + η1 u x xt t + η2t t θx + 2η2t θxt + η2 θxt t , η1t t = (u xt , θt )Hη1 (u xt , θt )τ + ∇η1 · (u xt t , θt t ), η1t = ∇η1 · (u xt , θt ), we have

eδt f t (t) ≤ C + δ)( vx x (t) + vx xt (t) + φx (t) + φt x (t) . 1 1 Here we only estimate the typical term in 0 f t t vt t t eδt d x, that is, 0 η1 u x xt t vt t t eδt d x. 1 Using (7.2.69) and (7.2.71)–(7.2.75), the other terms in 0 f t t vt t t eδt can be controlled by C( + δ)m(t; vt , φt ) in the same way. Noting that u x xt t eδt = vx xt t − 2δvx xt + δ 2 vx x ,

vt t t (t) 2 ≤ C( vt x x (t) 2 + vx x (t) 2 + φx (t) 2 + φxt (t) 2 ), and using integration by parts, we arrive at

1 0

1 d η1 u x xt t vt t t e d x ≤ C(δ + )m(t; vt , φt ) − 2 dt δt

1 0

η1 vt2t x d x.

Thus estimate (7.2.79) is valid. Similarly, we can prove estimates (7.2.80)–(7.2.84). The proof is complete.


355

Lemma 7.2.7. Under the same assumptions as in Theorem 7.1.1, the following inequalities hold for any t ∈ [0, t1 ), (1) When (V , ) = (v, φ) and (F, G) = (F, G),

1 1 2 ˜ (G + Gt t + Gt x x )d x ≤ C(δ + ) m(t; V, ) + (k ∗ x x ) d x , (7.2.85) 0

0

1

1 (Fx − GV Vt x )d x ≤ C(δ + ) m(t; V, ) + (k˜ ∗ x x )2 d x . (7.2.86)

0

0

˜ Ft , G t + γ k(t)φ (2) When (V, ) = (vt , φt ) and (F , G) = (F 0x x ),

1 1 2 ˜ ˜ (G t + γ k(t)φ )φ d x ≤ C(δ + ) m(t; v , φ ) + ( k ∗ φ ) d x 0x x t t t xx 0

μ + 8

0

1

1

1

2γ 2 ˜ 2 φt2x d x + (7.2.87) k (t) 2 , μ 0

1 1 2 ˜ ˜ (G t + γ k(t)φ0x x )t φt t d x ≤ C(δ + ) m(t; vt , φt ) + (k ∗ φ x x ) d x 0

μ 8

0

2γ 2

φt2t x d x + (7.2.88) k˜ 2 (t) 2 , μ 0

1 1 2 ˜ ˜ (G t + γ k(t)φ0x x )φt x x d x ≤ C(δ + ) m(t; vt , φt ) + (k ∗ φ x x ) d x +

0

μ + 8

0

2γ 2 ˜ 2 φt2x x d x + (7.2.89) k (t) 2 , μ 0

1 1 ˜ [F Ft φt x − (G t + γ k(t)φ (k˜ ∗ φx x )2 d x 0x x )vt t x ]d x ≤ C(δ + ) m(t; vt , φt ) + 0

0

β v2 d x + (7.2.90) k˜ 2 (t) 2 . 16 0 t t x β Proof. We only prove (7.2.88) here, the other estimates can be proved in the same way. By Cauchy’s inequality and (7.1.17), we get 1 μ 1 2 2γ 2 ˜ 2 ˜ γ k (t) φ0x x φt t d x ≤ φt t x d x + k (t) 2 . 8 0 μ 0 Differentiating (7.2.37) with respect to t and using (7.2.23)–(7.2.24), we get

φt t (t) ≤ C φx x (t) + φx xt (t) + vxt t (t) + (k˜ ∗ φx x )(t) + G t (t) . (7.2.91) Recalling the definition of G, using (7.2.21), (7.2.67), (7.2.69), (7.2.71)–(7.2.75) and the Poincaré´ inequality, we obtain

G t (t) ≤ C + δ)( φx x (t) + φx xt (t) + vx x (t)

+ vx xt (t) + (k˜ ∗ φx x )(t) ≤ C( + δ). (7.2.92) +

1

4γ 2

356


Inserting (7.2.92) into (7.2.91), we have

φt t (t) ≤ C φx x (t) + φx xt (t) + vx x (t) + vx xt (t)

+ vxt t (t) + (k˜ ∗ φx x )(t)

(7.2.93)

provided that + δ is small enough. 1 1 1 We only estimate two terms in 0 G t t φt t d x, i.e., 0 (W W1 θx x )t t eδt φt t d x and 0 (k ∗ (W W5 θx x ))t t eδt φt t d x. Using (7.2.69) and (7.2.71)–(7.2.75), the other terms can be bounded by C( + δ)m(t; vt , φt ) in the same way. By (7.2.30) and using integration by parts and the identities φx xt = θx xt eδt + δφx x , φx xt t = θx xt t eδt + 2δφx xt − δ 2 φx x ,

(7.2.94)

we easily deduce that

1 0

(W W1 θx x )t t eδt φt t d x =

1

0

(u xt , θt , θxt )HW1 (u xt , θt , θxt )τ θx x eδt φt t d x

+

1

∇W W1 · (u xt t , θt t , θxt t )θθx x eδt φt t d x +

0

+2

0

1

W1 θx xt t eδ φt t d x

0 1

∇W W1 · (u xt , θt , θxt )θθx xt eδt φt t d x

1

≤ C( + δ)m(t, vt , φt ) + ≤ C( + δ)m(t; vt , φt ).

W1 θx xt t eδt φt t d x

0

(7.2.95)

By (7.2.93)–(7.2.95), we finally arrive at

1 0

1 2 ˜ (k ∗ (W W5 θx x ))t t e φt t d x ≤ C( + δ) m(t; vt , φt ) + (k ∗ φ x x ) d x . δt

0

Thus the proof is complete. Let us introduce the functions 1 N3 1 N2 1 S(t; V , ) = η1 Vx2x d x − N3 F Vx x d x − η1 Vxt2 d x, 2 0 2 0 0 L(t; V, ) = K (t; V , ) − S(t; V, ), R(t) = R(t; v, φ) + R(t; vt , φt ), L(t) = L(t; v, φ) + L(t; vt , φt ), M(t) = m(t; v, φ) + m(t; vt , φt ), S(t) = S(t; v, φ) + S(t; vt , φt ).


357

Lemma 7.2.8. Under the same assumptions as in Theorem 7.1.1, the following inequalities hold for any t ∈ [0, t1 ): R(t) ≤

μα 1 d S(t) + (N N1 φt2x + N2 φt2t x + N3 φt2x x )d x dt 8β 0 1 1 β (k˜ ∗ φx x )2 d x + v2 d x + C(δ + ) M(t) + 16 0 t t x 0

4γ 2 ˜ 2 4αγ 2 ˜ 2 N1 k (t) + N2 (2k˜ 2 (t) + k˜ 2 (t)) + N3 k˜ 2 (t) 2 + + k (t) 2 μβ β

β 2 vt x (0, t) + vt2x (1, t) + vt2t x (0, t) + vt2t x (1, t) . + 24

Proof. Using Lemmas 7.2.3–7.2.7 and the definitions of R(t), M(t) and S(t), our conclusion follows. In order to estimate N (t; v, φ), we need to estimate two terms φx x (t) 2 and

φt x x (t) 2 in terms of sup N1 (τ ; v, φ) in the following lemma. τ ∈[0,t ]

Lemma 7.2.9. Under the same assumptions as in Theorem 7.1.1, we have for any t ∈ [0, t1 ),

φx x (t) 2 + φt x x (t) 2 ≤ γ2 sup N1 (τ ; v, φ) + C14 ( 2 + δ 2 ) 2 , (7.2.96) τ ∈[0,t ]

t

( k˜ ∗ φx x x 2 + φx x x 2 )(τ )dτ ≤ C

0

t

N1 (τ, v, φ) + φx x (τ ) 2 dτ.

(7.2.97)

0

Proof. Differentiating (7.2.28) with respect to t, using (7.2.92), Lemmas 7.2.3–7.2.7, and noting that l(0) = −γ k(0)/μ and 4 max(1, β 2 ) N1 (t; v, φ) + C G t (t) 2 , μ2 4 max(1, β 2 ) N1 (t; v, φ) + C G(t) 2 ,

g1 (t) 2 ≤ μ2 2 +∞ 2

(l ∗ g1 )(t) ≤ |l (τ )|dτ sup g1 (τ ) 2

g1t (t) 2 ≤

τ ∈[0,t ]

0

4C3

max(1, β 2 )

sup N1 (τ ; v, φ) + C sup G(τ ) 2 , μ2 τ ∈[0,t ] τ ∈[0,t ]

G(t) ≤ C( + δ) φx x (t) + vx x (t) + vxt (t) + (k˜ ∗ φx x )(t)

≤

≤ C( + δ),

(7.2.98)

358


we arrive at

φt x x (t) 2 ≤ 3 g1t (t) 2 + l 2 (0) g1 (t) 2 + (l ∗ g1 )(t) 2 ≤ γ3 N1 (t; v, φ) + γ4 sup N1 (τ ; v, φ) + C( 2 + δ 2 ) 2 τ ∈[0,t ]

γ2 sup N1 (τ ; v, φ) + C( 2 + δ 2 ) 2 , ≤ 2 τ ∈[0,t ]

φx x (t) 2 ≤ 2 g1 (t) 2 + (l ∗ g1 )(t) 2 γ2 sup N1 (τ ; v, φ) + C( 2 + δ 2 ) 2 ≤ 2 τ ∈[0,t ] k (0) ) with γ3 = 12 max(1,β (1 + γ μ ), γ4 = 2 μ2 (7.2.16) and (7.2.98), we easily obtain 2

2 2

12C 3 max(1,β 2 ) , μ2

which imply (7.2.96). Using

N1 (t; v, φ) + φx x (t) 2 )

(k˜ ∗ φx x )(t) 2 ≤ C(N

(7.2.99)

provided that +δ is small enough. Differentiating (7.2.27) with respect to x, multiplying it by −φx x x , integrating the result over [0, 1] × [0, t], and using (7.2.77), (7.2.99) and Lemma 7.2.2, we get t t 2 −1 μ

φx x x (τ ) dτ + γ (β0 k1 )

(k˜ ∗ φx x x )(τ ) 2 dτ 0 0 t μ t 2 ≤ N1 (τ ; v, φ) + φx x (τ ) 2 + ( 2 + δ 2 ) φx x x (τ ) 2

φx x x (τ ) dτ + C 2 0 0

+( 2 + δ 2 ) k˜ ∗ φx x x (τ ) 2 dτ which gives (7.2.97) if + δ is small enough.

Proof of Theorem 7.1.1. We will use a density argument to prove our results. To this end, we assume that S, N, Q are in C 4 , is in C 3 and the initial data belong to H 4(0, 1) satisfying compatibility conditions. From Lemmas 7.2.5–7.2.8, we easily obtain μα N1 1 2 μα N2 1 2 d 2 2 L(t) ≤ − (φ + φxt )d x − (φ + φxt (7.2.100) t )d x dt 16β 0 x 32β 0 xt 1 β μα N3 1 2 2 2 (φx x + φx2xt )d x − (v 2 + vxt + vx2 xt + vxt − t )d x 32β 0 32 0 x x αγ N1 1 ˜ (k ∗ φx φx + k˜ ∗ φxt φxt )d x − β 0 αγ N2 1 ˜ (k ∗ φxt φxt + k˜ ∗ φxt t φxt t )d x − β 0 αγ N3 1 ˜ (k ∗ φx x φx x + k˜ ∗ φx xt φx xt )d x − β 0


359

1 6γ 2 + C( + δ) (k˜ ∗ φx x )2 + (k˜ ∗ φx xt )2 d x β 0 4αγ 2 ˜ 2 4γ 2 ˜ 2 N1 k (t) + N2 (2k˜ 2 (t) + k˜ 2 (t)) + N3 k˜ 2 (t) 2 + + k (t) 2 μβ β

+

provided that + δ is small enough. On the other hand, it follows from the definitions of n(t; V , ), L(t; V , ), L(t) and Lemma 7.2.6 that 3γ1 γ0 n(t; V , ) ≤ L(t; V, ) ≤ n(t; V, ), 2 2 3γ1 γ0 (n(t; v, φ) + n(t; vt , φt )) ≤ L(t) ≤ (n(t; v, φ) + n(t; vt , φt )) 2 2

(7.2.101) (7.2.102)

if + δ is small enough. Note that there is δ2 < δ1 such that when δ ∈ (0, δ2 ), all ˜ is a strongly positive definite kernel, and max β0 (δ) ≤ estimates above hold and k(t) max β0 (δ) ≡

δ∈(0,δ1 )

β0∗

δ∈(0,δ2 )

ˆ = k(t) ˜ in Lemma 7.2.2. Integrating (7.2.100) with respect when k(t)

to t, and using Lemma 7.2.2, (7.2.61), (7.2.65) and (7.2.102), taking δ and small enough (say, δ < δ2 ), we conclude t 2 1 2 1

L(t) + C15

k˜ ∗ ∂ti φx 2 +

k˜ ∗ ∂ti φx x 2 +

∂ti φx 2 +

∂ti φx x 2 dτ +C15

0

t 0

i=0

i=0

N1 (τ ; v, φ)dτ ≤ L(0) + α1 2 ≤

i=0

3γ1 β4 + α1 2 2

i=0

(7.2.103)

which, combined with (7.2.102), leads to 2 N1 (t; v, φ) ≤ n(t; v, φ)+n(t; vt , φt ) ≤ γ0

3γ1 β4 + α1 2 , ∀t ∈ [0, t1 ). (7.2.104) 2

By Lemma 7.2.9, we easily obtain

φx x (t) 2 + φx xt (t) 2 ≤

4γ γ2 γ0

3γ1β4 + α1 2 2

(7.2.105)

if + δ is small enough. Thus we finally get from (7.2.104)–(7.2.105) 2(2γ γ2 + 1) 3γ1 β4 N (t; v, φ) = N1 (t; v, φ) + φx x (t) 2 + φx xt (t) 2 ≤ + α1 . γ0 2 (7.2.106) Now letting t → t1 in (7.2.106), we have 2(2γ γ2 + 1) 3γ1 β4 N (t1 ; v, φ) ≤ + α1 < α0 2 γ0 2

360


which is contradictory to the definition of t1 , (7.2.67). Thus we conclude that t1 = Tm = +∞ and all the estimates above are valid for any t > 0. Note that N1 (t; v, φ) is equivalent to the third-order full energy E(t; v, φ) := E 1 (t; v, φ) + E 2 (t; v, φ) + E 3 (t; v, φ) + E 2 (t; vt , φt ) + E 3 (t; vt , φt ), that is, C −1 N1 (t; v, φ) ≤ E(t; v, φ) ≤ CN N1 (t; v, φ), ∀t > 0

(7.2.107)

from which it is easy to verify that C −1 N1 (t; u, θ )e2δt ≤ N1 (t; v, φ) ≤ CN N1 (t; u, θ )e2δt , ∀t > 0.

(7.2.108)

By (7.2.48), (7.2.63) and (7.2.102)–(7.2.103), we have N1 (t; v, φ) ≤ n(t; v, φ) + n(t; vt , φt ) ≤ CL(t) ≤ C, ∀t > 0 which, together with (7.2.96) and (7.2.107)–(7.2.108), implies

φx x (t) + φx xt (t) ≤ C, ∀t > 0, 2

2

N1 (t; u, θ ) + E(t; u, θ ) + θθx x (t) + θθx xt (t) ≤ Ce

−2δt

,

∀t > 0.

(7.2.109) (7.2.110)

By equation (7.2.26) and (7.2.109), we get

vx x x (t) 2 ≤ C(N N1 (t; v, φ) + φx x (t) 2 ) ≤ C, ∀t > 0 which gives

u x x x (t) 2 ≤ Ce−2δt ,

∀t > 0.

(7.2.111)

By Lemma 7.2.9 and (7.2.103), we deduce t t ( k˜ ∗ φx x x 2 + φx x x 2 )(τ )dτ ≤ C (N N1 (τ ; v, φ) + φx x (τ ) 2 )dτ ≤ C. 0

0

(7.2.112) By (7.2.23)–(7.2.24), we get 2 1 d 2 i 2 i 2 ˜ ˜ ˜

(k ∗ φx x x )(t) +

(k ∗ ∂t φx )(t) +

(k ∗ ∂t φx x )(t)

dt ≤C

2

i=0

( (k˜ ∗ ∂ti φx )(t) 2 + ∂ti φx (t) 2 ) +

i=0

2 2 ˜ + (k ∗ φx x x )(t) + φx x x (t) .

i=0

1

( (k˜ ∗ ∂ti φx x )(t) 2 + ∂ti φx x (t) 2 )

i=0

(7.2.113)

Integrating (7.2.113) with respect to t, and exploiting (7.2.103) and (7.2.112), we finally obtain 2 i=0

(k˜ ∗ ∂ti φx )(t) 2 +

1

(k˜ ∗ ∂ti φx x )(t) 2 + (k˜ ∗ φx x x )(t) 2 ≤ C, i=0

(7.2.114)


361

which, together with (7.2.78) and (7.2.110), implies

φx x x (t) 2 ≤ C(N N1 (t; v, φ) + φx x (t) 2 + k˜ ∗ φx x (t) 2 ) + k˜ ∗ φx x x (t) 2 ) ≤ C, (7.2.115)

θθx x x (t) 2 ≤ Ce−2δt

(7.2.116)

provided that + δ is small enough. Thus (u(t), θ (t)) and (v(t), φ(t)) are bounded in H 3 × H 3, that is, the estimate (7.2.17) is valid and problem (7.1.1)–(7.1.4) admits a unique global solution (u(t), θ (t)) in H 3 × H 3. From (7.2.110)–(7.2.111), (7.2.114), (7.2.116) and the inequalities C

−1 −2δt

e

2 1 i 2 i 2 2 ˜ ˜ ˜

(k ∗ ∂t φx )(t) +

(k ∗ ∂t φx x )(t) + (k ∗ φx x x )(t)

i=0

≤

2

i=0

(k ∗ ∂ti θx )(t) 2 +

i=0

≤ Ce

−2δt

≤ Ce

−2δt

2 i=0

1

(k ∗ ∂ti θx x )(t) 2 + (k ∗ θx x x )(t) 2

i=0

(k˜ ∗ ∂ti φx )(t) 2 +

1 i 2 2 ˜ ˜

(k ∗ ∂t φx x )(t) + (k ∗ φx x x )(t)

i=0

,

we obtain the estimate (7.1.22) with C2 = 2δ. The proof of Theorem 7.1.1 is now com plete.

7.3 Bibliographic Comments For the classical 1D thermoelastic model (7.1.1)–(7.1.3) without any thermal memory (i.e., k(t) = 0), Slemrod [378] proved the global existence and asymptotic stability of small solutions with Neumann-Dirichlet (u x |x=0,1 = θ |x=0,1 = 0) or Dirichlet-Neumann (u|x=0,1 = θx |x=0,1 = 0) boundary conditions. Racke and Shibata [352] proved the global existence and polynomial decay of small smooth solutions with Dirichlet-Dirichlet ( u|x=0,1 = θ |x=0,1 = 0) boundary conditions, and later for this type of boundary conditions, Racke, Shibata and Zheng [353] further proved the exponential stability of small smooth solutions. Qin and Muñ˜ oz Rivera [340] established the global existence and asymptotic behavior of thermoelastic systems of type II with a thermal memory. We also refer the readers to Burns, Liu and Zheng [46], Dafermos [67], Dafermos and Hsiao [78], Hale and Perissinotto [136], Hansen [137], Hoffmann and Zochowski [153], Hrusa and Messaoudi [155], Hrusa and Tarabek [156], Jiang, Muñ˜ oz Rivera and Racke [172], Jiang and Racke [173], Kim [198], Kirane and Kouachi and Tatar [199], Kirane and Tatar [200], Lebeau and Zuazua [216], Liu and Zheng [238, 240], Messaoudi ˜ Rivera [274, 275], Muñoz ˜ Rivera and Barreto [277], Muñoz ˜ Rivera and [260], Munoz Oliveira [278], Muñ˜ oz Rivera and Qin [279], Qin [315], Qin and Muñoz ˜ Rivera [341],

362


Racke [348], Racke and Zheng [355], Slemord [378], Zheng [450] for classical thermoelastic models. We consult the works by Messaoudi [261], Racke [350, 351], Racke and Wang [354] for thermoelastic models with second sound. For thermoelastic models of type II, we refer to the works by Green and Naghdi [127, 128], Gurtin and Pipkin [133], and Qin and Muñ˜ oz Rivera [340]. For the thermoelastic models of type III, we refer to the works by Green and Naghdi [127, 128], Quintanilla and Racke [347], Reissig and Wang [360], and Zhang and Zuazua [444], and the references therein.

Chapter 8

One-dimensional Thermoelastic Equations of Hyperbolic Type In this chapter, we shall introduce some results on the global existence and exponential stability of solutions to a class of 1D thermoelastic equations of hyperbolic type, which models the thermoelastic system of type II with a thermal memory. The results of this chapter are chosen from Qin and Muñ˜ oz Rivera [340].

8.1 Global Existence This chapter is concerned with the global existence, uniqueness and exponential stability of solutions to thermoelastic equations of hyperbolic type u t t − σ (u x )x + αθθx = 0, in [0, 1] × [0, +∞), θt − k ∗ θx x + βu xt = 0, in [0, 1] × [0, +∞)

(8.1.1) (8.1.2)

subject to the initial conditions u(x, 0) = u 0 (x),

u t (x, 0) = u 1 (x),

θ (x, 0) = θ0 (x),

∀ ∈ [0, 1] ∀x

(8.1.3)

∀t ≥ 0.

(8.1.4)

and the boundary conditions u(0, t) = u(1, t) = 0, θ x (0, t) = θ x (1, t) = 0,

Here by u = u(x, t) and θ = θ (x, t) we denote the displacement and the temperature difference respectively. By σ = σ (s) we denote a nonlinear function and k = k(t) t is the relaxation kernel. The sign ∗ denotes the convolution product, i.e., k ∗ y(·, t) = 0 k(t − τ )y(·, τ )dτ . Finally α and β are constants satisfying αβ > 0.

364

Chapter 8. One-dimensional Thermoelastic Equations of Hyperbolic Type

The aim of this chapter is to establish the global existence and exponential stability of “small” solutions to problem (8.1.1)–(8.1.4). Now let us explain some difficulties in deriving our results. When deriving exponential decay (or stability) of solution (or energy), ˜ we usually strive to construct a functional L(t), equivalent to the energy, satisfying ˜ L˜ (t) + λ L(t) ≤ g(t)

(8.1.5)

where λ > 0 is a constant and g(t) is an exponential function. But in our case, due to involving the kernel terms, we conclude that the energy is not necessarily a decreasing function which in particular means that inequality (8.1.5) is not possible to achieve. To overcome this difficulty, we in advance make the following exponential transforms in t: ˜ k(t) := eδt k(t), and v(x, t) = eδt u(x, t), φ(x, t) = eδt θ (x, t) with a small parameter δ > 0 (see (8.2.5) and (8.2.9)), then we study the new transformed problem (see, e.g., (8.2.13)–(8.2.14)) and we prove that it admits a unique global solution which is uniformly bounded (the bounded constants are independent of any length of time). This implies the global existence and exponential stability of solutions to the original problem (8.1.1)–(8.1.4). To show the uniform bound of the new system, we use some multiplicative techniques and the fact that the relaxation kernel is a strongly positive definite. Throughout this chapter we assume that σ = σ (s) is a C 3 function in a neighborhood of s = 0, say, O = {s ∈ R : |s| < 1}, satisfying σ (0) > 0

(8.1.6)

and concerning the kernel we assume that k(t) ∈ C 1 (R+ ) and that k(t) is a strongly positive definite kernel; additionally we assume that there exist positive constants c0 ≤ c1 such that (8.1.7) k(t) > 0, k (t) + c0 k(t) ≤ 0 ≤ k (t) + c1 k(t), ∀t ≥ 0. By u 2 and θ1 we denote u 2 := [σ (u x )x − αθθx ]|t =0 = σ (u 0x )x − αθ0x , θ1 := −βu xt |t =0 = −βu 1,x satisfying the compatibility conditions u 0 = u 1 = u 2 = θ0x = θ1x = 0

at x = 0, x = 1.

