Attractors for Nonlinear Autonomous Dynamical Systems 9782759827039

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Current Natural Sciences

Yuming QIN and Keqin SU

Attractors for Nonlinear Autonomous Dynamical Systems

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EDP Sciences – ISBN(print): 978-2-7598-2702-2 – ISBN(ebook): 978-2-7598-2703-9 DOI: 10.1051/978-2-7598-2702-2 All rights relative to translation, adaptation and reproduction by any means whatsoever are reserved, worldwide. In accordance with the terms of paragraphs 2 and 3 of Article 41 of the French Act dated March 11, 1957, “copies or reproductions reserved strictly for private use and not intended for collective use” and, on the other hand, analyses and short quotations for example or illustrative purposes, are allowed. Otherwise, “any representation or reproduction – whether in full or in part – without the consent of the author or of his successors or assigns, is unlawful” (Article 40, paragraph 1). Any representation or reproduction, by any means whatsoever, will therefore be deemed an infringement of copyright punishable under Articles 425 and following of the French Penal Code. The printed edition is not for sale in Chinese mainland. Customers in Chinese mainland please order the print book from Science Press. ISBN of the China edition: Science Press 978-7-03-070250-0 Ó Science Press, EDP Sciences, 2022

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

CHAPTER 1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Some Useful Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Basic Theory of Infinite-Dimensional Dynamical Systems for Autonomous Nonlinear Evolutionary Equations . . . . . . . . . . 1.2.1 Uniformly Compact Semigroups . . . . . . . . . . . . . . . . . . . 1.2.2 Weakly Compact Semigroups . . . . . . . . . . . . . . . . . . . . . 1.2.3 X-Limit Compact Semigroups . . . . . . . . . . . . . . . . . . . . 1.2.4 Asymptotically Compact Semigroups . . . . . . . . . . . . . . . 1.2.5 Asymptotically Smooth Semigroups . . . . . . . . . . . . . . . . 1.2.6 Norm-to-Weak Continuous Semigroups . . . . . . . . . . . . . . 1.2.7 Closed Operator Semigroups . . . . . . . . . . . . . . . . . . . . . 1.3 Basic Theory of Finite-Dimensional Attractors . . . . . . . . . . . . . 1.3.1 The Fractal Dimension of Global Attractors . . . . . . . . . . 1.3.2 The Estimate on Fractal Dimension of Global Attractors

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1 1

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10 10 16 17 22 27 28 30 32 32 33

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37 37 43 43 44 46

Global Attractor and Its Upper Estimate on Fractal Dimension for the 2D Navier–Stokes–Voight Equations . . . . . . . . . . . . . . . . . . . . . . 3.1 Global Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Existence of Global Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Existence of Absorbing Sets . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Some Compactness and the Existence of Global Attractors

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47 47 55 55 56

CHAPTER 2 Global Attractors for the Navier–Stokes–Voight Equations with Delay 2.1 Global Wellposedness of Solutions . . . . . . . . . . . . . . . . . . . . . . . 2.2 Existence of Global Attractors . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Dissipation: Existence of Absorbing Sets . . . . . . . . . . . . 2.2.2 Asymptotical Compactness and Existence of Attractor . . 2.3 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 3

Contents

IV

3.3 3.4

Upper Estimate on the Fractal Dimension of Global Attractors . . . . . . Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58 64

CHAPTER 4 Maximal Attractor for the Equations of One-Dimensional Compressible Polytropic Viscous Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Our Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nonlinear Semigroup on H ð2Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . ð1Þ 4.3 Existence of an Absorbing Set in Hb . . . . . . . . . . . . . . . . . . . . . 4.4 4.5 4.6

.... .... ....

67 67 69

....

73

Existence of an Absorbing Set in ......................... Proof of Theorem 4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 86 88

ð2Þ Hb

CHAPTER 5 Universal Attractors for a Nonlinear System of Compressible One-Dimensional Heat-Conducting Viscous Real Gas . . . . . . 5.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i ði ¼ 1; 2Þ . . . . . . . . . . . 5.2 Nonlinear C0 -Semigroup on H þ 5.3 Existence of an Absorbing Set in Hd1 . . . . . . . . . . . . . . 5.4 Existence of an Absorbing Set in Hd2 . . . . . . . . . . . . . . 5.5 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . .

............ ............ ............ . . . .

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91 91 95 97 106 108 111

CHAPTER 6 Global Attractors for the Compressible Navier–Stokes Equations in Bounded Annular Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Nonlinear Semigroup on H ð2Þ . . . . . . . . . . . . . . . . . . . . . . ð1Þ 6.3 Existence of an Absorbing Set in Hd . . . . . . . . . . . . . . . . ð2Þ Hd ð4Þ Hd

. . . . . . . . . 115 . . . . . . . . . 115 . . . . . . . . . 119 . . . . . . . . . 120

6.4

Existence of an Absorbing Set in

6.5 6.6

Existence of an Absorbing Set in . . . . . . . . . . . . . . . . . . . . . . . . . 135 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

. . . . . . . . . . . . . . . . . . . . . . . . . 129

CHAPTER 7 Global Attractor for a Nonlinear Thermoviscoelastic System in Shape Memory Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 An Absorbing Set Bd in Hd . . . . . . . . . . . . . . . . . . . . . 7.3 Compactness of the Orbit in Hd . . . . . . . . . . . . . . . . . 7.4 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . .

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149 149 152 165 173

Contents

V

CHAPTER 8 Global Attractors for Nonlinear Reaction–Diffusion Equations and the 2D Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Global Attractor for Strong Solutions of Reaction–Diffusion Equations . . 8.1.1 Existence of Solutions and Uniqueness . . . . . . . . . . . . . . . . . . . 8.1.2 Global Attractor for the Semigroup in Lp ðXÞ . . . . . . . . . . . . . . 8.1.3 Global Attractor of System in Lp ðXÞ and H01 ðXÞ . . . . . . . . . . . . 8.2 Global Attractors for the 2D Navier–Stokes Equations in H01 ðXÞ . . . . .

175 175 176 176 177 183

CHAPTER 9 Global Attractors for an Incompressible Fluid Equation and a Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 An Incompressible Fluid Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 A Wave Equation with Nonlinear Damping . . . . . . . . . . . . . . . . . . . . 9.2.1 Wellposedness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Asymptotic Compactness and Existence of Global Attractor .

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187 187 193 194 196 200

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

In memory of Yuming’s father, Zhenrong QIN In memory of Keqin’s father, Shitou SU Dedicated to Yuming’s mother, Xilan XIA, Keqin’s mother, Yeling CHENG, Yuming’s wife and son, Yu YIN and Jia QIN, and Keqin’s wife and son, Hua LIU and Kejia SU

Preface

This book is based on the first author’s lecture “Infinite-dimensional autonomous dynamical systems on nonlinear evolutionary equations” given to graduate students in Donghua University since 2004. It aimed at presenting complete and systematic theories of infinite-dimensional dynamical systems and their applications in partial differential equations, especially in the models of fluid mechanics. Driven by the rapid advancement of science and technology, and the improvement of distributed function theory and Sobolev space theory in the first half of the 20th century, the theoretical framework for modern partial differential equations (PDEs) has been established almost completely. PDEs are one of the important branches of mathematics, which are widely used in natural science and engineering. An important purpose of the theory of partial differential equation is to reveal the existence of solutions and their asymptotic behavior in time, which is very significant for the stability of these systems. Based on the development, inspired by the theoretical and applied research into the dynamical system for ordinary differential equations in 1960s and 1970s, mathematicians began to investigate the asymptotic behavior in time of solutions of dissipative evolutionary equations, which led to the emergence of a new branch in evolutionary equations–the infinite-dimensional dynamical system (for short, idds) in theory and application. At the end of the 19th century, Hadamard discovered chaos when he was studying the Hamilton system. The Russian mathematician Lyapunov traced the source of research into dynamical systems and introduced the energy functions and characteristic functions, i.e., the concept of the Lyapunov function and index; this theory is the basis for the investigation into dissipative evolutionary equations, such as the LaSelle invariant principle, Morse structure and fractal dimension of global attractors. French mathematician Poincaré discussed sensitivity with initial data for celestial system using variational equation and integral inequality, proposed the concepts of stability, orbit periodic solutions, etc. Using the geometric method, he described the dynamical systems for ordinary differential equations, and initiated DOI: 10.1051/978-2-7598-2702-2.c901 Ó Science Press, EDP Sciences, 2022

X

Preface

research into the geometric theory for ordinary differential equations and for research into dynamical systems. Lyapunov and Poincaré were all devoted to study the case that how the solution orbits depend on time, which is the foundation for the dynamical systems. Birkhoff defined the ω-limit point and α-limit point of motion, non-wandering set, central motion, recovery motion, transition, minimal set and Hamilton system in his monograph “Dynamical Systems” and further proved their existence. Moreover, he also initiated the research into ergodic theory. These works are the significant basis for modern dynamical systems. In the meantime, the Ergodic theorem had been given, which built up a bridge between dynamical system and functional analysis. In 1931, Markov proposed an abstract framework and concepts of dynamical systems. Kolmogrov put forward the concepts of Kolmogrov entropy, fractal dimension, Lyapunov index and Kolmogrov entropy are the index system to measure chaos. Smale proved that the orbit for a dynamical system asymptotically approximates to a complex set called strange attractor. Based on the behavior of a dynamical system that depends on a certain parameter and changes essentially with this parameter, Feigenbaum discovered the period doubling bifurcation law. In 1963, Lorentz gave the existence and geometric structure of chaos for the Lorentz system. In 1975, Mandevbrot introduced the concept of fractal dimensions and proved their existence. Rull and Taken pointed out that the turbulence itself is chaos possessing strange attractors and Feigenbaum bifurcation, and pointed out the atmospheric motion itself is a complex chaos. The study of modern theory of finite-dimensional dynamical systems has established a modern framework for ordinary differential equations (such as ε-entropy, ε-capacity, bifurcation theory, differential dynamical system, symbolic dynamical system, etc.); these works are all based on differential equations with finite dimensional phase space, difference equations and integral equations. Later on, physicists, dynamicists and mathematicians found some irregular phenomena in the observation, water wave equations (KdV equations) have solitary, some hyperbolic equations (such as Burger’s equation) generate shock waves under certain conditions, and fluid flow (NS equations) contains turbulence (i.e., chaos phenomenon), etc., which inspired the study of infinite-dimensional dynamical systems (for short, idds). The idds is focused on the asymptotic behavior in time of solutions to nonlinear dissipative evolutionary equations in the science of nature such as physics, mechanics, biology, material science and atmospheric science, which is a widely used subject, very important in the reduction and computation of the asymptotical forms of nonlinear evolution equations, and its theoretical meaning is to provide some proving methods on compactness, and reduce and compute the long-time state of complex system such as Navier–Stokes system, MHD system and all kinds of nonlinear dissipative systems. The existence and properties of geometry and topology of global attractors to nonlinear evolution equation are always the key issue of concern in the infinite-dimensional dynamical system. For dynamical systems, the top concern is existence, geometric and topological properties of attractors; thus the existence problem of attractors is a hot topic in the infinite-dimensional dynamical systems. In the 1980s, the concept of attractors was presented in many papers [75, 146]. Thereafter, many scholars such as Babin, Vishik,

Preface

XI

Foias, Hale, Temam, Ladyzhenskaya, Sell, Miranville, Zelik, Grasselli, Boling Guo, Songmu Zheng, Chengkui Zhong and so on, devoted themselves to the study into the infinite-dimensional dynamical systems, and achieved momentous results [4, 5, 7, 18, 23, 24, 103, 133]. The attractor of autonomous systems is called the global attractor, and there has been a rapid development in the corresponding theories on the existence, dimension estimates, exponential attractor and inertial manifold [4, 54, 146]. Many problems in mathematical physics show that the asymptotic behavior in time of solutions could be described by the global attractor, the central issue of idds is to study the existence, geometric topological properties of global attractors to the dissipative evolutionary equations in suitable Banach spaces or complete metric spaces. In general, the global attractor is a nonempty, compact-invariant set, which is also the union of all bounded complete trajectories from its structure. Thus, the global attractor contains all the fixed points of a semigroup, all the equilibrium points of a system, or all the critical points of the Lyapunov function if it has. Meanwhile, it contains the periodic orbit, quasi-periodic orbit, the streamline from one equilibrium point to another equilibrium point, i.e., heteroclinic orbit, and also the homoclinic orbit. Generally, the more the equilibrium points, the worse the local dynamics properties of the equilibrium point. The multiplicity of equilibrium points is closely associated with the topological theory of nonlinear functional analysis and the theory of critical points. As to the autonomous evolution equations, the sufficient conditions to obtain the global attractor are as follows: (1) the solution semigroup of system is continuous for any fixed time in phase space; (2) a bounded global absorbing set of system exists; (3) the semigroup possesses “compactness” in some sense whose verification is relatively more difficult, and the usual methods to verify the compactness contain a uniformly compact semigroup method, weakly compact semigroup method, ω-limit compact method, asymptotically compact method, asymptotically smooth method, norm-to-weak continuous semigroup method and closed operator semigroup method. The uniformly compact method is the most basic and classical method, whose basic idea is to verify that the bounded absorbing set is bounded in a higher regular space, and it can be derived by energy estimate and Sobolev embeddings. The asymptotically compact method is the most widely applied method, and its basic idea is to show that any bounded sequence fSðtn Þxn g is relatively compact. Based on the existence of absorbing sets, asymptotical smoothness is equivalent to asymptotical compactness, and the ω-limit compact method is provided to show the asymptotical compactness (or smoothness) by combining the measure of noncompactness and the contraction of a semigroup. Since the continuity of a semigroup is hard to show under certain conditions, Zhong, Yang and Sun [163] introduced the definition of norm-to-weak continuous semigroup, and used it to prove the existence of global attractors. The closed operator semigroup method is first given by Pata and Zelik [103], who used the properties of closed operator to derive the existence of global attractor. At last, Ghidaglia [51] established the weak compactness induced by the compact embedding to study the global attractor, which is called the weakly compact semigroup method.

XII

Preface

This book aimed to present some recent results on complete, systematical basic existence theories of global attractors and some applications in some autonomous nonlinear evolutionary equations arising from physics, fluid mechanics and material science such as the Navier–Stokes equations, Navier–Stokes–Voight systems, a nonlinear thermoviscoelastic system. This book is essentially divided into two parts. The first part is chapter 1 which includes some useful inequalities and the survey on the basic existence theories of global attractors. The second part includes chapters 2–9 which are some applications of basic existence theories of attractors given in chapter 1. These applications include Navier–Stokes–Voight equations, 1D equations of polytropic viscous ideal gas and heat-conducting real gas, compressible Navier–Stokes equations of heat-conducting fluid in bounded annular domains in Rn , 1D thermoviscoelastic system, reaction–diffusion equations, an incompressible fluid equations and the Navier–Stokes equations. The materials of chapters 2–7 are based on the research carried out by the authors in recent years, while chapters 8 and 9 are chosen from others’ recent results in order to exhibit the applications of the methods of norm-to-weak continuous semigroups and energy equations. We sincerely wish that the reader will learn and understand main ideas and the essence of the basic existence theories and methods of proving global attractors. We also wish that the reader can deeply understand some ideas from this book and further undertake the related research after having learnt references of this book. We also want to take this opportunity to thank all the people who were concerned about us who include our parents, sisters, brothers, wives, sons and our teachers, colleagues and collaborators. Qin is supported in part by the NNSF of China with contract number 12171082 and by the Fundamental Research Funds for the Central Universities of China with contract number 2232022G-13.

Chapter 1 Preliminary In this chapter, we shall recall some basic knowledge in functional analysis (harmonic analysis) and idds for nonlinear evolutionary equations, most of which will be used in the subsequent chapters. The reader can easily find the detailed proofs in the related literature, see, e.g., Adams [1], Babin and Vishik [5], Chemin [20], Chepyzhov and Vishik [24], Constantin and Foias [27], Evans [30], Hale [54], Hille and Phillips [57], Kato [93], Ladyzhenskaya [74, 75], Lemarié-Rieusse [76], Lions [78], Liu and Zheng [81], Lorentz [82], Liu and Zheng [80], Maz’ja [89], Miao [90, 91], Nirenberg [97], Novotný and Strauskraba [98], Pazy [104], Qin [112], Robinson [125], Rudin [126], Sell and You [129], Serrin [130], Smoller [135], Sobolev [136], Sogge [137], Sohr [138], Stein [141], Temam, Babin and Vishik [5], Sell and You [129], Temam [144–146], Triebel [147, 148], Walter [150], Yosida [155], Zheng [156, 157], Zhong, Fan and Chen [161], etc.

1.1

Some Useful Inequalities

In this section, we shall recall some inequalities which will be used in the subsequent chapters. Throughout next chapters, we set Rs ¼ ½s; þ 1Þ; s 2 R þ , C will stand for a generic positive constant, depending on X and some constants, but independent of the choice of the initial time s 2 R þ and t. We introduce the Hausdorff semi-distance in X between two sets B1 and B2 , i.e., distX ðB1 ; B2 Þ ¼ sup inf kb1  b2 kX : b1 2B1 b2 2B2

We set E :¼ fuju 2 ðC01 ðXÞÞk ; divu ¼ 0g ðk ¼ 2; 3Þ, H is the closure of the set E in ðL2 ðXÞÞk topology, V is the closure of the set E in ðH 1 ðXÞÞk topology, W is the closure of the set E in ðH 2 ðXÞÞk topology, i.e.,
n o
V ¼ u 2 V
kukV ¼ kukH 1 ; uj@X ¼ 0 ; ð1:1:1Þ

DOI: 10.1051/978-2-7598-2702-2.c001 © Science Press, EDP Sciences, 2022

Attractors for Nonlinear Autonomous Dynamical Systems

2


n o
W ¼ u 2 W
kukW ¼ kukH 2 ; uj@X ¼ 0 :

ð1:1:2Þ

P is the Helmholz–Leray orthogonal projection in ðL2 ðXÞÞk onto the space H , A :¼ PD is the Stokes operator subject to the nonslip homogeneous Dirichlet boundary condition with the domain ðH 2 ðXÞÞk \ V , and A is a self-adjoint positively defined operator on H . A1 is a compact operator from H to H . The sequence fxj g1 j¼1 is an orthonormal system of eigenfunctions of A, 1 fkj gj¼1 ð0\k1  k2    Þ are the eigenvalues of the Stokes operator A corresponding to the eigenfunctions fxj g1 j¼1 . Let Vs :¼ DðA2 Þ; kV ks :¼ kA2 V k; s 2 R; s

s

ð1:1:3Þ

where V :¼ V1 ¼ ðH01 ðXÞÞk \ H is a Hilbert space, and kv k1 ¼ kv kV ¼ krv k. Clearly, V0 ¼ H , and V ,!H  H 0 ,!V 0 , H 0 and V 0 are dual spaces of H and V respectively, where the injection is dense, continuous. jj and ð; Þ denote the norm and inner product of H , respectively, i.e., 3 Z X ðu; vÞ ¼ uj ðxÞvj ðxÞdx; for all u; v 2 ðL2 ðXÞÞ3 ; ð1:1:4Þ j¼1

X

and kk and ðð; ÞÞ denote the norm and inner product in V , respectively, i.e., 3 Z X @uj @vj  dx; for all u; v 2 ðH01 ðXÞÞ3 ; ð1:1:5Þ ððu; vÞÞ ¼ @x @x i i X i;j¼1 and kru k2 :¼

3 X   @i uj 2 2 ; for all u ¼ ðu1 ; u2 ; u3 Þ: L ðXÞ

ð1:1:6Þ

i¼1;j¼1

The norm kk denotes the norm in V 0 , hi denotes the dual product in V and V 0 . We define the following bilinear form operator: Bðu; vÞ :¼ Pððu  rÞvÞ;

for all u; v 2 E

ð1:1:7Þ

and the trilinear form operator bðu; v; wÞ ¼

3 Z X i;j¼1

X

ui

@vj  wj dx ¼ ðBðu; vÞ; wÞ @xi

ð1:1:8Þ

and aðu; vÞ ¼ ððu; vÞÞ ¼ hAu; v i; where A is defined as A : V ! V 0 , for all u; v 2 V .

ð1:1:9Þ

Preliminary

3

Clearly, the trilinear operator satisfies bðu; v; vÞ ¼ 0; bðu; v; wÞ ¼ bðu; w; vÞ; 1

for all u; v; w 2 V ;

1

kbðu; v; wÞk  C ku k2 ku k21 kv k1 kw k1 ; for all u; v; w 2 V ; 3

1

kbðu; v; uÞk  C ku k2 ku k21 kv k1 ; for all u; v 2 V ; 1

1

kbðu; v; wÞk  C kuk1 kv k1 kw k2 kw k21 ; for all u; v; w 2 V ; 1

kbðu; v; wÞk  C k41 kuk1 kv k1 kw k1 ; for all u; v; w 2 V :

ð1:1:10Þ ð1:1:11Þ ð1:1:12Þ ð1:1:13Þ ð1:1:14Þ

Here, if the H01 norm and H02 norm replace V norm and W norm, respectively, the above inequalities also hold. There exists a positive constant C depending only on X such that 1

1

jbðu; v; wÞj ¼ jðBðu; vÞ; wÞj  C ku k2 kAu k2 kv kV kw k; for all ðu; v; wÞ 2 DðAÞ  V  H :

ð1:1:15Þ

Theorem 1.1.1 (Young’s Inequality). The following inequalities hold ab 

a p bq p ; 1\p\ þ 1; for all a; b [ 0; þ ; q¼ p1 p q

ð1:1:16Þ

and more specially, ab 

e p 1 q p a þ ; 1\p\ þ 1; for all a; b; e [ 0: 1 b ; q ¼ p p  1 qep1

ð1:1:17Þ

Theorem 1.1.2 (The Cauchy–Schwarz Inequality). There holds that jx  y j  jxjjy j; for all x; y 2 Rn ;

ð1:1:18Þ

here jxj ¼ ðx; xÞ1=2 ¼ ðRni¼1 xi2 Þ1=2 for all x 2 Rn . Theorem 1.1.3 (Hölder Inequality). Let XRn be a domain, assume that u 2 Lp ðXÞ; v 2 Lq ðXÞ with 1  p; q  þ 1 and p1 þ 1q ¼ 1. Then Z juvjdx  ku kLp ðXÞ kv kLq ðXÞ : ð1:1:19Þ X

Theorem 1.1.4 (Minkowski’s Inequality). Assume u; v 2 Lp ðXÞ,

1  p  þ 1. Then for any

ku þ v kLp ðXÞ  ku kLp ðXÞ þ kv kLp ðXÞ :

ð1:1:20Þ

Attractors for Nonlinear Autonomous Dynamical Systems

4

Theorem 1.1.5 (Jensen’s Inequality with Integration). Let gðxÞ be a function defined on ða; bÞ and a  gðxÞ  b, where a; b; a; b are bounded constants or can reach to þ 1, assume f ðxÞ is a continuous convex function defined on ða; bÞ, qðxÞ 2 L1 ða; bÞ and qðxÞ 0, then ! Rb Rb f ðgðxÞÞqðxÞdx a gðxÞqðxÞdx  a Rb : ð1:1:21Þ f Rb a qðxÞdx a qðxÞdx Theorem 1.1.6 (Poincaré’s Inequality). Assume that X Rn is a bounded smooth domain, then there holds 1

ku k  k12 ku k1 ; for all u 2 H01 ðXÞ;

ð1:1:22Þ

where k1 is the first eigenvalue of A under the homogeneous Dirichlet boundary condition, ku k1 is the norm of u in H01 , A ¼ D. Theorem 1.1.7 (Gronwall’s Inequality). Let aðtÞ 2 L1 ð0; T Þ, a 0, bðtÞ 2 L1 ð0; T Þ, Rs b0 2 R, bðsÞ ¼ b0 þ 0 bðtÞdt. Assume cðtÞ 2 L1 ð0; T Þ satisfies Z s cðsÞ  bðsÞ þ aðtÞcðtÞdt; for almost all ða:a:Þ s 2 ½0; T : ð1:1:23Þ 0

Then for a. a. s 2 ½0; T , we have Z s  Z cðsÞ  b0 exp aðsÞds þ 0

Z

s

s

bðtÞ exp

 aðsÞds dt:

0

ð1:1:24Þ

t

Theorem 1.1.8 (The Uniform Bellman–Gronwall Inequality). Let gðtÞ; hðtÞ and yðtÞ be three nonnegative locally integrable functions on ðt0 ; þ 1Þ such that y 0 ðtÞ is locally integrable on ð0; þ 1Þ and the following inequalities are satisfied y 0 ðtÞ  gðtÞyðtÞ þ hðtÞ; 8t t0 ; Z tþr Z tþr Z tþr gðsÞds  a1 ; hðsÞds  a2 ; yðsÞds  a3 ; 8t t0 ; t

t

t

where r; ai ði ¼ 1; 2; 3Þ are nonnegative constants. Then we have a  3 yðt þ rÞ  þ a2 ea1 ; 8t t0 : r Theorem 1.1.9 (Gronwall’s Inequality). Let E ! R satisfy   bkfkH0 m  EðfÞ  Q kfkH0 þ m; 8f 2 H0 ; for some b [ 0 and m 0. Let now n 2 C ðR þ ; H0 Þ be given. Suppose that the map t 7! EðnðtÞÞ is continuously differentiable and fulfills the differential inequality d EðnðtÞÞ þ eknðtÞk2H0  k; dt

Preliminary

5

for some e [ 0 and k [ 0. Then knðtÞkH0  Qðk þ m þ b1 Þ; 8t t0 ; where t0 ¼ QðknðtÞkH0 Þ þ QðkÞ [ 0: Theorem 1.1.10 (Gronwall’s Inequality). Let K : R þ ! R þ be an absolutely continuous function satisfying d KðtÞ þ 2eKðtÞ  hðtÞKðtÞ þ k; dt

where e [ 0; k 0 and Then

Rt s

hðsÞds  eðt  sÞ þ m, for all t s 0 and some m 0.

KðtÞ  Kð0Þem eet þ

kem ; 8t 0: e

Under assumptions of theorem 1.1.6, the following inequalities hold for dimension n ¼ 3. Theorem 1.1.11 (Ladyzhenskaya’s Inequality). 1

1

ku kL3  C ku k2 kru k2 ;

for all u 2 H01 ðXÞ;

3

1

ku kL4  C ku k4 ku k41 ;

for all u 2 H01 ðXÞ:

ð1:1:25Þ ð1:1:26Þ

Theorem 1.1.12 (Sobolev’s Inequality). Assume that X Rn is a bounded smooth domain, then for dimension n ¼ 3, there holds ku kL6  C ku k1 ;

for all u 2 H01 ðXÞ:

ð1:1:27Þ

Theorem 1.1.13 (The Gagliardo–Nirenberg Inequality). ku k

6 L32e

 C ku k1e ku ke1 ; 0  e  1; 2

12

ku kLp  C ku kp ku k32 p ; p 2 ½2; þ 1Þ;

for all u 2 H01 ðXÞ;

ð1:1:28Þ

for all u 2 V32 :

ð1:1:29Þ

Theorem 1.1.14 (Agmon’s Inequality). 1

1

ku kL1  C ku k21 kAu k2 ;

for all u 2 V2 ;

ð1:1:30Þ

where A ¼ PDu. P is the projection operator, D is the Laplace operator, V2 is defined in (1.1.3).

6

Attractors for Nonlinear Autonomous Dynamical Systems

Theorem 1.1.15 (Interpolation Inequality for Lp -Norms). Assume that u 2 Lr ðXÞ \ Ls ðXÞ with 1  r  s  þ 1. Then for any r  q  s and 0\h\1 such that 1 h 1h q ¼ r þ s , we have ku kLq ðXÞ  ku khLr ðXÞ ku k1h Ls ðXÞ

ð1:1:31Þ

for all u 2 Lq ðXÞ. Theorem 1.1.16 (Interpolation Inequality for the Sobolev Space). Assume that X Rn is a bounded Lipschitz domain, then we have ku kW a;r ðXÞ  C ku kkW b;p ðXÞ ku k1k W c;q ðXÞ

ð1:1:32Þ

for all 0  a; b; c  1, 1\p; q; r\ þ 1, where a ¼ kb þ ð1  kÞc,

1 r

¼ kp þ

1k q .

The following two theorems, the Hölder inequality and the Gagliardo–Nirenberg inequality in Lorentz spaces, are as follows. Theorem 1.1.17. Let f 2 Lp2 ;q2 ðR3 Þ and g 2 Lp3 ;q3 ðR3 Þ with 1  q2 ; q3  þ 1. Then fg 2 Lp1 ;q1 ðR3 Þ with

1  p2 ; p3  þ 1;

1 1 1 1 1 1 ¼ þ ; ¼ þ p1 p2 p3 q1 q2 q3 and for a positive constant C , the following Hölder inequality in Lorentz spaces holds kfg kLp1 ;q1  C kf kLp2 ;q2 kg kLp3 ;q3:

ð1:1:33Þ

Theorem 1.1.18. Let f 2 Lp;q ðR3 Þ with 1  p; q; p4 ; q4 ; p5 ; q5  þ 1. Then for a positive constant C , the following Gagliardo–Nirenberg inequality in Lorentz spaces holds kf kLp;q  C kf khLp4 ;q4 jkrf kj1h Lp5 ;q5

ð1:1:34Þ

with 1 h 1h 1 h 1h ¼ ¼ þ þ ; ; h 2 ð0; 1Þ: p p4 p5 q q4 q5 Theorem 1.1.19. For every s 2 R þ , every non-negative locally integrable function / on Rs ¼ ½s; þ 1Þ and every b [ 0, it holds that for all t s, Z hþ1 Z t 1 /ðsÞebðtsÞ ds  sup /ðsÞds: ð1:1:35Þ 1  eb h s h s Theorem 1.1.20. Let f : Rs ! R þ satisfy that for almost every t s, the differential inequality d fðtÞ þ /1 ðtÞfðtÞ  /2 ðtÞ; dt

ð1:1:36Þ

Preliminary

7

where, for every t s, the scalar functions /1 and /2 fulfill Z t Z t þ1 /1 ðsÞds bðt  sÞ  c; /2 ðsÞds  M ; s

ð1:1:37Þ

t

with some constants b [ 0; c 0 and M 0. Then we have fðtÞ  ec fðsÞebðtsÞ þ

Mec ; for all t s: 1  eb

ð1:1:38Þ

Theorem 1.1.21 (The Lemarié-Rieusset Inequality [76]). Let 1\p1  p2 \ þ 1; p3 2; a ¼ p32 þ p33  32 2 ð0; 1, then Z fghdx  C kf kM_ p ;p kg kp3 kh kH_ a ð1:1:39Þ 2 1

R3

_ p ;p ; g 2 L1 \ Lp ; h 2 L1 \ H_ a . for f 2 L1 \ M 2 1 3 Theorem 1.1.22 (The Kozono and Tamiuchi Inequality). Let 1\r\ þ 1, then we have kf  g kLr  C ðkf kLr kg kBMO þ kg kLr kf kBMO Þ

ð1:1:40Þ

for all f ; g 2 Lr \ BMO with a constant C ¼ C ðrÞ [ 0. Theorem 1.1.23 (The Lemarié-Rieusset Inequality [76]). For an integer k and reals 1  p  q  þ 1, the estimate  k    r Dj f   C 23jð1p1qÞ þ jk Dj f  ð1:1:41Þ q p holds, where Dj is the frequency localisation operator in the Littlewood–Paley decomposition and C is a positive constant independence of f and j. Theorem 1.1.24 There exists an absolute positive constant C , such that it holds that kf k2L4  C kf kL2 kf kBMO ;

ð1:1:42Þ

  1 ln2 ðe þ kf kH s1 Þ kf kB_ 0  C 1 þ kf kB_ 1;1 0

ð1:1:43Þ

1;2

for all functions f 2 H s1 ðR3 Þ with s [ 52. Theorem 1.1.25. Let X R3 be a bounded smooth domain, then for all vectors u 2 H 3 ðR3 Þ with r  u ¼ 0, it holds that  ð1:1:44Þ krm ðu  rvÞ  u  rrm v kL2  C kru kL1 krm v kL2 þ krv kL1 krm u kL2 where C [ 0 is a constant.

Attractors for Nonlinear Autonomous Dynamical Systems

8

Theorem 1.1.26 (The Gagliardo–Nirenberg Inequality). Let 0  j\m are integers, if

   1 j 1 m 1a j ¼ þa  ; a2 ;1 þ p n r n q m

1  p; q; r  þ 1, ð1:1:45Þ

where a\1 if r [ 1; m  j  nr ¼ 0, then there exists a constant C ¼ C ðn; m; j; a; q; rÞ [ 0 such that ku kW i;p ðXÞ  C ku kaW m;r ðXÞ ku k1a Lq ðXÞ

ð1:1:46Þ

holds for all u 2 DðXÞ. Theorem 1.1.27 (The Gagliardo–Nirenberg Interpolation Inequality). We introduce some interpolation inequalities which will be used in the following chapters, krf kL4  C kf kL4 kDf kL2 ; f 2 H 2 \ W 1;4 \ L4 ;

1=5

4=5

ð1:1:47Þ

1=4 3=4  krf kL3  C krf kL2 K3 f L2 ; f 2 H 3 \ W 1;3 \ W 1;2 ;

ð1:1:48Þ

 3   4 5=6 4 3 1 K f  3  C krf k1=6   L2 K f L2 ; f 2 H \ H \ H ; L

ð1:1:49Þ

where C [ 0 is a constant, every f in the above formulas belongs to the proper space, K is defined in theorem 1.2.23. Lemma 1.1.28 (The Bernstein Inequality). Let C be an annulus of center at 0, B is a ball of center on zero, then there exists a constant C [ 0 such that for all integer k 0 and function u 2 La ðRn Þ, b a 1, we have u kB; sup k@ a u kLa ðRn Þ  C k þ 1 kk þ nðabÞ ku kLa ðRn Þ ; supp^ 1

1

jaj¼k

C ðk þ 1Þ kk ku kLa ðRn Þ  sup k@ a u kLa ðRn Þ  C k þ 1 kk ku kLa ðRn Þ ; supp^ u kC: jaj¼k

ð1:1:50Þ

ð1:1:51Þ

Lemma 1.1.29 (The Special Bernstein Inequality). Let C be an annulus of center at 0, for any integer k 0, 1  p  q  þ 1 and function u 2 Lp ðRn Þ, we have c2km kDk u kLp ðRn Þ  krm Dk u kLp ðRn Þ  C 2km kDk u kLp ðRn Þ ; kDk u kLq ðRn Þ  C 2nðpqÞk kDk u kLp ðRn Þ ; supp^ u 2C; 1

1

where c and C are positive constants independent of u and k.

ð1:1:52Þ ð1:1:53Þ

Preliminary

9

Lemma 1.1.30 (The Gagliardo–Nirenberg Inequality). The following inequality  i  i=m r f  2m=i  C kf k1i=m ð1:1:54Þ krm f kL2 ; i 2 ½0; m L1 L holds for u 2 L1 ðRn Þ \ H m ðRn Þ. Lemma 1.1.31 (The Interpolation Inequalities). The following inequalities hold for three dimensional space 1=5

4=5

kru kL4  C ku kL4 kDu kL2 ; u 2 H 2 \ W 1;4 \ L4 ;

ð1:1:55Þ

1=3 2=3  kru kL2  C ku kL2 r3 u L2 ; u 2 H 1 \ H 3 \ L2 ;

ð1:1:56Þ

3=4 1=4  k u k L 1  C k u k L 2 r 2 u  L 2 ; u 2 L 2 \ H 2 \ L 1 ;

ð1:1:57Þ

1=4 3=4  ku kL4  C ku kL2 r3 u L2 ; u 2 L2 \ H 3 \ L4 :

ð1:1:58Þ

Lemma 1.1.32. There exists an absolute positive constant C [ 0, such that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð1:1:59Þ kru kL1  C 1 þ ku kL2 þ kr  u kBMO lnðe þ ku kH 3 Þ holds for all vectors u 2 H 3 ðR3 Þ with r  u ¼ 0. Proof. For more details of the proof, we can refer to Zhou and Lei [164]. For completeness, we only give a sketch here. From the Littlewood–Paley decomposition, ! þ1 0 A þ1 X X X X Dk ru ¼ þ þ ð1:1:60Þ Dk ru: ru ¼ k¼1

k¼1

k¼1

k¼A þ 1

lnðe þ ku k

Þ

From the Bernstein inequality, setting A ¼ ð2n=2Þ lnH 32 þ 1, we deduce that A  0 þ P  P P1  kru kL1  kDk ru kL1 þ  Dk ru  kDk ru kL1  þ k¼1 k¼1 L1  k¼A þ 1 1=2    P 0 A P  2 kð1 þ n=2Þ 1=2   2 jDk ruj kDk ru kL2 þ A    1  k¼1 k¼1 L þ  P1 kð2n=2Þ  3 Dk r u  2 þ 2 L k¼A þ 1   ð1:1:61Þ  C ðku kL2 þ A1=2 kru kBMO þ 2Að2n=2Þ r3 u L2 Þ: Denote Rj ¼ ð@=@xj ÞðDÞ1=2 ðR ¼ ðR1 ; R2 ; . . .; Rn ÞÞ be the Riesz transformation, from the Biot-Savard law, we can see uXj ¼ Rj ðR  ruÞ ðj ¼ 1; 2; . . .; nÞ. Since R is a bounded operator in BMO, we can derive that kru kBMO  C kr  u kBMO . Combining (1.1.61), we can complete the proof. h

10

1.2

Attractors for Nonlinear Autonomous Dynamical Systems

Basic Theory of Infinite-Dimensional Dynamical Systems for Autonomous Nonlinear Evolutionary Equations

It is well-known that the global well-posedness and asymptotic behavior of global solutions are hot topics for nonlinear evolutionary equations, the latter contains two categories: (i) the case of single trajectory, (ii) the case of multiple-trajectories. There is a rich literature related to the infinite-dimensional dynamical systems, we refer the reader to Babin and Vishik [3], Ball [6, 7], Cheban [19], Chepyzhov and Vishik [24], Hale [54], Haraux [56], Ladyzhenskaya [75], Ma, Wang and Zhong [83], Moise, Rosa and Wang [93], Pazy [104], Qin [112], Raugel [124], Robinson [125], Segal [127], Sell and You [129], Simon [134], Temam [146], Zheng [157], Zheng and Qin [159], Zhong, Yang and Sun [163], etc. The asymptotic behavior in time for global solutions of dynamical systems contains steady solutions, periodic solutions, quasi-periodic solutions, homoclinic orbit, heteroclinic orbit etc.; these solutions and orbits determine the asymptotic behavior in time of the system. From the compactness of semigroups, there are some methods for establishing the existence of global attractors which will be introduced in this section. As we know when proving the existence of a global attractor, the crucial and difficult step is to show the compactness of semigroups in some sense. There are some methods to show compactness in some sense (such as uniform compactness, weak compactness, asymptotic compactness, x-limit compactness, etc.). The first one is to show the uniform compactness of the semigroup which can be found in Temam [146]. The second one, proposed by Hale [54], is to prove asymptotic compactness of the semigroup. The third method, established by Ghidaglia [53], is to show the weak compactness of the semigroup. Ball first used the energy equation method together with asymptotic compactness to prove the existence of global attractors. Later on, Moise, Rosa and Wang [93] also used this approach to show the existence of global attractors. Recently, Ma, Wang and Zhong [83] devised a new method and used the measure of non-compactness to show x-limit compactness of the semigroup.

1.2.1

Uniformly Compact Semigroups

In this subsection, we recall some basic results on the global attractors of uniformly compact semigroups. Definition 1.2.1 (Semigroups). Let E be a Banach space or a closed subset of a Banach space, the parameter family T ðtÞ; 0  t  þ 1 from E to E is called a semigroup if (i) T ð0Þ ¼ Id (Identity operator on E), (ii) T ðt þ sÞ ¼ T ðtÞT ðsÞ; 8 s; t 2 R þ .

Preliminary

11

Definition 1.2.2 (C0-Semigroups). A semigroup T ðtÞ; 0  t  þ 1 from X to X is called a strong continuous semigroup of bounded linear operators if lim T ðtÞx ¼ x or limþ kT ðtÞx  x k ¼ 0; for all x 2 X;

t!0 þ

t!0

ð1:2:1Þ

with k  k ¼ k  kX , i.e., TðtÞ is a strongly continuous semigroup, short for C0 -semigroup. Definition 1.2.3 (Contraction Semigroups). The semigroup T ðtÞ is a contraction semigroup if there exists a constant a [ 0 ð0\a\1Þ such that for all t [ 0, kT ðtÞx  T ðtÞy k  akx  y k; for all x; y 2 X:

ð1:2:2Þ

Definition 1.2.4 (Analytic Semigroups). Let D ¼ fzju1 \argz\u2 ; u1 \0\u2 g and for all z 2 D, T ðzÞ be a bounded linear operator. The family T ðzÞ ðz 2 DÞ is an analytic semigroup in D if (i) z 7! T ðzÞ is analytic in D, (ii) T ð0Þ ¼ Id, i.e., limz!0 T ðzÞx ¼ x for every x 2 X, (iii) T ðz1 þ z2 Þ ¼ T ðz1 ÞT ðz2 Þ for all z1 ; z2 2 D. A semigroup T ðtÞ is called analytic if it is analytic in some sector containing the nonnegative real axis. Definition 1.2.5 (E−E Uniform Boundedness). We denote BðEÞ ¼ fall bounded sets in E g in view of the metric of E, the semigroup or semi-flow SðtÞ is called ðE; EÞ bounded if for any set BBðEÞ, we have SðtÞB 2 BðEÞ for any t 0. The semigroup or semi-flow SðtÞ is called ðE; EÞ uniformly bounded if for any BBðEÞ, there exists B1 2 BðEÞ such that SðtÞBB1 for any t 0. Definition 1.2.6 (Absorbing Sets). A bounded set B0 E is called an absorbing set if for any B 2 BðEÞ, there exists a time tB ¼ tðBÞ [ 0, such that SðtÞBB0 for any t tB , where fSðtÞgt 0 is a semigroup in the complete metric space E, BðEÞ is the collection of all bounded sets in E. An equivalent definition of an absorbing set is the point dissipation which was given by Hale [54]. Definition 1.2.7 (Point Dissipation). A semigroup SðtÞ on E is point dissipation semigroup if there is a bounded set B0 in E with property that for every w 2 E, there is a time tw ¼ tðwÞ such that SðtÞw 2 B0 , for all t [ tw . In this case, the set B0 is referred to as an absorbing set for the semigroup SðtÞ. Definition 1.2.8 (Attracting Sets). A set DE is said to be an absorbing set for a semigroup (semi-flow) if and only if for all BBðEÞ, we have distE ðSðtÞB; DÞ ! 0 as t ! þ 1, where distE ðX; Y Þ ¼ supx2X inf y2Y ky  x kE for X; Y E which is a Hausdorff semi-distance in the topology E.

12

Attractors for Nonlinear Autonomous Dynamical Systems

Remark 1.2.1. (i) The absorbing set is not unique. (ii) The existence of an absorbing set stands for the dissipative property of a system. (iii) For the dynamical system, there are three cases: energy conservation system, energy dissipation system, and energy compensated system; however, only and if only the energy dissipation system can generate absorbing set. Definition 1.2.9 (Compact Semigroups). The semigroup SðtÞ on E is said to be a compact semigroup for t [ 0 if for every B 2 BðEÞ and every t [ 0, the set SðtÞB lies in a compact set in E, or equivalently, SðtÞ admits a compact absorbing set. It is said to be an asymptotically compact semigroup if there exists a compact attracting set K ; K ,!,!E: Definition 1.2.10 (Uniformly Compact Semigroups). The semigroup SðtÞ on E is said to be a compact semigroup for all t [ 0 if for all bounded sets BE, there exists some time t0 ðBÞ [ 0, such that [ t t0 ðBÞ SðtÞB is relatively compact in E. Definition 1.2.11 (Omega-Limit Set). Similarly, we can define the x-limit set xðAÞ by xðAÞ ¼ \ [ SðtÞA s 0 t s

ð1:2:3Þ

where A is a set in E and the closure is taken in E. Equivalently, xðAÞ can be also defined as xðAÞ ¼ f/ : 9 tn ! þ 1 and a sequence /n 2 A such that Sðtn Þ/n ! / as n ! þ 1g:

ð1:2:4Þ

Definition 1.2.12 ((E, E) Continuity). The semigroup SðtÞ on E is called ðE; EÞ continuous if and only if SðtÞ is a continuous mapping, where SðtÞ : E ! E for all t 0 and kSðtÞkLðE;EÞ ¼ supkx kE ¼1 kSðtÞx kE . Definition 1.2.13 (Invariant Sets). A set A E is an invariant set for the semigroup SðtÞ if SðtÞA ¼ A for all t 2 R þ . Definition 1.2.14 (Attractors). A set A E is called an attractor if (i) A is an invariant set; (ii) A possesses an open neighborhood U such that for all u0 2 U , SðtÞu0 converges to A as t ! þ 1, i.e., distðSðtÞu0 ; AÞ ! 0 as t ! þ 1, where distðB0 ; B1 Þ ¼ supx2B0 inf y2B1 kx  y kE is the Hausdorff semi-distance. Definition 1.2.15 (Global Attractors). A set A E is called a global attractor if (1) A is an invariant set, i.e., SðtÞA ¼ A; for all t 0:

ð1:2:5Þ

Preliminary

13

(2) there exists a bounded absorbing set B0 E; (3) A is compact in E and attracts all the bounded sets in E, i.e., distðSðtÞu0 ; AÞ ¼ inf dðSðtÞu0 ; yÞ ! 0; as t ! þ 1: y2A

Then A ¼ xðB0 Þ 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. Moreover, for every fixed orbit uðtÞ ¼ SðtÞu0 and for every e [ 0, T [ 0, there exists a time s ¼ sðe; T Þ [ 0, v0 2 A such that for all 0  t  T , juðs þ tÞ  SðtÞv0 j  e:

ð1:2:6Þ

Definition 1.2.16. If A is a compact attractor and attracts bounded sets of E, then A is called a global attractor or universal attractor. Remark 1.2.2. The existence of global attractor needs three factors: (1) the dissipation of semigroups: the existence of an absorbing set; (2) the continuity of semigroup contains (a) weak continuous semigroup (weak-to-weak), (b) strong-to-strong semigroup, (c) norm-to-weak continuous semigroup, (d) closed operator semigroup, (e) C0 -semigroup, (f) C 0 -semigroup; (3) the compactness of the semigroup which can be obtained by the following methods: (a) uniformly compact method, (b) relatively compact method, (c) asymptotically compact method, (d) energy equation method, (e) asymptotically smooth method, (f) condition-(C) method, (g) semigroup decomposition method, (h) x-limit compact method, (i) contractive function method, and (j) j-contractive method. Moreover, the global attractor contains all limit states of solutions for the system: (a) equilibrium u  : SðtÞu  ¼ u  for all t 0; (b) periodic orbit: uðt þ pÞ ¼ uðtÞ, p [ 0, uðtÞ is the solutions for all t 2 R; quasiperiodic orbit: u^ ðtÞ ¼ Uða1 t; a2 t; . . .; an tÞ, Uðx1 ; x2 ; . . .; xn Þ is a 2p periodic coutinuous function with respect to every variable xi ði ¼ 1; 2; . . .; nÞ; (c) unstable orbits coming from equilibrium; (d) complete chaos solution, i.e., for every solution uðtÞ for the equation ut ¼ Au satisfies: (i) uðtÞ is bounded for all t 0, (ii) uðtÞ does not converge to equilibrium when t ! 1. We need the following lemma on the property of the analytic semigroup in chapter 7. ~ be the infinitesimal generator of an analytic semigroup SðtÞ Lemma 1.2.17. Let A ~ (the resolvent set of A), ~ then for every defined on a Banach space E. If 0 2 qðAÞ m ~ t [ 0 the operator A SðtÞ is bounded and

Attractors for Nonlinear Autonomous Dynamical Systems

14

 m  ~ SðtÞ A

LðEÞ

 C ðmÞt m edt ; 8m 0

where C ðmÞ and d are positive constants independent of t [ 0. ~ imply that A ~ is a densely defined closed linear Proof. The assumptions on A þ ~ R ¼ fk : 0\x\p=2g V ; where V is a neighborhood operator for which qðAÞ of zero, and     ~  M ; for k 2 R þ ; Rðk : AÞ  1 þ jkj ~ a for a 0. Since SðtÞ is analytic we have and therefore we have the existence of A 1

~ n Þ; for 8a 0: T ðtÞ : X ! \ DðA n¼0

~ a Þ, then x ¼ A ~ a y for some y 2 X If we let x 2 DðA Z 1 ~ a y ¼ 1 SðtÞx ¼ SðtÞA s a1 CðsÞCðtÞyds CðaÞ 0 ~ a SðtÞy ¼ A ~ a SðtÞA ~ a x; ¼A that is ~ a SðtÞx: ~ ax ¼ A SðtÞA ~ a is closed, A ~ a SðtÞ is bounded. Let n  1\a  n, from the fact that Since A  a  ~ SðtÞ  Mm t m edt , we have A  a   an n  ~ SðtÞ ~ SðtÞ ¼ A ~ A A Z 1  n  1 ~ Sðt þ sÞds  s na1 A Cðn  aÞ 0 Z 1   Mn s na1 ðt þ sÞn edðt þ sÞ ds  Cðn  aÞ 0 Z 1 Mn edt Ma  u na1 jjð1 þ uÞn du ¼ a edt ; Cðn  aÞ 0 t it thus follows that the conclusion holds, and we can also see theorem 6.13 in [104] for detailed process. h Now we introduce some definitions on a semiflow fSðtÞg. Definition 1.2.18 (Semi-flows). Let E be a completely metric space, SðtÞ be a parameter family. SðtÞ is a semi-flow on E if and only if SðtÞ is a mapping, Rðt; wÞ ¼ SðtÞw, where Rðt; wÞ : ½0; þ 1Þ  E ! E and SðtÞ satisfies (i) Sð0Þw ¼ w; for all w 2 E, (ii) the restricted mapping R : ½0; þ 1Þ  E ! E is continuous, (iii) SðsÞSðtÞw ¼ Sðs þ tÞw holds for every w 2 E; s; t 0.

Preliminary

15

Definition 1.2.19. We say that a semiflow fSðtÞg on E is compact for t [ 0, provided that for every bounded set B E and every t [ 0, the set SðtÞB lies in a compact set in E. Definition 1.2.20. A semiflow fSðtÞg on E is said to be point dissipative if there is a bounded set U in E with property that for every w 2 E, there is a time tw ¼ tðwÞ such that SðtÞw 2 U , for all t [ tw . In this case, the set U is referred to as an absorbing set for the semiflow fSðtÞg. Definition 1.2.21. We say that a set A in a phase space E is a global attractor of a semiflow fSðtÞg on E, provided the following properties are satisfied: (1) A is non-empty and compact. (2) A is invariant, i.e., SðtÞA ¼ A, for all t 0. (3) there is a bounded neigborhorhood U of A in E with the property that A attracts U , i.e., for every neighborhood V of A, there is a time T 0 such that SðtÞU V , for all t [ T . (4) A attracts every point in E. The basic theory of global attractors we shall use in this book is the following lemma. Lemma 1.2.22. ([128]). Let fSðtÞg be a point dissipative, compact semiflow on a complete metric space E. Then fSðtÞg 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 omega-limit set xðBÞ satisfies xðBÞA; (5) A is a connected set in E; (6) A is Lyapunov stable, i.e., for every neighborhood V of A ad every s [ 0, there is a neighborhood U of A with the property that SðtÞU V , for all t s. (7) A ¼ xðB0 Þ ¼ \ s 0 [ t s SðtÞB0 is invariant under SðtÞ, i.e., SðtÞA ¼ A; t 0. (8) A is compact. To our knowledge, there are five methods to prove the existence of global attractors in terms of some kind of compactness. Theorem 1.2.23 (Uniformly Compact Semigroup Method – Babin and Vishik [3], Hale [54], Ladyzhenskaya [75], Robinson [125], Sell and You [129], Temam [146]). Assume that E is a metric space and the operator SðtÞ is a semigroup on E. If the following properties hold: (1) SðtÞ is a continuous operator from E to E for all t 0; (2) there exists a bounded absorbing set B0 that attracts all bounded sets B in E; (3) the operator SðtÞ is uniformly compact. Then A ¼ xðB0 Þ is the global attractor for SðtÞ.

16

Attractors for Nonlinear Autonomous Dynamical Systems

Proof. We complete the proof in three steps. Step 1: Since A is nonempty, the sets [ t s SðtÞA are nonempty for every s 0. Hence, the set [ t s SðtÞA are nonempty compact sets which decrease as s increases, their intersection which is equal to xðAÞ is then a nonempty compact set. By the definition of the x-limit set, we know if w 2 SðtÞxðAÞ, then w ¼ SðtÞu; u 2 xðAÞ. Since the semigroup is continuous, we know there exist un 2 A; tn ! þ 1 such that Sðtn Þun ! u as n ! þ 1. So, we can get SðtÞSðtn Þun ¼ Sðt þ tn Þun ! SðtÞu ¼ w 2 xðAÞ; which shows that w 2 xðAÞ, and SðtÞxðAÞ xðAÞ. If u 2 xðAÞ, then there exist un 2 A; tn ! þ 1 such that Sðtn Þun ! u as n ! þ 1, and we observe that for tn t, the sequence Sðtn  tÞun is relatively compact in H . Thus, there exist tnk ! 1 and w 2 H such that Sðtnk Þunk ! u as t ! þ 1. From the definition of the x-limit set, we get w 2 xðAÞ. Since the semigroup is continuous, the following holds Sðtnk Þunk ¼ SðtÞSðtnk  tÞunk ! SðtÞu ¼ w; nk ! 1: So, u 2 SðtÞxðAÞ i.e., xðAÞ SðtÞxðAÞ. It follows that A is a nonempty, compact and invariant set. Since the set [ t t0 B SðtÞB is relatively compact, we show that xðBÞ is a nonempty compact invariant set. Step 2: We shall show that A ¼ xðBÞ is an attractor and that it attracts the bounded sets. We assume that for some bounded set B1 satisfying distðSðtÞB1 ; AÞ90 as t ! 1, thus, there exist d [ 0; tn ! 1 such that distðSðtÞB1 ; AÞ d [ 0; 8n: For each n, there exists bn 2 B1 satisfying distðSðtÞbn ; AÞ d=2 [ 0. Since B is absorbing, and hence Sðtn Þbn 2 B for n large enough (such that tn t1 ðB1 Þ). The sequence Sðtn Þbn is relatively compact and possesses at least one cluster point b, where b ¼ lim Sðtni Þbni ¼ lim Sðtni  t1 ÞSðt1 Þbni : ni !1

ni !1

Since Sðt1 Þbn 2 B, b 2 A ¼ xðBÞ, and this contradicts. Step 3: The attractor A is maximal. If A0 A is a large bounded attractor, then A0 B. Since SðtÞA0 ¼ A0 is □ included in B for t sufficiently large, xðA0 Þ ¼ A0 xðBÞ ¼ A.

1.2.2

Weakly Compact Semigroups

In this subsection, we introduce an abstract framework due to Ghidaglia [53], which is related to the existence of global attractors of weakly compact semigroups. This framework is very useful for some non-compact semigroups generated by some evolutionary equations such as the compressible Navier–Stokes equations.

Preliminary

17

The next abstract framework is due to Ghidaglia [53], which is summarized as the present form in Qin [112]. Theorem 1.2.24 (Weakly Compact Semigroup Method). Let E1 ; E2 ; E3 be three Banach spaces, satisfying the following conditions: (1) the embeddings E3 ,!E2 ,!E1 are compact; (2) there exists a C0 -semigroup SðtÞ on E2 and E3 , which map E2 ; E3 into E2 ; E3 respectively, for any t [ 0, SðtÞ are continuous and nonlinear operators on E2 ; E3 respectively, (this means SðtÞ : Ei ! Ei ; i ¼ 2; 3; t 0; are nonlinear operators); (3) the semigroup SðtÞ on E3 possesses a bounded absorbing set B3 in E3 . Then there is a weak universal attractor A3 in E3 . Furthermore, if the semigroup SðtÞ : E2 ! E2 satisfies the following conditions: (4) the semigroup SðtÞ on E2 possesses a bounded absorbing set B2 in E2 ; (5) for all t 0, SðtÞ is continuous on bounded sets of E2 for the topology of the norm of E1 ; then there is a weak attractor A2 ¼ xðB2 Þ in E2 and A3 A2 . 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 cannot generate compact semigroups. These models include the compressible Navier–Stokes equations and thermoviscoelastic equations, we refer to Qin [110–112], Qin, Liu and Song [119], Zheng and Qin [158, 159].

1.2.3

X-Limit Compact Semigroups

In this subsection, we mainly review some results on the existence of global attractors of x-limit compact semigroups for autonomous systems. Now, we introduce a compactness called x-limit compact, for more detail, we can refer to Hale [54], Qin [112], Robinson [125], Sell and You [129], Ma, Wang and Zhong [83]. The following is the Kuratowski measure of non-compactness. Definition 1.2.25 (The Kuratowski Non-Compactness Measure). K ðAÞ is called a non-compactness measure if cðAÞ ¼ inf fd [ 0jA has a finite open cover of sets of diameter\dg. If A is unbounded, then K ðAÞ ¼ þ 1. Lemma 1.2.26. cðAÞ is a non-compact measure, X is a complete metric space, then we have cðAÞ ¼ 0 if and only if A; the closure of A is compact in X. The next lemma concerns the properties of the non-compact measure. Lemma 1.2.27. Assume that X is a complete metric space, c is a non-compact measure, then for any B; B1 ; B2 2 BðEÞ, the non-compact measure c satisfies (1) If X is a Banach space, then cðB1 þ B2 Þ  cðB1 Þ þ cðB2 Þ for all B1 ; B2 2 X. (2) If B1 B2 , then cðB1 Þ  cðB2 Þ. (3) cðB1 \ B2 Þ  maxfcðB1 Þ; cðB2 Þg.

18

Attractors for Nonlinear Autonomous Dynamical Systems

cðB1 [ B2 Þ  maxfcðB1 Þ; cðB2 Þg. cðBÞ ¼ cðBÞ. cðBÞ ¼ 0 , cðN ðB; eÞÞ  2e , B is compact, N ðB; eÞ ¼ fxjdistðx; BÞ  eg. If B is a ball of radius e, then cðBÞ  2e. If fBt gt2R is a family of nonempty closed sets such that such that Bt2 Bt1 for t1 \t2 and limt! þ 1 cðBt Þ ¼ 0, then B ¼ \ t2R Bt is nonempty and compact. (9) If fBt gt2R and B are as above, given any tn ! þ 1 and any xn 2 Btn , there exists x 2 B and a subsequence xnk ! x as k ! þ 1. (10) Let X be an infinite-dimensional Banach space with the following decomposition X ¼ X1  X2 with dimX1 \ þ 1. Let P : X ! X1 , Q : X ! X2 be the canonical projectors, and A be a bounded subset of X. If the diameter of QA is less than e, then cðAÞ\e. (4) (5) (6) (7) (8)

Definition 1.2.28 (Ω-Limit Compactness). Assume X is a complete metric space, SðtÞ is a C0 -semigroup, if SðtÞ is said to be x-limit compact if and only if for any bounded set BM and for any e [ 0, there exists a time t0 ¼ t0 ðB; eÞ [ 0, such that cð [ t t0 SðtÞBÞ  e. Theorem 1.2.29 (X-Limit Compact Semigroup Method). We assume that X is a complete metric space, let SðtÞ be a C0 -semigroup on X. If (1) B0 X is a bounded absorbing set; (2) SðtÞ is x-limit compact, then the x-limit set of B0 , A ¼ xðB0 Þ is a compact attractor which attracts all bounded subset of X. Remark 1.2.3. A x-limit set xðBÞ is positively invariant set, i.e., SðtÞxðBÞxðBÞ for all t [ 0. Theorem 1.2.30. There is a global attractor for a C0 -semigroup if and only if (i) there exists an absorbing set, (ii) the semigroup is x-limit compact. In order to establish the x-limit compactness, we may prove that the semigroup SðtÞ possesses the property of set-contraction. Definition 1.2.31 (Set-Contraction). Assume X is a complete metric space, SðtÞ is a C0 -semigroup, we say SðtÞ to be set-contractive if and only if there exists a constant a 2 ½0; 1 and a time t0 [ 0, such that for any bounded BX, as t t0 , cðSðtÞBÞ  acðBÞ. Theorem 1.2.32 (Set-Contractive Semigroup Method). Assume that (1) SðtÞ is a C0 -semigroup on a complete metric space X; (2) B0 X is a bounded absorbing set; (3) SðtÞ is a set-contractive semigroup () x-limit compact), then SðtÞ possesses a global attractor A ¼ xðB0 Þ.

Preliminary

19

Remark 1.2.4. (1) The definition of x-limit set indicates that asymptotic compactness is a natural assumption. (2) In fact, asymptotic compactness implies that the x-limit set of any non-empty, bounded set is non-empty, compact, invariant, and attracts corresponding bounded set, but it is not a necessary maximal. (3) The existence of an absorbing set implies that the x-limit set of the absorbing set attracts any bounded set in the phase space X. (4) The method of proving asymptotic compactness strongly depends on the equation model. Definition 1.2.33 (C0-Semigroups – Ma, Wang and Zhong [83]). Let E be a complete metric space. A one parameter family fSðtÞgt 0 of maps SðtÞ : E ! E, t 0 is called a C 0 -semigroup if (1) Sð0Þ ¼ Id; (2) Sðt þ sÞ ¼ SðtÞSðsÞ for all t; s 0; (3) the function SðtÞx : ½0; þ 1Þ  E ! E is continuous at each point ðt; xÞ 2 ½0; þ 1Þ  E. Remark 1.2.5. The definitions of absorbing sets, global attractors, x-limit sets, x-limit compactness and set-contraction of the C 0 semigroups are similar to the definitions of general semigroups as before in subsections 1.2.1 and 1.2.3; here we omit the details. Clearly, a C 0 -semigroup must be a C0 -semigroup. Definition 1.2.34 (Uniformly Convex Space). A topology space X is called uniformly convex, if for all 0\e  2, there exists a constant dðeÞ [ 0, such  that for x 2 X and y 2 X satisfying kx k ¼ ky k ¼ 1, kx  y kX e, we have x þ2 yX  1  d. Definition 1.2.35 (Condition-(C) – Ma, Wang and Zhong [83]). For any bounded set B of X and for any e [ 0, there exists a time tðBÞ [ 0 and a finite dimensional subspace X1 of X such that fkPSðtÞB kg is bounded and kðI  PÞSðtÞx k  e for t tðBÞ; x 2 B; where P : X ! X1 is a bounded projector. Theorem 1.2.36. For the C 0 -semigroup SðtÞ, (1) SðtÞ satisfies condition-(C) implies that SðtÞ is x-limit compact; (2) in a uniformly convex space, condition-(C) is equivalent to the x-limit compactness. Proof. (1) If SðtÞ satisfies the condition-(C), from the properties of non-compact measure, we obtain         c [ SðtÞB  c P [ SðtÞB þ c ðI  P Þ [ SðtÞB t tðBÞ

t tðBÞ

 cðN ð0; eÞÞ ¼ 2e; which means that SðtÞ is x-limit compact.

t tðBÞ

20

Attractors for Nonlinear Autonomous Dynamical Systems

(2) If X is a uniformly convex Banach space and SðtÞ is x-limit compact, then for any bounded subset B X and any e [ 0, there exists tðBÞ [ 0 such that   c [ SðtÞB \e=2; t tðBÞ

which means that there exist subsets A1 ; A2 ; . . .; An with diamðAi Þ  e=2 satisfying n

[

t tðBÞ

SðtÞB [ Ai : i¼1

Let xi 2 Ai , then [ t tðBÞ SðtÞB [ ni¼1 N ðxi ; e=2Þ. Let X1 ¼ spanfx1 ; x2 ; . . .; xn g, since X is uniformly convex, we can get that there exists a projection P : X ! X1 such that for any x 2 X, kx  Px k ¼ distðx; X1 Þ; and kðI  PÞSðtÞx k  e=2\e, which means the condition-(C) is true.

□

Theorem 1.2.37 (X-Limit Compact Semigroup Method – Ma, Wang and Zhong [83]). We assume that X is a complete metric space, let SðtÞ be a C 0 -semigroup on X. If (1) B0 X is a bounded absorbing set; (2) SðtÞ is x-limit compact, then the x-limit set A ¼ xðB0 Þ is a compact attractor which attracts all bounded subset of X. Proof. We shall prove the theorem by three steps. Step 1: We must find the attractor xðBÞ. To this end, from the assumptions: (1) SðtÞ is x-limit compact, (2) B is bounded. We can get 8e [ 0, 9te [ 0, such that cð [ t te SðtÞBÞ\e. So, we take e ¼ 1=n; n ¼ 1; 2; . . ., we can get a sequence ftn g; t1 \t2 \    \tn \    such that   c [ SðtÞB \1=n: t tn

According to the definition of noncompact measure, we get     c [ SðtÞB ¼ c [ SðtÞB \1=n: t tn

t tn

Since we have obtained that \ 1 n¼1 [ t tn SðtÞB is nonempty compact set, and is x-limit set of B, i:e:; 1

A ¼ xðBÞ ¼ \ [ SðtÞB ¼ \ [ SðtÞB: n¼1 t tn

s 0 t s

Step 2: We want to prove xðBÞ is invariant. For this purpose, A is invariant, i.e., SðtÞxðBÞ ¼ xðBÞ. On one hand, if w 2 SðtÞxðBÞ, then w ¼ SðtÞu; u 2 xðBÞ, and there exists a sequence un 2 B and tn ! 1 such that Sðtn Þun ! u:

Preliminary

21

Thus, SðtÞSðtn Þun ¼ Sðt þ tn Þun ! SðtÞu ¼ w; we get w 2 xðBÞ; SðtÞxðBÞ xðBÞ. On the other hand, (1) if u 2 xðBÞ, then there exist un 2 B and tn ! 1 such that Sðtn Þun ! u. We want to get that fSðtn  tÞun g has a subsequence which converges in M , and u ¼ lim Sðtnj Þunj ¼ lim SðtÞSðtnj  tÞunj ¼ SðtÞw: j!1

j!1

(2) For any e [ 0, 9te , such that cð [ t 0 te Sðt 0 ÞBÞ\e; i:e:;   0 c 0 [ Sðt  tÞB \e: t te þ t

So we can get: 9N , [ n N Sðtn  tÞun [ t 0 te þ t Sðt 0  tÞB such that cð [ n N Sðtn  tÞun Þ\e. (3) Notice that [ N n¼N0 Sðtn  tÞun contains only a finite number of elements, such that when tn  t 0 as n N0 , we have     c [ Sðtn  tÞun ¼ c [ Sðtn  tÞun \e: n N0

n N

Let e ! 0, we know cð [ n N0 Sðtn  tÞun Þ ¼ 0, then fSðtn  tÞun g is relatively compact, i:e:; there exist tnj ! 1; w 2 M such that Sðtnj  tÞunj ! w as tnj ! 1, it is to say w 2 xðBÞ. Namely, u ¼ lim Sðtnj Þunj ¼ lim SðtÞSðtnj  tÞunj ¼ SðtÞw; j!1

j!1

so u 2 SðtÞxðBÞ and SðtÞxðBÞ ¼ xðBÞ. Step 3: We shall prove A ¼ xðBÞ attracts each bounded set of M . In fact, A ¼ xðBÞ is an attractor in M and attracts all bounded sets of M . Otherwise, there exists bounded set B0 M such that distðSðtÞB0 ; AÞ90 as t ! 1. i:e:; there exist d [ 0; tn ! 1 such that distðSðtn ÞB0 ; AÞ d [ 0; 8n 2 N : For each n, 9bn 2 B0 such that distðSðtn ÞB0 ; AÞ d=2 [ 0. By the assumption (2), B is an absorbing set, we know Sðtn ÞB0 ; Sðtn Þbn B for n large sufficiently. In Step 2, Sðtn Þbn is relatively compact, so there exists at least one cluster point b, where b ¼ lim Sðtnj Þbnj ¼ lim Sðtnj  t1 ÞSðt1 Þbnj ¼ Sðt1 Þw; nj !1

nj !1

and Sðt1 ÞB0 b. Thus, b 2 A ¼ xðBÞ, it is a contradiction.

□

Theorem 1.2.38 (Condition-(C) Method – Ma, Wang and Zhong [83]). X is a Banach space, SðtÞ is a C 0 -semigroup in X. If the following conditions hold: (1) there exists a bounded absorbing set B0 X; (2) (Condition-(C)) BX is bounded, and for all e [ 0, there exists a time tðBÞ [ 0 and a subspace X1 X with dimX1 \ þ 1 such that P : X ! X1 is a bounded projection, and we have

Attractors for Nonlinear Autonomous Dynamical Systems

22

ðaÞ fkPSðtÞB kg is bounded; ðbÞ kðI  PÞSðtÞx k\e for all t tðBÞ; x 2 B: Then the semigroup SðtÞ has a global attractor A ¼ xðB0 Þ.