(8.1.8)

For the initial data we assume that (u 0 , u 1 , u 2 ) ∈ H 3(0, 1)× H 2(0, 1)× H 1(0, 1), (θ0 , θ1 ) ∈ H 2(0, 1)× H 1(0, 1) (8.1.9) and

1 0

θ0 (x)d x = 0.

(8.1.10)

We will use the same notation as in Chapter 7. Our main results of this chapter read as follows.


365

Theorem 8.1.1. Under assumptions (8.1.6)–(8.1.10), there exists a small constant 0
0, u(t) 2H 3 + u t (t) 2H 2 + u t t (t) 2H 1 + u t t t 2 + θ (t) 2H 2 + θt (t) 2H 1 + θt t (t) 2 +

1 1 (k ∗ ∂ti θx )(t) 2 + ∂ti (k ∗ θ )(t) 2H 2−i + (k ∗ θx x )(t) 2 i=0

i=0

≤ C1 e−C2 t .

(8.1.15)

Remark 8.1.1. If equation (8.1.1) is replaced by the more general version of equation u t t − S(u x , θ )x = 0, we can rewrite this equation as u t t − au x x + bθθx = f where S = S(u x , θ ) is the Piola-Kirchhoff stress tensor and ∂S ∂S f = (u x , θ ) + b θx , (u x , θ ) − a u x x + ∂u x ∂θ ∂S ∂S a= (0, 0), b = − (0, 0). ∂u x ∂θ

(8.1.16)

(8.1.17) (8.1.18)

Assume that S = S(u x , θ ) ∈ C 3 in a neighborhood of (0, 0), say, |u x | ≤ 1, |θ | ≤ 1, and a > 0, b = 0 and (8.1.7)–(8.1.10) hold. Then the conclusion in Theorem 8.1.1 also holds.

8.2 Global Existence and Exponential Stability In this section we shall prove Theorem 8.1.1 whose proof is based on a priori estimates which we use to continue a local solution globally in time. The existence of a local solution to problem (8.1.1)–(8.1.4) under the assumptions in Theorem 8.1.1 can be established by a standard contraction mapping argument and we omit details here.

366


Theorem 8.2.1. Under the assumptions in Theorem 8.1.1, problem (8.1.1)–(8.1.4) admits a unique local solution (u(t), θ (t)) such that u(t) ∈

3

C j ([0, Tm ), H 3− j ), (k ∗ θ )(t), θ (t) ∈

j =0

1

C j ([0, Tm ), H 2− j ),

j =0 2

2

(k ∗ θ )(t), θ (t) ∈ C ([0, Tm ), L ), (k j

(k ∗ ∂t θx x )(t) ∈ L 2 ([0, Tm ), L 2 ),

∗ ∂ti θx )(t)

∈ L 2 ([0, Tm ), L 2 ), (i = 0, 1, 2),

( j = 0, 1)

where [0, Tm ) is the maximal existence interval of solution (u(t), θ (t)). Moreover, if sup

3

t ∈[0,T Tm ) j =0

j ∂t u(t) 2H 3− j

then

+

1 j =0

j ∂t θ (t) 2H 2− j

< ∞,

Tm = +∞.

Without loss of generality, we suppose that σ (0) = 1 and α > 0, β > 0. In order to obtain a global solution we need to show that u(t) H 3 (0,1) + u t (t) H 2 (0,1) + θ (t) H 2 (0,1) + θt (t) H 1 (0,1) ≤ C, ∀t ≥ 0. (8.2.1) To this end we reduce system (8.1.1)–(8.1.2) to u t t − u x x − αθθx = ηu x x θt − k ∗ θx x + βu t x = 0

in (0, 1) × [0, Tm ), in (0, 1) × [0, Tm )

(8.2.2) (8.2.3)

where η = σ (u x ) − σ (0) = σ (u x ) − 1. It follows from (8.1.7) that the kernel k(t) satisfies that for any t ≥ 0, k(0)e−c1 t ≤ k(t) ≤ k(0)e−c0 t ≤ k(0).

(8.2.4)

Thus we can choose δ ∈ δ0 ≡ (0, min(1, c0 /2)) such that for any t ≥ 0, c0

˜ := eδt k(t) ≤ k(0)e− 2 t k(0)e−c1 t ≤ k(t)

(8.2.5)

and

˜ ˜ ≤ 0 ≤ k˜ (t) + c1 k(t), ˜ > 0, k˜ (t) + c0 k(t) ∀t ≥ 0. (8.2.6) k(t) 2 By the Paley-Wiener theorem (e.g., pp. 149–150, [329]) and (8.2.5)–(8.2.6), there exists ˜ is a strongly positive definite kernel, and δ1 ∈ (0, δ0 ] such that for any δ ∈ (0, δ1 ), k(t) ˆ ˜ hence applying Lemma 7.2.2 to k(t) = k(t), using (8.2.5)–(8.2.6), there is a positive constant k1∗ , independent of δ, k1∗ ≥ k1 = k1 (δ) =

∞ 0

˜ |k(t)|dt

2

+4

0

∞

|k˜ (t)|dt

2 (8.2.7)


367

such that for any δ ∈ (0, δ1 ] and for any y(t) ∈ L 1loc (R+ ), t t |k˜ ∗ y(τ )|2 dτ ≤ β0 k1∗ y(τ ) k˜ ∗ y(τ )dτ. 0

Denote

(8.2.8)

0

v(x, t) = eδt u(x, t),

φ(x, t) = eδt θ (x, t).

(8.2.9)

Then v(x, t) and φ(x, t) satisfy u t eδt = vt − δv, u t t eδt = vt t − 2δvt + δ 2 v, θt eδt = φt − δφ, θt t eδt = φt t − 2δφt + δ 2 φ,

k˜ ∗ φx x = eδt k ∗ θx x .

(8.2.10) (8.2.11) (8.2.12)

Then using (8.2.10)–(8.2.12), we transfer equations (8.2.2)–(8.2.3) into in (0, 1) × [0, Tm ), (8.2.13) vt t − vx x + αφx = f ˜ in (0, 1) × [0, Tm ), (8.2.14) φt − k ∗ φx x + βvt x = g v(0, t) = v(1, t) = 0, φx (0, t) = φx (1, t) = 0 in (0, 1) × [0, Tm ) where

1

0

and

1

φ0 (x)d x =

θ0 (x)d x = 0

(8.2.15)

0

f = ηu x x eδt + 2δvt − δ 2 v,

g = δφ + δβvx .

(8.2.16)

To prove (8.2.1), it suffices to show that the solution (v(t), φ(t)) is bounded in H 3 × H 2. We easily get from (8.2.3) and (8.2.14)–(8.2.15) 1 1 θ (x, t)d x = φ(x, t)d x = 0, ∀t ≥ 0 (8.2.17) 0

0

which together with (8.2.16) gives 1 g(x, t)d x = 0, ∀t ≥ 0.

(8.2.18)

0

To continue our analysis let us introduce the linear problem Vt t − Vx x + α x = F in (0, 1) × [0, Tm ), ˜ Vt x = G in (0, 1) × [0, Tm ),

t − k ∗ x x + βV V (x, 0) = V0 , Vt (x, 0) = V1 , (x, 0) = 0 ,

(8.2.19) (8.2.20)

V (0, t) = V (1, t) = x (0, t) = x (1, t) = 0 with

1 0

0 (x)d x = 0.

(8.2.21)

368


In fact, it is obvious from (8.1.8), (8.2.9), (8.2.14)–(8.2.15) and (8.2.18) that (8.2.21) ˜ is satisfied if ( , G) = (φ, g) or ( , G) = (φt , gt + k(t)φ 0x x ). Thus it follows from (8.2.14), (8.2.16), (8.2.18) and (8.2.21) that when (V, ) = (v, φ), (F, G) = ( f, f g) or ˜ (V , ) = (vt , φt ), (F, G) = (φt , gt + k(t)φ 0x x ), 1 1 1

t (x, t)d x = G(x, t)d x = 0,

(x, t)d x = 0, ∀t ≥ 0. (8.2.22) 0

0

0

In the sequel we are going to study the linearized system (8.2.19)–(8.2.20). To we define the energy functions 1 1 2 E 1 (t; V, ) = (V V + Vx2 + αβ −1 2 )d x, 2 0 t 1 1 2 (V V + Vt2x + αβ −1 2t )d x, E 2 (t; V, ) = 2 0 tt 1 1 2 (V V + Vx2x + αβ −1 2x )d x. E 3 (t; V, ) = 2 0 tx

this end

(8.2.23) (8.2.24) (8.2.25)

Multiplying (8.2.19) and (8.2.20) by Vt and αβ −1 respectively, and summing up the product result, we have 1 1 d E 1 (t; V, ) = −αβ −1

x k˜ ∗ x d x + (F Vt + αβ −1 G )d x. (8.2.26) dt 0 0 Assuming regular initial data and noting that Vt and t satisfy the same boundary conditions, we get 1 1 d −1 −1 ˜ ˜ E 2 (t; V, ) = −αβ

t x k ∗ t x d x − αβ k(t)

0x t x d x dt 0 0 1 (F Ft Vt t + αβ −1 G t t )d x (8.2.27) + 0

1 1 d ˜ = −αβ −1

t x k˜ ∗ t x d x − αβ −1

0x x d x k(t) dt 0 0 1 1 −1 ˜

0x x d x + (F Ft Vt t + αβ −1 G t t )d x. + αβ k (t) 0

0

Similarly, multiplying (8.2.19) and (8.2.20) by −V Vx xt and −αβ −1 x x respectively, and summing up the product result, we obtain 1 1 d E 3 (t; V, ) = −αβ −1

x x k˜ ∗ x x d x − (F Vx xt + αβ −1 G x x )d x dt 0 0 1 1 d

x x k˜ ∗ x x d x − F Vx x d x = −αβ −1 dt 0 0 1 (F Ft Vx x − αβ −1 G x x )d x. (8.2.28) + 0


369

Now we introduce the functionals 1 x E 4 (t; V, ) = −

t d yV Vt t d x, E 5 (t; V, ) = 0

E 6 (t; V, ) =

1 0

0

1

E 8 (t; V, ) = −

Vt x d x,

0

Vt x Vx d x, E 7 (t; V, ) = −

1

1

0

t k˜ ∗ t d x,

x k˜ ∗ x d x.

0

Thus integrating (8.2.20) over (0, x) and using the boundary conditions, we derive x x

t d y − k˜ ∗ x + βV Vt = Gd y. (8.2.29) 0

0

By (8.2.19) and (8.2.29), we easily get β β 2 1 d E 4 (t; V, ) ≤ − V Vt t 2 + V Vt x 2 + (α + ) t 2 + [k 2 (0) x 2 dt 2 8 β β

x 1 x 2 ˜ + k ∗ x ] + G t d yV Vt t +

t d y Ft d x. (8.2.30) 0

Define

n(t; V, ) = 0

1

0

0

[V Vt2t + Vt2x + Vx2x + 2t + 2x ](t)d x

and L(t; V, ) ˜ = N E 1 (t; V, ) + E 2 (t; V, ) + αβ −1 k(t)

1

0x x d x + E 3 (t; V, )

0

β + E 4 (t; V, ) + E 5 (t; V , ) + E 6 (t; V, ) + a1 E 7 (t; V, ) + a2 E 8 (t; V, ) 4 where N > 0 is a parameter (specified later on) and a1 = 4+k 2 (0) β

+

α2 β 8

4 k(0) (α

+ β2 ), a2 =

4 k(0) (α

+

+ a1 ). Under the above notations, we have

Lemma 8.2.1. There exist positive constants β1 , β2 , β3 , C3 , C4 and a sufficiently large constant N > N0 := 2β0 k1∗ β(C C4 + a12 /8)/α such that for any t > 0, L(t; V, ) satisfies the inequality d L(t;V, ) ≤ −C3 n(t;V, ) + C4 ( k˜ ∗ x 2 + k˜ ∗ t x 2 + k˜ ∗ x x 2 ) (8.2.31) dt 1 − α Nβ N −1 ( x k˜ ∗ x + t x k˜ ∗ t x + x x k˜ ∗ x x )d x + R(t;V , ) 0

370


and

L(t; V, ) ≤ β2 n(t; V , ) + k˜ ∗ t 2 + k˜ ∗ x 2 + k˜ 2 (t) 0x 2 , L(t; V , ) ≥ β1 n(t; V, ) − β3 k˜ ∗ t 2 + k˜ ∗ x 2 + k˜ 2 (t) 0x 2

(8.2.32) (8.2.33)

where

1

R(t; V , ) = N

(F Vt + αβ −1 G + Ft Vt t + αβ −1 G t t + Ft Vx x − αβ −1 G x x )d x

0

1 1 1 d F Vx x d x + α Nβ N −1 k˜ (t)

0x x d x − a1 G t k˜ ∗ t d x dt 0 0 0

x 1 x 1 G t d yV Vt t +

t d y Ft d x + a2 G k˜ ∗ x x d x + −N

0

+

0

1

0

0

β Vt x G − F x − F Vx x d x. 4

0

(8.2.34)

Proof. By (8.2.19)–(8.2.20) and integration by parts, we get 1 1 d 2 2 E 5 (t;V, ) = −β V Vt x + α x − (V Vx x + F) x d x + Vt x (k˜ ∗ x x + G)d x dt 0 0

β β 4 1 2 2 ≤ − V Vt x + V Vx x + α + x 2 + k˜ ∗ x x 2 2 16 β 2β 1 + (V Vt x G − F x )d x (8.2.35) 0

and 1 1 d E 6 (t; V, ) = V Vt x 2 − V Vx x 2 + α

x Vx x d x − F Vx x d x dt 0 0 1 α2 1 2 2 2 Vx x + x + V Vt x − F Vx x d x. (8.2.36) ≤ − V 2 2 0 Thus it follows from (8.2.30) and (8.2.35)–(8.2.36) that β d E 4 (t; V, ) + E 5 (t; V, ) + E 6 (t; V, ) (8.2.37) dt 4 β β β Vt t 2 − V Vt x 2 − V Vx x 2 + (α + 2/β) t 2 ≤ − V 2 8 16 + (α + 4/β + α2 β/8 + k 2 (0)/β) x 2 + β −1 k˜ ∗ x 2 + (2β)−1 k˜ ∗ x x 2

1 1 x x β Vt x G − F x − F Vx x d x. + G t d yV Vt t +

t d y Ft d x + 4 0 0 0 0


371

On the other hand, differentiating (8.2.20) with respect to t, multiplying the resulting equation by k˜ ∗ t and integrating by parts, we deduce d E 7 (t; V, ) dt

1

= −k(0) t 2 + 0

−

( t k˜ ∗ t + k(0) x k˜ ∗ t x + k˜ ∗ x k˜ ∗ t x )d x

1

0

(βV Vt t k˜ ∗ t x + G t k˜ ∗ t )d x

k(0) β 1 k 2 (0) ˜ t 2 + k˜ ∗ t 2 + k ∗ t x 2 V Vt t 2 + x 2 + 2 4a1 2k(0) 4 1 1 + ( k˜ ∗ x 2 + k˜ ∗ t x 2 ) + βa1 k˜ ∗ t x 2 − G k˜ ∗ t d x. (8.2.38) 2 0

≤−

Similarly, differentiating (8.2.20) with respect to x, multiplying the resulting equation by k˜ ∗ x and integrating by parts, we infer 1 d E 8 (t; V, ) = −k(0) x 2 + k˜ ∗ x x 2 + G k˜ ∗ x x d x dt 0 1 ( x k˜ ∗ x + βV Vt x k˜ ∗ x x )d x − 0

k(0) β 1 x 2 + k˜ ∗ x 2 ≤− V Vt x 2 + 2 4a2 2k(0) 1 + (1 + βa2 ) k˜ ∗ x x 2 + G k˜ ∗ x x d x.

(8.2.39)

0

Combining (8.2.38) and (8.2.39) with (8.2.37) gives β d E 4 (t; V, ) + E 5 (t; V, ) + E 6 (t; V, ) + a1 E 7 (t; V, ) + a2 E 8 (t; V, ) dt 4 ≤ −C3 n(t; V, ) + (1/β + a1 /2 + a2 /(2k(0)) k˜ ∗ x 2 + 1/β + (1 + βa2 )a2 k˜ ∗ x x 2 + a1 /(2k(0)) k˜ ∗ t 2 + (k 2 (0)a1/4 + a1 /2 + βa12) k˜ ∗ t x 2 + R1 (t; V, )

(8.2.40)

where C3 = min {β/16, k(0)a1/4, k(0)a2 /4}. In view of (8.1.7), (8.2.22) and Poincaré´ ’s inequality, we have

1 0

k˜ ∗ t d x = 0, k˜ ∗ t ≤ C k˜ ∗ t x ≤ C k˜ ∗ t x .

(8.2.41)

Thus it follows from (8.2.26)–(8.2.28) and (8.2.40)–(8.2.41) that there is a constant C4 > 0 such that the inequality (8.2.31) holds. From the definition of L(t; V, ), we easily

372


know that there exist constants β1 , β2 , β3 > 0 and a sufficiently large constant N > N0 such that (8.2.32) and (8.2.33) hold. The proof is complete. Now we define M(t; v, φ) = n(t; v, φ) + n(t; vt , φt ) + φx x (t) 2 . Differentiating (8.2.20) with respect to t, we arrive at φt t − k(0)φx x − k˜ ∗ φx x + βvt t x = gt which, combined with (8.2.6) and (8.2.20), yields φx x (t) 2 ≤ C( φt t (t) 2 + vt t x (t) 2 + k˜ ∗ φx x (t) 2 + gt (t) 2 ) ≤ C φt t (t) 2 + vt t x (t) 2 + φt (t) 2 + vt x (t) 2 + g(t) 2 + gt (t) 2 ≤ C5 [n(t; v, φ) + n(t; vt , φt )].

(8.2.42)

Thus n(t; v, φ) + n(t; vt , φt ) ≤ M(t; v, φ) ≤ C6 (n(t; v, φ) + n(t; vt , φt ).

(8.2.43)

By (8.2.6) and noting that (k˜ ∗ φt )t = k(0)φt + k˜ ∗ φt , we easily obtain d ˜ k ∗ φt 2 + k˜ ∗ φt t 2 + k˜ ∗ φx 2 + k˜ ∗ φt x 2 (8.2.44) dt C3 ≤ C7 k˜ ∗ φt x 2 + k˜ ∗ φt t x 2 + k˜ ∗ φx 2 + φt t 2 + φx 2 + φt x 2 . 8C6 β3 By the smallness condition of initial data (8.1.11) and (8.1.1)–(8.1.2), there is a constant α1 > 1, independent of δ, such that M(0; u, θ ) < α1 2 .

(8.2.45)

Using equations (8.2.13)–(8.2.14) and (8.2.9)–(8.2.11), there exists a constant α2 > 1, independent of δ, such that n(0; v, φ) + n(0; vt , φt ) ≤ M(0; v, φ) ≤ α2 M(0; u, θ ) < α1 α2 2

(8.2.46)

which leads to v0 2H 2 + v1 2H 2 + v2 2H 1 + φ0 2H 2 + φ1 2H 1 ≤ α1 α2 2 .

(8.2.47)

We infer from (8.2.5)–(8.2.6) and (8.2.47) that there exists a constant η0 > 0, independent of δ, such that +∞ λ3 (k˜ (t))2 ( φ0x 2 + φ1x 2 ) + [λ1 (k˜ (t))2 + λ2 k˜ 2 (t)] φ0x x 2 dt < η0 2 0

(8.2.48)


373

where λ1 = 1 +

2C6 4N 2 α 2 C6 2C6 2N 2 α 2 C6 + , λ2 = + 2 C3 C3 β C3 C3 β 2

and λ3 =

2N 2 α 2 C6 . C3 β 2

Using the continuity of the solution it follows that there exists some t0 ∈ [0, Tm ) such that M(t; v, φ) ≤ α0 2 , ∀t ∈ [0, t0 ) (8.2.49) where α0 = α1 α2 + 4C6 α3 /β1 , α3 = α1 α2 (β1 + β2 )/2 + k 2 (0)α1 α2 [2(β2 + β3 ) + N 2 α 2 C5 /(β1 β 2 )] + η0 . t1 = sup τ1 > 0 : M(t; v, φ) ≤ α0 2 in [0, τ1 ) .

Define

(8.2.50)

Then we have either t1 = Tm or t1 < Tm . In the former case, (v(t), φ(t)) with its corresponding derivatives are bounded in the L 2 -norm for any t ∈ [0, Tm ). Thus, by Theorem 8.2.1, Tm = +∞. We will show that the latter case will not happen. To this end, we now assume that t1 < Tm . By Sobolev’s embedding theorem and (8.2.50), we obtain that for any (x, t) ∈ [0, 1] × [0, t1 ), |vx (x, t)| + |φ(x, t)| + |φx (x, t)| + |φt (x, t)| ≤ C9

(8.2.51)

which implies that for any (x, t) ∈ [0, 1] × [0, t1 ), |u x (x, t)| + |θ (x, t)| + |θθ x (x, t)| + |θt (x, t)| ≤ C10 e−δt .

(8.2.52)

Thus if is small enough, we have that for any (x, t) ∈ [0, 1] × [0, t1 ), |u x (x, t)| < 1, u x (x, t) ∈ O. Define

ν = sup |∂ ρ η(s)|; 0 ≤ |ρ| ≤ 2 |s|≤1

where

∂ρ

denotes the derivatives of order |ρ|. Recalling the definitions of η, we deduce |η| ≤ C11

(8.2.53)

with C11 = C11 (ν) > 0 being a constant. By (8.2.50)–(8.2.53), we easily derive that for any (x, t) ∈ [0, 1] × [0, t1 ), |vt (x, t)| + |vt x (x, t)| + |vt t (x, t)| ≤ C12

which, together with (8.2.10)–(8.2.12), implies that for any (x, t) ∈ [0, 1] × [0, t1 ), |u t (x, t)| + |u t x (x, t)| + |u t t (x, t)| ≤ C13 e−δt .

(8.2.54)

374


By equation (8.2.13), (8.2.51) and (8.2.53)–(8.2.54), we get |vx x (x, t)| ≤ | f (x, t)| + |φx (x, t)| + |vt t (x, t)| ≤ C + C |vx x (x, t)| which gives |vx x (x, t)| ≤ C14 , |u x x (x, t)| ≤ C14 e−δt , ∀(x, t) ∈ [0, 1] × [0, t1 ).