1.2.4

Asymptotically Compact Semigroups

In this section, we shall discuss the asymptotically compact semigroup. We shall introduce the energy equation approach to establish the existence of global attractors. In most applications, there are many equations which cannot generate a compact semigroup, so the compactness needed must be achieved in a different but weak sense. Definition 1.2.39 (Asymptotically Compact Semigroups). The semigroup SðtÞ is said to be a asymptotically compact semigroup if and only if there exists a compact attracting set K, K,!,!E, such that for every B 2 BðEÞ, we have distðSðtÞB; K Þ ! 0 as t ! þ 1. The next definition is equivalent to the above one. Definition 1.2.40. Let E be a complete metric Banach space and let SðtÞ be a semigroup of continuous (nonlinear) operator in E. We say that a semigroup SðtÞ is asymptotically compact in E if the following condition holds: If fuj gj2N E is bounded and ftj gj2N R þ ; tj ! þ 1, then fSðtj Þuj gj2N is precompact in E. Remark 1.2.6. In fact, a compact semigroup must be an asymptotically compact semigroup. The next result is stated in terms of asymptotic compactness. Theorem 1.2.41 (Asymptotically Compact Semigroup Method). Assume the semigroup SðtÞ on E satisfies the following properties (1) SðtÞ : E ! E is a continuous (nonlinear) mapping for any fixed t [ 0; (2) there exists a bounded absorbing set B0 E; (3) SðtÞ is asymptotically compact in E. Then SðtÞ possesses a unique global attractor A ¼ xðB0 Þ. In the next theorem, when semigroup SðtÞ possesses a splitting such that SðtÞ is asymptotically compact, the same conclusion also holds. Theorem 1.2.42 (Semigroup Decomposition Method). Assume that E is a metric space, the operator SðtÞ is a continuous semigroup, we also assume that (1) if E is a Banach space, and for every t, SðtÞ ¼ S1 ðtÞ þ S2 ðtÞ, where the operator S1 ðtÞ is uniformly compact for large t and S2 ðtÞ is a continuous mapping from E into itself such that the following holds: for all bounded set C E, rc ðtÞ ¼ sup/2C kS2 ðtÞ/kE ! 0 as t ! þ 1;

Preliminary

23

(2) there exists an open set U and a bounded set B0 of U such that B0 is absorbing in U, then the x-limit set A ¼ xðB0 Þ is the global attractor in U . In particular, if U ¼ E, then A is called a global attractor. Proof. At first, we show the conclusion: If the semigroup fSðtÞg is continuous, and [ t t0 SðtÞB is relatively compact or the semigroup decomposition is assumed, then for any bounded set B0 H , xðB0 Þ is nonempty, compact and invariant. If the semigroup decomposition is assumed, we make the following remark which will be used repeatedly: if un is bounded and tn ! 1, then S2 ðtn Þun ! 0 and S1 ðtn Þun is convergent if and only if Sðtn Þun converges. The norm of Sðtn Þun is indeed bounded by rc ðtÞ where c is the sequence un ; n 2 N: Thus S2 ðtn Þun ! 0 and Sðtn Þun ¼ S1 ðtn Þun þ S2 ðtn Þun converge if and only if S1 ðtn Þun converges. Step 1: We want to show xðBÞ is compact for SðtÞ. By the definition of an x-limit set, we show that the x-limit set of B0 for SðtÞ, xðB0 Þ is equal to the set \ [ S1 ðtÞB0 :

s 0 t s

So we can get the conclusion: u 2 x1 ðB0 Þ if and only if there exists un 2 B0 ; tn ! 1 such that S1 ðtn Þun ! u as n ! 1. Next, we want to show that xðB0 Þ ¼ x1 ðB0 Þ. Assume that u 2 xðB0 Þ, so it equals to that: 9un 2 B0 ; tn ! 1 such that Sðtn Þun ! u as n ! 1. Since S1 ðtn Þun ! u as n ! 1 and u 2 x1 ðB0 Þ, we know xðB0 Þ x1 ðB0 Þ. Similarly, we can prove that xðB0 Þ x1 ðB0 Þ, and xðB0 Þ ¼ x1 ðB0 Þ holds naturally. Before, we observed that x1 ðB0 Þ is nonempty and compact since the sets [ t s S1 ðtÞB0 are nonempty, closed and decreasing. By assumption, [ t t0 S1 ðtÞB0 is compact. Hence, xðB0 Þ is nonempty and compact, and we can show that xðB0 Þ is invariant of SðtÞ. Step 2: xðB0 Þ is invariant. On the one hand, we want to show SðtÞxðB0 Þ xðB0 Þ; 8t [ 0: Let w 2 SðtÞxðB0 Þ; w ¼ SðtÞu; u 2 xðB0 Þ. By the difinition of x-limit set, there exists un ; tn ! 1 such that Sðtn Þun ! u. Hence, SðtÞSðtn Þun ¼ Sðt þ tn Þun ! SðtÞ u ¼ w, and we get w 2 xðB0 Þ. On the other hand, we want to show SðtÞxðB0 Þ xðB0 Þ; 8t [ 0. Note that u 2 xðB0 Þ if and only if there exists un 2 B0 ; tn ! 1 such that Sðtn Þun ! u as n ! 1. For tn t, the sequence Sðtn  tÞ is of the form Sðtn  tÞun ¼ S1 ðtn  tÞun þ S2 ðtn  tÞun :

24

Attractors for Nonlinear Autonomous Dynamical Systems

The sequence S1 ðtn  tÞun is relatively compact in H and contains a converging subsequence S1 ðtni  tÞuni ! w as ni ! 1. Also, we know S1 ðtni  tÞuni ! 0, it follows that Sðtni  tÞuni ! w as ni ! 1, which implies that w 2 xðB0 Þ and SðtÞw ¼ u, u 2 SðtÞxðB0 Þ, and xðB0 Þ is invariant. In the following, we shall show the theorem. We have shown that the A ¼ xðBÞ is nonempty, compact and invariant. Because the convergence of SðtÞ is equivalent to S1 ðtÞ, use the forward technique to show that A attracts the bounded sets. Also, we can get the attractor is global. □ Definition 1.2.43 (γ-Contraction). c is a non-compact measure on the complete metric space E, fSðtÞgt 0 is a continuous semigroup, if there exists a continuous function j : R þ ! R þ with jðtÞ ! 0 as t ! þ 1, such that for all t [ 0 and B E is bounded, [ 0  s  t SðsÞB is bounded in E, then cðSðtÞBÞ  jðtÞcðBÞ as t ! þ 1. Remark 1.2.7. If SðtÞ ¼ T ðtÞ þ LðtÞ, T ðtÞ is compact for t [ 0 (but not necessarily uniformly compact), LðtÞ is linear with kLðtÞk continuous in t and vanishes for t ! þ 1, i.e., limt! þ 1 kLðtÞk ¼ 0, we say SðtÞ to be c-contractive with jðtÞ ¼ kLðtÞk. If the semigroup satisfies the c-contractive property, then SðtÞ has a unique global attractor. In order to establish the asymptotic compactness, we may prove that the semigroup SðtÞ is c-contractive. Theorem 1.2.44 (The c-Contraction Semigroup Method). If X is a complete metric space, the semigroup fSðtÞgt 0 satisfies (1) SðtÞ is a continuous mapping on X; (2) there exists a bounded absorbing set B0 X; (3) SðtÞ is c-contractive, then SðtÞ has a global attractor A ¼ xðB0 Þ in X. In the sequel, we shall introduce the energy equation method which results in the asymptotic compactness of the semigroup. We need the following assumptions: Assume that (1) SðtÞ is a continuous and nonlinear operator semigroup in E; (2) E is the phase space which is a reflexive Banach space (so that any bounded sequence is weakly precompact); (3) SðtÞ is weakly continuous in E for any t 0, the trajectories of SðtÞ are continuous in E, i.e., for all u0 2 E; SðtÞu0 ! u0 as t ! 0 þ ði:e:; limt!0 þ kSðtÞu0  u0 k ¼ 0Þ; (4) for the energy equation, we assume that for all u0 2 E, the following formula

Preliminary

25

d ½UðSðtÞu0 Þ þ J ðSðtÞu0 Þ þ c½UðSðtÞu0 Þ þ J ðSðtÞu0 Þ þ LðSðtÞu0 Þ ¼ K ðSðtÞu0 Þ; ð1:2:7Þ dt

holds in the distribution sense in R þ , where c is a positive constant, U; J ; K ; L are functions and satisfy: (A1): U : E ! R þ , U is continuous and bounded on bounded subset of E, if fuj g is bounded in E, ftj g R þ , tj ! þ 1, Sðtj Þuj *w weakly in E, lim supj! þ 1 UðSðtj Þuj Þ  UðwÞ, then Sðtj Þuj ! w strongly in E. (A2): J : E ! R is asymptotically weakly continuous in the sense that: if fuj g is bounded in E, ftj g 2 R þ , tj ! þ 1 and Sðtj Þuj *w weakly in E, then J ðSðtj Þuj Þ ! J ðwÞ. (A3): K : [ t 0 SðtÞE ! R is asymptotically weakly continuous in the sense that: if fuj g is bounded in E, ftj g ! þ 1, and Sðtj Þuj *w weakly in E, then for any t [ 0, Z t Z t cðtsÞ e K ðSðs þ tj Þuj Þds ¼ ecðtsÞ K ðSðtÞwÞds; ð1:2:8Þ lim j! þ 1

0

0

where s 7! K ðSðsÞuÞ 2 L ð0; tÞ for any u 2 E. (A4): L : [ t [ 0 SðtÞE ! R is any asymptotic weakly (lower) semi-continuous in the sense: if fuj g is bounded in E, ftj g 2 R þ , tj ! þ 1, Sðtj Þuj *w weakly in E, then for all t 0, Z t Z t cðtsÞ e LðSðsÞwÞds  lim inf ecðtsÞ LðSðt þ tj Þuj Þds; ð1:2:9Þ 1

0

j! þ 1

0

where s 7! LðSðsÞuÞ 2 L1 ð0; tÞ, for all t [ 0, u 2 E. The following is the main result of this section. Theorem 1.2.45 (Energy Equation Method – Moise, Rosa, Wang [93]). Let E be a reflexive Banach space or a closed, convex subset of such a space. Let SðtÞ be a semigroup of continuous (nonlinear) operators in E which are also weakly continuous in E. Assume that SðtÞ possesses a bounded absorbing set and that its trajectories are continuous. Assume also that the energy equation (1.2.7) holds where c is a positive constant and U; J ; K ; L are functions satisfying the above assumptions ðA1Þ  ðA4Þ, respectively. Then SðtÞ possesses a global attractor which is connected if E is connected. Proof. Step 1: Since fun gn E, ftn gn 2 R þ ; tn ! þ 1, we need toverify that fSðtn Þun gn is pre-compact in E. Let CoB ¼ fa1 b1 þ a2 b2 þ  þ an bn jbi 2 B; ai 2 ½0; 1; Rni¼1 ai ¼ 1g is closed convex hull of B. Obviously, BCoB. By the existence of bounded absorbing sets B, we get fSðtn Þun gn is bounded. Since E is reflexive, we deduce that B 3 Sðtn0 Þun0 *w

ð1:2:10Þ

Attractors for Nonlinear Autonomous Dynamical Systems

26

weakly in E for some w 2 CoB and some subsequence fn 0 g. Similarly, fSðtn0  T Þ un0 g has a weakly convergent subsequence for all T [ 0, if we restrict T to the countable set N, by a diagonalization process, we can obtain further subsequence (we also denote as fn 0 g) for which Sðtn0  T Þun0 *wT weakly in E; for all T 2 N; with wT 2 CoB. Then by the weak continuity of SðT Þ, we have Sðtn Þun ¼ Sðtn  T ÞSðT Þun ¼ SðT ÞSðtn  T Þun

ð1:2:11Þ

0

and SðtÞ is continuous. Let n ! þ 1, w ¼ SðT ÞwT holds for all T 2 N. Step 2: Since the trajectories of fSðtÞgt 0 are continuous, then we integrate the energy equation (1.2.7) from 0 to T with u0 ¼ Sðtn0  T Þun0 to get Z

T

d ðUðSðtÞu0 Þ þ J ðSðtÞu0 ÞÞ þ dt 0 Z T ¼ K ðSðtÞu0 Þdt:

Z

T

Z

T

cðUðSðtÞu0 Þ þ J ðSðtÞu0 ÞÞdt þ

0

LðSðtÞu0 Þdt

0

ð1:2:12Þ

0

Then we can get for all T 2 N and tn0 T , Z T UðSðtn 0 Þun0 Þ þ J ðSðtn 0 Þun 0 Þ þ c ecðT sÞ LðSðsÞSðtn 0  T Þun0 Þds 0 Z T cT 0 0 0 0 þ ecðT sÞ K ðSðsÞSðtn 0  T Þwn 0 Þds: ¼ ½UðSðtn  T Þun Þ þ J ðSðtn  T Þun Þe 0

ð1:2:13Þ Moreover, we can obtain the supremum limit of the last formula (1.2.13) Z T 0 0 lim supðUðSðtn  T Þun Þ þ J ðwÞ þ c ecðT sÞ LðSðsÞwT ÞdsÞ n0 ! þ 1

 ½CB þ J ðwT ÞecT þ

Z

0 T

ecðT sÞ K ðSðsÞwT Þds;

ð1:2:14Þ

0

where CB ¼ supfUðvÞjv 2 CoBg\ þ 1. Since u0 ¼ wT , by (1.2.11) and the energy equation, we can deduce Z T ecðT sÞ LðSðsÞwT Þds UðwÞ þ J ðwÞ þ c 0

¼ ½UðwT Þ þ J ðwT ÞecT þ

Z

T

ecðT sÞ K ðSðsÞwT Þds:

ð1:2:15Þ

0

Subtracting (1.2.15) from (1.2.14), then we have lim supðUðSðtn0 Þun0 ÞÞ  UðwÞ  ðCB þ UðwT ÞÞecT n0 ! þ 1

 2CB ecT :

ð1:2:16Þ

Preliminary

27

Let T ! þ 1, we derive the results from the next condition lim sup UðSðtn0 Þun0 Þ  UðwÞ: n0 ! þ 1

ð1:2:17Þ

Using assumptions (A1)–(A4), (1.2.17) and weak convergence (1.2.10), it is easy to see that the following strong convergence result holds Sðtn0 Þun0 ! w;

ð1:2:18Þ

which implies that fSðtÞgt 0 is asymptotically compact. This completes the proof of theorem. h

1.2.5

Asymptotically Smooth Semigroups

In this subsection, we shall introduce asymptotically smooth semigroups and the related method to establish the global attractors. Definition 1.2.46 (Asymptotic Smoothness Semigroups). Assume E is a metric space, the semigroup SðtÞ : E ! E is said to be asymptotically smooth if for all BE, B is an arbitrary closed, bounded, positively invariant set (i.e., SðtÞBB), then for all t 0, there exists a compact set K ¼ K ðBÞE that can attract the set B, i.e., dist ðSðtÞB; K ðBÞÞ ! 0 as t ! þ 1. Concerning the asymptotically smooth semigroup, we have the following theorem for the existence of global attractors. Theorem 1.2.47 (Asymptotically Smooth Semigroup Method). If the semigroup SðtÞ satisfies the following conditions: (1) SðtÞ is a (nonlinear) continuous mapping from E to E for any t [ 0; (2) there exists a bounded absorbing set B0 E; (3) SðtÞ is asymptotically smooth, then SðtÞ possesses a unique global attractor ~ ¼ [ t t SðtÞB. B 0

~ A ¼ xðBÞð¼ xðB0 ÞÞ, where

Remark 1.2.8. (1) Actually, asymptotic compactness and asymptotic smoothness are quite related. (2) That SðtÞ is asymptotically compact implies it is asymptotically smooth with K ðBÞ ¼ xðBÞ, but the converse is not true. (3) In particular, if SðtÞ is asymptotically smooth, then in general, a non-empty bounded set B which is further closed and positively invariant is such that its x-limit set is non-empty, compact, invariant and attracts B. (4) The asymptotically compact property is equivalent to the property that for every non-empty bounded set B (not necessary or positively invariant), there exists a compact set K ðBÞ ¼ K which attracts B. There are two methods to prove the existence of global attractors in terms of continuity of semigroups.

28

1.2.6

Attractors for Nonlinear Autonomous Dynamical Systems

Norm-to-Weak Continuous Semigroups

In this subsection, we shall introduce norm-to-weak continuous semigroups and related methods to establish global attractors. Definition 1.2.48 (Norm-to-Weak Continuous Semigroups – Zhong, Yang, Sun [163]). Let X be a Banach space and fSðtÞgt 0 be a family of operators on X. We say that fSðtÞgt 0 is a norm-to-weak continuous semigroup on X if fSðtÞgt 0 satisfies (1) Sð0Þ ¼ Id (the identity); (2) SðtÞSðsÞ ¼ Sðt þ sÞ for all t; s 0; (3) Sðtn Þxn *SðtÞx if tn ! t and xn ! x in X. Remark 1.2.9. The norm-to-weak continuous semigroup only satisfies weaker stability of solutions. In general, it is neither continuous (i.e., norm-to-norm) nor weak continuous (i.e., weak-to-weak), but continuous semigroups and weak continuous semigroups are both norm-to-weak continuous semigroups. Theorem 1.2.49 (Zhong, Yang and Sun [163]). Let X and fSðtÞgt 0 be a semigroup on X and Y respectively, and fSðtÞgt 0 is continuous or weak continuous on Y . norm-to-weak continuous semigroup on X if and only if subsets of X  R þ into bounded sets of X.

Y be two Banach spaces, assume furthermore that Then fSðtÞgt 0 is a fSðtÞgt 0 maps compact

Definition 1.2.50 (Zhong, Yang and Sun [163]). Semigroup SðtÞ is called to be x-limit compact if for every bounded subset B of X and for any e [ 0, there exists a time t0 [ 0 depending on B and e, such that cð [ t t0 SðtÞBÞ\e. Theorem 1.2.51 (Norm-to-Weak Continuous Semigroup Method – Zhong, Yang and Sun [163]). Assume X is a Banach space, SðtÞ is a norm to weak continuous semigroup. Then SðtÞ possesses a global attractor in the sense of the strong topology of X if (1) there exists a bounded absorbing set B0 in the strong topology of X; (2) SðtÞ is x-limit compact. In addition, if X is a uniform convex space, uniform condition-(C) is equivalent to ω-limit compactness. Proof. Step 1: We shall show the existence of attractors. ws Let A ¼ \ s 0 [ t s SðtÞB0 . On the one hand, if there exist tn ! 1; fxn g 2 B0 such that Sðtn Þxn *x, then from the definition we know ws ws x 2 SðtÞB0 [ t s SðtÞB0 , and it means that x 2 A.

Preliminary

29 ws

On the other hand, if x 2 A, then x 2 \ s 0 [ t s SðtÞB0 , and ws x 2 [ t s SðtÞB0 . From the definition, we deduce that there exists tnk n; fxnk g 2 B0 such that Sðtnk Þxnk *x as k ! 1. Let K ¼ fSðtnk Þxnk jk; n ¼ 1; 2; . . .g. From that fSðtÞg is x-limit compact ws (jð [ t t0 SðtÞB0 Þ  e), we can get that the closure of weak sequence K is weakly ws ws compact. So K is metrizable, the metric is the weak topology of K , it is to say d is the metric, i:e:; ws

ws

8fyn g K ; y 2 K ; yn *y $ dðyn ; yÞ ! 0; n ! 1: Hence, for any n 2 N and any sequence fSðtnkn Þxnkn g fSðtnk Þxnk g1 k¼1 , there holds dðSðtnkn Þxnkn ; xÞ\1=n. Then dðSðtnkn Þxnkn ; xÞ ! 0; tnkn ! 1 as n ! 1; Sðtnkn Þxnkn ! x and tnkn ! 1 as n ! 1: Step 2: We shall show that A is invariant set. Firstly, A is nonempty compact subset of X, from the assumption (2), we get ws

jð [ SðtÞB0 Þ ¼ jð [ SðtÞB0 Þ ! 0 as n ! 1: t s

t s

ws

Since [ t s SðtÞB0 is closed in X, we get the result. Secondly, for any x 2 A, there exists tn ! 1; xn 2 B0 such that Sðtn Þxn *x. From the fact that SðtÞ is x-limit compact, we know fSðtn Þxn g has a convergent subsequence fSðtnk Þxnk g such that Sðtnk Þxnk ! x. Since fSðtÞgt 0 is a norm-to-weak continuous semigroup, we have Sðt þ tnk Þxnk ¼ SðtÞSðtnk Þxnk *SðtÞx; and the result is that SðtÞx 2 A, which means SðtÞA A. At last, for any x 2 A, there exists tn ! 1; xn 2 B0 such that Sðtn Þxn *x. From the fact that SðtÞ is x-limit compact, we know fSðtn  tÞxn g has a convergent subsequence satisfying Sðtnk  tÞxnk ! y; and y 2 A. Because fSðtÞgt 0 is norm to weak continuous, we have Sðtnk Þxnk *x, i.e., Sðtnk Þxnk ¼ SðtÞSðtnk Þxnk *SðtÞy; and SðtÞy ¼ x, which means x 2 SðtÞA, it follows that A SðtÞA to sum up, A ¼ SðtÞA, and A is invariant. Step 3: We shall show that A attracts all bounded sets of X. We first assume that for any bounded subset B0 X, there holds distðSðtÞB0 ; AÞ ! 0 as t ! 1. Otherwise, there exist e0 [ 0; tn ! 1; xn 2 B0 such that distðSðtn Þxn ; AÞ e0 . Since fSðtÞgt 0 is norm to weak continuous, the sequence fSðtn Þxn g has a convergent subsequence

30

Attractors for Nonlinear Autonomous Dynamical Systems Sðtnk Þxnk *x;

x 2 A, and it is a contradiction, which means A attracts all bounded sets of X. In conclusion, the semigroup fSðtÞgt 0 has a global attractor.

1.2.7

Closed Operator Semigroups

In this subsection, we shall introduce the closed operator semigroups and related methods to establish the global attractors. Definition 1.2.52 (Closed Operator Semigroup – Pata and Zelik [103] ). Let K denote either R þ or N, X be a Banach space or a complete metric space. A semigroup SðtÞ ðt 2 K Þ on X is called a closed operator semigroup, if SðtÞ is a closed operator, i.e., for any fixed t [ 0, xn ! x and SðtÞxn ! y implies that y ¼ SðtÞx. When K ¼ N, SðtÞ is called a discrete semigroup. Theorem 1.2.53 (Closed Operator Semigroup Method – Pata and Zelik [103]). Let SðtÞ be a closed operator semigroup, and satisfy the following conditions: (1) there exists a bounded absorbing set B0 X; (2) there exists a sequence tn 2 K , K ¼ R þ , such that limn!1 cðSðtn ÞBÞ ¼ 0, c is a non-compactness measure, then A ¼ xðB0 Þ is the global attractor of SðtÞ. Proof. Step 1: We shall show that xðB0 Þ is compact and attracting. (I) Owing to that there exists an absorbing set B0 2 X, let t0 2 K such that SðtÞB0 B0 for all t t0 . For t t0 þ tn , we have the conclusion: SðtÞB0 ¼ Sðtn ÞSðt  tn ÞB0 Sðtn ÞB0 : Thus, the condition (2) implies that cðSðtn ÞB0 Þ ! 0 as t ! 1. Besides, if t t0 , Ut ¼ [ s t SðsÞB0 ¼ [ s2K Sðt  t0 ÞSðs þ t0 ÞB0 [ s2K Sðt  t0 ÞB0 ¼ Sðt  t0 ÞB0 . Hence, Ut 2 Sðt  t0 ÞB0 , and limt!1 cðUt Þ ¼ limt!1 cðUt Þ ¼ 0. Since the sets Ut are nested, we conclude that xðBÞ ¼ \ t 0 Ut is nonempty and compact. (II) Assume that xðB0 Þ is not attracting for SðtÞ, then there exist e [ 0 and sequence xn 2 B0 and s ! 1 such that inf jjSðsn Þxn  xjj e:

x2xðB0 Þ

If sn t0 , it follows that Sðsn Þxn 2 [ sn t0 . Appealing to the properties of c, the sequence Sðsn Þxn must have a cluster point in xðB0 Þ, which is a contradiction. Step 2: We shall show that xðB0 Þ is invariant set for SðtÞ. (I) Since xðB0 Þ is compact and attracting, given any sequence xn 2 B and sn ! 1, there exists y 2 xðB0 Þ such that Sðsn Þxn ! y. (II) From the attracting property of xðB0 Þ, we have lim distðSðsn ÞB0 ; xðB0 ÞÞ ¼ 0:

n!1

Preliminary

31

Thus, in particular, limn!1 ½inf x2xðB0 Þ jjSðsÞxn  xjj ¼ 0: (III) Exploiting the compactness of xðB0 Þ, there exists y 2 xðB0 Þ and a sequence nk such that ynk ! y, which implies that Sðsnk Þxnk ! y: (IV) Let x 2 xðB0 Þ, by the definition of x-limit set, we know that there exists sn ! 1 and xn 2 B satisfying Sðsn Þxn ! x. On the other hand, for any t 2 K, there exist y1 ; y2 2 xðB0 Þ such that Sðsn  tÞxn ! y1 and SðtÞSðsn Þxn ¼ Sðsn þ tÞxn ! y2 : Since SðtÞSðsn  tÞxn ¼ Sðsn Þxn , by the definition of closed operator semigroup, we conclude that x ¼ SðtÞy1 and y2 ¼ SðtÞx, which yields the invariant property: SðtÞxðB0 Þ ¼ xðB0 Þ. Thus the proof is complete. h Note that for some cases, we can recover the connectedness of global attractor A without requiring the continuity of semigroup SðtÞ. Lemma 1.2.54. Assume that there exist a sequence tn ! 1 and a connected set C with A C such that Sðtn ÞC is relatively compact for any n 2 N þ. Then A is connected. Based on theorem 1.2.53 and lemma 1.2.54, we have the next corollary. Corollary 1.2.1. Let SðtÞ have a connected compact attracting set K. Assume that SðtÞK K for any t large enough. Then SðtÞ possesses a connected global attractor. Remark 1.2.10. In conclusion, based on compactness and continuity of the semigroup, there are some methods to prove the existence of global attractors for the autonomous systems. In detail, there are five methods to prove it in terms of compactness of the associated semigroup. (1) Uniformly compact semigroup method (see theorem 1.2.23); (2) Asymptotically compact semigroup method (semigroup decomposition method, c-contraction semigroup, energy equation method, (see theorems 1.2.41, 1.2.42, 1.2.44 and 1.2.45)); (3) Asymptotically smooth semigroup method (see theorem 1.2.47); (4) x-limit compact method (set-contraction method, uniform condition-(C), (see theorems 1.2.29, 1.2.30, 1.2.32, 1.2.37, and 1.2.38)); (5) Weakly compact semigroup method (see theorem 1.2.24). There are two methods to prove the existence of global attractors in terms of continuity of the associated semigroup: (1) Norm-to-weak continuous semigroup method (see theorem 1.2.51); (2) Closed operator semigroup method (see theorem 1.2.53).

Attractors for Nonlinear Autonomous Dynamical Systems

32

1.3

Basic Theory of Finite-Dimensional Attractors

In section 1.2, we have given the basic existence theories of global attractors for autonomous system, and presented some techniques to derive the existence of global attractors. In this section, we shall show that the fractal dimension of global attractors is finite, even though these attractors are subsets of infinite-dimensional phase spaces. In the sequel, we focus on introducing the definition of fractal dimension and estimating its upper bound, in which the critical step is to find the sufficient conditions in the phase space such that any n-dimensional volumes are contracted, it is natural to expect that the dimension of global attractors is less than or equal to n. We mention here the content of this section is picked from [125]. To generalize the intuitive notion of dimension, we would expect that (1) dðXÞ  dðY Þ for any X Y ; (2) dðXÞ ¼ n for any nonempty open subset X Rn ; (3) dðX [ Y Þ ¼ maxðdðXÞ; dðY ÞÞ.

1.3.1

The Fractal Dimension of Global Attractors

The fractal dimension, written by df ðXÞ, is defined by counting the number of closed balls of a fixed radius e needed to cover the subsets X. By N ðX; eÞ, we denote the minimum number of balls in the cover. We note that one possible approach to obtain a general measure of dimensions would be to extract the exponent from N ðX; eÞ, which thus leads to the following definition. Definition 1.3.1 (Fractal dimension). Let the subset X be compact, then df ðXÞ ¼ lim sup e!0

logN ðX; eÞ ; logð1=eÞ

is called the fractal dimension of X, where the upper limit could been taken the value þ 1. Also, the fractal dimension has some properties as follows. Lemma 1.3.2. (1) df ð [ N n¼1 Xn Þ  maxn df ðXn Þ; (2) if g : H ! H is Hölder continuous with exponent h, that is jgðxÞ  gðyÞj  C jx  yjh ; then df ðgðXÞÞ  df ðXÞ=h; (3) df ðX  Y Þ  df ðXÞ þ df ðY Þ; (4) df ðXÞ ¼ df ðXÞ. Remark 1.3.1 There is another measure method on dimension of subsets, called the Hausdorff dimension and denoted as dH ðXÞ, which has the particular property   þ1 dH [ Xn  max dH ðXn Þ: n¼1

n

Preliminary

33

Since the Hausdorff dimension is always less than the fractal dimension, the corresponding definitions and theorems are all omitted here.

1.3.2

The Estimate on Fractal Dimension of Global Attractors

In this subsection, we shall introduce an analytical technique of estimating the fractal dimension of the global attractor. In this regard, we need to study the evolution of infinitesimal n-dimensional volumes as they evolve under the motion of fluid, and to find the smallest dimension n at which all such n-volumes contract asymptotically. We first give the definition of uniform differentiability, which is very important in estimating the fractal dimension of subsets. Definition 1.3.3 (Uniform Differentiability). If for any u0 2 A, there exists linear operator ^ðt; u0 Þ such that for any t 0, jSðtÞu  SðtÞu0  ^ðt; u0 Þðu  u0 Þj ! 0 ðas e ! 0 þ Þ ju  u0 j u;u0 2A;0\juu0 j  e sup

and sup k^ðt; u0 Þkop \1;

u0 2A

then SðtÞ is said to be uniformly differentiable on A. We consider the following abstract system ( du ¼ FðuðtÞÞ; dt uð0Þ ¼ u0 2 H ;

ð1:3:1Þ

where H is a Hilbert space. Let uðtÞ ¼ SðtÞu0 be the unique solution to the above system which is continuous, and A is the global attractor of system in H . At the initial point u0 2 A, we construct an orthogonal set of n displacements, obtain n vectors fdx ðjÞ g ð1  j  nÞ which form an n-dimensional body, and then we watch how the n-volume evolves. We write dx ð1Þ ^    ^ dx ðnÞ as the n-dimensional body whose volume is denoted by
ð1Þ

dx ^    ^ dx ðnÞ
; where jdx ð1Þ ^    ^ dx ðnÞ j2 ¼ detðdx ð1Þ ; . . .; dx ðnÞ Þ: To study the evolution of n-dimensional volume, we should consider the set of displacements dx ðiÞ about the trajectory and linearise the system under the condition of uniform differentiability. Let ^ðt; u0 Þn be the solution to the linearised system

Attractors for Nonlinear Autonomous Dynamical Systems

34

(

which can be also written as

dU ¼ F 0 ðSðtÞu0 ÞU ðtÞ; dt U ð0Þ ¼ n;

(

dU ¼ Lðt; u0 ÞU ðtÞ; dt U ð0Þ ¼ n:

ð1:3:2Þ

ð1:3:3Þ

Let Vn ðtÞ ¼ jdx ð1Þ ^    ^ dx ðnÞ j, then d 1d 1d 1d log Vn ðtÞ ¼ log Vn2 ¼ log½detðdx ð1Þ ; . . .; dx ðnÞ Þ ¼ log½detM ðtÞ: dt 2 dt 2 dt 2 dt Since log½detM  ¼ Tr½log M , we obtain

 d 1d 1 1 dM log Vn ðtÞ ¼ Tr½log M  ¼ Tr M : dt 2 dt 2 dt

Next, we shall provide a method to evaluate the trace in the above quality. Let f/ðjÞ ðtÞg be a set of orthonormal vectors dependent on time t which span the same space as fdx ðjÞ ðtÞg, then for any function u which is spanned by f/ðjÞ g there exists n X u¼ /ðjÞ ð/ðjÞ ; uÞ: j¼1

We define the matrix mðtÞ ¼ ðmij ðtÞÞ where mij ðtÞ ¼ ð/ðiÞ ðtÞ; dx ðjÞ Þ; and there hold M ¼ m T m; and M 1 ¼ m 1 ðm T Þ1 : According to the operator Lðt; u0 Þ, we construct aij ¼ ð/ðjÞ ; L/ðjÞ Þ;

and the matrix

dM ¼ dt

  dMij where dt

dMij d ¼ ðdx ðiÞ ; dx ðjÞ Þ dt dt     d ðiÞ d dx ; dx ðjÞ þ dx ðiÞ ; dx ðjÞ ¼ dt dt ¼ ðLdx ðiÞ ; dx ðjÞ Þ þ ðdx ðiÞ ; Ldx ðjÞ Þ n X ðdx ðiÞ ; /ðkÞ Þ½ðL/ðkÞ ; /ðlÞ Þ þ ð/ðkÞ ; L/ðlÞ Þð/ðlÞ ; dx ðjÞ Þ ¼ ¼

k;l¼1 n X k;l¼1

mki ðalk þ akl Þmlj ;

Preliminary

35

it follows that dM ¼ m T ða T þ aÞm: dt Therefore, d 1 dM log Vn ðtÞ ¼ Tr½M 1  dt 2 dt 1 ¼ Trðm 1 ðm T Þ1 m T ða T þ aÞmÞ 2 1 ¼ Trða T þ aÞ ¼ TrðaÞ; 2 and in the following way we shall discuss the trace of matrix a. Let P ðnÞ be the projection onto the subspace spanfdx ðnÞ g, from the fact that spanfdx ðnÞ g ¼ spanf/ðnÞ g we have P ðnÞ ¼

n X

/ðiÞ ð/ðiÞ ; Þ;

i¼1

it thus follows that TrðaÞ ¼

n X

aii ¼

i¼1

¼ ¼

ð/ðiÞ ; L/ðiÞ Þ

i¼1

n n X X i¼1 n X

n X

!

/ðjÞ ð/ðjÞ ; /ðiÞ Þ; L/ðiÞ

j¼1

ð/ðjÞ ð/ðjÞ ; /ðiÞ Þ; L/ðiÞ Þ ¼

i;j¼1

¼ Tr½LP

n X

ð/ðjÞ ; /ðiÞ Þð/ðjÞ ; L/ðiÞ Þ

i;j¼1 ðnÞ



which means d logVn ðtÞ ¼ Tr½LP ðnÞ : dt After integrating on time t, we have

Z t  Vn ðtÞ ¼ Vn ð0Þexp TrðLðs; u0 ÞP ðnÞ ðsÞÞds : 0

According to the upper asymptotic growth rate of n-dimensional body volume

Z t  1 ðnÞ lim exp TrðLðs; u0 ÞP ðsÞÞds ; t!1 t 0

36

Attractors for Nonlinear Autonomous Dynamical Systems

we define 1 T Rn ðAÞ ¼ sup sup lim sup ðnÞ u0 2A P ð0Þ t!1 t

Z

t

TrðLðs; u0 ÞP ðnÞ ðsÞÞds

0

¼ sup sup \TrðLðt; u0 ÞP ðnÞ ðtÞÞ [ ; u0 2A P ðnÞ ð0Þ

where \  [ denotes the time average operator. From the above discussion, we have the fact that the volume of n-dimensional body decay exponentially if T Rn ðAÞ\0, and it is natural to expect that df ðAÞ  n, it thus follows that Theorem 1.3.4. Assume that the semigroup SðtÞ of system on A is uniformly differential, and there exists a constant T such that the semigroup ^ðt; u0 Þ of linearized system is compact for any t T . If T Rn ðAÞ\0, then df ðAÞ  n.

Chapter 2 Global Attractors for the Navier–Stokes–Voight Equations with Delay In this chapter, we are concerned with the asymptotic behavior in time for the 2D incompressible Navier–Stokes–Voight equations with distributed delay on a non-smooth domain. Under some assumptions on initial data and delay data, we prove the existence of compact global attractors. The content of this chapter is adapted from [142], and the method used here is the asymptotically compact semigroup method (see theorem 1.2.41).