(8.2.55)

Similarly, differentiating (8.2.13) with respect to x, we conclude that vx x x (t) 2 ≤ C[M(t; v, φ) + f x (t) 2 ] ≤ CM(t; v, φ) + C 2 vx x x (t) 2 which gives that for any t ∈ [0, t1 ), vx x x (t) 2 ≤ CM(t; v, φ) ≤ C15 2 , u x x x (t) 2 ≤ C15 2 e−δt

(8.2.56)

provided is small enough. In the next two lemmas we will estimate each term in R(t; V, ) for the cases of both (V, ) = (v, φ), (F, G) = ( f, f g) and (V , ) = (vt , φt ), (F, G) = ( ft , gt + ˜ ), respectively. k(t)φ 0x x Lemma 8.2.2. Under the assumptions in Theorem 8.1.1, the following estimates hold for any t ∈ [0, t1 ): R(t; v, φ) ≤ C( + δ)M(t; v, φ) + Cδ( k˜ ∗ φt x 2 + k˜ ∗ φx x 2 ) 1 2N 2 α 2 C6 ˜ d C3 2 2 2 φx + (k (t)) φ0x − N f vx x d x, + 8C6 dt 0 C3 β 2 1 f vx x d x ≤ C( + δ)M(t; v, φ). N 0

Proof. The estimates in the lemma are easily proved from the definition of M(t; v, φ) and (8.2.16). . Lemma 8.2.3. Under the assumptions in Theorem 8.1.1, the following estimates hold for any t ∈ [0, t1 ): R(vt , φt ) ≤ C( + δ)M(t; v, φ) + (Cδ + a12 /4) k˜ ∗ φt x 2 + Cδ k˜ ∗ φt x x 2 ) C3 ( vt t t 2 + vt t x 2 + φt 2 + φt t 2 + φt x 2 + φx x 2 ) 8C6 + λ1 (k˜ (t))2 + λ2 k 2 (t) φ0x x 2 + λ3 (k˜ (t))2 φ1x 2 1 N d ˜ (ηvt2x x − vt2t x ) − Nαβ −1 k(t)φ + 0x x φ x x − N ft vt x x d x, dt 0 2 +

1

N

ft vt x x d x ≤ C M(t; v, φ),

0

˜ −Nαβ −1 k(t) ≤

1 0

(8.2.57) (8.2.58)

φ0x x φx x d x

N 2 α 2 C5 ˜ 2 β1 (n(t; v, φ) + n(t; vt , φt )) + k (t) φ0x x 2 . 2 2β1 β 2

(8.2.59)


375

Proof. For the proof of (8.2.57), we only prove the following estimate holds for some terms in R(t; vt , φt ), i.e., 1 ˜ [ f t t vt x x − αβ −1 (gt + k(t)φ N 0x x )φt x x ]d x 0

1 C3 d ˜ k(t) φx x 2 − Nαβ −1 φ0x x φx x d x ≤ C( + δ)M(t; v, φ) + 8C6 dt 0 1 2 2 N d 2N α C 6 + ηvt2x x d x + (k˜ (t))2 φ0x x 2 , (8.2.60) 2 dt 0 C3 β 2 while other terms in R(t; vt , φt ) can be proved in the same manner. In fact, it is obvious that 1 1 1 −1 −1 ˜ N f t t vt x x − αβ (gt + k(t)φ0x x )φt x x d x = N f t t vt x x d x + Nαβ φt x g t x d x 0 0 0 1 1 d ˜ k(t) φ0x x φx x d x + Nαβ −1 k˜ (t) φ0x x φx x d x. (8.2.61) − Nαβ −1 dt 0 0 By virtue of (8.2.10)–(8.2.12), (8.2.16), (8.2.52)–(8.2.56) and integration by parts, we get 1 f t t vt x x d x N 0

1

ηt t vx x + 2ηt (vt x x − δvx x ) + 2δηt vx x + η(vt t x x − 2δvt x x + δ 2 vx x ) + 2δη(vt x x − δvx x ) + δ 2 ηvx x + 2δvt t t − δ 2 vt t vt x x d x 1 N d ηvt2x x d x. (8.2.62) ≤ C( + δ)M(t; v, φ) + 2 dt 0

=N

0

Similarly, Nαβ −1 Nαβ

−1 ˜

k (t)

1

φt x gt x d x ≤ C( + δ)M(t; v, φ),

0 1

0

φ0x x φx x d x ≤

(8.2.63)

C3 2N 2 α 2 C6 ˜ φx x 2 + (k (t))2 φ0x x 2 . 8C6 C3 β 2

(8.2.64)

Thus (8.2.60) follows from (8.2.62)–(8.2.64). The proof is complete. Let us introduce the function

1

L1 (t) = L(t; v, φ) + L(t; vt , φt ) + N ˜ +Nαβ −1 k(t)

0

1 0

φ0x x φx x d x.

( f vx x + f t vt x x )d x +

N 2

0

1

η(vt2t x − vt2x x )d x

376


Then it follows from (8.2.43), (8.2.52)–(8.2.56) and Lemmas 8.2.1–8.2.3 that if + δ is small enough, β1 L1 (t) ≤ (β2 + ) n(t; v, φ) + n(t; vt , φt ) 2 N 2 α 2 C5 ˜ 2 + C( + δ)M(t; v, φ) + k (t) φ0x x 2 2β1 β 2 + β2 k˜ ∗ φt 2 + k˜ ∗ φt t 2 + k˜ ∗ φx 2 + k˜ ∗ φt x 2 + k˜ 2 (t) φ0x 2 + k˜ 2 (t) φ1x 2 ≤ (β2 + β1 )[n(t; v, φ) + n(t; vt , φt )]/2 N 2 α 2 C5 2 2 2 ˜2 φ0x x k (t) + β2 φ0x + β2 φ1x + 2β1 β 2 + β2 k˜ ∗ φt 2 + k˜ ∗ φt t 2 + k˜ ∗ φx 2 + k˜ ∗ φt x 2 and

(8.2.65)

L1 (t) ≥ β1 n(t; v, φ) + n(t; vt , φt ) /2 N 2 α 2 C5 ˜ 2 − C( + δ)M(t; v, φ) − k (t) φ0x x 2 2β1 β 2 − β3 k˜ ∗ φt 2 + k˜ ∗ φt t 2 + k˜ ∗ φx 2 + k˜ ∗ φt x 2 + k˜ 2 (t) φ0x 2 + k˜ 2 (t) φ1x 2 ≥ β1 n(t; v, φ) + n(t; vt , φt ) /4 N 2 α 2 C5 2 2 2 ˜2 φ0x x k (t) − β3 φ0x + β3 φ1x + 2β1 β 2 − β3 k˜ ∗ φt 2 + k˜ ∗ φt t 2 + k˜ ∗ φx 2 + k˜ ∗ φt x 2 .

(8.2.66)

Define N 2 α 2 C5 2 ˜2 k (t) φ L(t) = L1 (t) + β3 φ0x 2 + β3 φ1x 2 + 0x x 2β1 β 2 +β3 k˜ ∗ φt 2 + k˜ ∗ φt t 2 + k˜ ∗ φx 2 + k˜ ∗ φt x 2 .

(8.2.67)

Then it follows from (8.2.66), (8.2.6) and (8.2.43)–(8.2.44) that if + δ is small enough, β1 M(t; v, φ), (8.2.68) 4C6 d C3 d L(t) ≤ L1 (t) + n(t; v, φ) + n(t; vt , φt ) + C7 β3 k˜ ∗ φt x 2 + k˜ ∗ φt t x 2 dt dt 8 2 (8.2.69) + k˜ ∗ φx . L(t) ≥ β1 [n(t; v, φ) + n(t; vt , φt )]/4 ≥


377

Proof of Theorem 8.1.1. We will assume that the initial data u 0 , u 1 and θ0 belong to H 4 × H 3 × H 4. Our result will follow using the standard density argument. By virtue of Lemmas 8.2.2–8.2.3, we easily obtain d L1 (t) dt ≤ −Nαβ −1

1 φx k˜ ∗ φx + 2φt x k˜ ∗ φt x + φt t x k˜ ∗ φt t x + φx x k˜ ∗ φx x + φt x x k˜ ∗ φt x x d x 0

C3 − C3 (n(t,v,φ) + n(t;vt ,φt )) + M(t;v,φ) + C16 ( + δ)M(t;v,φ) 8C6 + C4 k˜ ∗ φx 2 + 2 k˜ ∗ φt x 2 + k˜ ∗ φt t x 2 + k˜ ∗ φx x 2 + k˜ ∗ φt x x 2 + (a12/4) k˜ ∗ φt x 2 + C16 δ k˜ ∗ φt x 2 + k˜ ∗ φx x 2 + k˜ ∗ φt x x 2 + λ3 (k˜ (t))2 ( φ0x 2 + φ1x 2 ) + λ1 (k˜ (t))2 + λ2 k˜ 2 (t) φ0x x 2 which together with (8.2.68)–(8.2.69) and (8.2.44) yields that if + δ is small enough, d L(t) dt ≤ −Nαβ −1

1

(φx k˜ ∗ φx + 2φt x k˜ ∗ φt x + φt t x k˜ ∗ φt t x + φx x k˜ ∗ φx x + φt x x k˜ ∗ φt x x )d x

0

C3 n(t,v,φ) + n(t;vt ,φt ) + (2C C4 + a12 /4 + C7 β3 2 + C16 δ) k˜ ∗ φx 2 + k˜ ∗ φt x 2 + k˜ ∗ φt t x 2 + k˜ ∗ φx x 2 + k˜ ∗ φt x x 2 + λ3 (k˜ (t))2 ( φ0x 2 + φ1x 2 ) + λ1 (k˜ (t))2 + λ2 k˜ 2 (t) φ0x x 2 . −

(8.2.70)

Integrating (8.2.70) with respect to t, using (8.2.7)–(8.2.8), (8.2.65), (8.2.67)–(8.2.68) and (8.2.46)–(8.2.47) and taking δ(≤ δ1 ) and small enough, we deduce C3 t [n(τ ; v, φ) + n(τ ; vt , φt )]dτ L(t) + 2 0 t 2 1 i 2 i 2 ˜ ˜ + C8 k ∗ ∂t φx + k ∗ ∂t φx x dτ 0

i=0

i=0

≤ (β1 + β2 )α1 α2 /2 + k (0) 2(β2 + β3 ) + N 2 α 2 C5 /(β1 β 2 ) α1 α2 2 2

2

+ η0 2 =: α3 2 where C8 = Nα/(ββ0 k1∗ ) − 2C C4 − a12 /4 /2 > 0 (see Lemma 8.2.1).

(8.2.71)

378


Thus it follows from (8.2.43), (8.2.68) and (8.2.71) that for any t ∈ [0, t1 ), t M(τ, v, φ)dτ M(t; v, φ) + (2C3 /β1 ) 0

t 2 1 i 2 i 2 ˜ ˜ k ∗ ∂t φx + k ∗ ∂t φx x dτ + (4C6 C8 /β1 ) 0

i=0

2

i=0 2

≤ (4C6 α3 )/β1 = (α0 − α1 α2 ) .

(8.2.72)

Letting t → t1 in (8.2.72), we have M(t1 ; v, φ) ≤ (α0 − α1 α2 ) 2 < α0 2 which is contradictory to the definition of t1 , (8.2.50). Thus we conclude that t1 = Tm = +∞ and all the estimates above are valid for any t > 0. Note that M(t; v, φ) is equivalent to the third-order full energy E(t; v, φ) := E 1 (t; v, φ) + E 2 (t; v, φ) + E 3 (t; v, φ) + E 2 (t; vt , φt ) + E 3 (t; vt , φt ), that is, −1 C17 M(t; v, φ) ≤ E(t; v, φ) ≤ C17 M(t; v, φ), ∀t > 0.

(8.2.73)

On the other hand, it is easy to verify that −1 C18 M(t; u, θ )e2δt ≤ M(t; v, φ) ≤ C18 M(t; u, θ )e2δt , ∀t > 0.

(8.2.74)

In fact, note that {E 1 (t; v, φ), E 2 (t; v, φ), E 3 (t; v, φ), E 2 (t; vt , φt ), E 3 (t; vt , φt )} is equivalent to e2δt E 1 (t; u, θ ), e2δt E 2 (t; u, θ ), e2δt E 3 (t; u, θ ), e2δt E 2 (t; u t , θt ), e2δt E 3 (t; u t , θt ) . Thus (8.2.74) follows from (8.2.73). By (8.2.72) and (8.2.56), we have M(t; u, θ ) ≤ CM(t; v, φ)e−2δt ≤ Ce−2δt , ∀t > 0, 2 −2δt

2

u x x x ≤ C vx x x e

≤ CM(t; v, φ)e

−2δt

≤ Ce

(8.2.75) −2δt

, ∀t > 0.

(8.2.76)

By (8.2.6), we deduce d ˜ (k ∗ ∂ti φx )(t) 2 + (k˜ ∗ φx x )(t) 2 ≤ C ( (k˜ ∗ ∂ti φx )(t) 2 + ∂ti φx (t) 2 ) dt i=0 i=0 2 2 + (k˜ ∗ φx x )(t) + φx x (t) ) 1

≤C

1

1

( (k˜ ∗ ∂ti φx )(t) 2 + (k˜ ∗ φx x )(t) 2 + M(t; v, φ) .

i=0

Integrating (8.2.77) with respect to t, and exploiting (8.2.72), we finally obtain 1 i=0

(k˜ ∗ ∂ti φx )(t) 2 + (k˜ ∗ φx x )(t) 2 ≤ C

(8.2.77)


379

which, together with (8.2.10)–(8.2.12), implies 1 (k ∗ ∂ti θx )(t) 2 + (k ∗ θx x )(t) 2 ≤ Ce−2δt .

(8.2.78)

i=0

Differentiating (8.1.1)–(8.1.2) with respect to t respectively and using (8.2.75) and (8.2.78), we have u t t t (t) 2 ≤ CM(t; u, θ ) ≤ Ce−2δt , θt t (t) 2 ≤ C(M(t; u, θ ) + k ∗ θx x 2 ) ≤ Ce−2δt .

(8.2.79) (8.2.80)

Similarly, by (8.2.75)–(8.2.76) and (8.2.78)–(8.2.80), 1 1 i 2 ∂t (k ∗ θ )(t) H 2−i ≤ C (k ∗ ∂ti θx )(t) 2 + (k ∗ θx x )(t) 2 + k 2 (t) ≤ Ce−2δt . i=0

i=0

(8.2.81) Thus (u(t), θ (t)) and (v(t), φ(t)) are uniformly bounded in H 3 × H 2, therefore problem (8.1.1)–(8.1.4) admits a unique global solution (u(t), θ (t)) in H 3 × H 2 and the estimate (8.1.15) follows from (8.2.75)–(8.2.76) and (8.2.78)–(8.2.81) with C2 = 2δ. The proof of Theorem 8.1.1 is now complete.

8.3 Bibliographic Comments Since the pioneer work of Dafermos [74] on the existence, differentiability and asymptotic stability of solutions to the system of linear thermoelasticity, significant progress has been made on the mathematical aspects in this direction. Muñ˜ oz Rivera [274] established the decay rate of energy in one-dimensional linear thermoelasticity obeying Fourier’s law without any memory effect. Concerning the nonlinear one-dimensional thermoelastic model obeying Fourier’s law without any thermal memory, Slemrod [378] proved the global existence and asymptotic stability of small solutions with NeumannDirichlet ( u x |x=0,1 = θ |x=0,1 = 0) or Dirichlet-Neumann (i.e. (8.1.4)) boundary conditions. Racke and Shibata [352] proved the global existence and polynomial decay of small smooth solutions with Dirichlet-Dirichlet ( u|x=0,1 = θ |x=0,1 = 0) boundary conditions, and later for this type of boundary conditions, Racke, Shibata and Zheng [353] showed the exponential stability of small global smooth solutions by using a similar idea as in [274]. Munoz ˜ Rivera and Barreto [277] improved the results in [353] for small initial data (u 0 , u 1 ) in the H 2 × H 1 norm. Recently, Muñ˜ oz Rivera and Qin [279] (see also Chapter 7) established the global existence and exponential stability of small solutions to a nonlinear one-dimensional thermoelastic model obeying Fourier’s law with thermal memory subject to Dirichlet and mixed boundary conditions at the endpoints. For other thermoelastic models, we refer the readers to Section 7.3.

Chapter 9

Blow-up for the Cauchy Problem in Nonlinear One-dimensional Thermoelasticity 9.1 Introduction In this chapter we study the blow-up phenomena of solutions in a finite time to the following Cauchy problem with a non-autonomous forcing term and a thermal memory: u t t = au x x + bθθx + du x − mu t + f (t, u), cθt = κθθx x + g ∗ θx x + bu xt + pu x + qθθx

(9.1.1) (9.1.2)

subject to the initial conditions u(x, 0) = u 0 (x),

u t (x, 0) = u 1 (x),

θ (x, 0) = θ0 (x),

∀ ∈ R. ∀x

(9.1.3)

Here by u = u(x, t) and θ = θ (x, t) we denote the displacement and the temperature difference respectively. The function g = g(t) isthe relaxation kernel and the sign ∗ t denotes the convolution product, i.e., g ∗ y(·, t) = 0 g(t − τ )y(·, τ )dτ . The coefficients a, b, c are positive constants, while d, κ, p, q, m are non-negative constants. The function f = f (t, u) is a non-autonomous forcing term. The aim of this chapter is to establish the blow-up results for a nonlinear onedimensional thermoelastic system with a non-autonomous forcing term and a thermal memory when the heat flux obeys both Fourier’s law and Gurtin and Pipkin’s law and hence the results in [200] have been extended. We organize the rest of this paper as follows: we deal with two cases in Sections 9.2 and 9.3 where the relaxation functions take the forms of (9.2.1) and (9.3.1) respectively

382

Chapter 9. Blow-up for the Cauchy Problem

and the results of α = 0 are also established. Throughout this chapter, we assume that for any fixed t > 0, f (t, u) is the Fré´ chet derivative of some functional F(t, u) such that d F(t, u) = Ft (t, u) + f (t, u)u t . dt

(9.1.4)

9.2 Main Results – Case I In this section, we suppose that there exists a constant α > 0 such that g(t) ˜ = eαt g(t)

(9.2.1)

is a positive definite kernel. First we should note that there indeed exists a function g(t) ˜ to satisfy (9.2.1). To this end, we need Lemma 7.2.1. In fact, we first take g(t) ∈ C 1 [0, +∞) verifying g (t) = −γ g(t) + c0 e−δt , g(0) > c0 γ −1

(9.2.2)

with δ > 0, γ > 0 and c0 ≥ 0 being constants and define G(t) = g(t) +

2c0 −δt e . δ

(9.2.3)

Then taking γ0 = min[γ , δ/2], we have G (t) = −γ g(t) − c0 e−βt ≤ −γ γ0 G(t) whence

g(t) ≤ G(t) ≤ G(0)e−γγ0 t ≡: c1 e−γγ0 t

with c1 = g(0) +

2c0 δ

> 0. Let us put ˆ˜ = J1 (ω) = Re g(t)

+∞ 0

eαt g(t) cos ωtdt,

+∞

ˆ˜ = J2 (ω) = I m g(t) eαt g(t) sin ωtdt, 0 +∞ (α−δ)t I1 (ω) = e cos ωtdt, 0 +∞ I2 (ω) = e(α−δ)t sin ωtdt. 0

Thus when 0 < α < δ, we easily obtain I1 (ω) =

ω2 1 ω − I1 (ω), I2 (ω), I2 (ω) = δ−α δ−α (δ − α)2

(9.2.4)

9.2. Main Results – Case I

383

i.e.,

δ−α ω , I2 (ω) = . 2 2 (δ − α) + ω (δ − α)2 + ω2 By virtue of (9.2.2), (9.2.4) and integration by parts, we infer I1 (ω) =

(9.2.5)

ω J1 (ω) = (γ − α)JJ2 (ω) − c0 I2 (ω),

(9.2.6)

ω J2 (ω) = g(0) − (γ − α)JJ1 (ω) + c0 I1 (ω).

(9.2.7)

Inserting (9.2.5) into (9.2.6)–(9.2.7) gives that for 0 < α < min[γ γ0 , δ], g(0)(γ − α) c0 ω c0 (γ − α) − I2 (ω) + I1 (ω) (9.2.8) 2 2 2 2 (γ − α) + ω (γ − α) + ω (γ − α)2 + ω2 [g(0)(γ − α) − c0 ]ω2 + g(0)(γ − α)(δ − α)2 + c0 (γ − α)(δ − α) . = [(γ − α)2 + ω2 ][(δ − α)2 + ω2 ]

J1 (ω) =

Thus picking α so small that

0 < α ≤ min δ, γ0 , [g(0)γ − c0 ]/g(0) ,

then we readily get from (9.2.8) J1 (ω) ≥

c > 0, ∀ω ∈ (−∞, +∞) 1 + ω2

which together with Lemma 7.2.1 implies that g(t) ˜ is a strongly positive definite kernel verifying (9.2.1). In order to prove our results, we need to use Theorem 1.3.1 due to Kalantarov and Ladyzhenskaya [181] which was also proved in [200, 201, 220]. The energy for the system (9.1.1)–(9.1.2) is +∞ E(t) = [u 2t /2 + au 2x /2 − F(t, u) + cθ 2 /2]d x. (9.2.9) −∞

Put

v = eαt u, w = eαt θ.

(9.2.10)

Then the problem (9.1.1)–(9.1.3) can be transformed into vt t = avx x + bwx + dvx − (m − 2α)vt + (m − α)αv + f˜(t, v), cwt = κwx x + g˜ ∗ wx x + bvxt + ( p − bα)vx + qwx + cαw, t = 0 : v = u 0 (x) ≡ v0 (x), vt = u 1 (x) + αu 0 (x) ≡ v1 (x), w = θ0 (x) ≡ w0 (x)

(9.2.11) (9.2.12) (9.2.13)

with f˜(t, v) = eαt f (t, e−αt v). The energy of problem (9.2.11)–(9.2.13) is defined as follows +∞ ˜ ˜ v) + cw2 /2]d x E(t) = [−(m − α)αv 2 /2 + vt2 /2 + avx2 /2 − F(t, (9.2.14) −∞

384


˜ v) = e2αt F(t, e−αt v). Our main idea is that we only prove the solution to with F(t, the problem (9.2.11)–(9.2.13) blows up in a finite time, which also implies the blowup of solutions of problem (9.1.1)–(9.1.3). The following lemmas concern the results on ˜ ˜ ˜ E(t) ≤ E(0) ≤ 0 when we assume that E(0) ≤ 0. ˜ Lemma 9.2.1. We assume that E(0) ≤ 0, and (9.2.1) holds. Then if the following assumptions hold, (i) when α =

m 2

=

p b

> 0, d = 0 and κ ≥ 0, it holds that for any u ∈ R and t > 0, αu f (t, u) ≤ (α − m)α 2 u 2 + Ft (t, u),

(9.2.15)

(ii) when d + | p − bα| > 0 and κ ≥ 0, there exists a constant 2 > 0 such that

2 ≥ 2+ −[4ac(2α − m)(α − m) − (cd 2 + ( p − bα)2 )] + ≡ 8ac(3α − 2m) 0≤

2−

2

2

2

≡ {−[4ac(2α − m)(α − m) − (cd + ( p − bα) )] −

≤ 2 ≤

(≥ 0), if 0 ≤ m ≤ α,

(9.2.16) 2 }/[8ac(3α − 2m)]

2+ ,

2 > m − α, if m > 3α/2, ˜ 2 −4ac(3α − 2m) + (> m − α), if α < m < 3α/2,

2 ≥ ˜2+ ≡ 8ac 1/2 1 acα 2 + cd 2 + ( p − bα)2

2 ≥ , if m = 3α/2, 2 ac

(9.2.17) (9.2.18) (9.2.19)

verifying that for any u ∈ R and t > 0, 2( 2 + α − m)F(t, u) + αu f (t, u) ≤ Ft (t, u) + ( 2 + 2α − m)(α − m)αu 2 (9.2.20) with 2 = [4ac(2α − m)(α − m) − cd 2 − ( p − bα)2 ]2 + 16ac2d 2 (3α − m)(α − m), ˜ 2 = 16ac[acα 2 + cd 2 + ( p − bα)2 ]2 . Then ˜ ˜ E(t) ≤ E(0) ≤ 0, ∀t > 0.

(9.2.21)

Proof. An easy computation from (9.2.20) yields ˜ v) + e2αt Ft (t, u) − αv f˜(t, v) F˜t (t, v) = 2α F(t, = e2αt [2α F(t, u) + Ft (t, u) − αu f (t, u)].

(9.2.22)


385

Also we get from (9.1.4), (9.2.10) and (9.2.22) that d d ˜ F(t, v) = 2αe2αt F(t, e−αt v) + e2αt F(t, u) dt dt ˜ v) + e2αt Ft (t, u) + e2αt f (t, u)u t = 2α F(t,

˜ v) + e2αt Ft (t, u) + e2αt f (t, u)[−αeαt v + eαt vt ] = 2α F(t, = e2αt [2α F(t, u) + Ft (t, u) − αu f (t, u)] + f˜(t, v)vt = F˜t (t, v) + f˜(t, v)vt .

(9.2.23)

Using (9.2.9), (9.2.11)–(9.2.13) and (9.2.23), we get E˜ (t) = d

−

+∞

−∞ +∞

vx vt d x − (m − 2α)

−∞

+∞ −∞

vt2 d x

g˜ ∗ wx wx d x + ( p − bα)

−

+∞ −∞

+∞ −∞

F˜t (t, v)d x − κ

vx wd x + cα

+∞ −∞

+∞ −∞

w2 d x.

w2x d x

(9.2.24)

For case (i), we infer from (9.2.15) that ˜ v) − (α − m)α 2 v 2 − F˜t (t, v) ≤ 0 2α F(t, which with (9.2.14) further implies

E˜ (t) = −κ ≤−

+∞

−∞ +∞

−∞

+ ≤−

+∞ −∞

g˜ ∗ wx wx d x + cα

˜ −α g˜ ∗ wx wx d x + 2α E(t)

+∞

−∞ +∞

−∞

w2x d x −

+∞ −∞

+∞ −∞

w2 d x −

+∞

−∞

F˜t (t, v)d x

(vt2 + avx2 )d x

˜ v) − (α − m)α 2 v 2 − F˜t (t, v)]d x [2α F(t,

˜ g˜ ∗ wx wx d x + 2α E(t).

Consequently, ˜ ≤ E(0) ˜ E(t) −

+∞ t −∞

˜ ≤ E(0) + 2α

0 t

˜ )dτ. E(τ

0

That is,

2αt ˜ ˜ ≤ 0, ∀t > 0 E(t) ≤ E(0)e

which with (9.2.25) gives (9.2.21).

t

g˜ ∗ wx wx dτ d x + 2α

˜ )dτ E(τ

0

(9.2.25)

386


For case (ii), we deduce from (9.2.24) that for any 3 > 0, ˜ E˜ (t) ≤ 2[ 2 − (m − 2α)] E(t) +∞ [ 2 − (m − 2α)](m − α)αv 2 + −∞

˜ v) − F˜t (t, v) d x + 2[ 2 − (m − 2α)] F(t, +∞ w2 d x + [ 3 + cα − c 2 + c(m − 2α)] −∞

( p − bα)2 + + − a[ 2 − (m − 2α)] 4 2 4 3 +∞ +∞ w2x d x − g˜ ∗ wx wx d x. −κ d2

−∞

+∞ −∞

vx2 d x

−∞

(9.2.26)

In what follows, we shall show that the conditions (9.2.16)–(9.2.19) in case (ii) verify the inequalities 2( 2 + α − m)F(t, u) + αu f (t, u) ≤ Ft (t, u) + ( 2 + 2α − m)(α − m)αu 2 , (9.2.27) ( p − bα)2 d2 + − a[ 2 − (m − 2α)] ≤ 0, 4 2 4 3

3 = c( 2 − m + α) > 0.

(9.2.28) (9.2.29)

First, (9.2.16) is just (9.2.27). Second, we know from (9.2.16)–(9.2.19) that 2 > m − α and now choose 3 > 0 to verify (9.2.27), then substitution for 3 > 0 in (9.2.28) implies 4ac 23 +4ac(3α −2m) 22 +[4ac(2α −m)(α −m)−cd 2 −( p −bα)2 ] 2 +(m −α)cd 2 ≥ 0. (9.2.30) Noting that for 0 ≤ m ≤ α or m > 3α/2, 2 = [4ac(2α − m)(α − m) − cd 2 − ( p − bα)2 ]2 + 16ac2d 2 (3α − 2m)(α − m) ≥ [4ac(2α − m)(α − m) − cd 2 − ( p − bα)2 ]2 .

(9.2.31)

We obtain from (9.2.16)–(9.2.17) that 2 verifies for 0 ≤ m ≤ α or m > 3α/2, 4ac(3α − 2m) 22 + 4ac(2α − m)(α − m) − cd 2 − ( p − bα)2 2 + (m − α)cd 2 ≥ 0 which further gives (9.2.30) and (9.2.28). Noting that ˜ 2 = 16a 2c2 α 2 + 16ac[cd 2 + ( p − bα)2 ] > 16a 2c2 α 2 > 0, (9.2.18) implies that for α < m < 3α/2,

˜2+ > m − α,

(9.2.32)

4ac 22 − 4ac(2m − 3α) 2 + 4ac(m − α)(m − 2α) − [cd 2 + ( p − bα)2 ] ≥ 0 (9.2.33)


387

which, again give (9.2.30) and (9.2.28). Similarly, (9.2.19) satisfies that for m = 3α/2, 4ac 22 − [acα 2 + cd 2 + ( p − bα)2 ] ≥ 0

(9.2.34)

which implies (9.2.30) and (9.2.28). Now inserting (9.2.22) into (9.2.27) gives ˜ v) − F˜t (t, v) ≤ 0. [ 2 − (m − 2α)](m − α)αv 2 + 2[ 2 − (m − 2α)] F(t, Thus it follows from (9.2.26)–(9.2.29), (9.2.35) and (9.2.1) that t ˜ ˜ ˜ )dτ. E(t) ≤ E(0) + 2( 2 + 2α − m) E(τ

(9.2.35)

(9.2.36)

0

Hence

2( 2 +2α−m)t ˜ ˜ ≤ 0, ∀t > 0 E(t) ≤ E(0)e

which with (9.2.36) gives (9.2.21). The proof is complete.

Remark 9.2.1. It follows from the proof of Lemma 9.2.1 that assumptions (9.2.16)– (9.2.19) in case (ii) verify (9.2.30), so if we assume that there exists a constant 2 > 0 to verify (9.2.30), then the same conclusions as in Lemma 9.2.1 hold. ˜ Lemma 9.2.2. We assume that (9.2.1) holds and E(0) ≤ 0, κ > 0. If it holds that for any u ∈ R, t ≥ 0, 2(˜ 2 + α − m)F(t, u) + (˜ 2 + 2α − m)(m − α)αu 2 + ( p − bα)2 u 2 /(4κ) − Ft (t, u) − αu f (t, u) ≤ 0

(9.2.37)

where if d > 0 and if 0 ≤ m ≤ α or if m 2 − d 2 /a ≥ 0, α + ≤ α < m or 0 < α ≤ α − or if m 2 − d 2 /a < 0, 0 < α < m,

˜2 = ˜2∗ , (9.2.38) or

˜2 = ˜2∗ + α, if d = 0

with α + = (m + and

˜2∗ = then

m 2 − d 2 /a)/2, α − = (m −

(9.2.39)

m 2 − d 2 /a)/2

(2α − m)2 + d 2 /a − (2α − m) /2,

˜ ˜ E(t) ≤ E(0) ≤ 0, ∀t > 0.

(9.2.40)

Proof. First of all, note that (9.2.22) and (9.2.37) yield 2 2 ˜ v) + [ ˜2 − (m − 2α)](m − α)αv2 + ( p − bα) v − F˜t (t, v) ≤ 0. 2[ ˜2 − (m − 2α)] F(t, 4κ (9.2.41)

388


By virtue of (9.2.24), we easily get +∞ ˜ vx vt d x − (m − 2α) E (t) = d −

−∞ +∞ −∞

+∞ −∞

−∞

vt2 d x

g˜ ∗ wx wx d x − ( p − bα)

≤ [ ˜2 − (m − 2α)] −

+∞

+∞ −∞

˜ + ≤ 2[ ˜2 − (m − 2α)] E(t)

+∞

4 ˜ 2

+∞

−∞

2 d

4 ˜2

+∞

−∞

w2 d x −

−∞

F˜t (t, v)d x − κ

vwx d x + cα

−∞ +∞ d2

vt2 d x +

F˜t (t, v)d x + cα

−

+∞

vx2 d x +

+∞ −∞

−∞

+∞ −∞

w2x d x

w2 d x

( p − bα)2 4κ

+∞

−∞

v2 d x

g˜ ∗ wx wx d x

− a[ ˜2 − (m − 2α)]

2

+∞

+∞

+∞ −∞

vx2 d x

w dx − g˜ ∗ wx wx d x + c(m − α − ˜2 ) −∞ −∞ +∞

˜ v) 2(˜ 2 − (m − 2α)) F(t, + −∞

+ [ ˜2 − (m − 2α)](m − α)αv 2 +

( p − bα)2 v 2 − F˜t (t, v) d x. 4κ

(9.2.42)

It is not hard to find that (9.2.36)–(9.2.37) verify d2 − a(˜ 2 + 2α − m) ≤ 0, 4 ˜2

˜2 ≥ m − α.

(9.2.43) (9.2.44)

In fact, if d > 0, by (9.2.38) and (9.2.39), we know that ˜2 ≥ ˜2∗ implies (9.2.43)–(9.2.44) for the cases in (9.2.38). Noting that d = 0 implies ˜2∗ = m − 2α if m > 2α and ˜2∗ = 0 if m ≤ 2α, we can conclude (9.2.43)–(9.2.44). In a word, in all cases stated in (9.2.38)– (9.2.39), we always have (9.2.43)– (9.2.44). Thus (9.2.35)–(9.2.36) yield +∞ ˜ ˜ E (t) ≤ 2(˜ 2 + 2α − m) E(t) − g˜ ∗ wx wx d x. −∞

Hence ˜ ≤ E(0) ˜ E(t) + 2(˜ 2 + 2α − m)

t

˜ )dτ − E(τ

0

˜ ≤ E(0) + 2(˜ 2 + 2α − m)

t 0

t

+∞ −∞

˜ )dτ, E(τ

g˜ ∗ wx wx d x dτ (9.2.45)

0

i.e.,

2( ˜2 +2α−m)t ˜ ˜ ≤0 E(t) ≤ E(0)e

which along with (9.2.45) yields (9.2.40). The proof is complete.


389

We put

(t) =

+∞ −∞

v 2 (x, t)d x + β(t + t0 )2

where β ≥ 0 and t0 > 0 are to be determined later on. In the next lemma, we shall show that (t) verifies the assumptions of Theorem 1.3.1 by choosing suitable β ≥ 0, t0 > 0 and initial data (u 0 , u 1 , θ0 ). ˜ ˜ Lemma 9.2.3. We suppose that for any t ≥ 0, E(t) ≤ E(0) ≤ 0 and initial data u 0 ∈ H 2(R), u 1 ∈ H 1(R), θ0 ∈ H 1(R)

(9.2.46)

and one of the following assumptions (I) and (II)(a)–(c) holds, (I) when α =

m 2

=

p b

> 0 and d = 0, there exists a positive constant γ ≥ ( 1 + b2c/a − 1)/(4c), ∀u ∈ R, t > 0

(9.2.47)

u f (t, u) − 2(1 + 2γ )F(t, u) ≥ 0, ∀u ∈ R, t > 0

(9.2.48)

verifying and initial data satisfy

+∞ −∞

+∞ −∞

u 0 u 1 d x > 0,

˜ u 0 u 1 d x > 0, if E(0) < 0,

+∞

−∞

˜ u 20 d x > 0, if E(0) = 0,

(9.2.49) (9.2.50)

(II) when d + | p − bα| > 0, (a) if m ≥ 2α, then there exists a constant γ such that (9.2.47)–(9.2.48) and the following conditions (9.2.51)–(9.2.52) hold

+∞ +∞ −1 ˜ γ2 γ −1 1 − γ2 γ −1 (−2 E(0)) u 20 d x + 2 u 0 (u 1 + αu 0 )d x ≥ 0, −∞

−∞

˜ if E(0) < 0, +∞ +∞ u 20 d x > 0, 2 u 0 (u 1 + αu 0 )d x + γ2 γ −1 −∞

−∞

˜ if E(0) = 0,

(9.2.51) +∞ −∞

u 20 d x > 0,

with (m − 2α)2 /4 + 4γ 2 (m − α)α, γ2 = −(m − 2α)/2 − (m − 2α)2 /4 + 4γ 2 (m − α)α,

γ1 = −(m − 2α)/2 +

(9.2.52)

390


√ ac(2α−m)2 (1+ 1+b2 /(ac)) (b) if α ≤ m < 2α, there exist constants δ2 : 0 < δ2 ≤ δ2+ ≡ 2b2 + + (δ ) ≡ {−[8ac+4(2α−m)2ac/δ ]+ and γ > 0 or there exist constants δ > δ , γ ≥ γ 2 2 2 2 γ }/(32ac) such that for any u ∈ R, t ≥ 0, u f (t, u) − [2(1 + 2γ ) + (2α − m)2 /(2δ2)]F(t, u) ≥ 0

(9.2.53)

and initial data satisfy −1 1 ˜ 1 − γ2 γ −1 − [2(1 + 2γ ) + (2α − m)2 /(2δ2 )] E(0) 1 + 2γ +∞ +∞ × 2 u 0 (u 1 + αu 0 ) + γ2 γ −1 u 20 d x ≥ 0, −∞

˜ if E(0) < 0, +∞ u 20 d x > 0, 2 −∞

−∞

+∞

−∞

u 0 (u 1 + αu 0 )d x + γ2 γ −1

˜ if E(0) = 0,

(9.2.54)

+∞ −∞

u 20 d x > 0, (9.2.55)

with γ = 64ac(ac + b2 ) > 0, γ1 = 2 γ A 1 , γ2 = − 2 γ A 1 , A1 = δ2 − 2γ (α − m)α − (2α − m)2 (α − m)α,

(9.2.56)

(c) (1) if m < α, then there exist δ2 : 0 < δ2 < δ2+ and γ > 0 or there exist δ2 > δ2+ and γ ≥ γ + (δ2 ) > 0 such that for any u ∈ R, t ≥ 0, u f (t, u) + 2γ (α − m)αu 2 + (2α − m)2 (α − m)αu 2 /(4δ2 ) − 2(1 + 2γ )F(t, u) ≥ 0 and initial data satisfy (9.2.54) and (9.2.55) with γ1 =

√

(9.2.57)

√ 2δ2 , γ2 = − 2δ2 .

(2) if m < α, then

√ (a) when δ2+ ≥ δ20 ≡ (m + 2α) (m + α)α/2, there exist constants δ2 > 0 and γ > 0 verifying δ20 ≤ δ2 ≤ δ2+ , 0 < γ ≤ γ + (δ2 ) ≤ γˆ (δ2 )

(9.2.58)

or there are constants δ2 > δ2+ with γ + (δ2 ) ≤ γˆ (δ2 ) and γ > 0 verifying γ + (δ2 ) ≤ γ ≤ γˆ (δ2 )

(9.2.59)


391

and initial data satisfy −1 1 ˜ [2(1 + 2γ ) + (2α − m)2 /2δ2 ] E(0) (9.2.60) 1 − γ2 γ −1 − 1 + 2γ +∞ +∞ ˜ u 0 (u 1 + αu 0 )d x + γ2 γ −1 u 20 d x > 0, if E(0) < 0, A1 > 0, × 2

+∞

−∞ +∞ −∞

−∞

−∞

˜ u 0 (u 1 + αu 0 )d x ≥ 0, if E(0) < 0, A1 = 0, u 20 d x > 0, 2

+∞

−∞

u 0 (u 1 + αu 0 )d x + γ2 γ −1

(9.2.61)

+∞ −∞

u 20 d x > 0,

˜ i f E(0) = 0, A1 > 0, +∞ +∞ 2 ˜ u 0 d x > 0, u 0 (u 1 + αu 0 )d x > 0, if E(0) = 0, A1 = 0

−∞

−∞

with γ1 =

(9.2.62) (9.2.63)

2 γ A 1 , γ2 = − 2 γ A 1 ,

A1 = δ2 − 2γ (α − m)α − (2α − m)2 (α − m)α

(9.2.64)

or (b) when δ2+ < δ20 , there exist constants δ2 > 0 and γ > 0 with γ + (δ2 ) ≤ γˆ (δ2 ) verifying γ + (δ2 ) ≤ γ ≤ γˆ (δ2 )

(9.2.65)

and initial data satisfy (9.2.60)–(9.2.63). Moreover, δ2 and γ in the cases of (c)(1)– (c)(2)(b) verify u f (t, u) − 2(1 + 2γ )F(t, u) − (2α − m)2 /(2δ2 )F(t, u) ≥ 0.

(9.2.66)

Then for β > 0 small enough and suitable t0 > 0 or β = 0, there exist constants C1 ≥ 0 and C2 ≥ 0 such that (1.3.1)–(1.3.5) or (1.3.1), (1.3.6)–(1.3.8) hold. Proof. An easy computation yields

(t) = 2 and

(t) = 2

+∞ −∞ +∞

−∞

vvt d x + β(t + t0 )

(9.2.67)

+ vvt t )d x + β .

(9.2.68)

(vt2

By the Cauchy inequality and the Hö¨ lder inequality, we derive from (9.2.37)

2 1 1 +∞ +∞ +∞ 2 2 2 2 2 vvt d x + β(t + t0 ) ≤ v dx vt d x + β(t + t0 ) β −∞ −∞ −∞ +∞ ≤ (t) vt2 d x + β −∞

392


which, together with (9.2.67) and (9.2.68), yields (t) (t) − (1 + γ )( (t))2

2 +∞ +∞ 2 = 2 (t) (vt + vvt t )d x + β − 4(1 + γ ) vvt d x + β(t + t0 ) −∞ −∞ +∞

+∞ 2 ≥ 2 (t) −(1 + 2γ ) vt d x + β + vvt t d x . (9.2.69) −∞

−∞

Inserting (9.2.11) into (9.2.69), integrating by parts and recalling that +∞ +∞ 2 ˜ ˜ v) − cw2 ]d x, vt d x = 2 E(t) + [(m − α)αv 2 − avx2 + 2 F(t, −∞

−∞

we have (t) (t) − (1 + γ )( (t))2 +∞ 2 vt d x + β − ≥ 2 (t) − (1 + 2γ ) − (m − 2α)

−∞ +∞

−∞

(9.2.70) +∞ −∞

vvt d x + (m − α)α

(avx2 + bvx w)d x +∞

−∞

2

v dx +

+∞

−∞

˜ v f (t, v)d x .

When (I) holds, noting that (1 + 2γ )c − 1 ≥ 0 for 1 = b2/8aγ > 0, we get from (9.2.70) (t) (t) − (1 + γ )( (t))2

˜ − (1 + 2γ )β − 2γ α(m − α) ≥ 2 (t) −2(1 + 2γ ) E(t)

+∞ −∞

v2 d x − b

+∞

−∞

vx wd x

+∞ +∞ +∞ ˜ vx2 d x − (m − 2α) v2 d x + [v f˜(t,v) − 2(1 + 2γ ) F(t,v)]d x + 2aγ −∞ −∞ −∞

+∞ ˜ − (1 + 2γ )β − 2γ α 2 (t) + [(1 + 2γ )c − 1 ] w2 d x ≥ 2 (t) −2(1 + 2γ ) E(0) −∞

+∞ +∞ b2 2 ˜ ˜ + 2aγ − v dx + [v f (t,v) − 2(1 + 2γ ) F(t,v)]d x 4 1 −∞ x −∞ ˜ − (1 + 2γ )β − 2γ α 2 (t)]. ≥ 2 (t)[−2(1 + 2γ ) E(0)

(9.2.71)

˜ If E(0) < 0, we pick β > 0 and t0 > 0 in (9.2.71) so small that ˜ 0 < β ≤ −2 E(0), 0 < t0 < [1 +

1 + 4αβ −1

+∞

−∞

u 0 u 1 d x]/2

(9.2.72)

which with (9.2.49) gives (1.3.1)–(1.3.5) with C1 = 0, C2 = 4γ α 2 and γ1 = 2γ α, γ2 = ˜ −2γ α. If E(0) = 0, then we take β = 0 in (9.2.71) and can use (9.2.50) to derive (1.3.1)–(1.3.5) with C1 = 0, C2 = 4γ α 2 and γ1 = 2γ α, γ2 = −2γ α.


393

When (II)(a) holds, we choose β to verify (9.2.72) to get from (9.2.70) for δ1 = b2 /(8aγ ), (t) (t) − (1 + γ )( (t))2

˜ ≥ 2 (t) −2(1 + 2γ ) E(0) − β(1 + 2γ ) + 2aγ

+∞

+∞

+∞ −∞

vx2 d x − b

+∞ −∞

vx wd x

˜ v)]d x vvt d x + [v f˜(t, v) − 2γ (m − α)αv 2 − 2(1 + 2γ ) F(t, −∞ −∞ +∞ 2 w dx + c(1 + 2γ ) −∞

+∞ ˜ w2 d x ≥ 2 (t) −2(1 + 2γ ) E(0) − β(1 + 2γ ) + [c(1 + 2γ ) − δ1 ] − (m − 2α)

−∞

+∞ +∞ b2 + 2aγ − vx2 d x + (2α − m) vvt d x − 2γ (m − α)α (t) 4δ1 −∞ −∞ +∞ ˜ v)]d x [v f˜(t, v) − 2(1 + 2γ ) F(t, + −∞

+∞ ˜ w2 d x ≥ 2 (t) −(1 + 2γ )(2 E(0) + β) + [c(1 + 2γ ) − δ1 ] −∞

+∞ b2 + 2aγ − vx2 d x − (m − 2α)/2 (t) − 2γ (m − α)α (t) δ1 −∞ +∞ ˜ v)]d x [vv f˜(t, v) − 2(1 + 2γ ) F(t, + −∞ ˜ ≥ 2 (t) − (1 + 2γ )(2 E(0) + β) − (m − 2α)/2 (t) − 2γ (m − α)α (t) . (9.2.73) ˜ If E(0) < 0, we may pick β > 0 and t0 > 0 in (9.2.73) so small that ˜ 0 < β ≤ −2 E(0), +∞ +∞ 1 + 1 − γ2 γ −1 β −1 [γ γ2 γ −1 −∞ u 20 d x + 2 −∞ u 0 (u 1 + αu 0 )d x 0 < t0 < (9.2.74) −γ γ2 γ −1 with γ2 in (9.2.51). Then (1.3.1)–(1.3.5) follow from (9.2.51) and (9.2.73) with C1 = ˜ (m − 2α)/2 ≥ 0, C2 = 4γ (m − α)α > 0. If E(0) = 0, then we take β = 0 in (9.2.73) and can derive (1.3.1)–(1.3.5) from (9.2.52) with C1 = (m − 2α)/2, C2 = 4γ (m − α)α > 0. When (II)(b) holds, noting that for any δ2 > 0, +∞ +∞ +∞ vvt d x ≤ δ2 v 2 d x + (2α − m)2 /(4δ2 ) vt2 d x (2α − m) −∞

−∞

≤ δ2 (t) + (2α − m)2 /(4δ2 )

+∞

−∞

−∞

vt2 d x,

394


we derive from (9.2.70) that (t) (t) − (1 + γ )( (t))2 ˜ ≥ 2 (t) − 2(1 + 2γ ) E(0) − β(1 + 2γ )

− [δ2 + 2γ (m − α)α] (t) − (2α − m)2 /(4δ2 ) + [c(1 + 2γ ) − δ1 ] +

+∞ −∞

+∞

−∞

+∞ −∞

w2 d x + (2aγ − b 2 /4δ1 )

˜ v)]d x [v f˜(t, v) − 2(1 + 2γ ) F(t,

vt2 d x +∞

−∞

vx2 d x

˜ − β(1 + 2γ ) ≥ 2 (t) − [2(1 + 2γ ) + (2α − m)2 /(2δ2 )] E(0) − [δ2 + 2γ (m − α)α + (2α − m)2 (m − α)α/(4δ2 )] (t) +∞ w2 d x + [c(1 + 2γ ) − δ1 + (2α − m)2 c/(4δ2)] + [2aγ − b2 /4δ1 + (2α − m)2 a/(4δ2)] +

+∞ −∞

−∞ +∞

−∞

vx2 d x

˜ v) − (2α − m)2 /(2δ2 ) F(t, ˜ v)]d x . [v f˜(t, v) − 2(1 + 2γ ) F(t,

(9.2.75)

Obviously, (9.2.53) amounts to ˜ v) − (2α − m)2 /(2δ2 ) F(t, ˜ v) ≥ 0. v f˜(t, v) − 2(1 + 2γ ) F(t,

(9.2.76)

Next we show that if we pick δ1 = b2 /[8aγ + (2α − m)2 a/δ2], which amounts to 2aγ + (2α − m)2 a/(4δ2) = b 2 /(4δ1 ),

(9.2.77)

then δ2 will satisfy c(1 + 2γ ) + (2α − m)2 c/(4δ2) − δ1 ≥ 0, i.e., 16acγ 2 + [8ac + 4(2α − m)2 ac/δ2 ]γ + (2α − m)2 ac/δ2 + (2α − m)4 ac/(4δ22) − b 2 ≥ 0. (9.2.78) In fact, we note that γ ≡ [8ac + 4(2α − m)2 ac/δ2]2 − 64ac[(2α − m)2 ac/δ2 + (2α − m)4 ac/(4δ22) − b2 ] = 64ac(ac + b2 ) > 0, γ + (δ2 ) = {−[8ac + 4(2α − m)2 ac/δ2] + and

γ − (δ2 ) = {−[8ac + 4(2α − m)2 ac/δ2] −

γ }/(32ac),

γ }/(32ac)


395

are the roots of (9.2.78) where the equality holds with γ − (δ2 ) < 0. Moreover, using δ2 ≡ 16a 2c2 (2α − m)4 [1 + b2/ac] > 16a 2c2 (2α − m)4 > 0, we easily deduce that δ2− = ac(2α − m)2 [1 − and δ2+ = ac(2α − m)2 [1 −

1 + b2 /ac]/(2b 2) < 0 1 + b2 /ac]/(2b 2) > 0

which along with (9.2.78) imply that γ + (δ2 ) ≤ 0, i f 0 < δ2 ≤ δ2+

(9.2.79)

γ + (δ2 ) > 0, i f δ2 > δ2+ .