2.1

Global Wellposedness of Solutions

In this chapter, we discuss the long-time behavior for the 2D Navier–Stokes–Voight equations with a distributed delay on a Lipschitz domain X, 8 @u > >
mDu
a2 Dut þ ðu  rÞu þ rp > > > @t > R0 > > ðx; tÞ 2 Xs ; < ¼ f ðxÞ þ
h Gðs; uðt þ sÞÞds; ð2:1:1Þ divu ¼ 0; ðx; tÞ 2 Xs ; > > uðt; xÞj ¼ u; u  n ¼ 0; ðx; tÞ 2 @X ; > s @X > > > > uðs; xÞ ¼ u0 ðxÞ; x 2 X; > : uðt; xÞ ¼ /ðt
s; xÞ; ðx; tÞ 2 Xsh ; where Xs ¼ X  ðs; þ 1Þ, @Xs ¼ @X  ðs; þ 1Þ, Xsh ¼ X  ðs
h; sÞ, s 2 R is the initial time. The function u ¼ uðt; xÞ ¼ ðu1 ðt; xÞ; u2 ðt; xÞÞ is the unknown velocity R0 field of the fluid, p is the pressure, u 2 L1 ð@XÞ,
h Gðs; uðt þ sÞÞds is the distributed delay, and h [ 0 is a constant. For any t 2 ðs; T Þ, we can define u : ðs
h; T Þ ! ðL2 ðXÞÞ2 , and u t is a function defined on ð
h; 0Þ satisfying u t ¼ uðt þ sÞ; s 2 ð
h; 0Þ.

DOI: 10.1051/978-2-7598-2702-2.c002 © Science Press, EDP Sciences, 2022

Attractors for Nonlinear Autonomous Dynamical Systems

38

Denote E :¼ fuju 2 ðC01 ðXÞÞ2 ; divu ¼ 0g, H denotes E in ðL2 ðXÞÞ2 topology, j  j and ð; Þ represent the norm and inner product in H , respectively, i.e., Z 1=2 2 Z X juj ¼ juj2 dx ; ðu; vÞ ¼ uj ðxÞvj ðxÞdx; 8 u; v 2 ðL2 ðXÞÞ2 ; ð2:1:2Þ X

j¼1

X

and V denotes E in ðH 1 ðXÞÞ2 topology, and k  k and ðð; ÞÞ denote the norm and inner product in V , respectively, i.e., Z 1=2 2 Z X @uj @vj kuk ¼ jruj2 dx ; ððu; vÞÞ ¼ dx; 8 u; v 2 V : ð2:1:3Þ @xi @xi X X i;j¼1 Also, V 0 is the dual space of V with norm k  k . Write CH ¼ C 0 ð½
h; 0; H Þ; CV ¼ C 0 ð½
h; 0; V Þ; with norms kukCH ¼ suph2½
h;0 juðt þ hÞj; and kukCV ¼ suph2½
h;0 kuðt þ hÞk respec1 tively. Similarly, L2H ¼ L2 ð
h; 0; H Þ, L2V ¼ L2 ð
h; 0; V Þ, L1 H ¼ L ð
h; 0; H Þ, 1 1 LV ¼ L ð
h; 0; V Þ. The sequence fxj g1 j¼1 is an orthonormal system of eigenfunctions to the eigenð0\k  k2    Þ of the Stokes operator A :¼
PD, where P is the value fkj g1 1 j¼1 orthogonal projection. Define for s 2 C as follows P s P As f ¼ kj aj xj ; s 2 C; j 2 R; f ¼ aj xj ; j j ( ) P P ReZ 2 s s DðA Þ ¼ ff : A f 2 H g ¼ f ¼ aj xj : kj jaj j \ þ 1 ; j

j

and DðAs Þ denotes E in DðAs Þ topology with norm kuk2s , and As satisfies ([10]) Z Z jAa uj2 1 1 dx  C jAa þ 4 uj2 dx; 8 u 2 DðAa þ 4 Þ; distðx; @XÞ X X 1

1

kukL4  C jA4 uj; 8u 2 DðA4 Þ: The following Hardy inequality will be used: Z Z juðxÞj2 dx  C jruðxÞj2 dx; 8u 2 V : 3 2 X ½distðx; @XÞ X Theorem 2.1.1. (Arzelà–Ascoli Theorem) Assume X is a compact subset of Rm1 , and ffn g is a sequence of continuous functions from X into Rm2 . If fn is uniformly bounded and equi-continuous, then fn has a subsequence that converges uniformly on X. To transform problem (2.1.1) into a homogeneous problem, we introduce the background flow function w solving

Global Attractors for the Navier–Stokes–Voight Equations with Delay 8 <
Du þ rq ¼ 0; divu ¼ 0; : u ¼ u a:e:

39

in X; in X; on @X in the sense of nontangential convergence;

and in [10] it is shown that sup jwðxÞj þ sup jrwðxÞjdistðx; @XÞ  C kukL1 ð@XÞ ; x2X

x2X

1

kjrwjdistð; @XÞ1
p kLp ðXÞ  C kukLp ð@XÞ ; 2  p  1: Let e 2 ð0; c  diamðXÞÞ, ge 2 C01 ðR2 Þ satisfies 8 < ge ¼ 1; in fx 2 R2 j distðx; @XÞ  C10 eg; g ¼ 0; in fx 2 R2 j distðx; @XÞ  C20 eg; : e 0  ge  1; otherwise, 1 and jra ge j  Ca0 =ejaj , where ge ¼ hðqðxÞ is e Þ, h is a standard bump function and q 2 C a regularized distance bump function to @X. Thus

w ¼ u; x 2 fx 2 X; distðx; @XÞ\C10 eg; Suppw fx 2 X; distðx; @XÞ\C20 eg; and Dw ¼ rðqge Þ þ F; where 3

kFkL2 ðXÞ  C =e2 kukL2 ð@XÞ ; rq ¼ Du; and F ¼ 0; if x 2 fxjdistðx; @XÞ\C10 e or distðx; @XÞ [ C20 eg: Suppose the external force G in (2.1.1) satisfies: (A1) G: ½
h; 0  R2 ! R2 is measurable, and Gðs; 0Þ ¼ 0; s 2 ½
h; 0; (A2) 9 Lg [ 0 such that jgw ðnÞ
gw ðgÞj2  Lg kn
gk2CH for n; g 2 CH ; (A3) for u; v 2 C 0 ð½s
h; T ; H Þ and t [ s, 9 Cg [ 0; m0  0 ðm 2 ½0; m0 Þ such that Z t Z t jgðs; u s Þ
gðs; v s Þjds  Cg2 juðsÞ
vðsÞjds; s

s
h

(A4) mk1 [ 3Cg . Let v ¼ u
w, then the system (2.1.1) reduces to the following problem

Attractors for Nonlinear Autonomous Dynamical Systems

40

8 @v > > >
mDv
a2 Dvt þ ðv  rÞv þ ðv  rÞw þ ðw  rÞv > > @t > > > ðx; tÞ 2 Xs ; < þ rðp
mqge Þ ¼ f
ðw  rÞw þ gw ðv t Þ; divv ¼ 0; ðx; tÞ 2 Xs ; ð2:1:4Þ > > v ¼ 0; ðx; tÞ 2 @X ; > s > > > vðs; xÞ ¼ u0
w ¼ v0 ðxÞ; x 2 X; > > : vðt; xÞ ¼ /ðt
s; xÞ
wðxÞ ¼ gðt
s; xÞ; ðx; tÞ 2 Xsh ; R 0 where f ¼ f þ mF, gw ðv t Þ ¼
h Gðs; vðt þ sÞ þ wÞds, / 2 L2H . Let RðvÞ ¼ Bðv; wÞ þ Bðw; vÞ. Then we arrive at the following equivalent abstract system of (2.1.4) 8 dv > < þ mAv þ a2 Avt þ BðvÞ þ RðvÞ ¼ Pf
BðwÞ þ gw ðv t Þ; dt ð2:1:5Þ > : vðsÞ ¼ v0 ; vðtÞ ¼ gðt
sÞ: Theorem 2.1.2. Let f 2 ðL2 ðXÞÞ2 ; v0 2 V ; g 2 L2H , and assume that ðA1Þ ðA4Þ hold, then there exists a unique solution vðtÞ to the problem of (2.1.1) satisfying vðtÞ 2 L1 ð0; T ; V Þ \ L2 ð0; T ; V Þ;

dv 2 L2 ð0; T ; V 0 Þ: dt

P Proof. Fix n  1, define an approximate solution vn ðtÞ ¼ nj¼0 anj ðtÞwj to (2.1.1) satisfying 8 dv > < n þ mAvn þ a2 Avnt þ Bðvn Þ þ Rðvn Þ ¼ Pn f
BðwÞ þ gw ðvnt Þ; dt ð2:1:6Þ > : vn ðsÞ ¼ vn0 ; vn ðtÞ ¼ gn ðt
sÞ; t 2 ðs
h; sÞ: Multiplying (2.1.6) by vn, we arrive at   dvn ; vn þ mðAvn ; vn Þ þ a2 ðAvnt ; vn Þ þ ðBðvn Þ; vn Þ þ ðRðvn Þ; vn Þ dt ¼ hPn f ; vn i
ðBðwÞ; vn Þ þ hgw ðvnt Þ; vn i;

ð2:1:7Þ

which gives 1d ðjvn j2 þ a2 kvn k2 Þ þ mkvn k2  jbðvn ; vn ; vn Þj þ jbðw; vn ; vn Þj þ jbðvn ; w; vn Þj 2 dt þ jhPn f ; vn ij þ jðBðwÞ; vn Þj þ jhgw ðvnt Þ; vn ij: ð2:1:8Þ Now in what follows, we can estimate each term on the right-hand side of (2.1.8). Indeed, using the Hardy inequality, the Hölder inequality and the properties of operators, choosing suitable e, we can derive (see [10, 113, 114])

Global Attractors for the Navier–Stokes–Voight Equations with Delay Z jbððvn Þ; w; vn Þj  C ekukL1 ð@XÞ

jvn j2

½distðx; @XÞ2 m  C ekukL1 ð@XÞ kvn k2  kvn k2 ; 4 Z

j\Pn f ; vn [ j  jf jjvn j þ m

41

dx

X

C1 e  distðx;@XÞ  C2 e

jFjjvn jdx

  jf j Cm  kvn k pffiffiffiffiffi þ pffiffi kukL2 ð@XÞ ; e k1 pffiffi jbðw; w; vn Þj  C ekuk2L1 ð@XÞ j@Xj1=2 kvn k e; j\gw ðvnt Þ; vn [ j 

jgw ðvnt Þj2 Cg k
1 1 kvn k2 : þ 2Cg 2

From the above estimates, (2.1.8) reduces to jgw ðvnt Þj2 d 2 2 ðjvn j2 þ a2 kvn k2 Þ þ ðm
Cg k
1 Þkv k  K þ ; ð2:1:9Þ n 1 0 dt Cg n o pffiffi where K02 ¼ Cm pjfffiffiffikj1ffi þ pmffiekukL2 ð@XÞ þ ekuk2L1 ð@XÞ j@Xj1=2 . Choosing m [ 0 so small
1 2 that m
3Cg k
1 1
mCg k1
ma [ 0, we derive

d mt ½e ðjvn j2 þ a2 kvn k2 Þ dt emt 2
1 2  K02 emt þ jgw ðvnt Þj2
emt ðm
Cg k
1 1
mk1
ma Þkvn k : Cg

ð2:1:10Þ

Integrating (2.1.10) in t, we conclude emt ðjvn j2 þ a2 kvn k2 Þ
ðjvn ð0Þj2 þ a2 kvn ð0Þk2 Þ Z t K02 mt 1 ðe
1Þ þ  ems jgw ðvns Þj2 ds Cg 0 m Z t 2
1 2 ms ðm
Cg k
1
1
mk1
ma Þe kvn ðsÞk ds 0 Z t K2 ems j/n j2 ds þ jvn ðsÞj2 ds  0 emt þ Cg m
h Z t 2
1 2 ms
ðm
Cg k
1 1
mk1
ma Þe kvn ðsÞk ds 0

Z 0 K2 2Cg mt 2  0 emt þ Cg e jwj ems j/n j2 ds þ m m
h Z t 2
1 2 ms
ðm
3Cg k
1 1
mk1
ma Þe kvn ðsÞk ds 

0 2 K0 mt

m

e

Z þ Cg

0


h

j/n j2 ds þ

2Cg 2 mt jwj e ; m

ð2:1:11Þ

Attractors for Nonlinear Autonomous Dynamical Systems

42

which implies the existence of a constant K1 [ 0 such that jvn ðtÞj2 þ a2 kvn ðtÞk2  K12 : Similarly, integrating (2.1.9) over ½t; t þ 1, we also get ðjvn ðt þ 1Þj2 þ a2 kvn ðt þ 1Þk2 Þ
ðjvn ðtÞj2 þ a2 kvn ðtÞk2 Þ þ ðm
Cg k
1 1 Þ Z  K02 þ Cg

tþ1

Z

and ðm
3Cg k
1 1 Þ

t

t
h

Z

jvn ðsÞj2 ds þ jwj2 h þ k
1 1

tþ1 t

tþ1

kvn k2 ds

t

jvn ðsÞ þ wj2 ds

t
h

 K02 þ 2Cg

Z

 Z  K02 þ 2Cg 2

Z

t þ1

Z


h

ð2:1:12Þ

t

kvn k2 ds  K02 þ 2Cg 0

 jvn ðsÞj2 ds þ jwj2

t

t
h

 jvn ðsÞj2 ds þ jwj2 h þ jwj2 þ K12

 j/n ðsÞj2 ds þ K12 h þ 3hC42 kuk2L1 ð@XÞ þ C42 kuk2L1 ð@XÞ þ K12 ;

Rt þ1 kvn k2 ds  IV2 , and vn ðtÞ 2 which yields the existence of IV [ 0 such that t L1 ð0; T ; V Þ \ L2 ð
h; T ; V Þ holds naturally. By Alaoglu compactness theorem, we may seek a subsequence vn such that vn ! v in L1 ð0; T ; V Þ; vn ! v in L2 ð
h; T ; V Þ; ð2:1:13Þ T i.e., v 2 L1 ð0; T ; V Þ 2L ð
h; T ; V Þ. Noting that it had been shown that Bðvn Þ; Rðvn Þ 2 L2 ð0; T ; V 0 Þ, and vn 2 L2 ð0; T ; V Þ, we deduce that
mAvn ; a2 vnt ; gw ðvnt Þ; BðwÞ 2 L2 ð0; T ; V 0 Þ: Thus the following equation holds in L2 ð0; T ; V 0 Þ, dvn ¼
mAvn
a2 Avnt
Bðvn Þ
Rðvn Þ þ Pn f
BðwÞ þ gw ðvnt Þ; dt which gives

ð2:1:14Þ

dvn 2 L2 ð0; T ; V 0 Þ. It thus follows from the compactness theorem that dt vn ! v; in L2 ð0; T ; V Þ; vn ð0Þ ¼ Pn v0 ! vð0Þ ¼ v0 :

ð2:1:15Þ

For the uniqueness of solutions, assume that u1 and u2 are two solutions for (2.1.1) with stream functions w1 and w2 , respectively. Letting v 2 C01 ðXÞ with divv ¼ 0, we get

Global Attractors for the Navier–Stokes–Voight Equations with Delay

43

d d \u1
u2 ; v [
m\u1
u2 ; Dv [
a2 \u1
u2 ; Dv [ dt dt Z X Z 0 2 @v i ¼ ðu1i u1j
u2i u2j Þ dx þ \ ½Gðs; u1t Þ
Gðs; u2t Þds; v [ : x j X i;j¼1
h Let w ¼ u1
u2 , we get d kwk2  C ðku1 k4L4 kwk2 þ kwkCH kwkÞ; dt which leads to w ¼ 0 and the uniqueness of solutions follows readily.

2.2

h

Existence of Global Attractors

In this dissipative system, from the discussion in section 2.1, we know that the solution generates a continuous semiflow SðtÞ : V  L2H ! C ðs; T ; V Þ  CV ; from the fundamental theory of attractors ([75, 125, 146]). However, we also need to establish some dissipation and some compactness for the semiflow SðtÞ to derive the existence of global attractors.

2.2.1

Dissipation: Existence of Absorbing Sets

Indeed, V  L2H is a Hilbert space with norm kðv0 ; gÞk2V L2 ¼ kv0 k2 þ H

Z

0


h

jgðsÞj2 ds:

Theorem 2.2.1. Assume f 2 ðL2 ðXÞÞ2 and ðv0 ; gÞ 2 V  L2H , the conditions ðA1Þ

ðA4Þ hold, then the semiflow SðtÞ possesses an absorbing set in CV for the system (2.1.4). Proof. Similarly to the proof of theorem 2.1.1, we get jgw ðvnt Þj2 d 2 2 ðjvn j2 þ a2 kvn k2 Þ þ ðm
Cg k
1 ; 1 Þkvn k  K0 þ dt Cg

ð2:2:1Þ

and d mt ½e ðjvj2 þ a2 kvk2 Þ dt emt 2
1 2  K02 emt þ jgw ðv t Þj2
emt ðm
Cgk
1 1
mk1
ma Þkvk : Cg

ð2:2:2Þ

Attractors for Nonlinear Autonomous Dynamical Systems

44

Integrating (2.2.2) in t, we arrive at emt ðjvðtÞj2 þ a2 kvðtÞk2 Þ  jvð0Þj2 þ a2 kvð0Þk2 þ Z

t


0

K02 mt 1 ðe
1Þ þ Cg m

Z

t

ems jgw ðv s Þj2 ds

0

2
1 2 ms ðm
Cg k
1 1
mk1
ma Þe kvðsÞk ds

Z 0 K2 2Cg mt  jvð0Þj2 þ a2 kvð0Þk2 þ 0 ðemt
1Þ þ Cg ðe
1Þjwj2 ems j/j2 ds þ m m
h Z t 2
1 2 ms
ðm
3Cg k
1 1
mk1
ma Þe kvðsÞk ds 0  2  K0 2Cg 2 mt þ jwj e þ C kðv0 ; gÞk2V L2 ;  ð2:2:3Þ H m m

yielding jvðtÞj2 þ a2 kvðtÞk2 

K02 2Cg 2 þ jwj þ Ce
mt kðv0 ; gÞk2V L2 : H m m

ð2:2:4Þ

Let t [ h; h 2 ½
h; 0, then we get

 2  K0 2Cg 2 jvðt þ hÞj þ a kvðt þ hÞk
þ jwj m m 2  Ce
mðt þ hÞ kðv0 ; gÞkV L2  Ce
mt emh kðv0 ; gÞk2V L2 ; 2

2

2

H

and kvk2CV
2K 2

H

 2  K0 2Cg 2 þ jwj  Ce
mt emh kðv0 ; gÞk2V L2 : H m m

ð2:2:5Þ

ð2:2:6Þ

4C

Let q2V ¼ m0 þ mg jwj2 , then for any ðv0 ; gÞ 2 D V  L2H , there exists a constant TV [ 0 such that for all t  TV , SðtÞD BV ð0; qV Þ; which leads to the existence of bounded absorbing balls in CV .

2.2.2

h

Asymptotical Compactness and Existence of Attractor

To obtain the compactness of SðtÞ in CV , we use the asymptotically compact semigroup method in chapter 1 (see theorem 1.2.36), and SðtÞ of system (2.1.1) can be decomposed in the following form SðtÞfv0 ; gg ¼ S1 ðtÞfv0 ; gg þ S2 ðtÞfv0 ; gg  v1 ðtÞ þ v2 ðtÞ;

Global Attractors for the Navier–Stokes–Voight Equations with Delay where S1 ðtÞ is the semigroup generated by the following system 8 dv d > < 1 þ mAv1 þ a2 Av1 ¼ 0; dt dt > : v1 ðsÞ ¼ v0 ; x 2 X; v1 ðtÞ ¼ gðt
sÞ; t 2 ð
h; 0Þ; and v2 ðtÞ is the solution to the system 8 dv d > < 2 þ mAv2 þ a2 Av2 ¼ Pf þ gw ðv t Þ
BðvÞ
RðvÞ
BðwÞ; dt dt > : v2 ðsÞ ¼ 0; x 2 X; v2 ðtÞ ¼ 0; t 2 ð
h; 0Þ:

45

ð2:2:7Þ

ð2:2:8Þ

Multiplying (2.2.7) by v1 ðtÞ, by the Galerkin method we derive that for all t [ s, jv1 ðtÞj2 þ a2 kv1 ðtÞk2  eCt ðjvðsÞj2 þ a2 kvðsÞk2 Þ:

ð2:2:9Þ

Using the Sobolev inequalities, we also conclude that for any u 2 V, Z kBðvÞkDðA
1=3 Þ ¼ sup ðv  rÞv  udx  sup kvkL3  kvk  kukL6 kukDðA1=3 Þ ¼1

C

X

sup kukDðA1=3 Þ ¼1

kukDðA1=3 Þ ¼1

kvk2 kukDðA1=3 Þ  C kvk2 ;

which gives that BðvÞ 2 L1 ðs; T ; DðA
1=3 ÞÞ:

ð2:2:10Þ

RðvÞ 2 L1 ðs; T ; DðA
1=3 ÞÞ:

ð2:2:11Þ

Also, we derive that

Multiplying (2.2.8) by A1=3 v2 and using the upward estimates, we obtain that Z T 2 2 2 jgw ðv t Þj2 dt jv2 ðtÞjDðA1=6 Þ þ a kv2 ðtÞkDðA2=3 Þ  C þ C s Z T ð2:2:12Þ jvðtÞj2 dt; C þC s
h

which means S2 ðtÞ maps V into DðA2=3 Þ, and the fact that DðA2=3 Þ,!,!V leads to that S2 ðtÞ is compact for t [ s. Combining (2.2.9) with theorem 1.2.36 in chapter 1, we obtain the asymptotical compactness of SðtÞ in CV . Since we have established the existence of continuous semigroup fSðtÞg to system (2.1.1) in CV , the existence of the bounded absorbing set together with the asymptotical compactness of fSðtÞg in CV can lead to the following theorem.

46

Attractors for Nonlinear Autonomous Dynamical Systems

Theorem 2.2.2. Assume f 2 ðL2 ðXÞÞ2 , ðv0 ; gÞ 2 V  L2H , and the conditions ðA1Þ

ðA4Þ hold, then the system (2.1.4) possesses a compact connect global attractor A0 in CV which contains all equilibriums and unstable manifolds, i.e., the x-limit set which contains all bounded complete trajectories.

2.3

Bibliographic Comments

The delay effect was investigated firstly for ordinary differential equations in physics, non-instant transmission phenomena, memory processes, specially biological model, control theory and engineer, such as the retard differential equations, which can be seen in Caraballo, Langa and Robinson [13], Caraballo, Marín-Rubio and Valero [14], and Caraballo and Kiss [12]. For the research on the Navier–Stokes equations with delays on bounded smooth domain, such as the existence of global solutions, global and pullback attractors, regularity of attractors, we can refer to Barbu and Sritharan [8], Caraballo and Han [11], Caraballo and Real [17], Garrido-Atienza and Marín-Rubio [49], García-Luengo, P. Marín-Rubio and G. Planas [48], and Marín-Rubio and Real [84]. The dynamical systems for nonlinear dissipative systems on the non-smooth domains (such as the Lipschitz domain ½0; 1  ½0; 1) especially the domains containing singularity is a difficult problem. For the 2D incompressible Navier–Stokes equation, Miranville and Wang [92] investigated the attractors for non-autonomous non-homogeneous Navier–Stokes equation, then Brown, Perry and Shen [10] introduced a background function which will be used later in our paper to solve the non-homogeneous boundary value problems and gave the existence, dimension of global attractor. Inspired by above ideas, the long-time behavior of 2D incompressible Navier– Stokes–Voight equation with distributed delay on a non-smooth domain is considered in this chapter, and the main features of this chapter can be summarized as follows. (1) Our problem is not similar to the problem with continuous delay or distributed delay on regular domain in [17], because we should use background function and Hardy inequality to deal with the non-homogeneous boundary and non-regular boundary, respectively, for wellposedness and existence of absorbing sets. Sometimes, the estimates of background function influence some energy estimates such that the regularity of solutions cannot reach to optimization. (2) Comparing with the conclusion in [10], our problem contains distributed delay that has to be estimated and some assumptions are also needed for the uniqueness of solutions.

Chapter 3 Global Attractor and Its Upper Estimate on Fractal Dimension for the 2D Navier–Stokes–Voight Equations In this chapter, we first establish the existence of global attractor for the 2D Navier–Stokes–Voight (NSV) equations with a distributed delay and then further give the upper estimate on its fractal dimension. The latter is a new result for this model. The content of this chapter is chosen from [121], and the asymptotically compact semigroup method is also used (see theorem 1.2.41).

3.1

Global Existence of Solutions

In this chapter, we shall establish the existence of global attractor and its upper estimates on the fractal dimension for the following 2D Navier–Stokes–Voight equations with a distributed delay in a bounded domain X with smooth boundary @X 8 d 2 d > > Du þ ðu  rÞu þ rp ¼ f ðxÞ þ gðut Þ; ðx; tÞ 2 X0 ; > u
mDu
a > dt dt > > > < divu ¼ 0; ðx; tÞ 2 X0 ; ð3:1:1Þ ðx; tÞ 2 @X0 ; uðt; xÞj@X ¼ u; u  n ¼ 0; > > > > > uð0; xÞ ¼ u 0 ðxÞ; x 2 X; > > : uðt; xÞ ¼ /ðtÞ; ðx; tÞ 2 Xh ; where X0 ¼ X  ½0; þ 1Þ, @X0 ¼ @X  ½0; þ 1Þ, Xh ¼ X  ½
h; 0, m is the kinematic viscosity, u ¼ ðu1 ðt; xÞ; u2 ðt; xÞÞ is the velocity field, p is the pressure, a [ 0 is a length scale parameter, u 2 L1 ð@XÞ denotes the boundary velocity, f ðxÞ is the steady external force, / is the initial datum in ½
h; 0 with h [ 0 being a fixed R0 constant. The function gðut Þ ¼
h Gðs; uðt þ sÞÞds is the distributed delay and the delay function ut ¼ uðt þ sÞ; s 2 ½
h; 0. DOI: 10.1051/978-2-7598-2702-2.c003 © Science Press, EDP Sciences, 2022

Attractors for Nonlinear Autonomous Dynamical Systems

48

ðL2 ðXÞÞ2

Let E :¼ fuju 2 ðC01 ðXÞÞ2 ; divu ¼ 0g, H ¼ E with norm jj and inner product ð; Þ respectively, where 2 Z X ðu; vÞ ¼ uj ðxÞvj ðxÞdx; 8 u; v 2 ðL2 ðXÞÞ2 : j¼1 ðH 1 ðXÞÞ2

And V ¼ E respectively, where

X

is a Hilbert space with norm kk and inner product ðð; ÞÞ

ððu; vÞÞ ¼

2 Z X @uj @vj dx; 8 u; v 2 ðH01 ðXÞÞ2 : @x @x i i i;j¼1 X

Define the following Hilbert space XK ¼ L2 ð½
h; 0; K Þ  K; K ¼ H ; V with the norm 1=2

kfa; bgkXK ¼ ðfa; bg; fa; bgÞXK ; for fa; bg 2 XK ; where the inner product is defined as Z 0 ðaðhÞ; fðhÞÞK dh þ ðb; gÞK ; for ða; bÞ; ðf; gÞ 2 XK ; ðfa; bg; ff; ggÞXK ¼
h

and define another space CK ¼ C 0 ð½
h; 0; K Þ with norm ku kCK ¼ sup kuðt þ hÞkK : h2½
h;0

The operator P is the Helmholz–Leray orthogonal projection of ðL2 ðXÞÞ2 onto H , and P1 ; P2 denote the projections of XK onto L2 ð½
h; 0; K Þ and K . A :¼
PD is a self-adjoint positively defined operator on H , A
1 is compact from H to H , and thus the orthonormal system of eigenfunctions fxj g1 j¼1 of A exists corresponding to the ð0\k  k    Þ with Ax eigenvalues fkj g1 1 2 j ¼ kj xj . j¼1 Now define s

Vs ¼ DðA2 Þ; with the norm

 s  kv ks ¼ A2 v ; s 2 R; v 2 Vs :

Clearly, V1 ¼ V , kv k1 ¼ jrv j, and V0 ¼ H . Let H 0 and V 0 be dual spaces of H and V , respectively, where kk denotes the norm in V 0 , and hi denotes the dual product between V and V 0 .

Global Attractor and Its Upper Estimate on Fractal Dimension

49

Define further the bilinear and trilinear form operators as (see [146]) Bðu; vÞ :¼ Pððu  rÞvÞ; 8 u; v 2 V ; 2 Z X @vj bðu; v; wÞ ¼ ðBðu; vÞ; wÞ ¼ ui wj dx; @xi i;j¼1 X which clearly satisfy (see [5]) ( bðu; v; vÞ ¼ 0; bðu; v; wÞ ¼
bðu; w; vÞ; jbðu; v; wÞj  C1 ju jkv kjw j;

8 u; v; w 2 V ; 8 u; v; w 2 V :

ð3:1:2Þ

Now we construct a background function w such that divw ¼ 0; x 2 X; w ¼ u; x 2 @X; kwk  C2 kukL1 ð@XÞ : Letting v ¼ u
w, then (3.1.1) reduces to the following problem 8 dv d >
mDv
a2 Dvt þ ðv  rÞv > > > dt dt > > > þ ðv  rÞw þ ðw  rÞv þ rp ¼ f þ gðvt þ wÞ; ðx; tÞ 2 X0 ; < divv ¼ 0; ðx; tÞ 2 X0 ; > > v ¼ 0; ðx; tÞ 2 @X0 ; > > > 0 > vð0; xÞ ¼ v ðxÞ; x 2 X; > : vðt; xÞ ¼ /
wðxÞ ¼ g; ðx; tÞ 2 Xh ;

ð3:1:3Þ

where f ¼ f þ mDw
ðw  rÞw. Let RðuÞ ¼ Bðu; wÞ þ Bðw; uÞ whose characteristics can be found in [146]. Particularly, (3.1.3) can be transformed into the abstract form: 8 dv d > < þ mAv þ a2 Av þ BðvÞ þ RðvÞ ¼ Pf þ gðvt þ wÞ; dt dt ð3:1:4Þ 0 > : vð0Þ ¼ v ; x 2 X; v0 ¼ g; x 2 X: Now assume that ðA1Þ g : L2 ð½
h; 0; H Þ ! H is twice continuously differentiable, gð0Þ ¼ 0, and sup U2L2 ð½
h;0;H Þ; i¼1;2

jDi gðUÞj  l0 ; 0\l0 \ þ 1;

ðA2Þ 9 Lg  0 s.t 8 t 2 R; ut ; vt 2 CH , jgðut Þ
gðvt Þj  Lg kut
vt kCH ; t  0; 9 Cg [ 0; m0  0 s.t 8 m 2 ½0; m0 ; ut ; vt 2 L2 ð½
h; 0; H Þ, Z t Z t ems jgðus Þ
gðvs Þj2 ds  Cg2 ems juðsÞ
vðsÞj2 ds; t  0; 0

ðA3Þ mk1 [ 2C1 C2 kwkL1 ð@XÞ þ 3Cg .


h

50

Attractors for Nonlinear Autonomous Dynamical Systems

Remark 3.1.1. The assumption (A1) is assumed to show the differentiability of the semigroup SðtÞ, and is first proposed in this chapter. (A2) appears in many references [15–17], which implies for ut 2 L2 ð½
h; 0; H Þ that gðut Þ : t 2 ½0; T  ! ðL2 ðXÞÞ2 is measurable ([9]), and (A3) is proposed to guarantee the global existence of solutions. In the sequel, we shall prove the global existence of solutions to our problem (3.1.4). To do so, we now need to introduce some lemmas given below. Lemma 3.1.1. Let f 2 H , conditions (A1) (A3) hold and fg; v 0 g 2 XV , then there exists a unique global solution vðtÞ to problem (3.1.4) such that vðtÞ 2 L1 ð½0; T ; V Þ \ L2 ð½
h; T ; V Þ: Proof. We first establish the global existence of solutions to (3.1.4) by the standard Faedo–Galerkin method. Indeed, fix n  1, we define an approximate solution vn to (3.1.4) as vn ðtÞ ¼

n X

anj ðtÞwj ;

j¼0

which satisfies 8 dv > < n þ mAvn þ a2 Avnt þ Bðvn Þ þ Rðvn Þ ¼ Pn f þ gðvnt þ wÞ; dt 0 > : vn ð0Þ ¼ vn ; x 2 X; vn0 ¼ gn ; x 2 X;

ð3:1:5Þ

and anj ðtÞ is to be determined later on. Multiplying (3.1.5) by vn, we have   dvn ; vn þ mðAvn ; vn Þ þ a2 ðAvnt ; vn Þ þ ðBðvn Þ; vn Þ þ ðRðvn Þ; vn Þ dt ð3:1:6Þ ¼ hPn f ; vn i þ hgðvnt þ wÞ; vn i; and 1d ðjvn j2 þ a2 kvn k2 Þ þ mkvn k2 2 dt  jðBðvn Þ; vn Þj þ jðRðvn Þ; vn Þj þ jhPn f ; vn ij þ jhgðvnt þ wÞ; vn ij    f kvn k þ jgðvnt þ wÞj  jvn j þ 0 þ C1 kvn kkvn kkwk  2 mkvn k2 f  jgðvnt þ wÞj2 Cg 2 þ þ  þ kvn k2 þ C1 k
1 1 kv n k kw k 2 2m 2Cg 2k1  2   f  jgðvnt þ wÞj2 m Cg þ C1 C2 k
1 þ þ u þ  kv n k2 ; k k 1 1 L ð@XÞ 2 2m 2Cg 2k1

ð3:1:7Þ

Global Attractor and Its Upper Estimate on Fractal Dimension

51

that is, d ðjvn j2 þ a2 kvn k2 Þ dt  2 f  jgðvnt þ wÞj2 2
1  þ
ðm
2C1 C2 k
1 1 kukL1 ð@XÞ
Cg k1 Þkvn k : m Cg

ð3:1:8Þ

Now choosing some constant m [ 0 so small that mk1 [ 2C1 C2 kukL1 ð@XÞ þ 3Cg þ m þ ma2 k1 , we obtain d mt ½e ðjvn j2 þ a2 kvn k2 Þ dt  2 emt f  emt jgðvnt þ wÞj2  þ m Cg m
1
emt ðm
2C1 C2 k
1
ma2 Þkvn k2 1 kukL1 ð@XÞ
Cg k1
k1  2 emt f  emt jgðvnt þ wÞj2 þ  m Cg
emt ðmk1
2C1 C2 kukL1 ð@XÞ
Cg
m
ma2 k1 Þjvn j2 :

ð3:1:9Þ

Integrating (3.1.9) over ½0; t, we conclude emt ðjvn j2 þ a2 kvn k2 Þ  jvn ð0Þj2 þ a2 kvn ð0Þk2 þ Z
0

t

1 m

Z

t 0

 2 1 ems f  ds þ Cg

Z

t

ems jgðvns þ wÞj2 ds

0

ems ðmk1
2C1 C2 kukL1 ð@XÞ
Cg
m
ma2 k1 Þjvn ðsÞj2 ds;

ð3:1:10Þ

and emt ðjvn ðtÞj2 þ a2 kvn ðtÞk2 Þ
ðjvn ð0Þj2 þ a2 kvn ð0Þk2 Þ Z 0 2 2Cg mt emt  2   ðe
1Þjwj þ Cg f þ ems j/n j2 ds  mm m
h Z t 2
ðmk1
2C1 C2 kukL1 ð@XÞ
3Cg
m
ma k1 Þ ems jvn ðsÞj2 ds 0

2  emt  f 2 þ 2C2 Cg emt kuk2 1 þ 2Cg kgk2 2  L LH mm mk1

Z


ðmk1
2C1 C2 kukL1 ð@XÞ
3Cg
m
ma k1 Þ 2

0

t

ems jvn ðsÞj2 ds;

ð3:1:11Þ

Attractors for Nonlinear Autonomous Dynamical Systems

52

which implies jvn ðtÞj2 þ a2 kvn ðtÞk2  2 f  2C22 Cg þ  kuk2L1 ð@XÞ þ 2Cg kgk2L2 þ jvn ð0Þj2 þ a2 kvn ð0Þk2 K 2 : H mm mk1

ð3:1:12Þ

Integrating (3.1.8) in t also gives ðjvn ðtÞj2 þ a2 kvn ðtÞk2 Þ
ðjvn ðsÞj2 þ a2 kvn ðsÞk2 Þ Z t
1 þ ðm
2C1 C2 k
1 u
C k Þ k k kvn ðrÞk2 dr 1 g 1 1 L ð@XÞ  2 f 

Z

s

t

1 ðt
sÞ þ jgðvnr þ wÞj2 dr Cg s m  2 Z t f  ðt
sÞ þ Cg  jvn ðrÞ þ wj2 dr m s
h  2 Z t f  ðt
sÞ þ 2Cg ðjvn ðrÞj2 þ jwj2 Þdr  m s
h  2 Z t 2 f  2C2 ðt
sÞ þ  ðt
s þ hÞCg kuk2L1 ð@XÞ þ 2Cg jvn ðrÞj2 ds m k1 s 

þ 8C22 hCg kuk2L1 ð@XÞ ;

ð3:1:13Þ

and ðm



1 2C1 C2 k
1 1 kukL1 ð@XÞ
3Cg k1 Þ

 2 f  m

þ

2C22 k1

!

Z

t

kvn k2 ds

s

Cg kuk2L1 ð@XÞ ðt
sÞ þ 10C22 hCg kuk2L1 ð@XÞ

þ jvn ðsÞj2 þ a2 kvn ðsÞk2 :

ð3:1:14Þ

Thus vn ðtÞ 2 L1 ð½0; T ; V Þ \ L2 ð½
h; T ; V Þ. By the Alaoglu compactness theorem, we may seek a subsequence fvn g such that vn ! v in L1 ð½0; T ; V Þ; vn ! v in L2 ð½0; T ; V Þ; i.e., v 2 L1 ð½0; T ; V Þ \ L2 ð½
h; T ; V Þ. dvn 2 L2 ð½0; T ; V 0 Þ. Since Next, we prove that dt dvn ¼
mAvn
a2 Avnt
Bðvn Þ
Rðvn Þ þ Pn f þ gðvnt þ wÞ; dt and vn 2 L2 ð0; T ; V Þ, we have
mAvn ; a2 Avnt ; gðvnt þ wÞ 2 L2 ð0; T ; V 0 Þ;

Global Attractor and Its Upper Estimate on Fractal Dimension

and

Z kBðvn Þk2L2 ð0;T ;V 0 Þ ¼

T

Z

T

ð sup jðvn  rÞvn ; u jÞ2 ds 

0

Z

C

ku k¼1

T

Z C

0

ðjðvn  rÞvn jju jÞ2 ds

0

Z

jðvn  rÞvn j2 ku k2 ds ¼ C

0

53

T

jðvn  rÞvn j2 ds

0 T

jvn j2 jrvn j2  C kvn k2L1 ð0;T ;V Þ kvn k2L2 ð0;T ;V Þ :

Similarly, we have kRðvn Þk2L2 ð0;T ;V 0 Þ  C kuk2L1 ð@XÞ ðkvn k2L1 ð0;T ;V Þ þ kvn k2L2 ð0;T ;V Þ Þ: Thus that

dvn 2 L2 ð½0; T ; V 0 Þ: By the compact embedding theorem, we conclude dt vn ! v; in L2 ð½0; T ; V Þ; vn ð0Þ ¼ Pn v0 ! vð0Þ ¼ v0 : ð3:1:15Þ

In the sequel, we shall show the uniqueness and continuity of global solutions. To this end, let v1 ; v2 be two solutions to problem (3.1.5) with initial data fg1 ; v1 g and fg2 ; v2 g, and let w ¼ v1
v2 , then dw þ mAw þ a2 Awt þ Bðv1 ; v1 Þ
Bðv2 ; v2 Þ þ RðwÞ dt ¼ gðv1t þ wÞ
gðv2t þ wÞ: ð3:1:16Þ Since Bðv1 ; v1 Þ
Bðv2 ; v2 Þ ¼ Bðw; v1 Þ þ Bðv2 ; wÞ; we get dw þ mAw þ a2 Awt þ Bðw; v1 Þ þ Bðv2 ; wÞ þ RðwÞ ¼ gðv1t Þ
gðv2t Þ: dt Multiplying (3.1.17) by w, we obtain 1d ðjw j2 þ a2 kw k2 Þ þ mkw k2 þ bðw; v1 ; wÞ þ bðv2 ; w; wÞ 2 dt þ ðRw; wÞ ¼ hgðv1t þ wÞ
gðv2t þ wÞ; wi; and 1d ðjw j2 þ a2 kw k2 Þ þ mkw k2 2 dt  jbðw; v1 ; wÞj þ jðRw; wÞj þ jhgðv1t þ wÞ
gðv2t þ wÞ; wij  C jw jkw kkv1 k þ C jw jkw kkwk þ jgðv1t þ wÞ
gðv2t þ wÞjjw j m C m C  kw k2 þ jw j2 kv1 k2 þ kw k2 þ jw j2 kwk2 2 2m 2 2m jgðv1t þ wÞ
gðv2t þ wÞj2 jw j2 þ ; þ 2 2

ð3:1:17Þ

Attractors for Nonlinear Autonomous Dynamical Systems

54

i.e.,

  d C C ðjwj2 þ a2 kwk2 Þ  kv1 k2 þ kwk2 þ 1 jwj2 dt m m þ jgðv1t þ wÞ
gðv2t þ wÞj2 :

ð3:1:18Þ

Integrating (3.1.8) over ½0; t, we get ðjwðtÞj2 þ a2 kwðtÞk2 Þ
ðjwð0Þj2 þ a2 kwð0Þk2 Þ  Z t Z t C C 2 2  kv1 k þ kwk þ 1 jw j2 ds þ jgðv1t Þ
gðv2t Þj2 ds m m 0 0  Z t Z t C C 2 2 2 2  kv1 k þ kwk þ 1 jw j ds þ Cg jv1 ðrÞ
v2 ðrÞj2 dr m m 0
h  Z 0  Z t Z t C C 2 2 2 2 ¼ kv1 k þ kwk þ 1 jw j ds þ Cg jg1
g2 j2 dr þ jwðrÞj2 dr m m 0
h 0  Z t C C 2 2 2 2 2 2 ¼ kv1 k þ kwk þ 1 þ Cg jw j ds þ Cg kg1
g2 kL2 H m m 0  Z t  1 1  C kv1 k2 þ kwk2 þ 1 þ Cg2 ðjw j2 þ a2 kw k2 Þds m m 0 þ Cg2 kg1
g2 k2L2 :

ð3:1:19Þ

H

Thus jwðtÞj2 þ a2 kwðtÞk2  jwð0Þj2 þ a2 kwð0Þk2 þ Cg2 kg1
g2 k2L2 H  Z t  1 1 2 2 2 þ C kv1 k þ kwk þ 1 þ Cg ðjwðtÞj2 þ a2 kwðtÞk2 Þds; m m 0 which, by Gronwall’s inequality, gives jwðtÞj2 þ a2 kwðtÞk2  ðjwð0Þj2 þ a2 kwð0Þk2 þ Cg2 kg1
g2 k2L2 Þe

Rt 0

C ð1mkv1 k2 þ 1mkwk2 þ 1 þ Cg2 Þds

H

;

which implies the global existence, uniqueness and continuity of solutions. Hence, we can define the semigroup in CV as SðtÞfg; v 0 g ¼ vt ð; fg; v 0 gÞ: Remark 3.1.2. In fact, we can also get kwðtÞk2  kfg1
g2 ; v1
v2 gk2XV eCT ; and

ð3:1:20Þ

Global Attractor and Its Upper Estimate on Fractal Dimension Z kwt k2L2 V

¼ Z ¼

0
h

Z 2

t

kwðsÞk2 ds Z kwðsÞk2 ds þ

kwðt þ sÞk ds 

55

t
h

½t
h;t \ ½
h;0

½t
h;tn½
h;0

kwðsÞk2 ds

 kfg1
g2 ; v1
v2 gk2XV eCT h; whence kwðtÞkXV  C kfg1
g2 ; v1
v2 gkXV :

ð3:1:21Þ

It thus follows that the continuous semigroup in XV can also be defined as SðtÞfg; v 0 g ¼ fvt ; vðtÞg: □

3.2

Existence of Global Attractors

3.2.1

Existence of Absorbing Sets

In order to obtain global attractors, we need to derive the existence of absorbing sets of SðtÞ in CV and XV , and main results are given as follows. Lemma 3.2.1. Assume f 2 H , fg; v 0 g 2 XV , and the conditions (A1) (A3) hold, then the semigroup SðtÞ to problem (3.1.4) possesses an absorbing set in XV . Proof. Assume fg; v 0 g 2 D which is any bounded set of radius d [ 0 in XV . Multiplying (3.1.4) by v, we obtain 1d ðjvj2 þ a2 kvk2 Þ þ mkvk2 þ bðv; v; vÞ þ ðRv; vÞ ¼ ðPf ; vÞ þ ðgðvt þ wÞ; vÞ; 2 dt and 1d ðjvj2 þ a2 kvk2 Þ þ mkvk2  jðRv; vÞj þ jðPf ; vÞj þ jðgðvt þ wÞ; vÞj 2 dt 2  kf kkvk þ jgjjvj þ C1 k
1 1 kvk kwk m 1 1 Cg 2 2 2 kf k2 þ jvj jgj2 þ  C1 k
1 1 kvk kwk þ kvk þ 2 2m 2Cg 2   1 1 m 1
1 þ C1 k
1 C  kf k2 þ jgj2 þ kwk þ k kvk2 : g 1 1 2m 2Cg 2 2

ð3:2:1Þ

Attractors for Nonlinear Autonomous Dynamical Systems

56

It thus follows that d ðjvj2 þ a2 kvk2 Þ dt 1 1  kf k2 þ jgj2
ðmk1
2C1 kwk
Cg Þjvj2 ; m Cg

ð3:2:2Þ

and d mt emt emt 2 ½e ðjvj2 þ a2 kvk2 Þ  kf k2 þ jgj
emt ðmk1
2C1 C2 kwkL1 ð@XÞ dt m Cg
Cg
m
ma2 k1 Þjvj2 :

ð3:2:3Þ

Integrating (3.2.3) over ½0; t, we arrive at jvðtÞj2 þ a2 kvðtÞk2 kf k2 2C22 Cg þ kuk2L1 ð@XÞ þ e
mt ðjvð0Þj2 þ a2 kvð0Þk2 þ 2Cg kgk2L2 Þ H mm mk1 2 2 kf k 2C2 Cg  þ kuk2L1 ð@XÞ þ Cd 2 e
mt mm mk1



and for any t [ 0; h 2 ½
h; 0, 2

2

jvðt þ hÞj þ a kvðt þ hÞk
2

ð3:2:4Þ

! kf k2 2c2 Cg 2 þ kukL1 ð@XÞ  e
mt emh Cd 2 : mm m

Thus there exist constants qV and TV such that for any fg; v 0 g 2 D and t  TV , kvk2CV  q2V ; which means BV ð0; qV Þ is an absorbing ball for SðtÞ in CV . Remark 3.2.1 Using the technique in (3.1.21), we derive kvt k2L2  C q2V ; kfvt ; vðtÞgkXV  C q2V ; V

and hence the existence of an absorbing ball in XV follows.

3.2.2

Some Compactness and the Existence of Global Attractors

To prove our results, we also need some compactness of SðtÞ in XV . Note that v ¼ SðtÞfg; v 0 g is the solution to (3.1.4) with initial data fg; v 0 g 2 XV , and SðtÞ can be decomposed in the following sum SðtÞfg; v 0 g ¼ Y ðtÞfg; v 0 g þ Z ðtÞfg; v 0 g yðtÞ þ zðtÞ

Global Attractor and Its Upper Estimate on Fractal Dimension where Y ðtÞ is the semigroup generated by the following system 8 dy d > < þ mAy þ a2 Ay ¼ 0; dt dt 0 > : yð0Þ ¼ v ; x 2 X; y0 ¼ g; x 2 X; and Z ðtÞfg; v 0 g is the solution to the problem 8 dz d > < þ mAz þ a2 Az ¼ Pf þ gðvt þ wÞ
BðvÞ
RðvÞ; dt dt > : zð0Þ ¼ 0; x 2 X; z0 ¼ 0; x 2 X:

57

ð3:2:5Þ

ð3:2:6Þ

Multiplying (3.2.5) by yðtÞ, and using the Galerkin method, we can easily derive that for all t [ 0 jyðtÞj2 þ a2 kyðtÞk2  eCt ðjvð0Þj2 þ a2 kvð0Þk2 Þ  eCt kfg; v 0 gk2XV :

ð3:2:7Þ

Using the Hölder and the Gagliardo–Nirenberg inequalities, we also conclude for any u 2 V, Z kBðvÞkDðA
1=3 Þ ¼ sup ðv  rÞv  udx  sup kvkL3 kvkkukL6 kukDðA1=3 Þ ¼1

C

kukDðA1=3 Þ ¼1

X

sup kukDðA1=3 Þ ¼1

kvk2 kuk2=3  C kvk2 ;

which gives kBðvÞk 2 L1 ð0; T ; DðA
1=3 ÞÞ:

ð3:2:8Þ

In a similar way, we can derive kRðvÞk 2 L1 ð0; T ; DðA
1=3 ÞÞ:

ð3:2:9Þ

Multiplying (3.2.6) by A1=3 z, using the condition (A2), (3.2.8), (3.2.9) and the Gronwall’s inequality, we can conclude Z T 2 2 2 jgðvt þ wÞj2 dt jyðtÞj1=3 þ a kyðtÞk4=3  C þ C Z C þC

0

T
h

jðvðtÞÞj2 dt

 C þ C kuk2L1 ð@XÞ h þ C

Z

T 0

jðvðtÞÞj2 dt;

ð3:2:10Þ

Attractors for Nonlinear Autonomous Dynamical Systems

58

which means the operator Z ðtÞ maps XV into XV4=3 , and from the fact that V ,!,!V4=3 , we derive the operator Z ðtÞ is compact for t [ 0. Using the decomposition method (see [75, 146]), from (3.2.7) and the compactness of Z ðtÞ, we derive that SðtÞ is asymptotically compact in XV . Since we have established the existence of continuous semigroup fSðtÞg to problem (3.1.4) in XV , and the existence of bounded absorbing set BV ð0; qV Þ in XV , the asymptotical compactness of fSðtÞg in XV , from the fundamental theory of global attractors (see theorem 1.2.37), the existence of global attractor A of fSðtÞg in XV follows naturally, that is, the following theorem holds. Theorem 3.2.2. Under assumptions in lemma 3.2.1, the problem (ref. [32]) or semigroup SðtÞ possesses a global attractor A 2 XV .

3.3

Upper Estimate on the Fractal Dimension of Global Attractors

In this section, we aim to estimate the upper bound of the fractal dimension of the global attractor obtained in the previous section. To do so, we need first to show the differentiability of SðtÞ with respect to initial data. Consider now the first variation equation of problem (3.1.4) 8 d < dU þ mAU þ a2 AU þ Bðv; U Þ þ RðU ; vÞ þ RðU Þ ¼ gðvt þ wÞUt ; ð3:3:1Þ dt dt : fU0 ; U ð0ÞÞg 2 XV : Using the usual energy estimates, we can prove a unique solution U ðtÞ to problem (3.3.1) exists such that U ðtÞ 2 L1 ð½0; T ; V Þ \ L2 ð½
h; T ; V Þ; Ut 2 ð½0; T ; V Þ; and the linear operator Kðt; fv0 ; vð0ÞgÞ : XV ! XV can be defined as Kðt; fv0 ; vð0ÞgÞfU0 ; U ð0Þg ¼ fUt ðtÞ; U ðtÞg: Theorem 3.3.1. Let A be the global attractors of SðtÞ to problem (3.1.4) in XV , then SðtÞ is uniformly differentiable on A, that is, if for every fg2 ; v2 g 2 A, there exists a linear operator Kðt; fg2 ; v2 gÞ such that for all t  0, kSðtÞfg1 ; v1 g
SðtÞfg2 ; v2 g
Kðt; fg2 ; v2 gÞfg1
g2 ; v1
v2 gkXV !0 kfg1
g2 ; v1
v2 gkXV fg1 ;v1 g;fg2 ;v2 g2A sup

as kfg1
g2 ; v1
v2 gkXV  e ! 0, and

Global Attractor and Its Upper Estimate on Fractal Dimension

sup fg2 ;v2 g2A

59

jjKðt; fg2 ; v2 gÞjjLðXV Þ \ þ 1; for all t  0:

Proof. Assume that v1 ðtÞ and v2 ðtÞ are solutions to problem (3.1.4) with initial data fg1 ; v1 g and fg2 ; v2 g respectively, U ðtÞ ¼ Kðt; fg2 ; v2 gÞfg1
g2 ; v1
v2 g is the solution to problem (3.3.1) with initial data fg1
g2 ; v1
v2 g. Let w ¼ v1
v2 , then h ¼ v1
v2
U ¼ w
U satisfies d ðh þ a2 AhÞ þ mAh þ Bðv2 ; hÞ þ Bðw; wÞ þ Bðh; v2 Þ þ RðhÞ dt ¼ gðv1t Þ
gðv2t Þ
g 0 ðv2t ÞUt ;

ð3:3:2Þ

where gðv1t þ wÞ
gðv2t þ wÞ
g 0 ðv2t þ wÞUt ¼ gðv1t þ wÞ
gðv2t þ wÞ
g 0 ðv2t þ wÞðv1t
v2t Þ þ g 0 ðv2t þ wÞht ¼ g 00 ðÞðv1t
v2t Þ2 þ g 0 ðv2t þ wÞht ; with  ¼ pv1t þ ð1
pÞv2t ; 0  p  1: Multiplying (3.3.2) by h, we obtain 1d ðjhj2 þ a2 khk2 Þ þ mkhk2 þ bðw; w; hÞ þ bðh; v2 ; hÞ þ ðRh; hÞ 2 dt ¼ ðg 00 ðÞðv1t
v2t Þ2 þ g 0 ðv2t þ wÞht ; hÞ; and 1d ðjhj2 þ a2 khk2 Þ þ mkhk2 2 dt  jbðw; w; hÞj þ bðh; v2 ; hÞ þ jðRh; hÞj þ Ljht jjhj þ Ljwt j2 jhj  C ðkwk4 þ kwt k4 Þ þ C ðkhk2 þ kht k2 Þ þ mkhk2 : Thus we can derive from (3.1.20) that d khk2  kfg1
g2 ; v1
v2 gk4XV eKT þ C ðkhk2 þ kht k2 Þ: dt From Gronwall’s inequality, it follows that Z khðtÞk2  kðv1 ; g1 Þ
ðv2 ; g2 Þk4XV TeKT þ C

t

ðkhðsÞk2 þ khs k2 Þds;

0

and Z sup khðsÞk2  kðv1 ; g1 Þ
ðv2 ; g2 Þk4XV TeKT þ C

s2½t
h;t

t

sup khðrÞk2 ds:

0 r2½s
h;s

Attractors for Nonlinear Autonomous Dynamical Systems

60

Thus sup khðsÞk2  kðv1 ; g1 Þ
ðv2 ; g2 Þk4XV TeKT ð1 þ eKT Þ;

s2½t
h;t

which implies khðtÞk2  C kðv1 ; g1 Þ
ðv2 ; g2 Þk4XV : Furthermore, we also get Z 0 Z t 2 2 khðt þ sÞk ds  khðsÞk2 ds kht kL2 ¼ V
h t
h Z Z khðsÞk2 ds þ ¼ Z 

½t
h;t \ ½
h;0 t 2

0

½t
h;tn½
h;0

ð3:3:3Þ

khðsÞk2 ds

khðsÞk ds  CT kðv1 ; g1 Þ
ðv2 ; g2 Þk4XV ;

ð3:3:4Þ

which, together with (3.3.3), leads to khðtÞkXV  C kðv1 ; g1 Þ
ðv2 ; g2 Þk2XV :

ð3:3:5Þ

From the expression of h, it follows kSðtÞfg1 ; v1 g
SðtÞfg2 ; v2 g
Kðt; fg2 ; v2 gÞfg1
g2 ; v1
v2 gkXV kðv1 ; g1 Þ
ðv2 ; g2 ÞkXV  C kðv1 ; g1 Þ
ðv2 ; g2 ÞkXV : Therefore that the differentiability of SðtÞ with respect to initial data follows naturally and SðtÞ is uniformly differentiable on A since we have proved that the linear operator Kðt; fg2 ; v2 gÞ is bounded. Finally, we plan to estimate the upper bound of the fractal dimension of A in XV . To do so, we rewrite (3.3.1) as follows dU m m ¼
2 U þ 2 G
2 U
G
1 Bðv; G
1 U Þ dt a a
G
1 BðG
1 U ; vÞ
G
1 RðG
1 U Þ þ G
2 g 0 ðvt ÞU t ; where G ¼ ðI þ a2 AÞ1=2 , and U ¼ GU . The corresponding form is written as d U ¼ LðtÞU þ G
2 g 0 ðvt þ wÞU t ; dt

ð3:3:6Þ

where LðtÞ ¼


m m  þ 2 G
2 
G
1 Bðv; G
1 Þ
G
1 BðG
1 ; vÞ
G
1 RðG
1 Þ: a2 a

Global Attractor and Its Upper Estimate on Fractal Dimension

61

Define L : DðLÞ  XV ! XV by DðLÞ ¼ ffa; bg 2 XV : a is absolutely continuous on ½
h; 0; a_ 2 L2 ð½
h; 0; V Þ and b ¼ að0Þ 2 DðLÞg; _ Lbg; fa; bg 2 DðLÞ; Lfa; bg ¼ fa; and DðLÞ is dense in XV . Let W ¼ fU t ; U ðtÞg, then equation (3.3.6) can be reformulated as an evolutionary system on the product space XV of the form 8 < d W ¼ RðtÞW ; dt ð3:3:7Þ : W ð0Þ 2 XV ; where GðtÞ : XV ! XV is defined by RðtÞW ¼ LðtÞW þ f0; G
2 g 0 ðvt þ wÞP1 W g: From (3.3.6) and (3.3.7), we can extend the generator L (in V ) to L (in XV ) naturally so that the variational equation can be reformulated as an evolution system on XV . Only in this way, can we apply directly the method in [146] to establish the upper estimates, whose idea comes from papers by Webb [153, 154] in which he constructed a generator for a nonlinear semigroup and we also find that ut ¼ P1 W ðtÞ if uðtÞ ¼ P2 W ðtÞ; where P1 ; P2 are defined at the beginning of section 3.1 of this chapter. Theorem 3.3.2. Under the same assumptions, the fractal dimension of A to problem (3.1.4) is less than or equal to 0 11=2 2C 2 kf k2 1=2 1=2 þ k12 Cg kuk2L1 ð@XÞ C3 ak1 jXj1=2 m 2 þ C2 kukL1 ð@XÞ A þ 1:  1=2 @
1
1 m
2C C k kuk 1=2 m 2l0 1 ð@XÞ
3Cg k1 2m 1 2 L 1 ð2pmÞ

a

a2 k1

k1

Proof. For any fUj0 ; Uj ð0Þgj¼1;2;...;m 2 XV , let fUjt ; Uj ðtÞg ¼ Kðt; fv0 ; vð0ÞgÞ 0 fUj0 ; Uj ð0Þg, and Qm ðsÞ and Qm ðsÞ denote the projectors of XH to the spans ffUjs ; Uj ðsÞgj¼1;2;...;m g and ffGUjs ; GUj ðsÞgj¼1;2;...;m g respectively. Let fujs ; uj ðsÞgj¼1;2;...;m be an orthonormal basis for span fUjs ; Uj ðsÞgj¼1;2;...;m . Since fUjs ; Uj ðsÞg 2 XV , fujs ; uj ðsÞg 2 XV , we conclude 0 TrðRðsÞ  Qm ðsÞÞ m P ¼ \GðsÞfRujs ; Guj ðsÞg; fGujs ; Guj ðsÞg [

¼ ¼

j¼1 m P j¼1 m P j¼1

\LfGujs ; Guj ðsÞg þ f0; G
2 g 0 ðvt þ wÞGujs g; fGujs ; Guj ðsÞg [
2 0 _ \fGu js ; LGuj ðsÞg þ f0; G g ðvt þ wÞGujs g; fGujs ; Guj ðsÞg [ ;

Attractors for Nonlinear Autonomous Dynamical Systems

62

where, from (3.3.6) and the assumption (A1), it follows
2 0 _ \fGu js ; LGuj ðsÞg þ f0; G g ðvt þ wÞGujs g; fGujs ; Guj ðsÞg [  Z 0 d Guj ðs þ sÞ; Guj ðs þ sÞÞds þ ðLGuj ðsÞ; Guj ðsÞ ¼
h ds

þ ðG
2 g 0 ðvt þ wÞGujs ; Guj ðsÞÞ m m ¼ jGuj ðsÞj2
jGuj ðs
hÞj2
2 jGuj ðsÞj2 þ 2 juj ðsÞj2 a a
bðuj ðsÞ; v; uj ðsÞÞ
ðRðuj ðsÞÞ; uj ðsÞÞ þ ðg 0 ðvt þ wÞujs ; uj ðsÞÞ m m  jGuj ðsÞj2
jGuj ðs
hÞj2
kuj ðsÞk2 þ 2 juj ðsÞj2 a a þ jbðuj ðsÞ; v; uj ðsÞÞj þ jbðuj ðsÞ; w; uj ðsÞÞj þ l0 jujs jjuj ðsÞj m m  jGuj ðsÞj2
jGuj ðs
hÞj2
kuj ðsÞk2 þ 2 kuj ðsÞk2 a a k1 l0 l0 þ jbðuj ðsÞ; v; uj ðsÞÞj þ jbðuj ðsÞ; w; uj ðsÞÞj þ kujs k2 þ kuj ðsÞk2 k k1  1 m m l0 2 2 2
 jGuj ðsÞj
jGuj ðs
hÞj

kuj ðsÞk a a2 k1 k1 l0 þ jbðuj ðsÞ; v; uj ðsÞÞj þ jbðuj ðsÞ; w; uj ðsÞÞj þ kujs k2 ; k1 and 0 ðsÞÞ TrðRðsÞ  Qm   m m m m l0 X l0 X
2
kuj ðsÞk2 þ ku k2 
a a k1 k1 j¼1 k1 j¼1 js

þ

m m X X ðjGuj ðsÞj2
jGuj ðs
hÞj2 Þ þ jbðuj ðsÞ; v; uj ðsÞÞj j¼1

þ

m X

j¼1

ð3:3:8Þ

jbðuj ðsÞ; w; uj ðsÞÞj:

j¼1

From the Lieb–Thirring inequality, j

m X j¼1

juj ðsÞj2 j2  C3

m X j¼1

kuj ðsÞk2 ;

Global Attractor and Its Upper Estimate on Fractal Dimension

63

and the last two terms in (3.3.8) are bounded, respectively, as follows m X jbðuj ðsÞ; v; uj ðsÞÞj j¼1

j

m X

juj ðsÞj2 j  kvk 

m pffiffiffiffiffiffi X C3 ð kuj ðsÞk2 Þ1=2  kvk

j¼1



m X

m 2a2 k1

j¼1 m X

kuj ðsÞk2 þ

j¼1

jbðuj ðsÞ; w; uj ðsÞÞj 

j¼1

2

C 3 a k1 kvk2 ; 2m

m m X C3 a2 k1 kwk2 : kuj ðsÞk2 þ 2 2a k1 j¼1 2m

ð3:3:9Þ

ð3:3:10Þ

Using (3.3.9), (3.3.10) and the variational principle in (3.3.8), we get 0 TrðRðsÞ  Qm ðsÞÞ   m m m 2m l0 X l0 X
2
kuj ðsÞk2 þ ku k2 
a a k1 k1 j¼1 k1 j¼1 js

þ

m X

ðjGuj ðsÞj2
jGuj ðs
hÞj2 Þ þ

j¼1

C 3 a2 k1 C 3 a2 k1 kwk2 þ kvk2 2m 2m



 m m m 2m l0 X l0 X
2

kj þ ku k2 a a k1 k1 j¼1 k1 j¼1 js þ

m X

ðjGuj ðsÞj2
jGuj ðs
hÞj2 Þ þ

j¼1

 
þ

C 3 a2 k1 C 3 a2 k1 kwk2 þ kvk2 2m 2m

 m m 2m l0 pm 2 l0 X
2
þ ku k2 a a k1 k1 jXj k1 j¼1 js

m X

ðjGuj ðsÞj2
jGuj ðs
hÞj2 Þ þ

j¼1

C 3 a2 k1 C 3 a2 k1 kwk2 þ kvk2 : 2m 2m

ð3:3:11Þ

Noting that Z t

1 0 qm ðtÞ ¼ sup sup TrðRðsÞ  Qm ðsÞÞds v2A fU0 ;U ð0Þg2XH ;j¼1;...;m t 0   Z m 2m 2l0 pm 2 C3 a2 k1 1 t 

2
þ sup ðkwk2 þ kvk2 Þds; a a k1 k1 jXj 2m v2A t 0

Attractors for Nonlinear Autonomous Dynamical Systems

64

we derive from (3.1.14) that qm lim sup qm ðtÞ t!1   Z m 2m 2l0 pm 2 C3 a2 k1 1 t
2
þ sup 
ðkwk2 þ kvk2 Þds a a k1 k1 jXj 2m v2A t 0   m 2m 2l0 pm 2
2

a a k1 k1 jXj 0 1 2C22 2 kf k2 C 3 a2 k1 @ m þ k1 Cg kukL1 ð@XÞ 2A þ kwk þ
1 2m m
2C1 C2 k
1 1 kukL1 ð@XÞ
3Cg k1   m 2m 2l0 pm 2
2

a a k1 k1 jXj 0 1 2C22 kf k2 2 C 3 a2 k1 @ m þ k1 Cg kukL1 ð@XÞ 2 þ C2 kukL1 ð@XÞ A: þ
1 2m m
2C1 C2 k
1 1 kukL1 ð@XÞ
3Cg k1

ð3:3:12Þ

Using the assumption ðA4Þ

m 2m 2l0 [ 2 þ ; a a k1 k1

and theorem 1.3.4, it thus follows from (3.3.12) that the fractal dimension of A is smaller than or equal to 0 11=2 2C22 kf k2 2 1=2 1=2 1=2 þ C kuk 1 g C3 ak1 jXj L ð@XÞ m k1 @ þ C2 kuk2L1 ð@XÞ A þ 1:
1
1 1=2 m 2l0 1=2 2m m
2C C k kuk
3C k 1 ð2pmÞ ða
a2 k
k Þ 1 2 1 g 1 L ð@XÞ 1

1

The proof is hence complete.

3.4

□

Bibliographic Comments

The NSV equations model the dynamics of a Kelvin–Voight viscoelastic incompressible fluid and was introduced by Oskolkov in [100], and its generated semigroup has a finite dimensional global attractor which can be seen in [65]. Under the homogeneous boundary condition, Kalantarov [66] derived the existence of global attractor for the NSV equations, and further showed that the attractor lays in a bounded subset of the Sobolev space H 1 ðXÞ whenever the forcing term f 2 L2 ðXÞ. In 1963, Krasovskii [73] first considered the stability problem for differential equations with time delay. Caraballo and Real obtained some results relating to the existence and asymptotic behavior of solution for the Navier–Stokes equations with variable delay and distributed delay in [17]. Taniguchi established the existence of absorbing sets of the non-autonomous Navier–Stokes equations with a continuous delay

Global Attractor and Its Upper Estimate on Fractal Dimension

65

in [143]. Li and Qin gave the existence of pullback attractors of the NSV equations with continuous delay on smooth domain in [77]. We also know that many nonlinear evolution equations for infinite-dimensional dynamical systems have indeed attractors with finite Hausdorff and fractal dimensions. In 2000, Brown, Perry and Shen were concerned with the 2D Navier–Stokes equations on a Lipschitz domain and established the upper bounds of the global attractors in [10]. Also, the upper bounds for the dimension of a global attractor for the 3D Navier–Stokes–Voight equations were derived by Kalantarov and Titi in [67]. However, to our knowledge, there are fewer results involving the dimension estimate on attractors for the NS equations, or NSV equations, with delay. In addition, the aim of this paper was to establish the upper estimates on the Hausdorff and fractal dimensions of global attractor for the 2D NSV equations with a distributed delay. Comparing with Li and Qin’s conclusion, the problem in this chapter has the following different features: (1) The existence of global attractors A of 2D NSV equations with a distributed delay in the product space XV is given; (2) Using the method of extention for genertor appeared in Webb’s paper (JDE, 1976), we establish the differentiability of the semigroup in the product space XV , and first obtain the upper estimates on the Hausdorff and fractal dimensions of A.

Chapter 4 Maximal Attractor for the Equations of One-Dimensional Compressible Polytropic Viscous Ideal Gas In this chapter, we shall prove the existence of maximal (global) attractor for the equations of compressible polytropic viscous ideal gas. One important feature is that our working metric spaces H ð1Þ and H ð2Þ are two incomplete ones, as can be seen from the constraints h [ 0 and u [ 0 with h and u being absolute temperature and specific volume, respectively. For any constants b1 ; b2 ; b3 ; b4 ; b5 with some ðiÞ constraint conditions, we establish two sequences of closed subspaces Hb  ð1Þ

H ðiÞ ði ¼ 1; 2Þ and the existence of two maximal (universal) attractors in Hb

and

ð2Þ Hb .

The content of this chapter is picked up from Zheng and Qin [159] and the method used here is the weakly compact semigroup method, see theorem 1.2.24.