(9.2.80)

and

Thus (9.2.77)–(9.2.78) follow from assumptions in II(b) and (9.2.79)–(9.2.80). Now inserting (9.2.76)–(9.2.78) into (9.2.75) gives ˜ (t) (t) + (1 + γ )[ (t)]2 ≥ 2 (t){−[2(1 + 2γ ) + (2α − m)2 /(2δ2 )] E(0) (9.2.81) −β(1 + 2γ ) − A1 (t)}. ˜ If E(0) < 0, then we can choose β > 0 and t0 > 0 so small that 1 ˜ [2(1 + 2γ ) + (2α − m)2 /2δ2 ] E(0), (9.2.82) 1 + 2γ +∞ +∞ 1 + 1 − γ2 γ −1 β −1 (γ γ2 γ −1 −∞ u 20 d x + 2 −∞ u 0 (u 1 + αu 0 )d x)

0 0. When (c)(1) holds, similarly to (9.2.75), we easily derive ˜ − β(1 + 2γ ) (t) (t) − (1 + γ )( (t))2 ≥ 2 (t) − 2(1 + 2γ ) E(0) +∞ +∞ v 2 d x − (2α − m)2 /(4δ2 ) vt2 d x − δ2 −∞

+ [c(1 + 2γ ) − δ1 ] +

+∞ −∞

+∞

−∞

−∞

w2 d x + (2aγ − b 2 /(4δ1 ))

+∞ −∞

˜ v)]d x [v f˜(t, v) + 2γ (α − m)αv 2 − 2(1 + 2γ ) F(t,

vx2 d x

396


˜ ≥ 2 (t) − [2(1 + 2γ ) + (2α − m)2 /(2δ2 )] E(0) − β(1 + 2γ ) − δ2 (t) +∞ w2 d x + [2aγ − b2 /(4δ1 ) + [c(1 + 2γ ) − δ1 + (2α − m)2 c/(4δ2)]

+∞

+∞

−∞

˜ v) [v f˜(t, v) − 2(1 + 2γ ) F(t, + 2γ (α − m)αv2 + (2α − m)2 (α − m)αv 2 /(4δ2 )]d x . (9.2.84)

+ (2α − m)2 a/4δ2 ]

−∞

vx2 d x +

−∞

First, note that (9.2.57) yields ˜ v) + 2γ (α − m)αv2 + +(2α − m)2 (α − m)αv2 /(4δ2 ) ≥ 0. v f˜(t, v) − 2(1 + 2γ ) F(t, (9.2.85) Second, similarly to case II(b), we choose δ1 = b2/(8aγ + (2α − m)2 a/δ2) to verify (9.2.77) and assumptions in (c)(1) and (9.2.79)–(9.2.80) also lead to (9.2.77)–(9.2.78). ˜ If E(0) < 0, then we choose β > 0 and t0 > 0 so small that (9.2.82)–(9.2.83) hold. ˜ If E(0) = 0, we take β = 0 and (1.3.1)–(1.3.5) follow from (9.2.54)–(9.2.55). When (c)(2)(a) holds, similarly to (9.2.75), we have ˜ (t) (t) − (1 + γ )( (t))2 ≥ 2 (t) − [2(1 + 2γ ) + (2α − m)2 /(2δ2 )] E(0) +∞ +∞ v 2 d x + [2aγ + (2α − m)2 a/(4δ2 ) − b2 /(4δ1 )] vx2 d x −β(1 + 2γ ) − A1 −∞

+[c(1 + 2γ ) − δ1 + (2α − m)2 c/(4δ2)] +

+∞ −∞

+∞ −∞

−∞

w2 d x

˜ v) − (2α − m)2 /(2δ2 ) F(t, ˜ v)]d x . [v f˜(t, v) − 2(1 + 2γ ) F(t,

(9.2.86)

We know from (9.2.79) and (9.2.80) that when 0 < δ2 ≤ δ2+ , γ > 0 or δ2 > δ2+ , γ ≥ γ + (δ2 ), (9.2.77)–(9.2.78) hold. The unique difference here from case II(b) is that we have to verify A1 ≥ 0 (9.2.87) which is automatically satisfied for case II(b). In fact, it is easy to verify that assump+∞ tions in (c)(2)(a) satisfy (9.2.87). Noting that −∞ v 2 d x ≤ (t) and using (9.2.87) and (9.2.77)–(9.2.78), we arrive at (t) (t) − (1 + γ )( (t))2 (9.2.88) ˜ − β(1 + 2γ ) − A1 (t) . ≥ 2 (t) − [2(1 + 2γ ) + (2α − m)2 /(2δ2 )] E(0) ˜ If E(0) < 0, then we choose β > 0 and t0 > 0 so small that (9.2.82) and +∞ +∞ 1 + 1 − γ2 γ −1 β −1 [γ γ2 γ −1 −∞ u 20 d x + 2 −∞ u 0 (u 1 + αu 0 )d x] 0 < t0 < , −γ γ2 γ −1 (9.2.89) if A1 > 0, 0 < t0 , if A1 = 0

(9.2.90)


397

˜ hold. Thus (1.3.1)–(1.3.5) follow from (9.2.58)–(9.2.59) and (9.2.86). If E(0) = 0, we take β = 0 and can derive (1.3.1), (1.3.6)–(1.3.8) from (9.2.62)–(9.2.63) and (9.2.88). When (c)(2)(b) holds, the assumptions in (c)(2)(b) also satisfy (9.2.87) and (9.2.77)– (9.2.78). The rest of the proof is similar to that of case (c)(2)(a). The proof is now complete. Now we are in a position to state one main result in this chapter. Theorem 9.2.1. We assume that assumptions in Lemma 9.2.1 or in Lemma 9.2.2 and assumptions in Lemma 9.2.3 hold, then the solution v(t) in L 2 (R) to problem (9.2.11)– (9.2.13) blows up in a finite time, that is, there exists some t1 > 0 such that

+∞

lim

t →t1− −∞

v 2 (x, t)d x = +∞

(9.2.91)

and further the solution u(t) in L 2 (R) to problem (9.1.1)–(9.1.3) blows up in a finite time, that is, +∞ lim u 2 (x, t)d x = +∞. (9.2.92) t →t1− −∞

Proof. By Lemma 9.2.1 or Lemma 9.2.2, we get ˜ ˜ E(t) ≤ E(0) ≤0 which along with Lemma 9.2.3 implies (1.3.1)–(1.3.5) or (1.3.1), (1.3.6)–(1.3.8) hold. Thus we can derive (9.2.91)–(9.2.92) from (9.2.1) and Theorem 1.3.1. The proof is complete. Now we study the problem (9.1.1)–(9.1.3), that is, the case of α = 0. To this end, we summarize assumptions in Lemmas 9.2.1–9.2.3 which are satisfied by α = 0. Lemma 9.2.4. We assume that E(0) ≤ 0, and g(t) is a positive definite kernel. Then if one of the following assumptions holds, (1) when α = m = p = d = 0 and κ ≥ 0, it holds that for any u ∈ R and t > 0, 0 ≤ Ft (t, u),

(9.2.93)

(2) when d + p > 0, m = α = 0 and κ ≥ 0, there exists a constant 2 > 0 such that

1 cd 2 + p 2 1/2

2 ≥ 2 ac

(9.2.94)

verifying that for any u ∈ R and t > 0, 2 2 F(t, u) ≤ Ft (t, u),

(9.2.95)

398


(3) when κ > 0, there exists a constant 2 > 0 such that m 2 + d 2 /a + m /2,

˜2 =

(9.2.96)

2( 2 − m)F(t, u) + p2 u 2 /(4κ) − Ft (t, u) ≤ 0

(9.2.97)

˜ 2 = 16ac[cd 2 + p 2 ]2 , then with 2 = [4acm 2 − cd 2 − p 2 ]2 + 16ac2d 2 m 2 , E(t) ≤ E(0) ≤ 0, ∀t > 0.

(9.2.98)

Proof. The conclusions follow from (9.2.15) in (i), (9.2.19)–(9.2.20) in (ii) of Lemma 9.2.1 and (9.2.37)–(9.2.39) of Lemma 9.2.2 where α = 0. Here we have used the fact that ˜ when α = 0, E(t) = E(t), ∀t ≥ 0. The proof is now complete. Lemma 9.2.5. We suppose that g(t) is a positive definite kernel and for any t ≥ 0, E(t) ≤ E(0) ≤ 0 and initial data u 0 ∈ H 2(R), u 1 ∈ H 1(R), θ0 ∈ H 1(R)

(9.2.99)

and one of the following assumptions (I) and (II) holds, (I) when α = m = p = d = 0, there exists a positive constant γ ≥ 1 + b2 c/a − 1 /(4c), ∀u ∈ R, t > 0

(9.2.100)

verifying u f (t, u) − 2(1 + 2γ )F(t, u) ≥ 0, ∀u ∈ R, t > 0 and initial data satisfy

+∞ −∞

+∞

−∞

u 0 u 1 d x > 0,

u 0 u 1 d x > 0, if E(0) < 0, +∞

−∞

u 20 d x > 0, if E(0) = 0,

(9.2.101)

(9.2.102) (9.2.103)

(II) when d + p > 0, α = 0, then there exists a constant γ such that (9.2.101)–(9.2.102) hold and

+∞ +∞ u 20 d x + 2 u 0 u 1 d x ≥ 0, 1 − γ2 γ −1 (−2E(0))−1 γ2 γ −1 −∞

if E(0) < 0, +∞ +∞ 2 −1 u 0 d x > 0, 2 u 0 u 1 d x + γ2 γ −∞

−∞

if E(0) = 0,

−∞

(9.2.104) +∞ −∞

u 20 d x > 0, (9.2.105)

with γ1 = 0, γ2 = −m. Then for β > 0 small enough and suitable t0 > 0 or β = 0, there exist constants C1 ≥ 0 and C2 ≥ 0 such that (1.3.1)–(1.3.5) or (1.3.1), (1.3.6)–(1.3.8) hold.

9.3. Main Results – Case II

399

Proof. The conclusions follow from (9.2.46)–(9.2.50) in (I) and (9.2.51)–(9.2.52) in (II) of Lemma 9.2.3. The proof is complete. Based on Lemmas 9.2.3–9.2.4, we easily prove the following result. Theorem 9.2.2. Assume that assumptions in Lemmas 9.2.4–9.2.5 hold. Then the solution u(x, t) in L 2 (R) to problem (9.1.1)–(9.1.3) blows up in a finite time. Remark 9.2.2. When g(t) ≡ 0 and α = 0, the problem (9.2.11)–(9.2.13) is reduced to problem (9.1.1)–(9.1.3). Thus our results extend those in [200].

9.3 Main Results – Case II In this section, we suppose that there exists a constant α > 0 such that g(t) ˆ = e−αt g(t)

(9.3.1)

is a positive definite kernel. Indeed there exists a kernel g(t) ˆ verifying (9.3.1). For example, if we take g(t) = e−λt with λ > α, then we can easily compute by integrating by parts, +∞ ˆ ˆ J1 (ω) = Re g(t) ˆ = e−αt g(t) cos ωtdt 0

c0 α+λ ≥ = 2 , ∀ω ∈ R ω + (λ − α)(λ + α) 1 + ω2

(9.3.2)

with 0 < c0 ≤ min[ λα+λ ˆ = 2 −α 2 , λ + α]. Thus it follows from Lemma 7.2.1 that g(t)

e−αt g(t) = e−(λ+α)t is a strongly positive definite kernel verifying (9.3.1). Corresponding to (9.3.1), we introduce v( ˆ x, t) = e−αt u(x, t), w( ˆ x, t) = e−αt θ (x, t).

(9.3.3)

Then vˆ and wˆ satisfy the system vˆt t = a vˆ x x + bwˆ x + d vˆ x − (m + 2α)vˆt − (m + α)α vˆ + fˆ(t, v), ˆ cwˆ t t = κ wˆ x x + gˆ ∗ wˆ x x + bvˆ xt + (αb + p)vˆ x + q wˆ x − αcw, t = 0 : vˆ = vˆ0 (x) = u 0 (x), vˆt = vˆ1 (x) = u 1 (x) − αu 0 (x), wˆ = wˆ 0 (x) = θ0 (x) ˆ The corresponding energy is with fˆ(t, v) = e−αt f (t, eαt v). +∞ ˆ ˆ v) E(t) = (m + α)α vˆ 2 /2 + vˆt2 /2 + a vˆ x2 /2 − F(t, ˆ + cwˆ 2 /2 d x −∞

with

(9.3.4) (9.3.5) (9.3.6)

(9.3.7)

ˆ v) ˆ (9.3.8) F(t, ˆ = e−2αt F(t, eαt v). ˆ ˆ The following lemmas concern the results on E(t) ≤ E(0) ≤ 0 when we suppose that ˆ E(0) ≤ 0.

400


ˆ Lemma 9.3.1. We suppose (9.3.1) holds and E(0) ≤ 0, κ ≥ 0. If one of the following conditions holds, (i) when κ ≥ 0, and there exists a constant 2 > 0 such that

2 ≥ ˆ2+ ≡

4ac(2m + 3α) +

ˆ 2

8ac

= m + 3α/2 +

ˆ 2

(9.3.9)

8ac

such that Ft (t, u) + αu f (t, u) + (m + α)α( 2 − m − 2α)u 2 ≥ 2( 2 − m − α)F(t, u) (9.3.10) ˆ 2 = 16a 2c2 α 2 + 16ac[cd 2 + ( p + bα)2 ] > 0

where

or

(ii) when κ > 0, and there exist a constant ˆ2 such that m + 2α +

ˆ2 ≥ ˆˆ2+ ≡

(m + 2α)2 + d 2 /a (≥ m + 2α) 2

(9.3.11)

such that 2(ˆ 2 − m − α)F(t, u) − (ˆ 2 − m − 2α)(m + α)αu 2 + ( p + αb)2 u 2 /(4κ) (9.3.12) −F Ft (t, u) − αu f (t, u) ≤ 0, then ˆ ˆ E(t) ≤ E(0) ≤ 0, ∀t > 0.

(9.3.13)

Proof. A direct computation from (9.3.8) gives Fˆt (t, v) ˆ = e−2αt [−2α F(t, u) + Ft (t, u) + αu f (t, u)]

(9.3.14)

which along with (9.1.4) yields d ˆ ˆ vˆt F(t, v) ˆ = e−2αt [−2α F(t, u) + Ft (t, u)] + fˆ(t, v) dt = Fˆt (t, v) ˆ + fˆ(t, v) ˆ vˆt .

(9.3.15)

Thus we use (9.3.4)–(9.3.5), (9.3.7) and (9.3.15) to get Eˆ (t) = d

−

+∞

−∞ +∞ −∞

vˆ x vˆt d x − (m + 2α)

+∞ −∞

gˆ ∗ wˆ x wˆ x d x + ( p + bα)

vˆt2 d x −

+∞ −∞

+∞ −∞

ˆ dx − κ Fˆt (t, v)

vˆ x wd ˆ x − cα

+∞

−∞

+∞ −∞

wˆ 2 d x.

wˆ 2x d x

(9.3.16)


401

If (i) holds, then we infer from (9.3.16) +∞ 2 2 2 ˆ E (t) ≤ [ 2 − (m + 2α)] vˆt d x + [d /(4 2 ) + ( p + bα) /(4 3 )] + ( 3 − αc)

+∞ −∞

−∞

+∞

2

wˆ d x − κ

−∞

wˆ 2x d x

−

+∞ −∞

ˆ dx − Fˆt (t, v)

ˆ = 2[ 2 − (m + 2α)] E(t) + d 2 /(4 2 ) + ( p + bα)2 /(4 3 ) − a[ 2 − (m + 2α)] + 3 − c[ 2 − (m + α)] +

+∞

−∞

+∞ −∞

2

wˆ d x − κ

+∞ −∞

+∞ −∞

wˆ 2x d x

−

+∞

−∞ +∞

−∞

vˆ x2 d x

gˆ ∗ wˆ x wˆ x d x

vˆ x2 d x +∞

−∞

gˆ ∗ wˆ x wˆ x d x

ˆ v) − (m + α)α[ 2 − (m + 2α)]vˆ 2 + 2[ 2 − (m + 2α)] F(t, ˆ − Fˆt (t, v) ˆ d x. (9.3.17)

In what follows, we prove that conditions (9.3.9)–(9.3.10) verify the inequalities ˆ v), ˆ + (m + α)α[ 2 − (m + 2α)] ˆ 2 ≥ 2[ 2 − (m + 2α)] F(t, ˆ Fˆt (t, v) 2

2

(9.3.18)

d /(4 2 ) + ( p + αb) /(4 3 ) − a[ 2 − (m + 2α)] ≤ 0,

3 = c[ 2 − (m + α)] > 0,

(9.3.19) (9.3.20)

2 ≥ m + 2α.

(9.3.21)

Clearly, (9.3.10) and (9.3.14) give (9.3.18). Second, pick 3 > 0 to satisfy (9.3.20), that is, substitution for 3 in (9.3.19) implies 4ac 23 −4ac(2m +3α) 22 +[4ac(m +α)(m +2α)−( p +αb)2 −cd 2 ] 22 +cd 2 (m +α) ≥ 0. (9.3.22) Let g( 2 ) = 4ac 22 − 4ac(2m + 3α) 2 + 4ac(m + α)(m + 2α) − ( p + bα)2 − cd 2 . Then since ˆ 2 = 16a 2c2 (2m + 3α)2 − 16ac[4ac(m + α)(m + 2α) − ( p + bα)2 − cd 2 ] = 16a 2c2 α 2 + 16ac[( p + bα)2 + cd 2 ] > 16a 2c2 α 2 > 0,

(9.3.23)

we infer from (9.3.9) and (9.3.23) g( 2 ) ≥ 0, ˆ2+ > m + 2α which implies (9.3.19)–(9.3.21). Thus it follows from (9.3.17)–(9.3.21) that +∞ ˆ − gˆ ∗ wˆ x wˆ x d x Eˆ (t) ≤ 2[ 2 − m − 2α] E(t) −∞

(9.3.24)

402


or, by (9.3.1), ˆ ˆ E(t) ≤ E(0) + 2[ 2 − m − 2α]

t

0

ˆ )dτ. E(τ

(9.3.25)

Hence

2( 2 −m−2α)t ˆ ˆ ≤ 0, ∀t > 0. E(t) ≤ E(0)e +∞ +∞ If (ii) holds, noting that −∞ vˆ x wd ˆ x = − −∞ vˆ wˆ x d x, we derive from (9.3.16) and (9.3.7), +∞ +∞ +∞ Eˆ (t) ≤ [ 2 − m − 2α] vˆt2 d x + d 2 /(4 ˆ2 ) vˆ x2 d x + ( p + αb)2 /(4κ) vˆ 2 d x

−

+∞ −∞

−∞

−∞

−∞

[ Fˆt (t, v) ˆ + cα wˆ 2 + gˆ ∗ wˆ x wˆ x ]d x

ˆ + [d 2 /(4 ˆ2 ) − a(ˆ 2 − m − 2α)] ≤ 2[ 2 − m − 2α] E(t) −

+∞ −∞

[c( ˆ2 − m − α)wˆ 2 + gˆ ∗ wˆ x wˆ x ]d x +

+∞ −∞

+∞

−∞

vˆ x2 d x

ˆ v) [2(ˆ 2 − m − 2α) F(t, ˆ

ˆ x. − ( 2 − m − 2α)(m + α)α vˆ 2 + ( p + αb)2 vˆ 2 /(4κ) − Fˆt (t, v]d

(9.3.26)

We will prove that (9.3.11)–(9.3.12) verify ˆ v) 2(ˆ 2 − m − 2α) F(t, ˆ − (ˆ 2 − m − 2α)(m + α)α vˆ 2 + ( p + αb)2 vˆ 2 /(4κ) − Fˆt (t, vˆ ≤ 0, 2

(9.3.27)

d /(4 ˆ2 ) − a(ˆ 2 − m − 2α) ≤ 0,

(9.3.28)

ˆ2 ≥ m + 2α.

(9.3.29)

In fact, (9.3.12) and (9.3.14) imply (9.3.27) and (9.3.11) implies (9.3.28) and (9.3.29). Hence similar to case (i) we can derive (9.3.13) from (9.3.26)–(9.3.29). The proof is now complete. We define ˆ (t) =

+∞ −∞

ˆ + tˆ0 )2 vˆ 2 (x, t)d x + β(t

(9.3.30)

where βˆ ≥ 0 and tˆ0 > 0 are to be determined later on. In the next lemma, we will show ˆ that (t) verifies assumptions of Theorem 1.3.1 by picking appropriate βˆ ≥ 0, tˆ0 > 0 and initial datum (u 0 , u 1 , θ0 ). ˆ ˆ Lemma 9.3.2. We assume that for any t ≥ 0, E(t) ≤ E(0) ≤ 0, u 0 ∈ H 2(R), u 1 ∈ H 1(R), θ0 ∈ H 1(R) and one of the following assumptions (i) and (ii) holds,

(9.3.31)


403

(i) there exists a positive constant γ verifying + 2 γ ≥ γ ≡ 1 + 1 + b /ac /4 > 0,

(9.3.32)

2γ (m + α)αu 2 − 2(1 + 2γ )F(t, u) + u f (t, u) ≥ 0, ∀u ∈ R, ∀t > 0 and initial data satisfy 1 − γ2 γ

−1

+2

+∞ −∞

ˆ [−2(1 + 2γ ) E(0)/(1 + 2γ )]−1 [γ γ2 γ −1 +∞

−∞

+∞ −∞

u 20 d x

ˆ u 0 (u 1 − αu 0 )d x] > 0, if E(0) < 0,

u 20 d x > 0, γ2 γ −1

+∞ −∞

u 20 d x + 2

+∞

−∞

(9.3.33)

(9.3.34)

u 0 (u 1 − αu 0 )d x > 0,

ˆ if E(0) =0

(9.3.35)

with γ1 = 0, γ2 = −2C1 and C1 = (m + 2α)/2 > 0, (ii) there exist constants γ > 0 verifying or

γ ≥ A0

(9.3.36)

max[0, A0 /2 − (m + 2α)2 /(8 A0 )] ≤ γ < A0

(9.3.37)

and 2 > 0 such that 0 < 2 ≤

2+

= γ ( 1 + (m + 2α)2 /(4γ 2 (m + α)α − 1),

(9.3.38)

2γ + 2 ≥ A0 ,

(9.3.39) 2

(2γ + 2 )(m + α)αu − 2(1 + 2γ − 2 )F(t, u) + u f (t, u) ≥ 0, ∀u ∈ R (9.3.40) with A0 = 1 + b2 /ac − 1 /2, and initial data satisfy +∞ ˆ 1 − γ2 γ −1 [−2(1 + 2γ + 2 ) E(0)/(1 + 2γ )]−1 (γ γ2 γ −1 u 20 d x +2

+∞

−∞ +∞ −∞

+∞

−∞

−∞

ˆ u 0 (u 1 − αu 0 )d x ≥ 0, if A2 > 0, E(0) < 0,

(9.3.41)

ˆ u 20 d x > 0, if A2 = 0 E(0) < 0, u 20 d x > 0, 2

+∞

−∞

u 0 (u 1 − αu 0 )d x + γ2 γ −1

(9.3.42)

+∞ −∞

u 20 d x > 0,

ˆ = 0, if A2 > 0, E(0) +∞ +∞ ˆ u 20 d x > 0, u 0 (u 1 − αu 0 )d x > 0, if A2 = 0, E(0) = 0,

−∞

−∞

(9.3.43) (9.3.44)

404


with A2 = (m + 2α)2 /(4 2 ) − (2γ + 2 )(m + α)α ≥ 0, γ1 =

2 A 2 γ , γ2 = − 2 A 2 γ .

Then for β > 0 small enough and suitable t0 > 0 or β = 0, there exist constants C1 ≥ 0 and C2 ≥ 0 such that (1.3.1)–(1.3.5) or (1.3.1), (1.3.6)–(1.3.8) hold. Proof. Similarly to (9.2.67)–(9.2.70), we have

+∞ ˆ ˆ ˆ ˆ (t) = 2 vˆ vˆt d x + β(t + t0 ) , (t) = 2 −∞

+∞ −∞

(vˆt2

ˆ + vˆ vˆt t )d x + β ,

ˆ (t)]2 ˆ ˆ (t) − (1 + γ )[ (t)

+∞ +∞ ˆ ≥ 2 (t) −(1 + 2γ ) vˆt2 d x + βˆ + vˆ vˆt t d x −∞ −∞ +∞

+∞ ˆ vˆt2 d x + βˆ − [a vˆ x2 + bvˆ x w]d ˆ x ≥ 2 (t) −(1 + 2γ ) − (m + 2α)

+∞ −∞

−∞

vˆ vˆt d x − (m + α)α

−∞ +∞ 2

−∞

vˆ d x +

+∞

−∞

ˆ ˆ ˆ ≥ 2 (t) − (1 + 2γ )β − 2(1 + 2γ ) E(t) + 2γ (m + α)α + 2γ a −b

+∞

−∞ +∞

−∞

vˆ x2 d x

− 2(1 + 2γ )

−∞ +∞

vˆ x wd ˆ x − (m + 2α)

+∞

+∞

−∞

+∞ −∞

vˆ 2 d x

ˆ v) F(t, ˆ d x + c(1 + 2γ )

vˆ vˆt d x +

−∞

vˆ fˆ(t, v) ˆ dx

+∞

−∞

wˆ 2 d x

vˆ fˆ(t, v) ˆ dx .