4.1

Our Problem

The Lagrangian form of the conservation laws of mass, momentum, and energy for this kind of gas with the reference density q0 ¼ 1 can take as (see [72]). ut  vx ¼ 0; vt þ

CV h t þ


   Rh lvx  ¼ 0; u x u x


 Rhvx Khx lv 2   x ¼ 0: u u x u

ð4:1:1Þ ð4:1:2Þ

ð4:1:3Þ

DOI: 10.1051/978-2-7598-2702-2.c004 © Science Press, EDP Sciences, 2022

Attractors for Nonlinear Autonomous Dynamical Systems

68

Here u; v; h denote the specific volume, velocity and absolute temperature, respectively. R; l; CV and K are the given positive constants. Consider problem (4.1.1)–(4.1.3) in the region f0  x  1; t  0g subject to the boundary conditions vð0; tÞ ¼ vð1; tÞ ¼ 0; hx ð0; tÞ ¼ hx ð1; tÞ ¼ 0;

ð4:1:4Þ

and the initial conditions uðx; 0Þ ¼ u0 ðxÞ; vðx; 0Þ ¼ v0 ðxÞ; hðx; 0Þ ¼ h0 ðxÞ

½0; 1:

on

ð4:1:5Þ

For our simplicity, we always assume that CV ¼ R ¼ l ¼ K ¼ 1: Now define the spaces H ð1Þ ¼ fðu; v; hÞ 2 H 1 ½0; 1  H 1 ½0; 1  H 1 ½0; 1 : uðxÞ [ 0; hðxÞ [ 0; x 2 ½0; 1; vjx¼0 ¼ vjx¼1 ¼ 0g and H ð2Þ ¼ fðu; v; hÞ 2 H 2 ½0; 1  H 2 ½0; 1  H 2 ½0; 1 : uðxÞ [ 0; hðxÞ [ 0; x 2 ½0; 1; vjx¼0 ¼ vjx¼1 ¼ hx jx¼0 ¼ hx jx¼1 ¼ 0g: Obviously, these two spaces are two metric spaces when equipped with the metrics induced from the usual norms. Here, H 1 ; H 2 are the usual Sobolev spaces. Assume that bi ði ¼ 1; :::; 5Þ are any given constants such that b1 2 R; e b1 b2 [ 0; b4  [ b3 [ 0; 0\b5 \b2 are arbitrarily given constants, and let b2  Z 1 Z 1 ðiÞ ðlnðhÞ þ lnðuÞÞdx  b1 ; b5  ðh þ v 2 =2Þdx  b2 ; Hb :¼ ðu; v; hÞ 2 H ðiÞ : Z b3 

0

1

0



udx  b4 ; b5 =2  h  2b2 ; b3 =2  u  2b4 ; i ¼ 1; 2:

0 ðiÞ

Indeed, Hb is a sequence of closed subspaces of H ðiÞ ði ¼ 1; 2Þ: Later on, we will show that the first three constraints are invariant, while the last two constraints are not invariant. These two constraints are just introduced to overcome the difficulty that the original spaces are incomplete. As we shall mention in section 4.6, it is very ðiÞ ðiÞ crucial to prove that the orbit starting from any bounded set of Hb will reenter Hb after a finite time. Our main theorem reads as follows. Theorem 4.1.1. The nonlinear semigroup SðtÞ defined by the solution to problem (7.1.1)–(7.1.5) maps H ðiÞ ði ¼ 1; 2Þ into itself. Moreover, for any bi ði ¼ 1; :::; 5Þ e b1 ðiÞ with b1 \0; b2 [ 0; b4  [ b3 [ 0; 0\b5 \b2 , it possesses in Hb a maximal b2 attractor Ai;b ði ¼ 1; 2Þ:

Maximal Attractor for the One-Dimensional Equations Ai ¼

Remark 4.1.1. The set

S ðiÞ

b1 ;b2 ;b3 ;b4 ;b5

69

Ai;b ði ¼ 1; 2Þ is a global noncompact

attractor in the metric space H in the following sense that it attracts any bounded sets of H ðiÞ with constraints u  g1 ; h  g2 with g1 ; g2 being any given positive constants. The notation in this chapter will be as follows: Lp ; 1  p  þ 1; W m;p ; m 2 N; H 1 ¼ W 1;2 ; H01 ¼ W01;2 denote the usual (Sobolev) spaces on ð0; 1Þ: In addition, ð;Þ stands for the inner product in L2 ; and k  kB denotes the norm in the space B; we also put k  k ¼ k  kL2 : We denote by C k ðI ; BÞ; k 2 N0 ; the space of k-times continuously differentiable functions from I R into a Banach space B, and likewise by Lp ðI ; BÞ; 1  p  þ 1 the corresponding Lebesgue spaces. Subscripts t and x denote the (partial) derivatives with respect to t and x, respecðiÞ tively. We use C0 ; i ¼ 1; 2 to denotes the universal constant depending only on the H ðiÞ norm of initial data and minx2½0;1 u0 ðxÞ. Cb denotes the universal constant ðiÞ

depending only on bi ði ¼ 1; :::; 5Þ, but independent of initial data. Cb denotes the universal constant depending on both bj ðj ¼ 1; :::; 5Þ, H ðiÞ norm of initial data and minx2½0;1 u0 ðxÞ.

4.2

Nonlinear Semigroup on H ð2Þ

As explained in section 4.1, for any initial data ðu0 ; v0 ; h0 Þ 2 H ð1Þ , the results on global existence, uniqueness and asymptotic behaviour of solutions to problem (4.1.1)– (4.1.5) have been established in [72, 106], that is, we have the following lemma. Lemma 4.2.1. SðtÞ defines a nonlinear C0 -semigroup on H ð1Þ such that for any ðu0 ; v0 ; h0 Þ 2 H ð1Þ ; SðtÞðu0 ; v0 ; h0 Þ ¼ ðuðtÞ; vðtÞ; hðtÞÞ 2 C ð½0; þ 1Þ; H ð1Þ Þ; ut ; hx ; vx ; ux ; vxx ; hxx 2 L2 ð½0; þ 1Þ; L2 Þ: Moreover, ð1Þ

0\hðx; tÞ  C0 ð1Þ

on ½0; 1  ½0; 1Þ;

ð4:2:1Þ

ð1Þ

ð4:2:2Þ

0\1=C0  uðx; tÞ  C0

on ½0; 1  ½0; 1Þ;

Z t kuðtÞk2H 1 þ khðtÞk2H 1 þ kvðtÞk2H 1 þ kux k2 þ kvx k2 0  ð1Þ þ khx k2 þ kvxx k2 þ khxx k2 ðsÞds  C0 ; 8t [ 0:

ð4:2:3Þ

kðuðtÞ  u0 ; vðtÞ; hðtÞ  hÞkH 1 ! 0;

ð4:2:4Þ

as t ! þ 1

where Z

1

u0 ¼ 0

Z

1

u0 ðxÞdx; h ¼ 0

ðv02 ðxÞ=2 þ h0 ðxÞÞdx:

ð4:2:5Þ

Attractors for Nonlinear Autonomous Dynamical Systems

70

Now we have the following theorem. Theorem 4.2.1. SðtÞ defines a nonlinear C0 -semigroup on H ð2Þ . Proof. The proof of theorem 4.2.1 is divided into a series of the following lemmas.h Lemma 4.2.2. ([105]) For each t  0, there exists a point x1 ¼ x1 ðtÞ 2 ½0; 1 such that the solution uðx; tÞ to problem (7.1.1)–(7.1.5) has the following expression:
 Z t hðx; sÞ ds ð4:2:6Þ uðx; tÞ ¼ Dðx; tÞZ ðtÞ 1 þ 0 Dðx; sÞZ ðsÞ where Dðx; tÞ ¼ u0 ðxÞ expðBðx; tÞÞ; Z Bðx; tÞ ¼

Z

x

x1 ðtÞ

vðy; tÞdy 


Z

þ

1

u0 ðxÞdx


Z Z ðtÞ ¼ exp  0

1

u0 ðxÞdx

x

v0 ðyÞdy 0 1 Z 1

Z

u0 ðxÞ

0

ð4:2:7Þ

0

1 Z t Z

x

v0 ðyÞdydx;

1

! ðv þ hÞðx; sÞdxds : 2

0

ð4:2:8Þ

0

ð4:2:9Þ

0

Lemma 4.2.3. For any ðu0 ; v0 ; h0 Þ 2 H ð2Þ , the unique global solution ðu; v; hÞ of problem (4.1.1)–(4.1.5) belongs to C ð½0; þ1Þ; H ð2Þ Þ. Moreover, ð2Þ

kðuðtÞ; vðtÞ; hðtÞÞkH ð2Þ  C0 ; 8t [ 0:

ð4:2:10Þ

Proof. By the dense argument, we need to (4.2.10) when ðu0 ; v0 ; h0 Þ 2 H ð2Þ and is smooth enough. From the results in [106], it follows that there exists a unique smooth solution ðuðtÞ; vðtÞ; hðtÞÞ to problems (4.1.1)–(4.1.5). By equation (4.1.3), lemma 4.2.1 and Sobolev’s imbedding theorem, we have Z t kht k2 ðsÞds 0 Z t ð1Þ  C0 ðkvx k2 þ khxx k2 þ khx kkhxx kkux k2 þ kvx k3 kvxx kÞðsÞds 0 Z t ð1Þ  C0 ðkvx k2 þ kvxx k2 þ khx k2 þ khxx k2 ÞðsÞds 0

ð1Þ

 C0 :

ð4:2:11Þ

Differentiating (4.1.2) with respect to t, then multiplying the resultant by vt and integrating over ð0; 1Þ, we get

Maximal Attractor for the One-Dimensional Equations

71

d 1 kvt k2 þ ð1Þ kvxt k2 dt C0 ð1Þ

 C0 ðkht k þ kvx k þ kvx2 kÞkvxt k 1 ð1Þ  kvxt k2 þ C0 ðkht k2 þ kvx k2 þ kvx k3 kvxx kÞ: ð1Þ 2C0 Then integrating with respect to t and applying lemma 4.2.1 yields Z t ð2Þ ð2Þ kvt ðtÞk2  C0 ; kvxt k2 ds  C0 ; 8t [ 0:

ð4:2:12Þ

ð4:2:13Þ

0

By equation (4.1.2), Sobolev’s imbedding theorem and Young’s inequality, we have ð1Þ

kvxx k  C0 ðkvt k þ khkH 1 þ kux k þ kvx ux kÞ ð1Þ

 C0 ðkvt k þ kvx k1=2 kvxx k1=2 kux k þ 1Þ 1 ð1Þ  kvxx k þ C0 ðkvt k þ 1Þ: 2

ð4:2:14Þ

Combining (4.2.13) with (4.2.14) yields ð2Þ

ð2Þ

kvxx k  C0 ; kvx kL1  C0 ; 8t [ 0:

ð4:2:15Þ

By equation (4.1.3), in the same manner, we have d 1 kht k2 þ ð1Þ khxt k2 dt C0 1 ð1Þ  khxt k2 þ C0 ðkhx k2 kvx kkvxx k þ kvx kkvxx kkvxt k2 ð1Þ 2C0 þ kht k2 þ kvx k2 þ kvx k1=2 kvxx k1=2 kht k2 þ kvxt k2 þ kvx k3 kvxx kÞ 1 ð2Þ  khxt k2 þ C0 ðkhx k2 þ kvxt k2 þ kht k2 þ kvx k2 Þ: ð1Þ 2C0

ð4:2:16Þ

Integrating (4.2.16) with respect to t, then applying (4.2.13) and lemma 4.2.1, we get Z kht k2 þ 0

t

ð2Þ

khxt k2 ðsÞds  C0 ; 8t [ 0:

ð4:2:17Þ

Therefore, it follows from equation (4.1.3) that ð2Þ

khxx k  C0 ; 8t [ 0: In the sequel, we shall estimate the norm of u in H 2 . By lemmas 4.2.1 and 4.2.2, we easily get

ð4:2:18Þ

Attractors for Nonlinear Autonomous Dynamical Systems

72

ð1Þ

ð1Þ

kBðx; tÞkL1  C0 ; j expðBðx; tÞÞj  C0 ; 1

ð1Þ

ð1Þ C0

 Dðx; tÞ; D1 ðx; tÞ  C0 ; Z 0

t

ð1Þ

ðZ ðtÞZ 1 ðsÞÞp ds  C0 ;

1 ð1Þ

C0

 Z ðtÞ  1;

0\p\ þ 1:

ð4:2:19Þ ð4:2:20Þ

ð4:2:21Þ

Hence, from lemma 4.2.1, it follows ð1Þ

kDx k  C0 ; ð2Þ

ð4:2:22Þ

ð1Þ

ð2Þ

kDxx k  C0 þ C0 ðkvkH 1 þ kv0 kH 1 Þ  C0 :

ð4:2:23Þ

A straightforward calculation gives uxx ¼ I1 þ I2 þ I3

ð4:2:24Þ

where


Z

I1 ¼ Dxx ðx; tÞZ ðtÞ 1 þ 0

Z

t

 hðx; sÞ ds ; Dðx; sÞZ ðsÞ

 Z 1 ðsÞ hx ðx; sÞD1 ðx; sÞ 0 Dx ðx; sÞhðx; sÞD2 ðx; sÞ ds; t

I2 ¼ 2Dx ðx; tÞZ ðtÞ

Z

ð4:2:25Þ

ð4:2:26Þ

 Z 1 ðsÞ hxx ðx; sÞD1 ðx; sÞ  2hx ðx; sÞDx ðx; sÞD 2 ðx; sÞ 0 ð4:2:27Þ hðx; sÞDxx ðx; sÞD2 ðx; sÞ þ 2hðx; sÞDx2 ðx; sÞD3 ðx; sÞ ds:

I3 ¼Dðx; tÞZ ðtÞ

t

Thus by lemmas 4.2.1, 4.2.2 and (4.2.3), (4.2.18) and (4.2.20)–(4.2.23), we get
Z t 2 ! Z 1 ð2Þ 2 2 2 1 1 kI1 k  2 kDxx Z k þ jDxx ðx; tÞj hðx; sÞD ðx; sÞZðtÞZ ðsÞds dx  C0 ; 0

0

ð4:2:28Þ ð2Þ

kI2 k2  C0 kDx k2


Z 0

t

Z ðtÞZ 1 ðsÞds

2

ð2Þ

 C0 ;

ð4:2:29Þ

Maximal Attractor for the One-Dimensional Equations

kI3 k

2

ð1Þ  C0

Z

Z

1

t

Z ðtÞZ 2

0

ð1Þ þ C0 ð1Þ

þ C0

Z 0

0 1Z t

2

Z

t

ðsÞds 0

Z ðtÞZ

1

Z 1
Z 0

ðh2xx ðx; sÞ þ h2x ðx; sÞÞdsdx

Z

t

ðsÞds

0

0

t

Z ðtÞZ 1 ðsÞds

73

2 Dxx ðx; sÞZ ðtÞZ 1 ðsÞdsdx

2 dx

0

ð2Þ

 C0 :

ð4:2:30Þ

Combining (4.2.28)–(4.2.30) with (4.2.24),we can derive ð2Þ

kuxx ðtÞk  C0 ; 8t [ 0:

ð4:2:31Þ

Thus (4.2.10) follows from (4.2.31), (4.2.18), (6.2.3) and lemma 4.2.1. h Similarly as in lemma 4.2.3, the continuity of SðtÞ with respect to ðu0 ; v0 ; h0 Þ in h H 2 follows. Thus, the proof of theorem 4.2.1 is complete.

4.3

ð1Þ

Existence of an Absorbing Set in Hb

ð1Þ

In this section, we shall prove the existence of an absorbing set in Hb . We always ð1Þ

assume in this section that initial data belong to a bounded set of Hb . As mentioned in ð1Þ

section 4.1, we first need to prove that the orbit starting from any bounded set in Hb ð1Þ

will reenter Hb after a finite time which should be uniform with respect to all orbits starting from that bounded set. The next two lemmas have been proved in [106, 139]. ð1Þ

Lemma 4.3.1. If ðu0 ; v0 ; h0 Þ 2 Hb , then Z 1 Z b3  uðx; tÞdx ¼ 0

b5 

Z 1
0

Z 

1

u0 ðxÞdx  b4 ; 8t [ 0;

1

¼ 0

ð4:3:2Þ

Z t Z 1
0

Z

ð4:3:1Þ

0

  Z 1
v02 v2 hþ h0 þ ðx; tÞdx ¼ ðxÞdx  b2 ; 8t [ 0; 2 2 0

ðln h þ ln uÞðx; tÞdx þ

0

1

0

 h2x vx2 þ ðx; sÞdxds uh2 uh

ðln h0 þ ln u0 ÞðxÞdx   b1 ; 8t [ 0:

ð4:3:3Þ

Attractors for Nonlinear Autonomous Dynamical Systems

74

ð1Þ

Lemma 4.3.2. If ðu0 ; v0 ; h0 Þ 2 Hb , then  Z t Z 1
2 hx vx2 þ ðx; sÞdxds  Cb ; uh2 uh 0 0

8t [ 0;

ð4:3:4Þ

kvðtÞk  Cb ; 8t [ 0; 0\Cb1 

Z

1

ð4:3:5Þ

hc ðx; tÞdx  Cb ; c 2 ½0; 1; 8t [ 0:

ð4:3:6Þ

0 ð1Þ

Lemma 4.3.3. If ðu0 ; v0 ; h0 Þ 2 Hb , then Cb1  uðx; tÞ  Cb ; 8ðx; tÞ 2 ½0; 1  ½0; þ 1Þ:

ð4:3:7Þ

Proof. It follows from lemmas 4.2.2, 4.3.1 and 4.3.2 that jBðx; tÞj  Cb ; Cb1  Dðx; tÞ  Cb ; 8ðx; tÞ 2 ½0; 1  ½0; þ 1Þ;

Cb1


Z

1



1 Z u0 ðxÞdx

0

1

ðv 2 þ hÞðx; sÞdx  Cb ; 8s  0;

ð4:3:8Þ

ð4:3:9Þ

0 1

eCb t  Z ðtÞ  eCb t ; 8t [ 0; 1

eCb ðtsÞ  Z ðtÞZ 1 ðsÞ  eCb

ðtsÞ

;

ð4:3:10Þ

t  s  0:

ð4:3:11Þ

Using lemma 4.3.2, we know that for any t  0, there exists a point aðtÞ 2 ½0; 1 such that Z 1 Cb1  hðx; tÞdx ¼ hðaðtÞ; tÞ  Cb : ð4:3:12Þ 0

Therefore,
Z jh1=2 ðx; tÞ  h1=2 ðaðtÞ; tÞj  0

1

h2x dx uh2

1=2
Z

1=2

1

uhdx 0

1

 Cb2 V 1=2 ðtÞMu1=2 ðtÞ ð4:3:13Þ

with V ðtÞ ¼

R1

h2x 2

0 uh

dx and Mu ðtÞ ¼ maxx2½0;1 uðx; tÞ: Thus, for all x 2 ½0; 1,

Cb  Cb Mu ðtÞV ðtÞ  hðx; tÞ  Cb þ Cb Mu ðtÞV ðtÞ:

ð4:3:14Þ

Maximal Attractor for the One-Dimensional Equations It follows from (4.2.6) and (4.3.8)–(4.3.14) that Z t 1 1 uðx; tÞ  Cb eCb t þ Cb hðx; sÞeCb ðtsÞ ds:

75

ð4:3:15Þ

0

Therefore, Z

t

Mu ðtÞ  Cb þ Cb

Mu ðsÞV ðsÞds:

ð4:3:16Þ

0

Using Gronwall’s inequality and (6.3.6), we can obtain
Z t  Mu ðtÞ  Cb exp Cb V ðsÞds  Cb :

ð4:3:17Þ

0

On the other hand, by (4.2.6) and (4.3.8)–(4.3.11), we have uðx; tÞ  Cb1 eCb t :

ð4:3:18Þ

In the sequel, we will use a contradiction argument to prove that uðx; tÞ  Cb1 ; 8ðx; tÞ 2 ½0; 1  ½0; þ 1Þ:

ð4:3:19Þ

Indeed, if the above assertion does not hold, then there exists a sequence of ð1Þ solutions ðun ; vn ; hn Þ with initial data ðun0 ; vn0 ; hn0 Þ 2 Hb converging weakly in H 1 , ð1Þ

strongly in C ½0; 1 to ðu0 ; v0 ; h0 Þ 2 Hb such that for the corresponding solution ðu; v; hÞ to ðu0 ; v0 ; h0 Þ, inf x2½0;1;t  0 u ¼ 0: Thus there exists ðxn ; tn Þ 2 ½0; 1  ½0; þ 1Þ such that as n ! þ 1; uðxn ; tn Þ ! 0:

ð4:3:20Þ

If the sequence tn has a subsequence, still denoted by tn, converging to infinity, then by the result on the asymptotic behaviour in lemma 4.2.2, as n ! þ 1; Z 1 uðxn ; tn Þ ! u0 ðxÞdx  b3 [ 0 ð4:3:21Þ 0

which contradicts (4.3.20). If the sequence tn is bounded, i.e., there exists a constant M [ 0, independent of n, such that 0\tn  M , then by (4.3.18), uðxn ; tn Þ  Cb1 eCb tn  Cb1 eCb M [ 0 which again contradicts (4.3.20). Thus the proof is complete. The next lemma concerns the boundedness of h from below.

ð4:3:22Þ h

ð1Þ

Lemma 4.3.4. If ðu0 ; v0 ; h0 Þ 2 Hb , then Cb1  hðx; tÞ; 8ðx; tÞ 2 ½0; 1  ½0; þ 1Þ:

ð4:3:23Þ

76

Attractors for Nonlinear Autonomous Dynamical Systems

Proof. Let w ¼ 1h : Then equation (4.1.3) can be rewritten as wt ¼ ðqwx Þx  ½2qhwx2 þ qw 2 ðvx  h=2Þ2  þ q=4

ð4:3:24Þ

with q ¼ u1 : Multiplying (4.3.24) by 2rw 2r1 with r being an arbitrary natural number, integrating the result over X ¼ ð0; 1Þ; noting that the expression in the squarebracket is nonnegative and using Hölder inequality, we can obtain Z 1 d kwðtÞk kwðtÞk2r1  1=4 qw 2r1 dx; 2r 2r L L dt 0 ð4:3:25Þ  Cb kwðtÞk2r1  Cb kqk 2r kwðtÞk2r1 2r 2r ; L

L

L

which yields, by taking r ! þ 1; that kwðtÞkL1  k1=h0 kL1 þ Cb t  Cb ð1 þ tÞ:

ð4:3:26Þ

Thus, for all x 2 ½0; 1; t  0, hðx; tÞ 

1 : Cb ð1 þ tÞ

ð4:3:27Þ

Noting the asymptotic behaviour of h in lemmas 4.2.1, 4.3.1 and (4.3.27), similarly as in the proof of (4.3.19), we can easily derive (4.3.23). h ð1Þ

Lemma 4.3.5. For initial data belonging to a bounded set of Hb there is t0 [ 0 depending only on boundedness of this bounded set such that for all t  t0 , x 2 ½0; 1, b5  hðx; tÞ  2b2 ; 2

b3  uðx; tÞ  2b4 : 2

ð4:3:28Þ

Proof. We derive from (4.2.4) and (4.3.2) that as t ! þ 1,



 Z 1 Z 1
Z 1



v02

hðx; tÞdx ! h0 þ hdx

dx; h 

1 ! 0: 2 0 0 0 L

ð4:3:29Þ

Now we use a contradiction argument to prove (4.3.28). Suppose that it is not true. Then there exists a sequence tn ! þ 1 such that for all x 2 ½0; 1, sup hðx; tn Þ [ 2b2 ;

ð4:3:30Þ ð1Þ

where sup is taken for all initial data in a given bounded set of Hb . Then similarly as for the proof of lemma 4.3.3, there is ðu0 ; v0 ; h0 Þ belonging to this bounded set such that for the corresponding solution ðu; v; hÞ, we have hðx; tn Þ  2b2 ; 8x 2 ½0; 1:

ð4:3:31Þ

This contradicts (4.3.29) and (4.3.2). Similarly, we can derive other parts of (4.3.28). Thus the proof is complete. h

Maximal Attractor for the One-Dimensional Equations

77

It follows from lemmas 4.3.1 and 4.3.5 that for initial data belonging to a given ð1Þ ð1Þ bounded set of Hb , the orbit will reenter Hb after a finite time. Next, we shall ð1Þ

prove that there is an absorbing set in Hb . Let Eðu; v; SÞ ¼

v2 @e @e ðu; SÞðu  uÞ  ðu; SÞðS  SÞ; ð4:3:32Þ þ eðu; SÞ  eðu; SÞ  @u @S 2 S ¼ log h þ log u;

eðu; SÞ ¼ h ¼ hðu; SÞ ¼

expðSÞ ; u

(entropy),

ð4:3:33Þ

(internal energy),

ð4:3:34Þ

where S; u; and h are constants and defined as follows. Z 1 Z S ¼ log h þ log u; u ¼ uðx; tÞdx ¼ 0

1

u0 ðxÞdx;

ð4:3:35Þ

0

  Z 1
Z 1
v02 v2 hþ h0 þ b5  h ¼ ðx; tÞdx ¼ ðxÞdx  b2 : 2 2 0 0

ð4:3:36Þ

Since we assume that initial data ðu0 ; v0 ; h0 Þ belong to an arbitrarily bounded set ð1Þ of Hb , there is a positive constant B such that kðu0 ; v0 ; h0 ÞkH 1  B. We use Cb;B to denote universal positive constants depending on B and bi ; ði ¼ 1; :::; 5Þ. Then we have the next lemma. Lemma 4.3.6. The following inequalities hold. v2 1 v2 ðju  uj2 þ jS  Sj2 Þ  Eðu; v; SÞ  þ þ Cb;B ðju  uj2 þ jS  Sj2 Þ: ð4:3:37Þ Cb 2 2

Proof. Using the mean value theorem and (4.3.32), we know that there exists a e Þ between ðu; SÞ and ðu; SÞ such that point ðe u; S
2 v2 1 @2e e Þðu  uÞðS  SÞ e Þðu  uÞ2 þ 2 @ e ðe u; S Eðu; v; SÞ ¼ þ ðe u; S 2 2 @u @u@S 2  @2e e ÞðS  SÞ2 þ ðe u ; S @S 2
v2 1 e e 2e u 3 e S ðu  uÞ2  2e þ ¼ u 2 e S ðu  uÞðS  SÞ 2 2  2 1 e S ð4:3:38Þ e þ u e ðS  SÞ ;

Attractors for Nonlinear Autonomous Dynamical Systems

78

where e ¼ ku þ ð1  kÞu; 0  k  1; u

ð4:3:39Þ

e ¼ kS þ ð1  kÞS; 0  k  1: S

ð4:3:40Þ

It follows from lemma 4.2.1 that khkL1  Cb;B . Thus we deduce from lemma 4.3.3 that 1 e  Cb ; u Cb

e je S j  Cb;B ;

  2  @ e e Þðu  uÞ2   Cb;B ðu  uÞ2 ;  ðe u ; S  @u 2  2     @ e  e    Cb;B ðu  uÞ2 þ ðS  SÞ2 ; ðe u ; S Þðu  uÞðS  SÞ @u@S   2  @ e  2 e  u ; S ÞðS  SÞ   Cb;B ðS  SÞ2 ; @S 2 ðe

ð4:3:41Þ

ð4:3:42Þ

ð4:3:43Þ

ð4:3:44Þ

which, together with (4.3.38), yield Eðu; v; SÞ 

  v2 þ Cb;B ðu  uÞ2 þ ðS  SÞ2 : 2

ð4:3:45Þ

On the other hand, we can derive from lemmas 4.3.3 and 4.3.4 that S e  Cb ; e e  Cb : S

ð4:3:46Þ

Thus, it follows from Young’s inequality that  2  3 3 1 1 u e ðu  uÞðS  SÞ  u e ðu  uÞ2 þ u e ðS  SÞ2 4 3

ð4:3:47Þ

and it follows from (4.3.38) that v 2 1 3 e 1 1 e e e S ðu  uÞ2 þ u e e S ðS  SÞ2 þ u 4 6 2  v2 1  þ  ðu  uÞ2 þ ðS  SÞ2 : Cb 2

Eðu; v; SÞ 

Thus the proof is complete.

ð4:3:48Þ h

Lemma 4.3.7. There exists a positive constant a1 ¼ a1 ðCb;B Þ [ 0 such that for any fixed a 2 ð0; a1 ; the following estimates hold.

Maximal Attractor for the One-Dimensional Equations eat ðkvðtÞk2 þ kuðtÞ  uk2 þ khðtÞ  hk2 þ kux k2 Þ Z t 1 1 þ eas ðkh2 ux k2 þ kh2 qx k2 þ khx k2 þ kvx k2 ÞðsÞds  Cb;B ; 8t [ 0:

79

ð4:3:49Þ

0

Proof. Using q ¼ u1 and equations (4.1.1)–(4.1.3), we easily verify that ðq; v; SÞ satisfies
 v2 hþ þ ðqhv  qhx  qvvx Þx ¼ 0; ð4:3:50Þ 2 t St  ðqhx =hÞx  qðhx =hÞ2  qvx2 =h ¼ 0: Since u t ¼ 0; ht ¼ 0, we have


qh 2 h2x 1 vx þ Et ðq ; v; SÞ þ ¼ qvvx þ ð1  h=hÞqhx  ðqh  qhÞv x ; h h ½ðqx =qÞ2 =2 þ qx v=qt þ hq2x =q ¼ qvx2  qx hx  ðqvvx Þx

ð4:3:51Þ

ð4:3:52Þ

ð4:3:53Þ

with q ¼ 1=u: Multiplying (4.3.52), (4.3.53) by eat ; geat respectively, and then adding the results up, we get  @  at e ðE þ gðqx =qÞ2 =2 þ gqx v=qÞ þ eat hqðvx2 þ h2x =hÞ=h @t þ geat ðhq2x =q þ qx hx  qvx2 Þ ¼ aeat ðE þ gðqx =qÞ2 =2 þ gqx v=qÞ þ eat ðð1  gÞqvvx þ ð1  h=hÞqhx  ðqh  qhÞvÞx :

ð4:3:54Þ

Integrating (4.3.54) over Qt ¼ ½0; 1  ½0; t, using lemmas 4.3.3, 4.3.4 and 4.3.6, Young’s inequality and boundary conditions (4.1.4), we conclude Z tZ 1 Z 1  L eat E þ gðqx =qÞ2 =2 þ gqx v=q dx þ eas hqðvx2 þ h2x =hÞ=hðx; sÞdxds 0

þg

Z tZ

0

1

e 0

0

as

ðhq2x =q þ qx hx



0

qvx2 Þðx; sÞdxds


2  v 2 2 2  Cb;B þ a þ Cb;B ðju  uj þ jS  Sj Þ þ g=2ðqx =qÞ þ gqx v=q dxds e 2 0 0  Z tZ 1
v2 2 2 2 as  Cb;B þ a e ð1 þ gÞ þ Cb;B ðju  uj þ jS  Sj Þ þ gðqx =qÞ dxds: 2 0 0 Z tZ

1

as

ð4:3:55Þ

Attractors for Nonlinear Autonomous Dynamical Systems

80

On the other hand, using Young’s inequality and lemma 4.3.6, we can deduce Z 1
 1  2 2 2 2 Le ð1=2  gÞv þ ðu  uÞ þ ðS  SÞ þ gðqx =qÞ =4 dx Cb 0 Z tZ 1  þ eas ðh=h  gÞqvx2 þ ðh=h  g=2Þqh2x =h þ ghq2x =2q dxds at

0



1 at e Cb

0

Z 1

1 þ Cb;B

0

 v 2 þ ðu  uÞ2 þ ðS  SÞ2 þ gq2x dx

Z tZ 0

0

1

eas ðvx2 þ h2x þ hq2x þ hux2 Þdxds;

ð4:3:56Þ

where we take g so small that 0\g\1=2; and h=h  g  b5 =Cb;B  g [ 0: Using the mean value theorem, we have S S ¼

1 1 ðh  hÞ þ ðu  uÞ; h1 u1

ð4:3:57Þ

where C1b  minðu; uÞ  u1  maxðu; uÞ  Cb ; C1b  minðh; hÞ  h1  maxðh; hÞ  Cb;B : By Poincaré’s inequality, we get kS  Sk  Cb ðku  uk þ kh  hkÞ  Cb ðkux k þ khx k þ kvx kÞ;

ð4:3:58Þ

and kh  hk  Cb;B ðkS  Sk þ ku  ukÞ:

ð4:3:59Þ

Thus it follows from (4.3.58), (4.3.59) and lemma 4.3.3 that eat ðkvk2 þ ku  uk2 þ kS  Sk2 þ kqx k2 þ kux k2 Þ Z t þ eas ðkvx k2 þ khx k2 þ kh1=2 qx k2 þ kh1=2 ux k2 Þds 0 Z t 0 a eas ðkvx k2 þ khx k2 þ kux k2 Þds:  Cb;B þ Cb;B

ð4:3:60Þ

0

Using lemma 4.3.4, we know that kux k2  Cb kh1=2 ux k2 ;

ð4:3:61Þ

kqx k2  Cb kh1=2 qx k2 :

ð4:3:62Þ

Maximal Attractor for the One-Dimensional Equations

81

Thus, inserting (4.3.61) and (4.3.62) into (4.3.60) yields eat ðkvk2 þ ku  uk2 þ kS  Sk2 þ kqx k2 þ kux k2 Þ Z t 1 1 þ eas ðkvx k2 þ khx k2 þ kh2 qx k2 þ kh2 ux k2 Þds 0 Z t 1 0 eas ðkvx k2 þ khx k2 þ kh2 ux k2 Þds;  Cb;B þ Cb;B a

ð4:3:63Þ

0

which implies the existence of a positive constant a1 ¼ a1 ðCb;B Þ  a 2 ð0; a1 ; it holds that

1 0 2Cb;B

so that when

eat ðkvk2 þ ku  uk2 þ kS  Sk2 þ kqx k2 þ kux k2 Þ Z t 1 1 þ eas ðkvx k2 þ khx k2 þ kh2 qx k2 þ kh2 ux k2 Þds 0

ð4:3:64Þ

 Cb;B ;

h

which, together with (4.3.59), yields the desired result.