(9.3.45)

If (i) holds, then noting that +∞ +∞ +∞ +∞ 1 b2 ˆ (t), b vˆ vˆt d x ≥ vˆ x wd ˆ x ≤ 1 wˆ 2 d x + vˆ x2 d x, 2 4

1 −∞ −∞ −∞ −∞ (9.3.46) +∞ +∞ +∞ 2 (m + 2α) vˆ vˆt d x ≤ 2 vˆt2 d x + vˆ 2 d x, (9.3.47) (m + 2α) 4 2 −∞ −∞ −∞ we derive ˆ (t)]2 ˆ ˆ (t) − (1 + γ )[ (t) ˆ ˆ ≥ 2 (t) − (1 + 2γ )βˆ − 2(1 + 2γ ) E(0) + 2γ (m + α)α + 2γ a −b

+∞

−∞ +∞

−∞

vˆ x2 d x − 2(1 + 2γ )

vˆ x wd ˆ x − (m + 2α)

+∞

−∞ +∞

−∞

+∞ −∞

ˆ v) F(t, ˆ d x + c(1 + 2γ )

vˆ vˆt d x +

+∞ −∞

vˆ 2 d x

vˆ fˆ(t, v) ˆ dx

+∞

−∞

wˆ 2 d x


405

m + 2α ˆ ˆ ˆ (t) ≥ 2 (t) − (1 + 2γ )βˆ − 2(1 + 2γ ) E(0) − 2 +∞ +∞ b2 + 2aγ − vˆ x2 d x + [c(1 + 2γ ) − 1 ] wˆ 2 d x 4 1 −∞ −∞ +∞ ˆ v) [vˆ fˆ(t, v) ˆ − 2(1 + 2γ ) F(t, ˆ + 2γ (m + α)α vˆ 2 ]d x . + −∞

(9.3.48)

Now choose 1 = b2 /8aγ to verify

or equivalently

c(1 + 2γ ) − 1 ≥ 0

(9.3.49)

16acγ 2 + 8acγ − b2 ≥ 0

(9.3.50)

which can be implied by assumption (9.3.32). Clearly, (9.3.33) implies ˆ v) fˆ(t, v) ˆ − 2(1 + 2γ ) F(t, ˆ + 2γ (m + α)α vˆ 2 ≥ 0, ∀t > 0.

(9.3.51)

Thus it follows from (9.3.48)–(9.3.51) that ˆ ˆ (t) − (1 + γ )[ ˆ(t)]2 (t) (m + 2α) ˆ ˆ ˆ ≥ 2 (t) − (1 + 2γ )βˆ − 2(1 + 2γ ) E(0) − (t) . 2

(9.3.52)

ˆ If E(0) < 0, then we take βˆ > 0 and tˆ0 > 0 so small that ˆ 0 < βˆ ≤ −2(1 + 2γ ) E(0)/(1 + 2γ ), (9.3.53) +∞ 2 +∞ 1 + 1 − γ2 γ −1 βˆ −1 (γ γ2 γ −1 −∞ u 0 d x + 2 −∞ u 0 (u 1 − αu 0 )d x) 0 < tˆ0 < −γ γ2 γ −1 (9.3.54) with γ1 = 0, γ2 = −2C1 and C1 = (m + 2α)/2. Thus (9.3.34)–(9.3.35) and (9.3.52)– (9.3.54) give (9.2.9)–(9.2.13) with C1 = −(m + 2α)/2, C2 = 0 and γ1 = 0, γ2 = 2C1 = ˆ −(m + 2α). If E(0) = 0, we pick βˆ = 0 and so (9.3.35) and (9.3.52) yield to (1.3.1)– (1.3.5) with C1 = −(m + 2α)/2, C2 = 0 and γ1 = 0, γ2 = 2C1 = −(m + 2α). If (ii) holds, then using (9.3.44)–(9.3.45), we derive from (9.3.45) that ˆ (t)]2 ˆ ˆ (t) − (1 + γ )[ (t)

ˆ ˆ ≥ 2 (t) −(1 + 2γ )βˆ − 2(1 + 2γ ) E(0) + (2aγ − b 2 /(4 1 )) + (c + 2cγ − 1 ) − A2

+∞ −∞

+∞

wˆ 2 d x − 2

−∞ +∞

vˆ 2 d x +

−∞

+∞ −∞

vˆt2 d x

ˆ v) [ ˆ fˆ(t, v) ˆ − 2(1 + 2γ ) F(t, ˆ ]d x

+∞ −∞

vˆ x2 d x

406


ˆ ˆ ≥ 2 (t) −(1 + 2γ )βˆ − 2(1 + 2γ ) E(0) + (2aγ − b 2 /(4 1 )) + (c + 2cγ − 1 ) + 2

+∞

−∞ +∞

+∞ −∞

+∞ −∞

vˆ x2 d x

ˆ wˆ d x − 2 2 E(t) 2

ˆ v) [(m + α)α vˆ 2 + a vˆ x2 − 2 F(t, ˆ + cwˆ 2 ]d x

(m + 2α)2 2 ˆ ˆ 2γ (m + α)α vˆ − + vˆ − 2(1 + 2γ ) F(t, v) ˆ + ˆ f (t, v) ˆ dx 4 2 −∞

+∞ ˆ ˆ + (2aγ − b 2 /(4 1 ) + a 2 ) vˆ x2 d x ≥ 2 (t) −(1 + 2γ )βˆ − 2[1 + 2γ + 2 ] E(0)

2

+ (c + 2cγ − 1 + c 2 ) +∞ vˆ 2 d x . − A2

+∞ −∞

wˆ 2 d x +

+∞ −∞

−∞

ˆ v) [ ˆ fˆ(t, v) ˆ − 2(1 + 2γ − 2 ) F(t, ˆ ]d x (9.3.55)

−∞

We will show that (9.3.36)–(9.3.39) verify 2aγ − b 2 /(4 1 ) + a 2 ≥ 0, c(1 + 2γ ) − 1 + c 2 ≥ 0,

(9.3.56) (9.3.57)

ˆ v) vˆ fˆ(t, v) ˆ − 2(1 + 2γ − 2 ) F(t, ˆ ≥ 0,

(9.3.58)

2

In fact, we take

A2 = −(2γ + 2 )(m + α)α + (m + 2α) /(4 2 ) ≥ 0.

(9.3.59)

1 = b2/[4a(2γ + 2 )]

(9.3.60)

which satisfies (9.3.56) and so (9.3.57) amounts to (2γ + 2 + 1/2)2 ≥ (b 2 + ac)/(4ac).

(9.3.61)

Clearly, (9.3.39) implies (9.3.61) and further (9.3.57), (9.3.38) and (9.3.40) verify (9.3.59). Obviously, (9.3.36) or (9.3.37) implies A0 ≤ 2γ + 2 ≤ 2γ + 2+ .

(9.3.62)

Thus it follows from (9.3.55)–(9.3.61) that ˆ ˆ (t)]2 ≥ 2 (t) ˆ ˆ ˆ ˆ (t) − (1 + γ )[ − (1 + 2γ )βˆ − 2(1 + 2γ + 2 ) E(0) − A2 (t) . (t) (9.3.63) ˆ If E(0) < 0, then we choose βˆ > 0 and tˆ0 > 0 so small that ˆ + 2γ ), (9.3.64) 0 < βˆ ≤ −2(1 + 2γ + 2 ) E(0)/(1 +∞ 2 +∞ 1 + 1 − γ2 γ −1 βˆ −1 γ2 γ −1 −∞ u 0 d x + 2 −∞ u 0 (u 1 − αu 0 )d x 0 < tˆ0 < , −γ γ2 γ −1 if A2 > 0, (9.3.65)


0 < tˆ0 , if A2 = 0,

407

+∞

−∞

u 0 (u 1 − αu 0 )d x ≥ 0,

ˆ tˆ0 > −[−2(1 + 2γ + 2 ) E(0)/(1 + 2γ )]−1 if A2 = 0,

+∞ −∞

+∞ −∞

(9.3.66) u 0 (u 1 − αu 0 )d x ≥ 0,

u 0 (u 1 − αu 0 )d x < 0.

(9.3.67)

ˆ Thus (1.3.1)–(1.3.5) follows from (9.3.63)–(9.3.67) and (9.3.41)–(9.3.42). If E(0) = 0, ˆ we take β = 0 and can use (9.3.43)–(9.3.44) and (9.3.63) to deduce (1.3.1)–(1.3.5) or (1.3.1), (1.3.6)–(1.3.8). The proof is complete. Thus exploiting Theorem 1.3.1 and Lemmas 9.3.1–9.3.2, we readily prove the following result. Theorem 9.3.1. Assume that assumptions in Lemmas 9.3.1–9.3.2 hold. Then the solution v( ˆ t) in L 2 (R) to problem (9.3.4)–(9.3.6) blows up in a finite time and further the solution u(x, t) in L 2 (R) to problem (9.1.1)–(9.1.3) blows up in a finite time. From the proofs of Lemmas 9.3.1–9.3.2 and Theorem 3.1, we easily prove the following results. Lemma 9.3.3. Assume that g(t) is a positive definite kernel and E(0) ≤ 0, κ ≥ 0 and assumptions (i)–(ii) (or (9.3.9)–(9.3.12)) with α = 0 in Lemma 9.3.2 hold; then E(t) ≤ E(0) ≤ 0, ∀t > 0.

(9.3.68)

ˆ Proof. Noting that when α = 0, E(t) = E(t), ∀t ≥ 0, we easily obtain the result. The proof is complete. Lemma 9.3.4. We suppose that for any t > 0, E(t) ≤ E(0) ≤ 0, and assumptions (9.3.31)–(9.3.37) and (9.3.39)–(9.3.44) with α = 0 hold. Then for β > 0 small enough and suitable t0 > 0 or β = 0, there exist constants C1 ≥ 0 and C2 ≥ 0 such that (1.3.1)–(1.3.5) or (1.3.1), (1.3.6)–(1.3.8) hold. Proof. Noting that when α = 0, (9.3.59) is automatically satisfied and so assumption (9.3.38) should be cancelled. Thus the result in this lemma readily follows from the proof of Lemma 9.3.2. The proof is complete. Based on Lemmas 9.3.3–9.3.4, we can show the following result. Theorem 9.3.2. Assume that assumptions in Lemmas 9.3.3–9.3.4 hold, then the solution u(x, t) in L 2 (R) to problem (9.1.1)–(9.1.3) blows up in a finite time. Remark 9.3.1. The results in Theorems 9.3.1–9.3.2 are new.

408


9.4 Bibliographic Comments Concerning the non-existence results in the related literature, we refer to the works by Messaoudi [260], Kirane, Kouachi and Tatar [199], Kirane and Tatar [200] for the onedimensional case and Racke [348] for the three-dimensional case. In [200], the authors established the blow-up of weak solutions in L 2 (R) to the equations (9.1.1)–(9.1.2) where the kernel term g ∗ θ x x disappears and κ > 0. It should be noted that the method used in [200] depends heavily on a lemma (see, e.g., Theorem 1.3.1) due to Kalantarov and Ladyzhenskaya [181], which is in fact a compact version of the concavity method of Levine, Park, Pucci, Sacks and Serrin [220–226]. To the authors’ knowledge, there have been no new blow-up results on nonlinear models when the heat flux obeys Gurtin and Pipkin’s law [133] mentioned above prior to the results of this chapter being established. For other thermoelastic models of types II and III, and those with second sound, we refer the readers to Section 7.3.

Chapter 10

Large-Time Behavior of Energy in Multi-Dimensional Elasticity In this chapter, we shall establish the large-time behavior of energy in multi-dimensional nonhomogeneous anisotropic elastic systems. The results of this chapter are picked from Qin and Muñ˜ oz Rivera [338] and Qin, Deng and Su [327].

10.1 Polynomial Decay of Energy In this section we investigate the large-time behavior of energy for the n-dimensional linear nonhomogeneous anisotropic elastic system. We assume here that the boundary surface is nonporous and locally reacting in the sense that wave motion along the boundary is negligible. We also suppose that a small part of the boundary reacts to the excess pressure due to the wave like a resistant harmonic oscillator (see, e.g., Morse and Ingard [272], p. 263). Such a model can be regarded as an extension of the “classical” elastic theory. We denote by an open bounded domain of Rn with sufficiently smooth boundary ∂ = . If φ = φ(x, t) is the displacement vector field, then the system in question reads φi − [ Ai j kl φk,l ], j = 0 i n × (0, +∞) (10.1.1) where i, j, k, l = 1, 2, . . . , n and φ(x, t) = (φ1 , . . . , φn ), f = ∂ f /∂ t, f, j = ∂ f /∂ x j , and we use Einstein’s convention on summing over repeated lower indices. We assume that the boundary ∂ = is divided into two parts, = 0 ∪ 1 with 0 ∩ 1 = ∅, 0 = ∅.

410

Chapter 10. Large-Time Behavior of Energy in Multi-Dimensional Elasticity

We suppose that 0 is not rigid and that each point reacts to excess pressure of a resistive harmonic oscillator, and that the different parts of the boundary do not influence each other, that is, the surface is locally reacting but subject to small oscillations. Under these conditions the normal displacement of 0 into the domain satisfies m i (x)i (x, t) + di (x)i (x, t) + ki (x)(x, t) = −ρφi (x, t) on

0

(10.1.2)

where ρ is the density of the medium, m i , di and ki are mass per unit area, resistivity and spring constant on 0 , respectively. If we also assume that 0 is impenetrable, we obtain from the continuity of the velocity at the boundary 0 that i (x, t) = Ai j kl φk,l ν j on

0

(10.1.3)

where ν(x) = (ν1 , . . . , νn ) denotes the outward normal vector at x ∈ . We assume that 1 is rigid and on it φ satisfies φ(x, t) = 0

1 .

on

(10.1.4)

Moreover, we assume that there are a point x 0 ∈ Rn and a constant a > 0 such that 1 = {x ∈ : q(x) · ν(x) ≤ 0}, 0 = {x ∈ : q(x) · ν(x) ≥ a > 0} with q(x) = x − x 0 . As a typical example for the existence of the point x 0 we can see the domain 0

' 0

$ 1 '$ ' $

= 0 \ 1 , &% & % &

%

The initial conditions of the system is given by φ(x, 0) = φ 0 (x), φ (x, 0) = φ 1 (x) 0

1

(x, 0) = (x), (x, 0) = (x)

∀ ∈ , ∀x ∀ ∈ 0 . ∀x

(10.1.5) (10.1.6)

We assume that m i (x), di (x) and ki (x) are positive sufficiently smooth functions on 0 , i = 1, 2, . . . , n and that Ai j kl (x) is a sufficiently smooth function satisfying Ai j kl = A j ikl = Akli j

(10.1.7)

10.1. Polynomial Decay of Energy

411

¯ and there are two positive constants α and β such that on , Ai j kl φi, j φk,l ≥ αφi, j φi, j , (Ai j kl − qμ Ai j kl,μ )ξi j ξkl ≥ β Ai j kl ξi j ξkl

(10.1.8) (10.1.9)

where μ = 1, 2, . . . , n. When n = 3, the boundary conditions (10.1.2)–(10.1.3) are called “acoustic boundary conditions” for the linear homogeneous wave equation for which Beale [26, 27] and Beale and Rosencrans [28] proved the global existence and regularity of solutions in a Hilbert space of data with finite energy by means of semi-group methods. The asymptotic behavior was obtained in [27, Theorem 2.6] but no decay rate was given there. This model is also used in [272, p. 263] for waves assumed to be at a definite frequency. The aim of this section is to establish the polynomial decay of the energy of problem (10.1.1)–(10.1.6) and extends the results in [280] to the case of a non-homogeneous anisotropic elastic system.

10.1.1 Main Results In this subsection we introduce the notation and main results of this section. Let us define the space H = (H H11 ())n × (L 2 ())n × (L 2 (0 ))n × (L 2 (0 ))n with H11 () = {u : u ∈ H 1(), u|1 = 0}. It is not difficult to see that H together with the inner product u, w = (ρ Ai j kl u k,l wi, j + ρu n+ j wn+ j )d x + (kk j u 2n+ j w2n+ j + m j u 3n+ j w3n+ j )d0 0

(10.1.10)

is a Hilbert space, where u = (u 1 , u 2 , . . . , u 4n )τ , w = (w1 , w2 , . . . , w4n )τ ∈ H. Thus it follows from (10.1.8) that the induced norm on H by the above inner product 2 |u|H = (ρ Ai j kl u k,l u i, j + ρu n+ j u n+ j )d x + (kk j u 2n+ j u 2n+ j + m j u 3n+ j u 3n+ j )d0

0

is equivalent to the usual norm on H, u2H = (u i, j u i, j + u n+ j u n+ j )d x + (u 2n+ j u 2n+ j + u 3n+ j u 3n+ j )d0

for any u = (u 1 , . . . , u 4n )τ ∈ H.

0

412


Introduce an operator A on H so that for smooth U = (φ, φt , , t )τ = (φ1 , . . . , φn , φ1 , . . . , φn , 1 , . . . , n , 1 , . . . , n ) ∈ R4n , (10.1.1)–(10.1.4) are equivalent to U(t) ∈ D(A) and Ut = AU where U = (u 1 , . . . , u 4n )τ satisfies u i = φi , u n+i = φi , u 2n+i = i , u 3n+i = i , i = 1, 2, . . . , n and

(10.1.11)

D(A) = U = (u 1 , . . . , u 4n )τ : u n+i ∈ H11 (), (Ai j kl u k,l ), j ∈ L 2 (), u 3n+i ∈ L 2 (0 ), u 3n+i = Ai j kl u k,l ν j 0 , i, j, k, l = 1, . . . , n .

In the definition of D(A), u 3n+i = Ai j kl u k,l ν j is in the weak sense that [(Ai j kl u k,l ), j + Ai j kl u k,l , j ]d x = u 3n+i d0 , ∀ ∈ H11 ()

0

L 2 ()

is equivalent to the condition of u 3n+i = Ai j kl u k,l ν j on 0 and (Ai j kl u k,l ), j ∈ as a trace. It follows from (10.1.1)–(10.1.3) and (10.1.11) that u i = u n+i , u n+i = (Ai j kl φk,l ), j ,

u 2n+i = u 3n+i , u 3n+i = −(ρu n+i + di u 3n+i + ki u 2n+i )/m i

(10.1.12) (10.1.13) (10.1.14) (10.1.15)

where i = 1, 2, . . . , n and −(ρu n+i +di u 3n+i +ki u 2n+i )/m i is understood as the trace in H 1/2(0 ). For any U ∈ D(A), it is easy to get from (10.1.10)–(10.1.15) and the definition of D(A) that AU, U = (ρ Ai j kl u n+k,l u i, j + ρ(Ai j kl u k,l ), j u n+i )d x + [ki u 3n+i u 2n+i − m i (ρu n+i + di u 3n+i + ki u 2n+i )u 3n+i /m i ]d0 0 =ρ Ai j kl u n+k,l u i, j d x + ρ Ai j kl u k,l ν j u n+i d0 − ρ Ai j kl u k,l u n+i, j d x 0 + ki u 3n+i u 2n+i d0 − (ρu n+i + di u 3n+i + ki u 2n+i )u 3n+i d0 0 0 =− di u 3n+i u 3n+i d0 ≤ 0 (10.1.16) 0

which yields that A is dissipative on H. Thus, similar to the proofs in [26–28], we are able to get the following results on the global existence and regularity of solutions.


413

Theorem 10.1.1. The operator A defined on H is closed, densely defined, and dissipative. It generates a C0 -semigroup on H. Theorem 10.1.2. Assume that U0 ∈ H is C ∞ and vanishes near ∂; let U(t) be the ¯ solution of U (t) = AU(t), t ≥ 0, with U(0) = U0 . Then u 1 (t), . . . , u 2n (t) ∈ C ∞ () ∞ and u 2n+1 (t), . . . , u 4n (t) ∈ C (0 ) for any t ≥ 0. Remark 10.1.1. “U vanishes near ∂ ” means u 2n+1 = · · · = u 4n = 0 on 0 , and u 1 , . . . , u 2n vanish near ∂. We introduce the energy functions 1 E0 (t; φ, ) = (ρφi φi + ρ Ai j kl φk,l φi, j )d x 2 1 + (ki (x)i i + m i (x)i i )d0 , 2 0 Eh (t) ≡ Eh (t; φ, ) = E0 (t; ∂th φ, ∂th ),

h = 1, 2, . . . .

(10.1.17) (10.1.18)

The main result of the paper is summarized in the following theorem. Theorem 10.1.3. Under the above assumptions and taking smooth initial data (φ 0 , φ 1 , 0 , 1 ) such that m+1 Eh (0) < ∞ (10.1.19) h=0

for some integer m ≥ 0, then constant C˜ such that

m

Eh (t) decays polynomially, that is, there is a positive

h=0 m h=0

Eh (t) ≤

m+1 C˜ Eh (0), t

∀t > 0.

(10.1.20)

h=0

The notation in this chapter is standard. We also put · = · L 2 . We use C (sometimes C1 , C2 , . . . ) to stand for the universal positive constant independent of time t > 0.

10.1.2 Proof of Theorem 10.1.3 In this section we prove Theorem 10.1.3. To this end, we first establish some energy estimates. First of all, multiplying (10.1.1) by φi , using (10.1.2)–(10.1.7) and Green’s formula, and noting that 2 Ai j kl φk,l φi, j = ∂(Ai j kl φk,l φi, j )/∂t, we deduce that d E0 (t; φ, ) = − dt

0

di (x)i i d0 .

(10.1.21)

414


Similarly, keeping in mind that equation (10.1.1) and boundary conditions (10.1.2)– (10.1.4) are all linear in t, we have that for h = 0, 1, . . . , m + 1, d Eh (t; φ, δ) = − di (x)∂th+1 i ∂th+1 i d0 ≤ 0. (10.1.22) dt 0 Let us denote

(n − 1) φi φi )d x, 2

(10.1.23)

Fh (t) ≡ Fh (t; φ, ) ≡ F0 (t; ∂th φi , ∂th i ),

h = 1, 2, . . . , m. (10.1.24)

F0 (t; φ, δ) =

(φi qμ φi,μ +

Under the above conditions, we have Lemma 10.1.1. For h = 0, 1, . . . , m, there holds that 1 (n − 1) d Fh (t) ≤ qμ νμ ∂th+1 φi ∂th+1 φi d0 + Ai j kl ∂th φk,l ∂th φi ν j d0 dt 2 0 2 0 1 1 h+1 h+1 h h − [∂t φi ∂t φi + Ai j kl ∂t φk,l ∂t φi, j ]d x + Ai j kl,μ ∂th φk,l ∂th φi, j qμ d x 2 2 1 + Ai j kl ν j qμ ∂th φk,l ∂th φi,μ d0 − Ai j kl νμ qμ ∂th φk,l ∂th φi, j d0 2 0 0 a1 − Ai j kl ∂th φk,l ∂th φi, j d1 , (10.1.25) 2 1 where a1 = min1 [−qμ νμ ] ≥ 0. Proof. Since (10.1.1)–(10.1.4) are all linear in t, it suffices to prove (10.1.25) for the case of h = 0. From (10.1.1)–(10.1.4) it follows that d φi φi d x = (φi φi − Ai j kl φk,l φi, j )d x + Ai j kl φk,l φi ν j d0 . (10.1.26) dt 0 Similarly, by (10.1.1), d φi qμ φi,μ + φ qμ φi,μ d x = dt i = [ Ai j kl φk,l ], j qμ φi,μ d x +

1 qμ (φi φi ),μ d x (10.1.27) 2 1 1 qμ νμ φi φi d0 − qμ,μ φi φi d x. 2 0 2

By (10.1.27), we deduce (Ai j kl φk,l φi, j ),μ = Ai j kl,μ φk,l φi, j + 2 Ai j kl φk,l φi, j μ or Ai j kl φk,l φi, j μ =

1 [(Ai j kl φk,l φi, j ),μ − Ai j kl,μ φk,l φi, j ]. 2

(10.1.28)


415

By (10.1.28), we arrive at [ Ai j kl φk,l ], j qμ φi,μ d x = Ai j kl ν j qμ φk,l φi,μ d − [ Ai j kl qμ, j φk,l φi,μ + Ai j kl qμ φk,l φi, j μ ]d x = Ai j kl ν j qμ φk,l φi,μ d − Ai j kl qμ, j φk,l φi,μ d x 1 1 − (Ai j kl φk,l φi, j ),μ qμ d x + Ai j kl,μ φk,l φi, j qμ d x 2 2 = Ai j kl ν j qμ φk,l φi,μ d − Ai j kl qμ, j φk,l φi,μ d x 1 1 + Ai j kl,μ φk,l φi, j qμ d x − Ai j kl φk,l φi, j qμ νμ d 2 2 1 + Ai j kl φk,l φi, j qμ,μ d x 2 which, combined with (10.1.27), yields d φ qμ φi,μ d x dt i 1 1 = qμ νμ φi φi d0 − qμ,μ φi φi d x + Ai j kl φk,l φi,μ ν j qμ d 2 0 2 1 − Ai j kl φk,l φi,μ qμ, j d x + Ai j kl,μ φk,l φi, j qμ d x 2 1 1 − Ai j kl φk,l φi, j νμ qμ d + Ai j kl φk,l φi, j qμ,μ d x 2 2 1 (n − 1) = qμ νμ φi φi d0 − [φi φi − Ai j kl φk,l φi, j ]d x 2 0 2 1 − [φi φi + Ai j kl φk,l φi, j ]d x + Ai j kl φk,l φi,μ qμ ν j d 2 1 1 − Ai j kl φk,l φi, j qμ νμ d + Ai j kl,μ φk,l φi, j qμ d x 2 2 1 + Ai j kl φk,l φi, j qμ,μ d x. (10.1.29) 2 Multiplying (10.1.26) by (n − 1)/2 and then adding the resulting equation to (10.1.29), we get d 1 (n − 1) F0 (t; φ, ) = qμ νμ φi φi d0 + Ai j kl φk,l φi ν j d0 dt 2 0 2 0 1 − [φ φ + Ai j kl φk,l φi, j ]d x + Ai j kl φk,l φi,μ qμ ν j d 2 i i

416


−

1 2

Ai j kl φk,l φi, j qμ νμ d +

1 2

Ai j kl,μ φk,l φi, j qμ d x. (10.1.30)

Noting that φ|1 = 0, we have that on 1 , ν j φi,μ = νμ φi, j which implies

1 Ai j kl φk,l φi,μ qμ ν j d − Ai j kl φk,l φi, j qμ νμ d 2 1 = Ai j kl φk,l φi,μ qμ ν j d0 + Ai j kl φk,l φi, j qμ νμ d1 2 1 0 1 − Ai j kl φk,l φi, j qμ νμ d0 2 0 a1 ≤ Ai j kl φk,l φi,μ qμ ν j d0 − Ai j kl φk,l φi, j d1 2 1 0 1 − Ai j kl φk,l φi, j qμ νμ d0 2 0

(10.1.31)

where, by the definition of 1 , a1 ≥ 0. Operating ∂th on (10.1.1)–(10.1.4), repeating the same process as above, we easily derive the desired estimate (10.1.25). The proof is complete. We now denote G0 (t; φ, ) =

0

(m i (x)i i + di (x)i i /2 + ρφi i )d0 ,

Gh (t) ≡ Gh (t; φ, ) ≡ G 0 (t; ∂th φ, ∂th ), h = 1, 2, . . . , m

(10.1.32) (10.1.33)

then we have Lemma 10.1.2. For h = 0, 1, 2, . . . , m, there holds that d h h G h (t) = − ki (x)∂t i ∂t i d0 + [m i (x)∂th+1 i ∂th+1 i + ρ∂th φi ∂th+1 i ]d0 . dt 0 0 (10.1.34) Proof. We prove the case of h = 0. Multiplying (10.1.2) by i and integrating the resulting equation on 0 give the desired estimate (10.1.34). Noting that (10.1.1) and boundary conditions (10.1.2)–(10.1.4) are all linear in t, we operate ∂th on (10.1.1)–(10.1.4) and repeat the same process as above to derive the desired estimate (10.1.34) for the general cases h = 1, 2, . . . , m. The proof is complete. Finally, let us define L(t) = N 3

m+1 h=0

Eh (t) + N

m h=0

Gh (t) +

m h=0

where N is a large positive number to be determined later on.

Fh (t)

(10.1.35)


417

Lemma 10.1.3. For N large enough, there are positive constants C0 , C1 , C2 and C3 such that m+1 m+1 0 ≤ C0 Eh (t) ≤ L(t) ≤ C1 Eh (t), ∀t ≥ 0 (10.1.36) h=0

h=0

and m+1 m d h+1 h+1 h h L(t) ≤ −C C2 ∂t i ∂t i d0 + ∂t i ∂t i d0 dt h=0 0 h=0 0 m min(1, β) h+1 h+1 − ∂t φi ∂t φi + Ai j kl ∂th φk,l ∂th φi, j d x 4 h=0 m m a a1 − Ai j kl ∂th φk,l ∂th φi, j d0 − Ai j kl ∂th φk,l ∂th φi, j d1 4 2 0 1 h=0 m

≤ −C3

h=0

Eh (t),

∀t > 0.

(10.1.37)

h=0

Proof. By (10.1.4), (10.1.8) and the trace theorem, we infer that for h = 0, 1, . . . , m ∂th φi ∂th φi d0 ≤ C∂t φ2H 1/2 () ≤ C∂t φ2H 1 () ≤ C Ai j kl ∂th φk,l ∂th φi, j d x 0

and

(10.1.38)

∂th φi ∂th φi d x ≤ C ≤C

∂th φi,μ ∂th φi,μ d x ≤ C∂t φ2H 1 () Ai j kl ∂th φk,l ∂th φi, j d x.

(10.1.39)

Thus by (10.1.17)–(10.1.18), (10.1.23)–(10.1.24), (10.1.32)–(10.1.33), (10.1.35) and (10.1.36)–(10.1.39), we infer that for N large enough there are two positive constants C0 and C1 such that L(t) = N 3

m+1 h=0

1 + 2

0

1 2

(ρ∂th+1 φi ∂th+1 φi + ρ Ai j kl ∂th φk,l ∂th φi, j )d x

(ki ∂th i ∂th i + m i ∂th+1 i ∂th+1 i )d0

m

(m i ∂th+1 i ∂th i + di ∂th i ∂th i /2 + ρ∂th φi ∂th i )d0 0 h=0 m + (∂th+1 φi qμ ∂th φi,μ + (n − 1)∂th+1 φi ∂th φi /2)d x h=0 +N

418


≥ C N3 +

m+1 h=0

0

(∂th+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j )d x

(∂th i ∂th i + ∂th+1 i ∂th+1 i )d0

− CN

m

(∂th i ∂th i + ∂th+1 i ∂th+1 i + ∂th φi ∂th φi )d0

h=0 0 m −C (∂th+1 φi ∂th+1 φi h=0

≥ C0

m+1

+ ∂th φi,μ ∂th φi,μ + ∂th φi ∂th φi )d x

Eh (t)

(10.1.40)

h=0

and L(t) ≤ C1

m+1

Eh (t)

h=0

which, along with (10.1.30), gives (10.1.36). On the other hand, by (10.1.22)–(10.1.25) and (10.1.34), we obtain m m+1 d 3 h+1 h+1 L(t) = −N di ∂t i ∂t i d0 − N ki ∂th i ∂th i d0 dt 0 0 h=0 h=0 m +N (m i ∂th+1 i ∂th+1 i + ρ∂th φi ∂th+1 i )d0

−

1 2

h=0 0 m


h=0 m

(n − 1) 1 h+1 h+1 h h qμ νμ ∂t φi ∂t φi + Ai j kl ∂t φk,l ν j ∂t φi d0 + 2 0 2 h=0 1 + Ai j kl,μ qμ ∂th φk,l ∂th φi, j d x + Ai j kl ν j ∂th φk,l ∂th φi,μ qμ d0 2 0 1 a 1 h h h h (10.1.41) − qμ νμ ∂t φk,l ∂t φi, j d0 − Ai j kl ∂t φk,l ∂t φi, j d1 . 2 0 2 1 We need to estimate some terms on the right-hand side of (10.1.41). In fact, by (10.1.21), we get that on 0 , ∂th+1 φi = −ρ −1 [m i ∂th+2 i + di ∂th+1 i + ki ∂th i ]


419

which implies 1 qμ νμ ∂th+1 φi ∂th+1 φi d0 ≤ C (∂th+2 i ∂th+2 i + ∂th+1 i ∂th+1 i + ∂th i ∂th i )d0 . 2 0 0 (10.1.42) Using Ai j kl ∂th φk,l ν j = ∂th+1 i on 0 , (10.1.8), (10.1.38), the Young inequality and the definition of 0 , we deduce that (n − 1) (n − 1) Ai j kl ∂th φi ∂th φk,l ν j d0 = ∂th φi ∂th+1 i d0 2 2 0 0 min(1, β) ≤ Ai j kl ∂th φi, j ∂th φk,l d x + C ∂th+1 i ∂th+1 i d0 , (10.1.43) 8 0 Ai j kl ∂th φk,l ν j qμ ∂th φi,μ d0 = ∂th+1 i qμ ∂th φi,μ d0 0 0 α ≤ qμ νμ ∂th φi, j ∂th φi, j d0 + C ∂th+1 i ∂th+1 i d0 4 0 0 1 ≤ qμ νμ Ai j kl ∂th φk,l ∂th φi, j d0 + C ∂th+1 i ∂th+1 i d0 (10.1.44) 4 0 0 and

N

ρ∂th φi ∂th+1 i d0 (10.1.45) min(1, β) Ai j kl ∂th φk,l ∂th φi, j d x + C N 2 ∂th+1 i ∂th+1 i d0 . ≤ 8 0

0

In view of (10.1.9), we easily derive 1 1 h h Ai j kl,μ qμ ∂t φk,l ∂t φi, j d x − (∂ h+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j )d x 2 2 t 1 β ≤− ∂th+1 φi ∂th+1 φi d x − Ai j kl ∂th φk,l ∂th φi, j d x 2 2 min(1, β) ≤− (∂th+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j )d x. (10.1.46) 2 Thus it follows from (10.1.41)–(10.1.46) that for N > 1 large enough, there are constants C2 , C3 > 0 such that m m+1 d 3 h+1 h+1 L(t) ≤ −C N ∂t i ∂t i d0 − C N ∂th i ∂th i d0 dt 0 0 h=0 h=0 m + CN ∂th+1 i ∂th+1 i d0 h=0 0

420


min(1, β) + 4 m

h=0 m

Ai j kl ∂th φk,l ∂th φi, j d x

+ CN

2

m h=0 0

∂th+1 i ∂th+1 i d0

min(1, β) (∂th+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j )d x 2 h=0 m +C (∂th+2 i ∂th+2 i + ∂th+1 i ∂th+1 i + ∂th i ∂th i )d0 −

−

1 4

h=0 0 m h=0 0

≤ −C C2

m+1 h=0

a1 2 m

qμ νμ Ai j kl ∂th φk,l ∂th φi, j d0 −

0

∂th+1 i ∂th+1 i d0 +

m

m

h=0 0

h=0 1

Ai j kl ∂th φk,l ∂th φi, j d1

∂th i ∂th i d0

min(1, β) (∂th+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j )d x 4 h=0 m m a a1 − Ai j kl ∂th φk,l ∂th φi, j d0 − Ai j kl ∂th φk,l ∂th φi, j d1 4 2 0 1 −

h=0 m

≤ −C3

h=0

Eh (t).

h=0


Proof of Theorem 10.1.3. By assumption (10.1.19) and inequalities (10.1.36)–(10.1.37), we get t m m+1 −1 −1 Eh (τ )dτ ≤ C3 (L(0) − L(t)) ≤ C3 C1 Eh (0) < ∞. 0 h=0

(10.1.47)

h=0

Also, we easily obtain from (10.1.22) that for any t > 0, d d [t Eh (t)] = Eh (t) ≤ Eh (t) + t Eh (t). dt dt m

h=0

m

m

h=0

m

h=0

(10.1.48)

h=0

Thus integrating (10.1.48) over (0, t) and using (10.1.47), we finally derive m m+1 C˜ Eh (t) ≤ Eh (0) t h=0

with C˜ = C3−1 C1 . The proof is now complete.

(10.1.49)

h=0

10.2. Exponential Decay of Energy

421

10.2 Exponential Decay of Energy 10.2.1 Main Results In this section we study the exponential decay of energy for a multi-dimensional dissipative non-homogeneous anisotropic elastic system. We assume here that the boundary surface is nonporous and locally reacting in the sense that wave motion along the boundary is negligible. We also suppose that a small part of the boundary reacts to the excess pressure due to the wave like a resistant harmonic oscillator (see, e.g., [272], p. 263). Such a model can be regarded as an extension of the “classical” elastic theory. We denote by an open bounded domain of Rn with sufficiently smooth boundary ∂ = . If φ = φ(x, t) is the displacement vector field, then the system in question reads φi − [ Ai j kl φk,l ], j + fî φi = 0

i n × (0, +∞)

(10.2.1)

where i, j, k, l = 1, 2, . . . , n and φ(x, t) = (φ1 , . . . , φn ), w = ∂w/∂t, w, j = ∂w/∂ x j , and we use Einstein’s convention on summing over repeated lower indices, and fî = fî (x) (i = 1, 2, . . . , n) is a continuous function in verifying ∀ ∈ . fî (x) ≥ 0, i = 1, 2, . . . , n, fˆ(x) = ( fˆ1 (x), . . . , fˆn (x)) ≡ 0, ∀x

(10.2.2)

We assume that the boundary ∂ = is divided into two parts, = 0 ∪ 1 with 0 ∩ 1 = ∅, 0 = ∅. We suppose that 0 is not rigid and that each point reacts to excess pressure of a resistive harmonic oscillator, and that the different parts of the boundary do not influence each other, that is, the surface is locally reacting but subject to small oscillations. Under these conditions the normal displacement of 0 into the domain satisfies m i (x)i (x, t) + di (x)i (x, t) + ki (x)(x, t) = −ρφi (x, t) on

0

(10.2.3)

where ρ is the density of the medium, m i , di and ki are mass per unit area, resistivity and spring constant on 0 , respectively. If we also assume that 0 is impenetrable, we obtain from the continuity of the velocity at the boundary 0 that i (x, t) + gi = Ai j kl φk,l ν j on

0

(10.2.4)

where ν(x) = (ν1 , . . . , νn ) denotes the outward normal vector at x ∈ , and gi (x, t) = −gˆ i φi (x, t), i = 1, 2, . . . , n

(10.2.5)

represents a function of boundary damping satisfying ˆ = (gˆ 1 (x), . . . , gˆ n (x)) ≡ 0, ∀x ∀ ∈ 0 . gˆ i (x) ≥ 0, i = 1, 2, . . . , n, g(x)

(10.2.6)

422


We assume that 1 is rigid, on which φ satisfies φ(x, t) = 0

1 .

on

(10.2.7)

Moreover, we assume that there are a point x 0 ∈ R and a constant a0 > 0 such that 1 = {x ∈ : q(x) · ν(x) ≤ 0}, 0 = {x ∈ : q(x) · ν(x) ≥ a0 > 0} with q(x) = x − x 0 . As a typical example for the existence of the point x 0 we can see that example in Section 10.1.1. The initial conditions of the system is given by φ(x, 0) = φ 0 (x), φ (x, 0) = φ 1 (x), 0

1

(x, 0) = (x), (x, 0) = (x),

∀ ∈ , ∀x ∀ ∈ 0 . ∀x

(10.2.8) (10.2.9)

We assume that m i (x), di (x) and ki (x) are positive sufficiently smooth functions on 0 , i = 1, 2, . . . , n and that Ai j kl (x) is a sufficiently smooth function satisfying Ai j kl = A j ikl = Akli j

(10.2.10)

¯ and there are two positive constants α and β such that on Ai j kl φi, j φk,l ≥ αφi, j φi, j , (Ai j kl − qμ Ai j kl,μ )ξi j ξkl ≥ β Ai j kl ξi j ξkl

(10.2.11) (10.2.12)

where μ = 1, 2, . . . , n. When n = 3 and fî = gˆ i ≡ 0 (i = 1, 2, . . . , n), the boundary conditions (10.2.2)– (10.2.3) are called “acoustic boundary conditions” for the linear homogeneous wave equation for which Beale [26,27], and Beale and Roscrans [28] proved the global existence and regularity of solutions in a Hilbert space of data with finite energy by means of semi-group methods. The asymptotic behavior for this model was obtained in [27, Theorem 2.6] but no decay rate was given there. This model was also used in [272, p. 263] for waves assumed to be at a definite frequency. When fî = gˆ i ≡ 0 (i = 1, 2, . . . , n) and under some reasonable assumptions, the polynomial decay of energy for the problem (10.2.1), (10.2.3)–(10.2.4) and (10.2.7) was established in Section 10.2.1 (see also, ˜ Rivera and Qin [338]). Munoz In this section, under the above assumptions including dissipative conditions (10.2.2) and (10.2.5) on fî and gˆ i , we establish the exponential decay of energy. It is noteworthy that the system (10.2.1) is a dissipative equation and the boundary condition (10.2.4) is a damping boundary, which can be easily seen from (10.2.2) and (10.2.5)–(10.2.6). Note that when fî ≡ gˆ i = 0 (i = 1, 2, . . . , n), only polynomial decay of energy for the problem (10.2.1), (10.2.3)–(10.2.4) and (10.2.7) could be obtained (see Section 10.2.1, see also Qin and Muñ˜ oz Rivera [338]). Thus we naturally anticipate the exponential decay of energy with such dissipative effects from the system (10.2.1)


423

and from the boundary 0 . It follows from the proofs of our results that the interaction between two kinds of dissipative effects from the system (10.2.1) and the boundary 0 (see (10.2.4)) results in the exponential decay of energy; if one of these two effects vanishes, that is, if fˆ(x) ≡ 0 or g(x) ˆ ≡ 0, then one of the two terms 0 ∂th+1 φi ∂th+1 φi d0

and 0 ∂th+1 i ∂th+1 i d0 , which are very important to guarantee the exponential decay of energy, will disappear. Thus to establish the exponential decay of energy does not seem feasible. The method we will use here is based on the construction of a Lyapunov functional L(t) for which an inequality of the form d L(t) ≤ −CL(t) dt holds with C > 0 being a constant. To construct such a functional L(t) we start from the energy identity. Then we look for other functions whose derivatives introduce negative terms such as [∂th+1 φi ∂th+1 φi + Ai j kl ∂th φi, j ∂th φk,l ]d x, (∂th+1 φi ∂th+1 φi + ∂th+1 i (t)∂th+1 i (t))d0 , 0

etc., until we are able to construct the whole energy on the right-hand side of the energy identity. Finally we take L(t) as the sum of such functions. Unfortunately, such a process above also produces some terms without definite signs. To overcome this difficulty, we have to introduce a new multiplier which allows us to derive appropriate estimates. Finally we should carefully choose the coefficients of each term of L(t) so that the resulting sum can satisfy the required inequality. Define the space H = (H H11 ())n × (L 2 ())n × (L 2 (0 ))n × (L 2 (0 ))n with

H11 () = {u : u ∈ H 1(), u|1 = 0}.

It is easy to see that H together with the inner product u, w = (ρ Ai j kl u k,l wi, j + ρu n+ j wn+ j )d x + (kk j u 2n+ j w2n+ j + m j u 3n+ j w3n+ j )d0 0

(10.2.13)

is a Hilbert space, where u = (u 1 , u 2 , . . . , u 4n )τ , w = (w1 , w2 , . . . , w4n )τ ∈ H. Thus it follows from (10.2.10) that the induced norm on H by the above inner product |u|2H = (ρ Ai j kl u k,l u i, j + ρu n+ j u n+ j )d x + (kk j u 2n+ j u 2n+ j + m j u 3n+ j u 3n+ j )d0

0

424


is equivalent to the usual norm on H, u2H = (u i, j u i, j + u n+ j u n+ j )d x + (u 2n+ j u 2n+ j + u 3n+ j u 3n+ j )d0

0

for any u = (u 1 , . . . , u 4n )τ ∈ H. Define an operator A on H so that for smooth U = (φ, φt , , t )τ = (φ1 , . . . , φn , φ1 , . . . , φn , 1 , . . . , n , 1 , . . . , n ) ∈ R4n , the equations (10.2.1)–(10.2.4) are equivalent to U(t) ∈ D(A) and Ut = AU where U = (u 1 , . . . , u 4n )τ satisfies u i = φi , u n+i = φi , u 2n+i = i , u 3n+i = i , i = 1, 2, . . . , n

(10.2.14)

and D(A) = U = (u 1 , . . . , u 4n )τ ∈ H : u n+i ∈ H11 (), (Ai j kl u k,l ), j − fî u n+i ∈ L 2 (), u 3n+i ∈ L 2 (0 ), u 3n+i = Ai j kl u k,l ν j + gˆ i u n+i on 0 , i, j, k, l = 1, . . . , n . In the definition of D(A), u 3n+i = Ai j kl u k,l ν j + gˆ i u n+i is in the weak sense that

[(Ai j kl u k,l ), j + Ai j kl u k,l , j ]d x =

0

(u 3n+i − gˆ i u n+i ) d0 , ∀ ∈ H11 ()

and (Ai j kl u k,l ), j − fî u n+i ∈ L 2 () is equivalent to the condition of u 3n+i = Ai j kl u k,l ν j + gˆ i u n+i on 0 as a trace. Obviously, we derive from (10.2.1), (10.2.3)–(10.2.4) and (10.2.2) that u i = u n+i , u n+i u 2n+i u 3n+i

(10.2.15) î

= (Ai j kl u k,l ), j − f u n+i , = u 3n+i ,

(10.2.16) (10.2.17)

= −(ρu n+i + di u 3n+i + ki u 2n+i )/m i

(10.2.18)

where i = 1, 2, . . . , n and −(ρu n+i + di u 3n+i + ki u 2n+i )/m i is understood as the trace in H 1/2(0 ).