Lemma 4.3.8. There exists a positive constant a2 ¼ a2 ðCb;B Þ  a1 such that for any fixed a 2 ð0; a2 ; it holds that Z 2

t

2

e ðkvx ðtÞk þ khx ðtÞk Þ þ at

eas ðkvxx k2 þ khxx k2 ÞðsÞds  Cb;B ;

8t [ 0;

ð4:3:65Þ

0

which, together with the previous lemma, implies that when a 2 ð0; a2 ; it holds that kðu  u; v; h  hÞkH 1  Cb;B eat ; 8t [ 0:

ð4:3:66Þ

Proof. We know that system (4.1.1)–(4.1.3) can be rewritten as follows qt þ q2 vx ¼ 0;

ð4:3:67Þ

vt  ðqvx Þx þ ðqhÞx ¼ 0;

ð4:3:68Þ

ht  ðqhx Þx þ qhvx  qvx2 ¼ 0:

ð4:3:69Þ

Multiplying (4.3.68) and (4.3.69) by eat vxx and eat hxx respectively, then integrating them over Qt , and adding the resultants up, by Young’s inequality, the imbedding theorem and lemma 4.3.3, we get

Attractors for Nonlinear Autonomous Dynamical Systems

82

Z tZ 1 1 at 2 2 2 e ðkvx k þ khx k Þ þ eas qðvxx þ h2xx Þðx; sÞdxds 2 0 0 Z t  Cb;B þ a=2 ðkvx k2 þ khx k2 Þeas ds þ

0

Z tZ 0

1 0

eas ðvxx ðqhÞx  qhvx hxx  qvx2 hxx  qx vx vxx  qx hx hxx Þdxds Z

t

 Cb;B þ a=2 þ

Z tZ

0

Z tZ 0

ðkvx k2 þ khx k2 Þeas ds þ 3=4

1 0

0

1

0

2 eas qðvxx þ h2xx Þdxds

eas q1 ðh2 q2x þ q2 h2x þ q2 h2 vx2 þ q2 vx4 þ q2x vx2 þ q2x h2x Þdxds

Z t Z tZ 1 2  Cb;B þ a=2 ðkvxx k2 þ khxx k2 Þeas ds þ 3=4 eas qðvxx þ h2xx Þdxds 0 0 0 Z t  1 2 2 2 3 as e kh2 qx k þ khx k þ kvx k þ kvx k kvxx k þ kvx kkvxx kkqx k2 þ Cb;B 0  þ kqx k2 khx kkhxx k ds Z t Z tZ 1 2 ðkvxx k2 þ khxx k2 Þeas ds þ 3=4 eas qðvxx þ h2xx Þdxds  Cb;B þ a 0

þ Cb e

Z tZ 0

þ Cb;B =e

Z

0

1

0 t

0

2 eas qðvxx þ h2xx Þdxds

 1  eas kh2 qx k2 þ khx k2 þ kvx k2 ds:

ð4:3:70Þ

0

Choosing e small enough, and using (4.3.70), lemmas 4.3.3 and 4.3.7, we can derive Z t 2 2 at e ðkvx k þ khx k Þ þ eas ðkvxx k2 þ khxx k2 Þds 0 Z t 00 ð4:3:71Þ  Cb;B þ Cb;B a ðkvxx k2 þ khxx k2 Þeas ds: 0

00 Þ, we get Choosing a in (4.3.71) small enough such that 0\a  minða1 ; Cb;B Z t eat ðkvx k2 þ khx k2 Þ þ eas ðkvxx k2 þ khxx k2 Þds  Cb;B ; ð4:3:72Þ 0

which, together with lemma 4.3.8, yields (4.3.65). Thus the proof is complete. h Finally, we conclude the following result on the existence of an absorbing ball in ð1Þ Hb . Theorem 4.3.9. Let

R1 ¼ R1 ðbÞ ¼ 4ðb22 þ b24 Þ and ð1Þ

v; hÞkH ð1Þ  R1 g. Then B1 is an absorbing ball in Hb .

ð1Þ

B1 ¼ fðu; v; hÞ 2 Hb ; kðu;

Maximal Attractor for the One-Dimensional Equations

83

Proof. From lemmas 4.3.1 and 4.3.8, it follows that for any initial data belonging to a bounded set with kðu0 ; v0 ; h0 ÞkH ð1Þ  B, there is t1 depending on B and b such that when t  t1 ; kðu; v; hÞðtÞk2H ð1Þ ¼ kuðtÞk2H 1 þ kvðtÞk2H 1 þ khðtÞk2H 1   2  2 u 2 þ h þ ku  uk2H 1 þ kvk2H 1 þ kh  hk2H 1  2ðb22 þ b24 Þ þ Cb;B ðbÞeat  R21 : h

The proof is thus complete.

4.4

ð2Þ

Existence of an Absorbing Set in Hb

ð2Þ

In this section, we shall prove the existence of an absorbing set in Hb . We always ð2Þ

assume in this section that initial data belong to a bounded set in Hb , i.e., kðu0 ; v0 ; h0 ÞkH ð2Þ  B with B being any given positive constant. We first obtain the uniform estimates on H 2 norms of v and h. Lemma 4.4.1. There exists a positive constant a0 ¼ aðCb;B Þ  a2 ðCb;B Þ such that for any fixed a 2 ð0; a0 ; and for all t [ 0, khk2H 2 þ kvk2H 2  b22 þ Cb;B eat :

ð4:4:1Þ

Proof. Differentiating equation (4.1.2) with respect to t, then multiplying the resulting equation by vt eat, integrating the resultant over Qt , and using lemma 4.3.3, we arrive at Z t 1 at 1 2 e kvt ðtÞk þ eas kvxt k2 ds ~b 0 2 C Z tZ 1 Z t eas kvt k2 ds þ Cb ðvx2 þ jht jÞjvxt jeas dxds  Cb;B þ a=2 0 0 0 Z Z t Cb t ð4:4:2Þ eas kvxt k2 ds þ kht k2 eas ds:  Cb;B þ a a 0 0

From equation (4.1.3), we know that ð4:4:3Þ kht k  Cb;B ðkhxx k þ khx k þ kvx k þ kvxx kÞ:  Let a0 ¼ min a2 ðCb;B Þ; 2C1~ and let a be a positive constant such that a 2 ð0; a0 . 

b

Then we deduce from (6.4.2), (4.4.3) and lemma 4.2.1 that

Attractors for Nonlinear Autonomous Dynamical Systems

84 Z 2

e kvt ðtÞk þ at

t

Z

t

2

e kvxt k ds  Cb;B þ Cb;B;a as

0

kht k2 eas ds  Cb;B :

ð4:4:4Þ

0

By equation (4.1.2), we easily get kvxx k  Cb;B ðkux k þ khx k þ kvx k þ kvt kÞ  Cb;B ðkux k þ khx k þ kvx k þ kvxt kÞ which, together with (4.4.4), lemmas 4.3.7 and 4.3.8, yields Z t eat ðkvt ðtÞk2 þ kvxx ðtÞk2 Þ þ eas kvxt k2 ds  Cb;B :

ð4:4:5Þ

ð4:4:6Þ

0

Similarly, we can obtain Z 2

t

2

e ðkht ðtÞk þ khxx ðtÞk Þ þ at

eas khxt k2 ds  Cb;B :

ð4:4:7Þ

0

Thus, by lemma 4.3.8, (4.4.6) and (4.4.7), for any fixed a 2 ð0; a0 , we have kh  hk2H 2 þ kvk2H 2  Cb;B eat ;

8t [ 0:

ð4:4:8Þ

 b22 þ Cb;B eat ; 8t [ 0:

ð4:4:9Þ

It turns out that 2

khk2H 2 þ kvk2H 2  h þ Cb;B eat

Thus, the proof is complete.

h

Remark 4.4.1. Let t2 ¼ t2 ðCb;B Þ  maxðt1 ðCb;B Þ; a1 lnðb22 =Cb;B ÞÞ. Then estimate (4.4.1) implies that for any t  t2 ðCb;B Þ; khðtÞk2H 2 þ kvðtÞk2H 2  2b22 :

ð4:4:10Þ

Lemma 4.4.2. There exists a constant t3 depending only on b; B with t3  t2 such that when t  t3 , the following estimate holds. Z t kuðtÞk2H 1 þ kvðtÞk2H 2 þ khðtÞk2H 2 þ kvðtÞkL1 þ khðtÞkL1 þ kux k2 þ kvk2H 2 t3  2 2 2 2 2 ð4:4:11Þ þ khx k þ khxx k þ kvx kL1 þ kvxt k þ khxt k ðsÞds  3ðb22 þ 2b24 Þ: Proof. The assertion easily follows from lemma 4.3.8, (4.4.1), (4.4.4), (4.4.6) and (4.4.7). h Next, we shall estimate kuxx k. Lemma 4.4.3. For any given constant d [ 0, there exists a constant t4 depending only on b; B and d with t4  t3 such that when t  t4 , the following estimate holds,

Maximal Attractor for the One-Dimensional Equations kuxx k  d:

85 ð4:4:12Þ

Proof. Differentiating equation (4.1.2) with respect to x, then multiplyig the resulting equation by uxx , and integrating with respect to x, using equation (4.1.1), i.e., uxxt ¼ vxxx , we get Z 1 1d h 2 kuxx k2 þ uxx dx 2 dt 0 u  Z 1
2hx ux 2hux 2vxx ux vx uxx 2vx ux2 ¼ þ 2 þ þ vxt u þ hxx   uxx dx: ð4:4:13Þ u u u u2 u2 0 ~ b ¼ b5 . Then it follows from lemma 4.3.5 that when t  t2  t0 , Let C 4b 4

Z 0

1

h 2 ~ b kuxx k2 : u dx  C u xx

ð4:4:14Þ

On the other hand, by Young’s inequality, lemmas 4.3.8 and 4.4.1, there is t5  t2 such that when t  t5 , we have Z 1       ekuxx k2 þ Ce;b kvxt k2 ; v uu dx ð4:4:15Þ xt xx   0

Z   

  e 1 uxx hxx dx   kuxx k2 þ khxx k2 ; 2 2e

ð4:4:16Þ

  2hx ux e uxx dx   kuxx k2 þ Ce;b khx k2 ; 2 u

ð4:4:17Þ

  2hux e uxx dx   kuxx k2 þ Ce;b kux k2 ; 2 2 u

ð4:4:18Þ

  vxx ux e uxx dx   kuxx k2 þ Ce;b kvxx k2 ; 2 u

ð4:4:19Þ

 ~b vx 2  C kuxx k2 ; uxx dx   Cb;B eat kuxx k2  2 u 4

ð4:4:20Þ

  2vx ux2 e uxx dx   kuxx k2 þ Ce;b kvx k2 : 2 2 u

ð4:4:21Þ

1

0

Z   

1

0

Z   

1

0

Z   

1

0

Z   

1

0

Z   

1 0

Attractors for Nonlinear Autonomous Dynamical Systems

86

Taking e small enough, and using (4.4.13)–(4.4.21), we can derive that when t  t5 , ~b 1d C kuxx k2 þ kuxx k2  Cb ðkvxt k2 þ khxx k2 þ kux k2 þ kvxx k2 þ kvx k2 Þ: 2 dt 2 Now choose a is such that a  (4.4.5)–(4.4.7) that when t  t5 ,

~b C 2.

ð4:4:22Þ

It follows from (6.4.15), lemma 4.2.3 and

~

~

kuxx ðtÞk2  kuxx ðt5 Þk2 eCb ðtt5 Þ þ Cb;B eat  Cb;B eCb ðtt5 Þ þ Cb;B eat :

ð4:4:23Þ

Thus, we can derive from (4.4.23) that there exists t4  t5 such that when t  t4 , (4.4.12) holds. h Let R2 ¼ 2ðb2 þ 2b4 Þ. Then the following theorem follows immediately from lemmas 4.4.2 and 4.4.3. ð2Þ

Theorem 4.4.4. The ball B2 ¼ fðu; v; hÞ 2 Hb ; kðu; v; hÞk2H 2  R22 g is an absorbing ð2Þ

ball in Hb .

4.5

Proof of Theorem 4.2.1 ð2Þ

ð1Þ

Recall that we have shown the existence of absorbing balls in Hb and Hb , then following exactly the abstract framework established in [51], we can conclude the next lemma. Lemma 4.5.1. The set xðB2 Þ ¼

\ [

SðtÞB2 ;

ð4:5:1Þ

s0 t s

where the closures are taken with respect to the weak topology of H ð2Þ , is included in B2 and nonempty. It is invariant by SðtÞ, i.e., SðtÞxðB2 Þ ¼ xðB2 Þ;

8t [ 0:

ð4:5:2Þ

ð2Þ

Remark 4.5.1. If B a bounded set in Hb , we can also define xðBÞ by (4.5.1) and when B is nonempty, xðBÞ is also included in B2 ; nonempty and invariant. Since B2 is an absorbing ball, it is obvious that xðBÞ xðB2 Þ, which implies that xðB2 Þ is maximal in the sense of inclusion. Theorem 4.5.2. The set A2;b ¼ xðB2 Þ

ð4:5:3Þ

satisfies A2;b

is bounded and weakly closed in

ð2Þ

Hb ;

ð4:5:4Þ

Maximal Attractor for the One-Dimensional Equations

SðtÞA2;b ¼ A2;b ;

87

8t  0;

ð4:5:5Þ

lim d w ðSðtÞB; A2;b Þ ¼ 0:

ð4:5:6Þ

ð2Þ

for every bounded set B in Hb , t! þ 1

Moreover, it is the maximal set in the sense of inclusion that satisfies (4.5.4)– (4.5.6). Proof. For the proofs of lemma 4.5.1 and theorem 4.5.2, we only exactly follow the same arguments as in [51], pp. 387–389, using the facts that SðtÞ is continuous in ð2Þ ð1Þ ð2Þ ð1Þ Hb , Hb , respectively, Hb is compactly embedded in Hb , and B2 , B1 are ð2Þ

ð1Þ

h

absorbing balls in Hb , Hb , respectively. ð2Þ

Note that A2;b is called the universal attractor of SðtÞ in Hb . In order to discuss ð1Þ

the existence of an universal attractor in Hb , we first need to prove the next lemma. Lemma 4.5.3. For every t  0, the mapping SðtÞ is continuous on bounded sets of ð1Þ Hb for the topology of the norm in L2  L2  L2 . ð1Þ

Proof. Let ðu0i ; v0i ; h0i Þ 2 Hb ; ði ¼ 1; 2Þ, kðu0i ; v0i ; h0i ÞkH 1  R; ði ¼ 1; 2Þ, ðui ; vi ; hi Þ ¼ SðtÞðu0i ; v0i ; h0i Þ, and ðu; v; hÞ ¼ ðu1 ; v1 ; h1 Þ  ðu2 ; v2 ; h2 Þ. Subtracting the corresponding equations (4.1.1)–(4.1.3) satisfied by ðu1 ; v1 ; h1 Þ and ðu2 ; v2 ; h2 Þ, then multiplying the resulting equations by u; v; h, respectively, adding together and integrating the final resulting equations over ½0; 1, we obtain   Z 1
v 2 1d h2x 2 2 2 x kuk þ kvk þ khk þ þ dx 2 dt u1 u1 0 
 Z 1

v2x  h2 v1x þ v2x þ 1  h2 þ 1 uvx þ vx h ¼ u1 u2 u1 0

 2 h2 v2x  v2x h2x v1x 2 þ uh þ uhx  h dx: ð4:5:7Þ u1 u2 u1 u2 u1

Using lemma 4.3.3, Young’s inequality, the imbedding theorem and kðui ; vi ; hi ÞkH 1  R; ði ¼ 1; 2Þ; we deduce from (4.5.7) that d ðkuk2 þ kvk2 þ khk2 Þ þ Cb ðkvx k2 þ khx k2 Þ dt Cb ðkvx k2 þ khx k2 Þ þ CR ðkv1x k2L1 þ kv2x k2L1 þ kh2 k2L1 þ 1Þðkuk2 þ khk2 Þ; ð4:5:8Þ  2

where CR is a positive constant depending also on R. Then the conclusion of this lemma follows from the Gronwall’s inequality. h

Attractors for Nonlinear Autonomous Dynamical Systems

88

Now following again exactly the same arguments as those in [51], we can obtain ð1Þ the existence of an universal attractor in Hb . Theorem 4.5.4. The set A1;b ¼

\ [

SðtÞB1

ð4:5:9Þ

s0 ts ð1Þ

is the universal attractor in Hb . Here the closures are taken with respect to the weak topology of H ð1Þ . Remark 4.5.2. Noting that A2;b is bounded in H ð2Þ , we know that it is bounded in H ð1Þ and using the invariance property (4.5.5), we can conclude A2;b A1;b :

ð4:5:10Þ

On the contrary, if we knew that A1;b is bounded in H ð2Þ , then the opposite inclusion would hold.

4.6

Bibliographic Comments

In this section, we shall recall the results on global existence; uniqueness and asymptotic behaviour of solutions to problem (4.1.1)–(4.1.5) have been established in [72, 106]; see also [21, 70] and the references therein. We shall explain some mathematical difficulties in this chapter, which include the following four points: (1) From physical viewpoint, the special volume u and the absolute temperature h must be positive for all times, which will result in some severe mathematical difficulties. For example, we must work on incomplete metric spaces H ð1Þ and H ð2Þ , H ð2Þ  H ð1Þ with these constraints. (2) The nonlinear semigroup SðtÞ maps each H ð1Þ and H ð2Þ into itself. However, it follows from equations (4.1.2) to (4.1.3) that the semigroup SðtÞ cannot be extended to the closure of H ð1Þ and H ð2Þ . On the other hand, there indeed exist great differences between the existence of global solutions and maximal attractors, the requirement on completeness of spaces is needed. To overcome this non-compactness of spaces, we need to restrict ourselves to a sequence of closed subspaces of H ð1Þ and H ð2Þ . Consequently, it is very 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 bounded set; otherwise, there is no ground to talk about the existence of absorbing set and maximal attractor in this subspace. The proof of this point becomes an essential part of this chapter. (3) Two quantities, i.e., the total mass and energy are conserved (see also (4.3.1) and (4.3.2)), that is, we have

Maximal Attractor for the One-Dimensional Equations Z

1 0

Z

1

uðx; tÞdx ¼

u0 ðxÞdx;

8t [ 0

89

ð4:6:1Þ

0

and   Z 1
Z 1
v02 v2 CV h þ CV h0 ðxÞ þ ðxÞ dx: dx ¼ 2 2 0 0

ð4:6:2Þ

Thus, these two conservations indicate obviously that there can be no absorbing set for initial data varying in the whole space. Instead, we should rather consider our problem in a sequence of closed subspaces defined by some parameters. Therefore, one of key issues in this chapter is how to choose these closed subspaces. (4) The system (4.1.1)–(4.1.3) is a hyperbolic–parabolic coupled system. Consequently, in general the orbit is not compact. In order to prove the existence of a maximal (global) attractor, the known classical existence theory of global attractors (see, Temam [144]), will do not work. To this end, we have used an approach motivated by the ideas in [51, 112, 158, 159].

Chapter 5 Universal Attractors for a Nonlinear System of Compressible One-Dimensional Heat-Conducting Viscous Real Gas This chapter is concerned with the existence of universal (global) attractors for a nonlinear system of compressible one-dimensional heat-conducting viscous real gas in bounded domain X ¼ ð0; 1Þ using some new ideas and more delicate estimates. Note that the model in this chapter we shall deal with is a 1D nonlinear system of heat-conducting viscous real gas with more general constitutive relations which is more complicated than that of 1D system of compressible polytropic ideal gas in chapter 4. The content of this chapter is adopted from Qin [120] and the method used here is the weakly compact semigroup method, see theorem 1.2.24.

5.1

Main Results

This chapter investigates the existence of universal (global) attractor for a nonlinear one-dimensional compressible heat-conductive viscous real gas in bounded domain X ¼ ð0; 1Þ. We know that the Lagrangian form of the conservation laws of mass, momentum, and energy for a one-dimensional gas with the reference density q0 ¼ 1 is (see [62, 70, 94, 99, 105, 108])




v2 eþ 2

ut  vx ¼ 0;

ð5:1:1Þ

vt  rx ¼ 0;

ð5:1:2Þ

 ðrvÞx þ Qx ¼ 0;

ð5:1:3Þ

t

DOI: 10.1051/978-2-7598-2702-2.c005 © Science Press, EDP Sciences, 2022

Attractors for Nonlinear Autonomous Dynamical Systems

92

and the Clausius–Duhem inequality satisfies the second law of thermodynamics in the form
 Q  0: ð5:1:4Þ gt þ h x Here subscripts indicate partial differentiations, u; v; r; e; Q; g and h denote the specific volume, velocity, stress, internal energy, heat flux, specific entropy and absolute temperature, respectively. Here u; h and e may only take positive values. Consider the problem (5.1.1)–(5.1.3) in the region f0  x  1; t  0g under the initial conditions uðx; 0Þ ¼ u0 ðxÞ;

vðx; 0Þ ¼ v0 ðxÞ;

hðx; 0Þ ¼ h0 ðxÞ on

½0; 1;

ð5:1:5Þ

subject to the boundary conditions vð0; tÞ ¼ vð1; tÞ ¼ 0;

Qð0; tÞ ¼ Qð1; tÞ ¼ 0; 8t  0;

ð5:1:6Þ

or vð0; tÞ ¼ vð1; tÞ ¼ 0;

hð0; tÞ ¼ hð1; tÞ ¼ T0 :¼ const [ 0; 8t  0:

ð5:1:7Þ

For a one-dimensional homogeneous real gas, the constitutive relations of e; r; g and Q take the following forms: (see [62, 105]) e ¼ eðu; hÞ;

r ¼ rðu; h; vx Þ; g ¼ gðu; hÞ; Q ¼ Qðu; h; hx Þ;

ð5:1:8Þ

which in order to be consistent with (5.1.4), must satisfy rðu; h; 0Þ ¼ Wu ðu; hÞ;

gðu; hÞ ¼ Wh ðu; hÞ;

ðrðu; h; wÞ  rðu; h; 0ÞÞw  0;

Qðu; h; gÞg  0;

ð5:1:9Þ ð5:1:10Þ

where W ¼ e  hg is the Helmholtz free energy function. Obviously, an ideal gas takes the form, e ¼ CV h;

r ¼ R

h vx hx þ l ; Q ¼ k ; u u u

ð5:1:11Þ

with some positive constants CV ; R; l and k. As it is known, the constitutive equations of a real gas are well approximated within moderate ranges of u and h by the model of an ideal gas (5.1.11). However, under very high temperatures and densities, (5.1.11) becomes inadequate. Thus a more realistic model would be a linearly viscous gas (or Newtonian fluid) rðu; h; vx Þ ¼ pðu; hÞ þ

lðu; hÞ vx u

ð5:1:12Þ

satisfying Fourier’s law of heat flux Qðu; h; hx Þ ¼ 

kðu; hÞ hx u

ð5:1:13Þ

Universal Attractors for a Nonlinear Compressible System

93

where the internal energy e and the pressure p are coupled by the standard thermodynamical relation eu ðu; hÞ ¼ pðu; hÞ þ hph ðu; hÞ

ð5:1:14Þ

to comply with (5.1.4). We assume that e; p; r and k are C 3 functions on 0\u\ þ 1 and 0  h\ þ 1. Let q and r be two positive constants (exponents of growth) satisfying one of the following relations 0  r  1=3; 1=3\r\4=7;

ð5:1:15Þ

1=3\q; ð2r þ 1Þ=5\q;

ð5:1:16Þ

4=7  r  1;

ð5r þ 1Þ=9\q;

ð5:1:17Þ

1\r  13=3;

ð9r þ 1Þ=15\q

ð5:1:18Þ

13=3\r;

ð11r þ 3Þ=19\q:

ð5:1:19Þ

We assume that there exist positive constants m; p0 ; p1 k0 ; and for any u [ 0; there are positive constants N ðuÞ; p2 ðuÞ; p3 ðuÞ and k1 ðuÞ such that for any u  u and h  0 the following conditions hold. 0  eðu; 0Þ; mð1 þ hr Þ  eh ðu; hÞ  N ðuÞð1 þ hr Þ;

ð5:1:20Þ

p0 hr þ 1  upðu; hÞ  p1 ð1 þ hr þ 1 Þ;

ð5:1:21Þ

 p2 ðuÞ½l þ ð1  lÞh þ hr þ 1   pu ðu; hÞ   p3 ðuÞ½l þ ð1  lÞh þ hr þ 1 ;

l ¼ 0 or

1;

ð5:1:22Þ

jph ðu; hÞj  p3 ðuÞð1 þ hr Þ;

ð5:1:23Þ

k0 ð1 þ hq Þ  kðu; hÞ  k1 ðuÞð1 þ hq Þ;

ð5:1:24Þ

jku ðu; hÞj þ jkuu ðu; hÞj  k1 ðuÞð1 þ hq Þ:

ð5:1:25Þ

For the viscosity lðu; hÞ, we assume that lðu; hÞ ¼ l0 [ 0; where l0 is a constant.

ð5:1:26Þ

Attractors for Nonlinear Autonomous Dynamical Systems

94

Now define two spaces as follows 1 Hþ ¼ fðu; v; hÞ 2 H 1 ½0; 1  H 1 ½0; 1  H 1 ½0; 1 : uðxÞ [ 0; hðxÞ [ 0; x 2 ½0; 1;

vjx¼0 ¼ vjx¼1 ¼ 0; hjx¼0 ¼ hjx¼1 ¼ T0 for ð5:1:7Þg and 2 Hþ ¼ fðu; v; hÞ 2 H 2 ½0; 1  H 2 ½0; 1  H 2 ½0; 1 : uðxÞ [ 0; hðxÞ [ 0; x 2 ½0; 1; vjx¼0 ¼ vjx¼1 ¼ 0; hx jx¼0 ¼ hx jx¼1 ¼ 0 for ð5:1:6Þ or hjx¼0 ¼ hjx¼1 ¼ T0 for ð5:1:7Þg

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. Let ( Z Hdi : ¼

i ðu; v; hÞ 2 H þ :

Z

1

d6 

1

ðEðu; hÞ þ v 2 =2Þdx  d1 ;

0

ðeðu; hÞ þ v 2 =2Þdx  d7 for ð5:1:6Þ; )

0

Z

1

d2 

udx  d3 ; d4  h  d5 ; d2 =2  u  2d3 ; i ¼ 1; 2;

0

where Eðu; hÞ ¼: Wðu; hÞ  Wð1; HÞ  Wu ð1; HÞðu  1Þ  Wh ðu; hÞðh  HÞ

ð5:1:27Þ

with H ¼ 1 for (5.1.6) or H ¼ T0 for (5.1.7), while di ði ¼ 1; . . .; 7Þ are any given constants satisfying d1 2 R; 0\d2 \d3 ; 0\d4 \d5 ; 0\d6 \d7

ð5:1:28Þ

with the following constraints 0\d4 \T0 \d5 for

ð5:1:7Þ;

ð5:1:29Þ

or 0\d4 \

min

n2½d2 ;d3 ;e2½d6 ;d7 

^ hðn; eÞ;

max

n2½d2 ;d3 ;e2½d6 ;d7 

^hðn; eÞ\d5 for ð5:1:6Þ:

ð5:1:30Þ

Here ^ h¼^ hðn; eÞ is the unique inverse function of the function e ¼ eðn; hÞ for any fixed n 2 ½d2 ; d3 , which is a monotone increasing function in e for any fixed h¼^ hðn; eÞ follows from our assumption (5.1.20). n 2 ½d2 ; d3 . The unique existence of ^ i ði ¼ 1; 2Þ: We Obviously, Hdi ði ¼ 1; 2Þ is a sequence of closed subspaces of H þ shall see later on that the first three constraints are invariant, while the last two constraints are not invariant, which are just introduced to overcome the difficulty i are incomplete. As stated above, it is very crucial to that the original spaces H þ prove that the orbit starting from any bounded set of Hdi will re-enter Hdi and stay there after a finite time.

Universal Attractors for a Nonlinear Compressible System

95

The notation in this chapter will be as follows: Lp ; 1  p  þ 1; W m;p ; m 2 N; H 1 ¼ W 1;2 ; H01 ¼ W01;2 denote the usual (Sobolev) spaces on ð0; 1Þ: In addition, k  kB denotes the norm in the space B; we also put k  k ¼ k  kL2 : We denote by C k ðI ; BÞ; k 2 N0 ; the space of k-times continuously differentiable functions from I R into a Banach space B, and likewise by Lp ðI ; BÞ; 1  p  þ 1 the corresponding Lebesgue spaces. Subscripts t and x denote ðiÞ the (partial) derivatives with respect to t and x, respectively. We use C0 ; i ¼ 1; 2 to i denote the universal constant depending only on the H norm of initial data, minx2½0;1 u0 ðxÞ and minx2½0;1 h0 ðxÞ. Cd (sometimes Cd0 ) denotes the universal constant ðiÞ

depending only on di ði ¼ 1; 2; . . .; 7Þ, but independent of initial data. Cd denotes the universal constant depending on both dj ðj ¼ 1; 2; . . .; 7Þ, H i norm of initial data, minx2½0;1 h0 ðxÞ and minx2½0;1 u0 ðxÞ. C denotes the generic absolute positive constant independent of d and initial data. 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., pðu; hÞ, and pðuðx; tÞ; hðx; tÞÞ and pðx; tÞ. Our main theorem reads as follows. Theorem 5.1.1. Assume that ( 5.1.8) and (5.1.12)–(5.1.26) hold, then the solution to problem (5.1.1)–(5.1.3), (5.1.5)–(5.1.6) or (5.1.1)–(5.1.3), (5.1.5), (5.1.7) defines a i i nonlinear C0 -semigroup SðtÞ on H þ ði ¼ 1; 2Þ, which maps H þ ði ¼ 1; 2Þ into itself. Moreover, for any di ði ¼ 1; 2; . . .; 7Þ satisfying (5.1.30)–(5.1.32), it possesses in Hdi a universal (maximal) attractor Ai;d ði ¼ 1; 2Þ: Remark 5.1.1. See section 5.6 of this chapter for the definition of (maximal) universal attractor. S Remark 5.1.2. The set Ai ¼ d1 ;...;d5 or d1 ;...;d7 Ai;d ði ¼ 1; 2Þ is a global noncompact i attractor in the metric space H þ in the following sense that it attracts any bounded i with constraints u  u; h  h with u; h being any given positive sets of H þ constants. Remark 5.1.3. Theorem 5.1.1 also holds for the polytropic viscous ideal gas (5.1.11) with the boundary conditions (5.1.7), while for the polytropic viscous ideal gas (5.1.11) with (5.1.6), the similar results can also be obtained.

5.2

i Nonlinear C0 -Semigroup on H þ ði ¼ 1; 2Þ

i As known in section 5.1, for any initial data ðu0 ; v0 ; h0 Þ 2 H þ ði ¼ 1; 2Þ, the results on global existence, uniqueness and asymptotic behaviour of solutions to problem (5.1.1)–(5.1.3), (5.1.5)–(5.1.6) or (5.1.1)–(5.1.3), (5.1.5), (5.1.7) have been established in [105]. It has been proved in [108] that the operator SðtÞ defined by the i ; ði ¼ 1; 2Þ, respectively. More precisely, solution is a nonlinear C0 -semigroup on H þ we have the following lemma.

Attractors for Nonlinear Autonomous Dynamical Systems

96

1 Lemma 5.2.1. Under the assumptions in theorem 5.1.1, for any ðu0 ; v0 ; h0 Þ 2 H þ , global solution ðuðtÞ; vðtÞ; hðtÞÞ to problem (5.1.1)–(5.1.3), (5.1.5)–(5.1.6) or 1 such (5.1.1)–(5.1.3), (5.1.5), (5.1.7) defines a nonlinear C0 -semigroup SðtÞ on H þ 1 1 that for any ðu0 ; v0 ; h0 Þ 2 H þ; SðtÞðu0 ; v0 ; h0 Þ ¼ ðuðtÞ; vðtÞ; hðtÞÞ 2 C ð½0; þ 1Þ; H þ Þ, 2 2 ut ; vt ; ht ; hx ; vx ; ux ; vxx ; hxx 2 L ð½0; þ 1Þ; L Þ. Moreover, ð1Þ

ð1Þ

on ½0; 1  ½0; þ 1Þ;

ð5:2:1Þ

ð1Þ

ð1Þ

on ½0; 1  ½0; þ 1Þ;

ð5:2:2Þ

0\1=C0  hðx; tÞ  C0

0\1=C0  uðx; tÞ  C0 Z kuðtÞk2H 1 þ khðtÞk2H 1 þ kvðtÞk2H 1 þ

t

½kux k2 þ kvk2H 2 þ khx k2H 1 þ kvt k2 þ kht k2 ðsÞds

0

ð1Þ

ð5:2:3Þ

 C0 ; 8t [ 0;

ð1Þ ð1Þ and there exist constants ~c1 ¼ ~c1 ðC0 Þ; C0 [ 0 such that for any fixed ~c 2 ð0; ~c1  and for any t [ 0, it follows ð1Þ

kðuðtÞ; vðtÞ; hðtÞÞ  ðu; 0; hÞk2H 1þ ¼ kSðtÞðu0 ; v0 ; h0 Þ  ðu; 0; hÞk2H þ1  C0 e~ct ; ð5:2:4Þ 1 which means that the semigroup SðtÞ is exponentially stable on H þ . Here Z 1 u¼ u0 ðxÞdx; h ¼ T0 for ð5:1:7Þ; ð5:2:5Þ 0

or for (5.1.6) h [ 0 is uniquely determined by Z

1

eðu; hÞ ¼ 0

ðeðu0 ; h0 Þ þ v02 =2ÞðxÞdx:

ð5:2:6Þ

The next lemma has been proved in [108]. 2 , Lemma 5.2.2. Under the assumptions in theorem 5.1.1, for any ðu0 ; v0 ; h0 Þ 2 H þ global solution ðuðtÞ; vðtÞ; hðtÞÞ to problem (5.1.1)–(5.1.3), (5.1.5)–(5.1.6) or 2 (5.1.1)–(5.1.3), (5.1.5), (5.1.7) defines a nonlinear C0 -semigroup SðtÞ on H þ 2 such that SðtÞðu0 ; v0 ; h0 Þ ¼ ðuðtÞ; vðtÞ; hðtÞÞ 2 C ð½0; þ 1Þ; H þÞ. In addition to lemma 5.2.1, we have hxt ; hxxx ; vxt ; vxxx 2 L2 ð½0; þ 1Þ; L2 ð0; 1ÞÞ and

Z kuðtÞk2H 2 þ khðtÞk2H 2 þ kvðtÞk2H 2 þ ð2Þ

 C0 ; 8t [ 0:

t

½kvxt k2 þ kvxxx k2 þ khxt k2 þ khxxx k2 ðsÞds

0

ð5:2:7Þ

Universal Attractors for a Nonlinear Compressible System

5.3

97

Existence of an Absorbing Set in Hd1

This section is concerned with the existence of an absorbing ball in Hd1 . We always assume in this chapter that initial data belong to a bounded set of Hd1 . Now, we have to show that the orbit starting from any bounded set in Hd1 will re-enter Hd1 and stay there after a finite time which should be uniform with respect to all orbits starting from that bounded set. Lemma 5.3.1. If ðu0 ; v0 ; h0 Þ 2 Hd1 , then, for any t [ 0, Z 1 Z 1 d2  uðx; tÞdx ¼ u0 ðxÞdx  d3 ; 0

Z

1

d6 

Z ðeðu; hÞ þ v =2Þðx; tÞdx ¼ 2

0

ð5:3:1Þ

0

0

1

ðeðu0 ; h0 Þ þ v02 =2ÞðxÞdx  d7 ; for ð5:1:6Þ; ð5:3:2Þ

Z

1

ðEðu; hÞ þ v 2 =2Þðx; tÞdx þ H

0

Z t Z 1
0

Z

1

¼ 0

0

 kðu; hÞh2x l0 vx2 þ dxds uh uh2

ðEðu0 ; h0 Þ þ v02 =2ÞðxÞdx  d1 :

ð5:3:3Þ

Proof. Estimates (5.3.1) and (5.3.2) have already been obtained in (5.1.43) and (5.1.44). Note that Wðu; hÞ ¼ eðu; hÞ  hgðu; hÞ is the Helmholtz free energy function. Recalling (5.2.1), the definition of E ¼ Eðu; hÞ, noting that eh ðu; hÞ ¼ hWhh ðu; hÞ, by (5.1.1)–(5.1.3), (5.1.8)–(5.1.10) and (5.1.12)–(5.1.14), we arrive at by a delicate calculation that    2    lv kðu; hÞh2x ðh  HÞkðu; hÞhx ¼ ðrvÞ @t Eðu; hÞ þ v 2 =2 þ H 0 x þ þ pð1; HÞv þ : ð5:3:4Þ x x uh uh uh2 x