425

For any U ∈ D(A), it is easy to verify from (10.2.1)–(10.2.6) and the definition of D(A) that AU, U = (ρ Ai j kl u n+k,l u i, j + ρ[(Ai j kl u k,l ), j − fî u n+i ]u n+i d x + [ki u 3n+i u 2n+i − m i (ρu n+i + di u 3n+i + ki u 2n+i )u 3n+i /m i ]d0 0 =ρ Ai j kl u n+k,l u i, j d x + ρ Ai j kl u k,l ν j u n+i d0 0 i ˆ −ρ Ai j kl u k,l u n+i, j d x − ki u 3n+i u 2n+i d0 f u n+i u n+i d x + 0 − (ρu n+i + di u 3n+i + ki u 2n+i )u 3n+i d0 0 =− (di u 3n+i u 3n+i + ρ gˆ i u n+i u n+i )d0 − ρ fî u n+i u n+i d x ≤ 0 0

(10.2.19) which implies that A is a dissipative operator on H. Thus, similar to the proofs of Theorems 10.1.1–10.1.2, we are able to obtain the following results on the global existence and regularity of solutions. Theorem 10.2.1. The operator A defined on H is closed, densely defined, and dissipative. It generates a C0 -semigroup on H. Theorem 10.2.2. Assume that U0 ∈ H is C ∞ and vanishes near ∂; let U(t) be the ¯ solution of U (t) = AU(t), t ≥ 0, with U(0) = U0 . Then u 1 (t), . . . , u 2n (t) ∈ C ∞ () and u 2n+1 (t), . . . , u 4n (t) ∈ C ∞ (0 ) for any t ≥ 0. Remark 10.2.1. “U vanishes near ∂” means u 2n+1 = · · · = u 4n = 0 on 0 , and u 1 , . . . , u 2n vanish near ∂. Similarly to (10.1.17)–(10.1.18), we introduce the energy functions 1 (ρφi φi + ρ Ai j kl φk,l φi, j )d x E 0 (t; φ, ) = 2 1 + (ki (x)i i + m i (x)i i )d0 , 2 0 E h (t) ≡ E h (t; φ, ) = E 0 (t; ∂th φ, ∂th ),

h = 1, 2, . . . , m.

(10.2.20) (10.2.21)

The following is our main result on the large-time behavior of energy. Theorem 10.2.3. Under the above assumptions and taking smooth initial datum (φ 0 , φ 1 , 0 , 1 ) such that m E h (0) < +∞ (10.2.22) h=0

426


for some integer m ≥ 0. Then

m

E h (t) decays exponentially, i.e., there exists a positive

h=0

constant C ∗ such that m

E h (t) ≤ C ∗

h=0

m

∗

E h (0)e−C t , ∀t > 0.

(10.2.23)

h=0

We also put · = · L 2 and use C (sometimes C1 , C2 , . . . ) to stand for the universal positive constant independent of time t > 0.

10.2.2 Proof of Theorem 10.2.3 In this section we are going to prove Theorem 10.2.3. To this end, we first establish some energy estimates. First, multiplying equation (10.2.1) by φi , and using Green’s formula, we arrive at ρ d φi φi d x − ρ Ai j kl φk,l ν j φi d + ρ Ai j kl φk,l φi, j d x + ρ fî φi φi d x = 0. 2 dt (10.2.24) Noting that 2 Ai j kl φk,l φi, j = ∂(Ai j kl φk,l φi, j )/∂t and using (10.2.4)–(10.2.7), we deduce that d ρ (φ φ + Ai j kl φk,l φi, j )d x = ρ Ai j kl φk,l ν j φi d0 − ρ fî φi φi d x. dt 2 i i 0 (10.2.25) By (10.2.3)–(10.2.4), we have ρ Ai j kl φk,l ν j φi d0 = ρ φi (i − gˆ i φi )d0 0 0 1 d =− (m i i i + ki i i )d0 − (di i i + ρ gˆ i φi φi )d0 2 dt 0 0 which along with (3.2) gives d i ˆ E 0 (t; φ, ) = −ρ (di (x)i i + ρ gˆ i φi φi )d0 . f φi φi d x − dt 0

(10.2.26)

Similarly, keeping in mind that equation (10.2.25) and boundary conditions (10.2.3)– (10.2.4) and (10.2.7) are all linear in t, we have that for h = 0, 1, . . . , m, d E h (t; φ, δ) = − ρ fî ∂th+1 φi ∂th+1 φi d x dt (di (x)∂th+1 i ∂th+1 i + gˆ i ∂th+1 φi ∂th+1 φi )d0 . (10.2.27) − 0


427

Let us put F0 (t; φ, δ) =

(φi qμ φi,μ +

(n − 1) (n − 1) î φi φi + f (x)φi φi )d x, 2 4

Fh (t) ≡ Fh (t; φ, ) ≡ F0 (t; ∂th φi , ∂th i ),

h = 1, 2, . . . , m.

(10.2.28) (10.2.29)

Lemma 10.2.1. For h = 0, 1, . . . , m, we have 1 (n − 1) d Fh (t) ≤ qμ νμ ∂th+1 φi ∂th+1 φi d0 + Ai j kl ∂th φk,l ∂th φi ν j d0 dt 2 0 2 0 1 1 − [∂th+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j ]d x + Ai j kl,μ ∂th φk,l ∂th φi, j qμ d x 2 2 1 + Ai j kl ν j qμ ∂th φk,l ∂th φi,μ d0 − Ai j kl νμ qμ ∂th φk,l ∂th φi, j d0 2 0 0 a1 − Ai j kl ∂th φk,l ∂th φi, j d1 − (10.2.30) fî ∂th+1 φi qμ ∂th φi,μ d x, 2 1 where a1 = min1 [−qμ νμ ] ≥ 0. Proof. Since (10.2.1), (10.2.3)–(10.2.4) and (10.2.7) are all linear in t, it suffices to prove (10.2.30) for the case of h = 0. First we can derive from (10.2.1)–(10.2.7) that d φi φi d x = (φi φi − Ai j kl φk,l φi, j )d x + Ai j kl φk,l φi ν j d0 − fî φi φi d x dt 0 which gives 1 d φi φi + fî φi φi d x = (φi φi − Ai j kl φk,l φi, j )d x + Ai j kl φk,l φi ν j d0 . dt 2 0 (10.2.31) Similarly, by (10.2.1), d 1 φi qμ φi,μ + qμ (φi φi ),μ d x (10.2.32) φi qμ φi,μ d x = dt 2 1 1 = ([ Ai j kl φk,l ], j − fî φi )qμ φi,μ d x + qμ νμ φi φi d0 − qμ,μ φi φi d x. 2 0 2 By (10.2.10), we deduce (Ai j kl φk,l φi, j ),μ = Ai j kl,μ φk,l φi, j + 2 Ai j kl φk,l φi, j μ or Ai j kl φk,l φi, j μ =

1 [(Ai j kl φk,l φi, j ),μ − Ai j kl,μ φk,l φi, j ]. 2

(10.2.33)

428


By (3.10), we arrive at [ Ai j kl φk,l ], j qμ φi,μ d x = Ai j kl ν j qμ φk,l φi,μ d − [ Ai j kl qμ, j φk,l φi,μ + Ai j kl qμ φk,l φi, j μ ]d x 1 = Ai j kl ν j qμ φk,l φi,μ d − Ai j kl qμ, j φk,l φi,μ d x − (Ai j kl φk,l φi, j ),μ qμ d x 2 1 + Ai j kl,μ φk,l φi, j qμ d x 2 1 = Ai j kl ν j qμ φk,l φi,μ d − Ai j kl qμ, j φk,l φi,μ d x + Ai j kl,μ φk,l φi, j qμ d x 2 1 1 − Ai j kl φk,l φi, j qμ νμ d + Ai j kl φk,l φi, j qμ,μ d x 2 2 which, combined with (10.2.32), yields d φ qμ φi,μ d x dt i 1 1 = qμ νμ φi φi d0 − qμ,μ φi φi d x + Ai j kl φk,l φi,μ ν j qμ d 2 0 2 1 1 − Ai j kl φk,l φi,μ qμ, j d x + Ai j kl,μ φk,l φi, j qμ d x − Ai j kl φk,l φi, j νμ qμ d 2 2 1 + Ai j kl φk,l φi, j qμ,μ d x − fî φi qμ φi,μ d x 2 1 (n − 1) = qμ νμ φi φi d0 − [φi φi − Ai j kl φk,l φi, j ]d x 2 0 2 1 − [φi φi + Ai j kl φk,l φi, j ]d x + Ai j kl φk,l φi,μ qμ ν j d 2 1 1 − Ai j kl φk,l φi, j qμ νμ d + Ai j kl,μ φk,l φi, j qμ d x 2 2 1 + [ Ai j kl φk,l φi, j qμ,μ − fî φi qμ φi,μ ]d x. (10.2.34) 2 Multiplying (10.2.31) by (n − 1)/2 and then adding the resulting equation to (10.2.34), we deduce d 1 (n − 1) F0 (t; φ, ) = qμ νμ φi φi d0 + Ai j kl φk,l φi ν j d0 (10.2.35) dt 2 0 2 0 1 − [φ φ + Ai j kl φk,l φi, j ]d x + Ai j kl φk,l φi,μ qμ ν j d 2 i i 1 1 − Ai j kl φk,l φi, j qμ νμ d + Ai j kl,μ φk,l φi, j qμ d x − fî φi qμ φi,μ d x. 2 2


429

Noting that φ|1 = 0, it holds that on 1 , ν j φi,μ = νμ φi, j which implies 1 Ai j kl φk,l φi,μ qμ ν j d − Ai j kl φk,l φi, j qμ νμ d = Ai j kl φk,l φi,μ qμ ν j d0 2 0 1 1 + Ai j kl φk,l φi, j qμ νμ d1 − Ai j kl φk,l φi, j qμ νμ d0 2 1 2 0 a1 ≤ Ai j kl φk,l φi,μ qμ ν j d0 − Ai j kl φk,l φi, j d1 2 1 0 1 − Ai j kl φk,l φi, j qμ νμ d0 . (10.2.36) 2 0 Thus (10.2.30) with h = 0 follows from (10.2.35)–(10.2.36). Operating ∂th on (10.2.1)–(10.2.7), repeating the same argumentation as above, we easily derive the desired estimate (10.2.30). The proof is complete. Now if we define G 0 (t; φ, ) = [m i (x)i i + di (x)i i /2 + ρφi i ]d0 , 0

G h (t) ≡ G h (t; φ, ) ≡ G 0 (t; ∂th φ, ∂th ), h = 1, 2, . . . , m

(10.2.37) (10.2.38)

then we have Lemma 10.2.2. For h = 0, 1, 2, . . . , m, we have d h h G h (t) = − ki (x)∂t i ∂t i d0 + [m i (x)∂th+1 i ∂th+1 i + ρ∂th φi ∂th+1 i ]d0 . dt 0 0 (10.2.39) Proof. It suffices to prove the case of h = 0. Multiplying (10.2.3) by i and integrating the resulting equation on 0 give the desired estimate (10.2.39). Noting that Eq. (10.2.1) and boundary conditions (10.2.3)–(10.2.4), (10.2.7) are all linear in t, we operate ∂th on (10.2.1)–(10.2.4) and repeat the same process as above to be able to derive the desired estimate (10.2.39) for the general cases h = 1, 2, . . . , m. The proof is complete. Now put L(t) = N 3

m h=0

E h (t) + N

m

G h (t) +

h=0

m

Fh (t)

(10.2.40)

h=0

where N is a large positive number to be determined later on. Lemma 10.2.3. For N large enough, there are positive constants C0 , C1 and C2 such that m m 0 ≤ C0 E h (t) ≤ L(t) ≤ C1 E h (t), ∀t ≥ 0 (10.2.41) h=0

h=0

430


and

d L(t) ≤ −C C2 E h (t), dt m

∀t > 0.

(10.2.42)

h=0

Proof. By (10.2.7), (10.2.10) and the trace theorem, we infer that for h = 0, 1, . . . , m, ∂th φi ∂th φi d0 ≤ C∂th φ2H 1/2 () ≤ C∂th φ2H 1 () 0 ≤C Ai j kl ∂th φk,l ∂th φi, j d x (10.2.43)

and

∂th φi ∂th φi d x

≤C

≤C

∂th φi,μ ∂th φi,μ d x ≤ C∂t φ2H 1 ()

Ai j kl ∂th φk,l ∂th φi, j d x.

(10.2.44)

Thus by (10.2.20)–(10.2.21), (10.2.26)–(10.2.27), (10.2.35)–(10.2.39) and (10.2.43)– (10.2.44), we infer that for N large enough there are two positive constants C0 and C1 such that m 1 3 L(t) = N (ρ∂ h+1 φi ∂th+1 φi + ρ Ai j kl ∂th φk,l ∂th φi, j )d x 2 t h=0 1 h h h+1 h+1 + (ki ∂t i ∂t i + m i ∂t i ∂t i )d0 2 0 m +N (m i ∂th+1 i ∂th i + di ∂th i ∂th i /2 + ρ∂th φi ∂th i )d0 +

h=0 0 m

∂th+1 φi qμ ∂th φi,μ + (n − 1)∂th+1 φi ∂th φi /2 +

h=0 m 3

≥ CN

h=0


+ m

n − 1 î h f ∂t φi ∂th φi d x 4

0

(∂th i ∂th i + ∂th+1 i ∂th+1 i )d0

(∂th i ∂th i + ∂th+1 i ∂th+1 i + ∂th φi ∂th φi )d0 0 h=0 m −C (∂th+1 φi ∂th+1 φi + ∂th φi,μ ∂th φi,μ + ∂th φi ∂th φi )d x h=0 − CN

≥ C0

m

E h (t)

h=0

(10.2.45)


431

and L(t) ≤ C N 3

0

+C N

h=0

+

m

(∂th i ∂th i m

(∂th+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j )d x + ∂th+1 i ∂th+1 i )d0

(∂th i ∂th i + ∂th+1 i ∂th+1 i + ∂th φi ∂th φi )d0

h=0 0 m +C (∂th+1 φi ∂th+1 φi h=0

+ ∂th φi,μ ∂th φi,μ + ∂th φi ∂th φi )d x ≤ C1

m

E h (t)

h=0

which, along with (10.2.45), gives (10.2.41). On the other hand, by (10.2.27)–(10.2.30) and (10.2.39), we obtain m d L(t) ≤ −N 3 di ∂th+1 i ∂th+1 i d0 dt 0 h=0 +ρ fî ∂th+1 φi ∂th+1 φi d x + ρ

−N −

1 2

m

h=0 0 m

h=0 m

ki ∂th i ∂th i d0 + N

m h=0 0

0

gˆ i ∂th+1 φi ∂th+1 φi d0

(m i ∂th+1 i ∂th+1 i + ρ∂th φi ∂th+1 i )d0


(n − 1) 1 h+1 h+1 h h qμ νμ ∂t φi ∂t φi + Ai j kl ∂t φk,l ν j ∂t φi d0 + 2 0 2 h=0 1 + Ai j kl,μ qμ ∂th φk,l ∂th φi, j d x + Ai j kl ν j ∂th φk,l ∂th φi,μ qμ d0 2 0 1 a1 h h − qμ νμ Ai j kl ∂t φk,l ∂t φi, j d0 − Ai j kl ∂th φk,l ∂th φi, j d1 2 0 2 1 i h+1 h ˆ − (10.2.46) f ∂t φi qμ ∂t φi,μ d x .

Now we need to estimate some terms on the right-hand side of (10.2.46). Using (10.2.4), (10.2.33), (10.2.43), the Young inequality and the definition of 0 , we deduce that (n − 1) (n − 1) Ai j kl ∂th φi ∂th φk,l ν j d0 = [∂th φi ∂th+1 i + gˆ i ∂th φi ∂th+1 φi ]d0 2 2 0 0

432


≤

min(1, β) 16

Ai j kl ∂th φi, j ∂th φk,l d x + C

0

[∂th+1 i ∂th+1 i + ∂th+1 φi ∂th+1 φi ]d0 ,

(10.2.47) h h h+1 h i h+1 h ∂t i qμ ∂t φi,μ − gˆ qμ ∂t φi ∂t φi,μ d0 Ai j kl ∂t φk,l ν j qμ ∂t φi,μ d0 = 0 0 α min(1, β) h h ≤ qμ νμ ∂t φi, j ∂t φi, j d0 + Ai j kl ∂th φi, j ∂th φk,l d x 4 0 16 +C [∂th+1 i ∂th+1 i + ∂th+1 φi ∂th+1 φi ]d0 0 1 min(1, β) h h ≤ qμ νμ Ai j kl ∂t φk,l ∂t φi, j d0 + Ai j kl ∂th φi, j ∂th φk,l d x 4 0 16 +C [∂th+1 i ∂th+1 i + ∂th+1 φi ∂th+1 φi ]d0 (10.2.48)

0

and

N

ρ∂th φi ∂th+1 i d0 min(1, β) ≤ Ai j kl ∂th φk,l ∂th φi, j d x + C N 2 ∂th+1 i ∂th+1 i d0 . 16 0 0

(10.2.49)

In view of (10.2.12), we easily derive 1 1 h h Ai j kl,μ qμ ∂t φk,l ∂t φi, j d x − (∂ h+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j )d x 2 2 t 1 β h+1 h+1 ≤− ∂ φ i ∂t φ i d x − Ai j kl ∂th φk,l ∂th φi, j d x 2 t 2 min(1, β) ≤− (∂th+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j )d x, (10.2.50) 2 − fî ∂th+1 φi qμ ∂th φi,μ d x min(1, β) h+1 h+1 ≤ Ai j kl ∂t φk,l ∂t φi, j d x + C ∂th+1 φi ∂th+1 φi d x. (10.2.51) 16 Thus it follows from (10.2.45)–(10.2.51) that for N > 1 large enough, there are constants C2 , C3 > 0 such that m d 3 h+1 h+1 h+1 h+1 h+1 h+1 L(t) ≤ −C N (∂t i ∂t i + ∂t φi ∂t φi )d0 + ∂t φi ∂t φi d x dt 0 h=0 m m − CN ∂th i ∂th i d0 + C N ∂th+1 i ∂th+1 i d0 h=0 0

h=0 0


min(1, β) + 4 m

433

h=0 m

Ai j kl ∂th φk,l ∂th φi, j d x

+ CN

2

m h=0 0

∂th+1 i ∂th+1 i d0

min(1, β) (∂th+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j )d x 2 h=0 m m +C (∂th+1 φi ∂th+1 φi + ∂th+1 i ∂th+1 i )d0 + C ∂th+1 φi ∂th+1 φi d x −

−

h=0 0 m a0

4

≤ −C3 + +

h=0 0 m

0

h=0

0

≤ −C2

a1 2 m

Ai j kl ∂th φk,l ∂th φi, j d0 −

h=0

h=0 1


(∂th+1 i ∂th+1 i + ∂th i ∂th i )d0

(∂th+1 φi ∂th+1 φi + Ai j kl ∂th φk,l ∂th φi, j )d0

m

(∂th+1 φi ∂th+1 φi

+

Ai j kl ∂th φk,l ∂th φi, j )d x

+ a1

1


E h (t).

h=0


Proof of Theorem 10.2.3. By assumption (10.2.22) and inequalities (10.2.41)–(10.2.42), we get m d L(t) ≤ −C C2 E h (t) ≤ −C1−1 C2 L(t) dt h=0

which gives

−1

L(t) ≤ L(0)e−C1 Thus the estimate (10.2.23) with proof is now complete.

C∗

=

C1−1 C2

C2 t

, ∀t > 0.

(10.2.52)

follows from (10.2.41) and (10.2.52). The

10.3 Bibliographic Comments Besides [26–28], we refer the readers to Morse and Ingard [272] for the theory of theoretical acoustics. For the viscoelastic models and wave equations, we would like to refer to the works by Li and Chen [227, 229], Muñ˜ oz Rivera and Andrade [276], Nakao [288–292], Zuazua [464, 465], Qin [314], and the references therein.

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Index attractors global ∼, 37 universal ∼, 154 in H i (i = 1, 2, 4), 332 Bessel potential, 15 blow-up of solutions, 381 bounded annular domains, 167 Bunyakovskii inequality, 28 C0 -semigroups of linear operators, 31 Cauchy-Schwarz inequality, 28 clamped and constant temperature boundary conditions, 71 classical Bellman-Gronwall inequality, 18 classical thermoelastic model, 361 Clausius-Duhem inequality, 47, 293 closed operator, 32 compact semigroups (semiflows) for autonomous systems, 39 weakly ∼, 42 compactness theorem, 14 continuous spectrum, 34 cylinder symmetry, 245 decay exponential ∼, 421 polynomial ∼, 409 density theorem, 15 differential inequalities for nonexistence of global solutions, 25 distribution, 2

eigenvalues, 32 embedding theorem, 9, 14 equation compressible Navier-Stokes ∼, xi thermoelastic ∼ of hyperbolic type, 363 Volterra ∼, 346 equicontinuous, 14 existence of an absorbing set in Hδ2, 335 in Hδ1, 332 in Hδ4, 336 exponential decay, 421 exponential stability and maximal attractors, 325 in H 1 and H 2, 331 in H 4, 332 fixed and thermally insulated boundary conditions, 46 Gagliardo-Nirenberg inequality, 270 generalized Bellman-Gronwall inequality, 19 global attractors, 37 global existence and asymptotic behavior of solutions, 293 Gronwall’s inequality, 259 Helmholtz free energy function, 47, 294 Hille-Yosida theorem for the infinitesimal generators of C0 contraction semigroups, 35 Holder ¨ inequality, 28

464

inequality Bellman-Gronwall ∼ classical ∼, 18 generalized ∼, 19 uniform ∼, 20 Bunyakovskii ∼, 28 Cauchy-Schwarz ∼, 28 Clausius-Duhem ∼, 47, 293 differential ∼ for nonexistence of global solutions, 25 Gagliardo-Nirenberg ∼, 270 Gronwall’s ∼, 259 Holder ¨ ∼, 28 Jensen ∼, 30 Krejci-Sprekels ∼, 21 Minkowski ∼, 29 Nakao ∼, 23 Poincaré ∼, 17 Schwarz ∼, 28 Shen-Zheng ∼, 21 Sobolev ∼, 9 Troisi ∼, 10 Young ∼, 27, 270 Zheng ∼, 22 infinitesimal generator, 31 Jensen inequality, 30 Krejci-Sprekels inequality, 21 Laplace transform, 342 large-time behavior of solutions, 159 Leray-Schauder fixed point theorem, xi Lumer-Phillips theorem, 36 Lyapunov functional, 423 Minkowski inequality, 29 multi-dimensional elasticity, 409 Nakao inequalities, 23 Navier-Stokes equation compressible ∼, xi

Index

Paley-Wiener theorem, 366 Piola-Kirchhoff stress tensor, 340 Plancherel identity, 343 Poincaré´ inequality, 17 polynomial decay, 409 polytropic viscous and heat-conductive gas, 143 positive definite kernel strongly, 340 potential Bessel ∼, 15 Riesz ∼, 15 Rellich-Kondrachov compactness embedding theorem, 16 theorem, 12 rest spectrum, 34 Riesz potential, 15 Schwarz inequality, 28 Shen-Zheng inequality, 21 Sobolev embedding theorem, 38 function, 8 inequalities, 9 space, 5 homogeneous ∼, 5 of fractional order, 15 spectral set, 33 spectrum continuous ∼, 34 rest ∼, 34 theorem compactness ∼, 14 density ∼, 15 embedding ∼, 9, 14 Hille-Yosida ∼ for the infinitesimal generators of C0 contraction semigroups, 35 Leray-Schauder fixed point ∼, xi Lumer-Phillips ∼, 36 Paley-Wiener ∼, 366 Rellich-Kondrachov

Index

compactness ∼, 12 compactness embedding ∼, 16 Sobolev embedding ∼, 38 trace ∼, 9 thermoelastic equations of hyperbolic type, 363 thermoelastic system, xiv of type II, xiv, 361 of type III, xiv, 362 with a thermal memory, 339 thermoelasticity, 339 thermoviscoelastic system (Model), xiii thermoviscoelasticity one-dimensional nonlinear ∼, 293 trace theorem, 9 Troisi inequality, 10 uniform a priori estimates, 175 uniform Bellman-Gronwall inequality, 20 universal attractors, 154 universal attractors in H i (i = 1, 2, 4), 332 viscoelastic model, xiii viscous and heat-conductive real gas, 45 Volterra equations, 346 weak derivatives, 4 well-posedness global ∼, xi local ∼, xi Young inequality, 27, 270 Zheng inequality, 22

465

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