Integrating (5.3.4) over Qt :¼ ð0; 1Þ  ð0; tÞ and using (5.1.6)–(5.1.7), we obtain  Z 1 Z t Z 1 2 l0 vx kðu; hÞh2x þ ðEðu; hÞ þ v 2 =2Þðx; tÞdx þ H dxds 2 uh uh 0 0 0 Z 1 ðEðu0 ; h0 Þ þ v02 =2Þdx; ¼ 0

which gives (5.3.3). Lemma 5.3.2. If ðu0 ; v0 ; h0 Þ 2 Hd1 , then, for any t [ 0,  Z 1 Z t Z 1
ð1 þ hq Þh2x vx2 r þ1 2 ðh þ v Þðx; tÞdx þ þ ðx; sÞdxds  Cd ; uh uh2 0 0 0

□

ð5:3:5Þ

Attractors for Nonlinear Autonomous Dynamical Systems

98

0\Cd1

Z 

1

hðx; tÞdx  Cd :

ð5:3:6Þ

0

Proof. By (5.1.9) and (5.1.12), we have Wuu ðu; HÞ ¼ pu ðu; HÞ [ 0 for any u [ 0. Therefore, by the Taylor theorem and (5.1.29), we obain Eðu; hÞ  Wðu; hÞ þ Wðu; HÞ þ ðh  HÞWh ðu; hÞ ¼ Wðu; HÞ  Wð1; HÞ  Wu ð1; HÞðu  1Þ Z 1 2 ð1  nÞWuu ð1 þ nðu  1Þ; HÞdn  0: ¼ ðu  1Þ 0

Thus Eðu; hÞ  Wðu; hÞ  Wðu; HÞ  ðh  HÞWh ðu; hÞ Z 1 ð1  sÞWhh ðu; h þ sðH  hÞÞds ¼ ðH  hÞ2 0 1

Z  mðH  hÞ

2

ð1  sÞf1 þ ½h þ sðH  hÞr g ds; h þ sðH  hÞ

0

i.e.,

(

Eðu; hÞ 

mHðh=H  logðh=HÞ  1Þ þ 2mHðh=H  logðh=HÞ  1Þ;

mH½ðHÞr hr  r

r þ1

 mH½ðHÞr þ 1h

r þ1



;

for r [ 0; for r ¼ 0;

 mHðh=H  logðh=HÞ  1Þ þ Cd0 hr þ 1  Cd which, combined with (5.3.3) and (5.1.24), gives Z 1 ½ðh=H  logðh=HÞ  1Þ þ hr þ 1 þ v 2 dx 0  Z t Z 1
ð1 þ hq Þh2x vx2 þ þ ðx; sÞdxds  Cd ; 8t [ 0: uh uh2 0 0

ð5:3:7Þ

Moreover, from (5.3.7) and Jensen’s inequality, we derive
Z 1  Z 1 h=Hdx  log h=Hdx  1  Cd ; 8t [ 0; 0

0

which leads to Z

1

r1 

hðx; tÞ=Hdx  r2 ;

ð5:3:8Þ

0

where ri ¼ ri ðdÞ ði ¼ 1; 2Þ are two positive roots of the equation y  log y  1 ¼ Cd . Thus (5.3.5) and (5.3.6) follow from (5.3.7)–(5.3.8). The proof is now complete. □

Universal Attractors for a Nonlinear Compressible System

99

Lemma 5.3.3. If ðu0 ; v0 ; h0 Þ 2 Hd1 , then h  h  h 0\Cd1  hðx; tÞ;

ð5:3:9Þ

8ðx; tÞ 2 ½0; 1  ½0; þ 1Þ;

ð5:3:10Þ

where h ¼ min½T0 ; minu2½d2 ;d3 ;e2½d6 ;d7  ^ hðu; eÞ and h ¼ max½T0 ; maxu2½d2 ;d3 ;e2½d6 ;d7  ^hðu; eÞ. Proof. We first prove that for (5.1.6), min

u2½d2 ;d3 ;e2½d6 ;d7 

^ hðu; eÞ  h 

max

u2½d2 ;d3 ;e2½d6 ;d7 

^hðu; eÞ:

ð5:3:11Þ

In fact, it follows from (5.3.1)–(5.3.2) that d6  e :¼ eðu; hÞ  d7 ; d2  u  d3 which yields that h ¼ ^ hðu; eÞ and (5.3.11). Thus (5.3.9) follows. We use the ideas in [159] and [108] to prove this lemma. In fact, if the conclusion in (5.3.10) does not hold, then there exists a sequence of solutions ðun ; vn ; hn Þ with initial data ðun0 ; vn0 ; hn0 Þ 2 Hd1 converging weakly in H 1 , strongly in C ½0; 1 to ðu0 ; v0 ; h0 Þ 2 Hd1 such that for the corresponding solution ðu; v; hÞ to ðu0 ; v0 ; h0 Þ, inf x2½0;1;t  0 h ¼ 0. Thus there is ðxn ; tn Þ 2 ½0; 1  ½0; þ 1Þ such that as n ! þ 1, hðxn ; tn Þ ! 0:

ð5:3:12Þ

If the sequence ftn g has a subsequence, denoted also by tn, converging to þ 1, then from lemma 5.3.1 and (5.3.9) that it follows that as n ! þ 1, hðxn ; tn Þ ! h  h [ 0 which contradicts (5.3.12). If the sequence ftn g is bounded, i.e., there exists a constant M [ 0, independent of n, such that for any n ¼ 1; 2; 3; . . .; 0\tn  M. Thus there exists a point ðx ; t Þ 2 ½0; 1  ½0; M  such that ðxn ; tn Þ ! ðx ; t Þ as n ! þ 1. On the other hand, by (5.3.12) and the continuity of solutions in lemma 5.2.1, we conclude that hðxn ; tn Þ ! hðx ; t Þ ¼ 0 as n ! þ 1, which contradicts (5.3.1). Thus the proof is complete. □ In what follows we will estimate the pointwise positive lower bound and upper bound for u. To this end, we need the following expression of u derived in [105]. Lemma 5.3.4. For each t  0, there exists a point x0 ¼ x0 ðtÞ 2 ½0; 1 such that the specific volume uðx; tÞ has the following representation:   Z t uðx; sÞpðx; sÞ 1 ds ; 8x 2 ½0; 1 uðx; tÞ ¼ Dðx; tÞZ ðtÞ 1 þ l0 0 Dðx; sÞZ ðsÞ

Attractors for Nonlinear Autonomous Dynamical Systems

100

where

(

Dðx; tÞ ¼ u0 ðxÞ exp " Z ðtÞ ¼ exp 

Z

l1 0

1 l0 u

x0 ðtÞ

Z tZ 0

Z

x

1

vðy; tÞdy 

x

0

1 v0 ðyÞdy þ u #

Z

1

Z u0 ðxÞ

0

x

!# v0 ðyÞdydx

;

0

ðv 2 þ upÞðy; sÞdyds

0

and u is defined in (5.2.5). Lemma 5.3.5. If ðu0 ; v0 ; h0 Þ 2 Hd1 , then 0\Cd1  uðx; tÞ  Cd ; 8ðx; tÞ 2 ½0; 1  ½0; þ 1Þ:

ð5:3:13Þ

Proof. Let Mu ðtÞ ¼ max uðx; tÞ; mu ðtÞ ¼ min uðx; tÞ; x2½0;1

x2½0;1

Mh ðtÞ ¼ max hðx; tÞ; mh ðtÞ ¼ min hðx; tÞ: x2½0;1

x2½0;1

By (5.1.21), (5.3.1), (5.3.5), (5.3.10) and lemma 5.3.4, we have 0\Cd1  Dðx; tÞ  Cd ;

8ðx; tÞ 2 ½0; 1  ½0; þ 1Þ;

expðCd ðt  sÞÞ  Z ðtÞZ 1 ðsÞ  expðCd1 ðt  sÞÞ; 0\Cd1

1  l0 u

Z

1

t  s  0;

ðup þ v 2 Þðx; sÞdx  Cd :

0

On the other hand, we have

 Z   x   m1 h hx dy  jh ðx; tÞ  h ðaðtÞ; tÞj  C    aðtÞ m

m


Z C 0

1

h2x ð1 þ hq Þ dx uh2

12
Z

1 0

uh2m dx 1 þ hq

12  CV 1=2 ðtÞMu1=2 ðtÞ;

where Z V ðtÞ ¼ 0

1

h2x ð1 þ hq Þ=uh2 dx;

and, by lemma 5.3.2, Z t Z V ðsÞds  Cd ; 0

0

1

m ¼ ðq þ r þ 1Þ=2

Z

1

h2m =ð1 þ hq Þdx  C 0

ð1 þ h1 þ r Þdx  Cd

ð5:3:14Þ ð5:3:15Þ

ð5:3:16Þ

Universal Attractors for a Nonlinear Compressible System

101

and for any t  0, there is a point aðtÞ 2 ½0; 1 such that Z 1 Cd1  hðaðtÞ; tÞ ¼ hðx; tÞdx  Cd : 0

Thus Cd00  Cd V ðtÞMu ðtÞ  h2m ðx; tÞ  Cd ð1 þ V ðtÞMu ðtÞÞ:

ð5:3:17Þ

Using lemmas 5.3.1–5.3.4, (5.3.14)–(5.3.16) and noticing that uðx; sÞpðx; sÞ  p1 ð1 þ hr þ 1 Þ  C ð1 þ h2m Þ  C ð1 þ V ðsÞMu ðsÞÞ; we obtain

"

Z

uðx; tÞ  Cd 1 þ "

t

# V ðsÞ expðCd ðt  sÞÞMu ðsÞds

0

Z

 Cd 1 þ

t

# V ðsÞMu ðsÞds ;

0

i.e.,

"

Z

Mu ðtÞ  Cd 1 þ

t

# V ðsÞMu ðsÞds :

ð5:3:18Þ

0

Thus it follows from Gronwall’s inequality and lemma 5.3.2 that Mu ðtÞ  Cd :

ð5:3:19Þ

By lemma 5.3.4 and (5.3.14)–(5.3.15), we have uðx; tÞ  Dðx; tÞZ ðtÞ  Cd1 expðCd tÞ:

ð5:3:20Þ

Similarly to the proof of (5.3.10) in lemma 5.3.3, using (5.3.20), we can prove uðx; tÞ  Cd1 ; 8ðx; tÞ 2 ½0; 1  ½0; þ 1Þ: □

The proof is thus complete.

Lemma 5.3.6. For initial data belonging to an arbitrary fixed bounded set B of Hd1 , there is t0 [ 0 depending only on boundedness of this bounded set B such that for all t  t0 , x 2 ½0; 1, d4  hðx; tÞ  d5 ; d2 =2  uðx; tÞ  2d3 :

ð5:3:21Þ

Proof. We use the ideas in [159] and [108] to prove this lemma. Suppose that the conclusion in lemma 5.3.6 does not hold. Then there is a sequence tn ! þ 1 such that for all x 2 ½0; 1, sup hðx; tn Þ [ d5

ð5:3:22Þ

Attractors for Nonlinear Autonomous Dynamical Systems

102

where sup is taken for all initial data in a given bounded set B of Hd1 . In the same manner as for the proof of lemma 5.3.3, there exists ðu0 ; v0 ; h0 Þ 2 B such that for the corresponding solution ðu; v; hÞ, we have hðx; tn Þ  d5 ; 8x 2 ½0; 1 which, with (5.2.4), yields h  d5 :

ð5:3:23Þ

This contradicts (5.2.3) or (5.2.4) and (5.3.9). Similarly, we can prove other parts of (5.3.21). □ Remark 5.3.1. From lemmas 5.3.1 and 5.3.6 we derive that for initial data in a given bounded set B of Hd1 , the orbit will re-enter Hd1 and stay there after a finite time. In the sequel, we shall prove the existence of an absorbing ball in Hd1 . Now, we introduce the density of the gas, q ¼ 1=u, then from (5.1.8)–(5.1.9) and (5.1.12)– (5.1.14) it follows that the entropy g ¼ gð1=q; hÞ satisfies @g=@q ¼ ph =q2 ;

@g=@h ¼ eh =h:

ð5:3:24Þ

Consider the transform A : ðq; hÞ 2 Dq;h ¼ fðq; hÞ : q [ 0; h [ 0g ! ðu; gÞ 2 ADq;h ;

ð5:3:25Þ

where u ¼ 1=q and g ¼ gð1=q; hÞ. Since the Jacobian j@ðu; gÞ=@ðq; hÞj ¼ eh =q2 h\0 on Dq;h , there is a unique inverse function h ¼ hðu; gÞ as the smooth function of ðu; gÞ 2 ADq;h . Therefore the functions e; p can be also regarded as the smooth functions of ðu; gÞ. Denote by e ¼ eðu; gÞ : eðu; hðu; gÞÞ ¼ eð1=q; hÞ;

p ¼ pðu; gÞ : pðu; hðu; gÞÞ ¼ pð1=q; hÞ:

Then from (5.1.8)–(5.1.9), (5.1.12)–(5.1.14) and (5.3.24)–(5.3.25), it is obvious that e; p and h satisfy eu ¼ p; eg ¼ h; pu ¼ ðq2 pq þ hp2h =eh Þ; pg ¼ hph =eh ; hu ¼ hph =eh ; hg ¼ h=eh :

ð5:3:26Þ

Define the following energy form Eðu; v; gÞ ¼

v2 @e @e ðu; gÞðu  uÞ  ðu; gÞðg  gÞ; þ eðu; gÞ  eðu; gÞ  @u @g 2

ð5:3:27Þ

where q ¼ 1=u; g ¼ gð1=q; hÞ and u; h are given in (5.2.5) and (5.2.6). Note that initial data ðu0 ; v0 ; h0 Þ belong to an arbitrarily bounded set B of Hd1 , then there is a 0 00 positive constant B such that kðu0 ; v0 ; h0 ÞkH 1  B. We now use Cd;B or Cd;B ; Cd;B to denote generic positive constants depending on B and di ; ði ¼ 1; . . .; 7Þ.

Universal Attractors for a Nonlinear Compressible System

103

Lemma 5.3.7. For any initial data ðu0 ; v0 ; h0 Þ 2 Hd1 , the unique generalized global solution ðuðtÞ; vðtÞ; hðtÞÞ to problem (5.1.1)–(5.1.3), (5.1.5)–(5.1.6) or (5.1.1)– (5.1.3), (5.1.5), (5.1.7) satisfies v2 v2 1 þ CB;d þ CB;d ðju  uj2 þ jg  gj2 Þ: ð5:3:28Þ ðju  uj2 þ jg  gj2 Þ  Eðu; v; gÞ  2 2 Proof. Using the mean value theorem, there exists a point ðe u; e g Þ between ðu; gÞ and ðu; gÞ such that Eðu; v; gÞ ¼

  v2 1 @2e @2e @2e 2 2 e e e ðe u ; g Þðu  þ ; ðe u ; g Þðu  uÞ þ 2 uÞðg  gÞ þ ðe u ; g Þðg  gÞ 2 @u 2 @u@g @g2 2

ð5:3:29Þ

where e ¼ k0 u þ ð1  k0 Þu; u

e g ¼ k0 g þ ð1  k0 Þg; 0  k0  1:

From lemmas 5.2.1, 5.3.3 and 5.3.5, we can derive that e  Cd ; 0\Cd1  h  CB ; 0\Cd1  u which implies

je g j  CB ;

  2  2   2   @ e @ e  @ e       ðe u; e g Þ þ  2 ðe u; e g Þ þ  u; e g Þ  CB;d : @u 2 ðe @u@g @g

ð5:3:30Þ

Thus it follows from (5.3.29) and (5.3.30) and the Cauchy’s inequality that Eðu; v; gÞ 

v2 þ CB;d ½ðu  uÞ2 þ ðg  gÞ2 : 2

ð5:3:31Þ

Moreover, from (5.3.26) we can deduce that euu ¼ pu ¼ q2 pq þ hp2h =eh ; eug ¼ pg ¼ hu ¼ hph =eh ; egg ¼ hg ¼ h=eh ; which gives that the Hessian of eðu; gÞ is positive definite for any u [ 0 and h [ 0. Thus it follows from (5.3.29) that Eðu; v; gÞ 

v2 v2 1 þ kmin ðe þ CB;d u; e g Þ½ðu  uÞ2 þ ðg  gÞ2   ½ðu  uÞ2 þ ðg  gÞ2 ; 2 2 ð5:3:32Þ

where kmin ðu; gÞ is the smaller characteristic root of the Hessian of eðu; gÞ, which is a 1 u; e g Þ  CB;d . Thus the left part of (5.3.28) smooth function of u and g and hence kmin ðe can be proved using (5.3.31) and (5.3.32). □ Lemma 5.3.8. There exists a positive constant c01 ¼ c01 ðCB;d Þ [ 0 such that for any fixed c 2 ð0; c01 ; there holds that ect ðkvðtÞk2 þ kuðtÞ  uk2 þ khðtÞ  hk2 þ kux ðtÞk2 þ kqx ðtÞk2 Þ Z t þ ecs ðkux k2 þ kqx k2 þ khx k2 þ kvx k2 ÞðsÞds  CB;d ; 8t [ 0: 0

ð5:3:33Þ

Attractors for Nonlinear Autonomous Dynamical Systems

104

Proof. Using equations (5.1.1)–(5.1.3), we can verify that ðq; v; gÞ satisfies
 v2 eþ ¼ ½pv þ l0 qvvx þ qkhx x ; ð5:3:34Þ 2 t gt ¼ ðkqhx =hÞx þ kqðhx =hÞ2 þ l0 qvx2 =h:

ð5:3:35Þ

Since u t ¼ 0; ht ¼ 0, we infer from (5.3.34)–(5.3.35) and (5.1.1)–(5.1.2) that E t ð1=q; v; gÞ þ ðh=hÞ½l0 qvx2 þ kqh2x =h ¼ ½l0 qvvx þ kð1  h=hÞqhx  ðp  pð1=q; hÞÞvx ;

ð5:3:36Þ

½l20 ðqx =qÞ2 =2 þ l0 qx v=qt þ l0 pq q2x =q ¼ l0 ph qx hx =q  l0 ðqvvx Þx þ l0 qvx2 : ð5:3:37Þ Multiplying (5.3.36)–(5.3.37) by ect ; bect respectively, and then adding the results up, we get @ GðtÞ þ ect ½ðh=hÞðl0 qvx2 þ kqh2x =hÞ=h þ bðl0 pq q2x =q  l0 qvx2 þ l0 ph qx hx =qÞ @t ¼ cect ½Eð1=q; v; gÞ þ bðl20 ðqx =qÞ2 =2 þ l0 qx v=qÞ þ ect ½ð1  bÞl0 qvvx þ kð1  h=hÞqhx  ðp  pðq; hÞÞvx ;

ð5:3:38Þ

where GðtÞ ¼ ect ½Eð1=q; v; gÞ þ bðl20 ðqx =qÞ2 =2 þ l0 vqx =qÞ. Integrating (5.3.38) over ½0; 1  ½0; t, using lemmas 5.2.1, 5.2.2, 5.3.1–5.3.3, 5.3.5, Cauchy’s inequality and Poincaré’s inequality, we can deduce that for small b [ 0 and for any c [ 0, Z t ect ½kqðtÞ  qk2 þ kvðtÞk2 þ kgðtÞ  gk2 þ kqx ðtÞk2  þ ecs ½kqx k2 þ kvx k2 þ khx k2 ðsÞds 0 Z t ð5:3:39Þ ecs ðkvk2 þ kq  qk2 þ kh  hk2 þ kqx k2 ÞðsÞds:  CB;d þ CB;d c 0

Considering the boundary conditions (5.1.7), we easily obtain Z x khðx; tÞ  hkL1 ¼ k hy ðy; tÞdykL1  khx ðtÞk; kvðtÞk  kvx ðtÞk:

ð5:3:40Þ

0

While for the boundary conditions (5.1.6), integrating (5.1.3) over ð0; 1Þ and using (5.2.6), we have   Z 1
Z 1
v2 v02 eðu; hÞ þ eðu0 ; h0 Þ þ dx ¼ dx ¼ eðu; hÞ 2 2 0 0 which, with Poincaré’s inequality, lemmas 5.2.1, 5.2.2, 5.3.1–5.3.3, 5.3.5 and the mean value theorem, implies

Universal Attractors for a Nonlinear Compressible System Z keðu; hÞ  eðu; hÞk  keðu; hÞ 

1

105

eðu; hÞdxk þ kvðtÞk2 =2

0

 C ðkex ðtÞk þ kvx ðtÞkÞ  CB;d ðkux ðtÞk þ kvx ðtÞk þ khx ðtÞkÞ:

ð5:3:41Þ

Moreover, using lemmas 5.2.1 and 5.2.2, (5.1.1), the mean value theorem and the Poincaré inequality, we deduce kuðtÞ  uk  C kux ðtÞk; khðtÞ  hk  CB;d ðkeðu; hÞ  eðu; hÞk þ kuðtÞ  ukÞ ð5:3:42Þ

 CB;d ðkeðu; hÞ  eðu; hÞk þ kux ðtÞkÞ; which, together with (5.3.41), gives khðtÞ  hk  CB;d ðkux ðtÞk þ kvx ðtÞk þ khx ðtÞkÞ:

ð5:3:43Þ

Similarly, we infer that Cd1 kuðtÞ  uk  kqðtÞ  qk  Cd kuðtÞ  uk;

ð5:3:44Þ

khðtÞ  hk  CB;d ðkgðtÞ  gk þ ku  ukÞ:

ð5:3:45Þ

Thus from (5.3.39)–(5.3.45), we can derive that there exists a constant c01 ¼ such that for any fixed c 2 ð0; c01  (5.3.33) holds. The proof is thus complete. □ c01 ðCB;d Þ [ 0

Lemma 5.3.9. There exists a positive constant c1 ¼ c1 ðCB;d Þ  c01 such that for any fixed c 2 ð0; c1 ; there holds that Z ect ðkvx ðtÞk2 þ khx ðtÞk2 Þ þ

t

ecs ðkvxx k2 þ khxx k2 þ kvt k2 þ kht k2 ÞðsÞds  CB;d ;

8t [ 0; ð5:3:46Þ

0

which, with lemma 5.3.9, yields that for any fixed c 2 ð0; c1 , kðuðtÞ  u; vðtÞ; hðtÞ  hÞk2H 1þ  CB;d ect ; 8t [ 0:

ð5:3:47Þ

Proof. Using (5.1.2), (5.1.3) and lemmas 5.2.1, 5.2.2, 5.3.1–5.3.3, 5.3.5 and Poincaré’s inequality, we can deduce kvx ðtÞk  C kvxx ðtÞk;

kvt ðtÞk  CB;d ðkux ðtÞk þ khx ðtÞk þ kvxx ðtÞkÞ;

khx ðtÞk  C khxx ðtÞk;

kht ðtÞk  CB;d ðkhxx ðtÞk þ kvxx ðtÞkÞ:

ð5:3:48Þ ð5:3:49Þ

Multiplying (5.1.2), (5.1.3) by ect vxx ; ect hxx respectively, integrating the results over ½0; 1  ½0; t, and adding them up, using Young’s inequality, the embedding theorem, lemmas 5.2.1, 5.2.2, 5.3.1–5.3.3, 5.3.5 and 5.3.9, we can derive

Attractors for Nonlinear Autonomous Dynamical Systems

106

Z t pffiffiffi pffiffiffiffiffi 2 1 e ðkvx ðtÞk þ k eh hx ðtÞk Þ þ CB;d ecs ðkvxx k2 þ k k hxx k2 ÞðsÞds 0 Z t ecs ½ðkux k þ khx k þ kux kkvx k1=2 kvxx k1=2 Þkvxx k  CB;d þ CB;d ct

2

0

þ ðkvx k þ kvx k3=2 kvxx k þ kux kkhx k1=2 khxx k1=2 Þkhxx kds Z t þ CB;d ecs ½kvx k2 þ kux k2 þ khx k2 þ ðkvx k þ kht kÞkhx k1=2 khxx k1=2 ðsÞds 0 Z t  CB;d þ 1=ð2CB;d Þ ecs ðkvxx k2 þ khxx k2 ÞðsÞds; 0

which, with lemmas 5.2.1, 5.2.2, 5.3.1–5.3.3, 5.3.5, equations (5.1.1)–(5.1.3) and (5.3.48)–(5.3.49), leads to (5.3.46). □ Thus from lemma 5.3.9, we conclude the following results on the existence of an absorbing set in Hd1 . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Theorem 5.3.10. Let R1 ¼ R1 ðdÞ ¼ 2 d23 þ ðh Þ2 and B1 ¼ fðu; v; hÞ 2 Hd1 ; kðu; v; hÞkH þ1  R1 g. Then B1 is an absorbing ball in Hd1 , i.e., there exists some

2 2 t1 ¼ t1 ðCB;d Þ ¼ maxfc1 1 log½2ðd3 þ ðh Þ Þ=CB;d ; t0 g  t0 such that when kðuðtÞ; vðtÞ; hðtÞÞk2H 1þ  R21 .

5.4

t  t1 ;

Existence of an Absorbing Set in Hd2

This section concerns the existence of an absorbing set in Hd2 . We always assume in this chapter that initial data belong to an arbitrarily fixed bounded set B in Hd2 , i.e., kðu0 ; v0 ; h0 ÞkH 2  B where B is a given positive constant. The following two lemmas are concerned with the existence of an absorbing set in Hd2 . Lemma 5.4.1. There exists a positive constant c02 ¼ c02 ðCB;d Þ  c1 such that for any fixed c 2 ð0; c02 , there holds that ect ðkht ðtÞk2 þ kvt ðtÞk2 þ khðtÞ  hk2H 2 þ kvðtÞk2H 2 Þ Z t þ ecs ðkvxt k2 þ khxt k2 ÞðsÞds  CB;d ; 8t [ 0:

ð5:4:1Þ

0

Proof. Differentiating equation (5.1.2) in t, multiplying the result by vt ect, integrating the resulting equation over ½0; 1  ½0; t, and using lemmas 5.2.1–5.3.2, 5.3.1–5.3.3, 5.3.5 and Young’s inequality, we can conclude

Universal Attractors for a Nonlinear Compressible System

107

Z t Z t pffiffiffi 2 1 ct 2 cs e kvt ðtÞk þ l0 e kvxt = u k ðsÞds  CB2 ;d þ c=2 ecs kvt k2 ðsÞds 2 0 0 Z t Z t pffiffiffi 2 2 4 cs e ðkvx k þ kht k þ kvx kL4 ÞðsÞds þ l0 =2 ecs kvxt = u k2 ðsÞds þ CB;d 0 0 Z t p ffiffiffi  CB;d þ ðCB;d c þ l0 =2Þ ecs kvxt = u k2 ðsÞds 0 Z t ecs ðkht k2 þ kvx k2 þ kvxx k2 ÞðsÞds; þ CB;d 0

which, along with lemmas 5.3.1–5.3.3, 5.3.5, 5.3.9, theorem 5.3.10 and equations (5.1.2) and (5.1.3), results in the existence of a constant c02 ¼ c02 ðCB;d Þ  c1 such that for any fixed c 2 ð0; c02 ; Z t ecs kvxt k2 ðsÞds  CB;d ; 8t [ 0: ð5:4:2Þ ect ðkvt ðtÞk2 þ kvxx ðtÞk2 Þ þ 0

Multiplying (5.1.3) by eh1, differentiating the resulting equation in t, we obtain htt ¼ I1 ðu; v; hÞ þ I2 ðu; v; hÞ þ I3 ðu; v; hÞ þ I4 ðu; v; hÞ þ I5 ðu; v; hÞ;

ð5:4:3Þ

where I1 ðu; v; hÞ ¼ ðeht =eh2 Þðkhx =uÞx ; I2 ðu; v; hÞ ¼ ðkhx =uÞxt =eh ; I3 ðu; v; hÞ ¼ hph vx eht =eh2 ; I4 ðu; v; hÞ ¼ ðht ph vx þ hpht vx þ hph vxt Þ=eh ; I5 ðu; v; hÞ ¼ l0 ½2vx vxt =eh u  vx2 ðeht u þ eh vx Þ=eh2 u 2 : Multiplying (5.4.3) by ht ect, integrating the result over ½0; 1  ½0; t and using lemmas 5.2.1, 5.2.2, 5.3.1–5.3.9, theorem 5.3.10, and (5.4.2), we can infer that Z t ect ðkht ðtÞk2 þ khxx ðtÞk2 Þ þ ecs khxt k2 ðsÞds  CB;d 0

which, with (5.4.2) and lemmas 5.3.8, 5.3.9, yields (5.4.1). The proof is now complete. □ Lemma 5.4.2. There exists a positive constant c2 ¼ c2 ðCB;d Þ  c02 such that for any fixed c 2 ð0; c2 , there holds that kuðtÞ  uk2H 2  CB;d ect ;

ð5:4:4Þ

which, with lemma 5.4.1, yields that for any fixed c 2 ð0; c2  and for all t [ 0, kuðtÞk2H 2 þ khðtÞk2H 2 þ kvðtÞk2H 2  2ðd23 þ ðh Þ2 Þ þ CB;d ect :

ð5:4:5Þ

Proof. Differentiating (5.1.2) in x, using (5.1.1) (utxx ¼ vxxx ), we can know @ uxx  l0  pu uxx ¼ vtx þ ðpuu ux2 þ 2phu hx ux þ phh h2x Þ þ ph hxx  2l0 vx ux2 =u 3 : @t u ð5:4:6Þ

Attractors for Nonlinear Autonomous Dynamical Systems

108

Multiplying (5.4.6) by uxx =u, and using Young’s inequality, lemmas 5.2.1, 5.2.2, 5.3.1–5.3.9, theorem 5.3.10 and (5.1.22), we can deduce that d uxx uxx k ðtÞk2 þ Cd1 k ðtÞk2 dt u u 1 uxx 2  k k þ CB;d ðkhx ðtÞk4L4 þ kux ðtÞk4L4 þ kvxt ðtÞk2 þ khxx ðtÞk2 þ kvx ux2 ðtÞk2 Þ 4Cd u 1 uxx k ðtÞk2 þ CB;d ðkhxx ðtÞk2 þ kux ðtÞk2 þ kvxt ðtÞk2 Þ;  2Cd u

ð5:4:7Þ

Multiplying (5.4.7) by et=2Cd and choosing c so small that c  minðc02 ; 1=4Cd Þ ¼ c2 ðCB;d Þ, and using lemma 5.3.9, theorem 5.3.10 and lemma 5.4.1, we conclude that kuxx ðtÞk2  CB;d et=2Cd þ CB;d ect  CB;d ect ; which, together with lemma 5.3.9, theorem 5.3.10, gives estimate (5.4.4). The proof is complete. □ 2 2 Defining t2 ¼ t2 ðCB;d Þ  maxðt1 ðCB;d Þ; c1 2 logð2ðd3 þ ðh Þ Þ=CB;d Þ, then we know that estimate (5.4.5) implies that for any t  t2 ðCB;d Þ,

kuðtÞk2H 2 þ khðtÞk2H 2 þ kvðtÞk2H 2  4ðd23 þ ðh Þ2 Þ: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Taking R2 ¼ 2 d23 þ ðh Þ2 , we immediately infer the next theorem. Theorem 5.4.3. The ball

ð5:4:8Þ

B2 ¼ fðu; v; hÞ 2 Hd2 ; kðuðtÞ; vðtÞ; hðtÞÞk2H þ2  R22 g is an

absorbing ball in Hd2 , i.e., when t  t2 ; kðuðtÞ; vðtÞ; hðtÞÞk2H þ2  R22 .

5.5

Proof of Theorem 5.1.1

In this section, we shall prove theorem 5.1.1. Noting that existence of absorbing balls in Hd2 and Hd1 has been established, we can use the abstract framework (see also chapter 4) given in [53] by Ghidaglia to conclude that Lemma 5.5.1. The set xðB2 Þ ¼

\ [

SðtÞB2 ;

ð5:5:1Þ

s0 t s 2 where the closures are taken in the weak topology of H þ , is included in B2 and nonempty, and also invariant by SðtÞ, i.e.,

SðtÞxðB2 Þ ¼ xðB2 Þ;

8t [ 0:

ð5:5:2Þ

Universal Attractors for a Nonlinear Compressible System

109

Remark 5.5.1. Similarly as in chapter 4, for a bounded set B in Hd2 , we can also define xðBÞ by (5.5.1) and when B is nonempty, xðBÞ is also included in B2 ; nonempty and invariant. Since B2 is an absorbing ball, we conclude at once that xðBÞxðB2 Þ, which indeed implies that xðB2 Þ is maximal in the sense of inclusion. Theorem 5.5.2. The set A2;d ¼ xðB2 Þ

ð5:5:3Þ

satisfies A2;d

is bounded and weakly closed in SðtÞA2;d ¼ A2;d ;

Hd2 ;

ð5:5:4Þ

8t  0;

ð5:5:5Þ

lim d w ðSðtÞB; A2;d Þ ¼ 0:

ð5:5:6Þ

for every bounded set B in Hd2 , t! þ 1

Moreover, it is the maximal set in the sense of inclusion that satisfies (5.5.4)– (5.5.6). Proof. The proof of this theorem is the same as that of lemma 4.5.3, also see the arguments in [51]. Thus we omit the details here. □ Similarly to chapter 4, following [51], we also call A2;d the universal attractor of SðtÞ in Hd2 . For the existence of an universal attractor in Hd1 , we need to show the following lemma, see also chapter 4. Lemma 5.5.3. For every t  0, the mapping SðtÞ is continuous on bounded sets of Hd1 for the topology induced by the norm in L2  L2  L2 . Proof. We suppose that ðu0j ; v0j ; h0j Þ 2 Hd1 ; kðu0j ; v0j ; h0j ÞkH 1  R; ðuj ; vj ; hj Þ ¼ SðtÞ ðu0j ; v0j ; h0j Þ ðj ¼ 1; 2Þ, and ðu; v; hÞ ¼ ðu1 ; v1 ; h1 Þ  ðu2 ; v2 ; h2 Þ. Subtracting the corresponding equations (5.1.1)–(5.1.3) satisfied by ðu1 ; v1 ; h1 Þ and ðu2 ; v2 ; h2 Þ, we have ut ¼ vx ; vt ¼ pu ðu1 ; h1 Þux  ðpu ðu1 ; h1 Þ  pu ðu2 ; h2 ÞÞu2x  ph ðu1 ; h1 Þhx 
  vxx vx u1x v2x u  ðph ðu1 ; h1 Þ  ph ðu2 ; h2 ÞÞh2x þ l0  2  ; u1 u2 x u1 u1

ð5:5:7Þ

ð5:5:8Þ

eh ðu1 ; h1 Þht ¼ ðeh ðu1 ; h1 Þ  ðeh ðu2 ; h2 ÞÞh2t  ðeu ðu1 ; h1 Þ  eu ðu2 ; h2 ÞÞv2x  eu ðu1 ; h1 Þvx  pðu1 ; h1 Þvx  ðpðu1 ; h1 Þ  pðu2 ; h2 ÞÞv2x þ ½kðu1 ; h1 Þhx þ ðkðu1 ; h1 Þ  kðu2 ; h2 Þh2x x :

ð5:5:9Þ

Attractors for Nonlinear Autonomous Dynamical Systems

110

t ¼ 0 : u ¼ u0 ; v ¼ v0 ; h ¼ h0 ; x ¼ 0; 1 : v ¼ 0; hx ¼ 0 or h ¼ 0:

ð5:5:10Þ

By lemmas 5.2.1, 5.3.1–5.3.3, and 5.3.5, we deduce that for any t [ 0 and j ¼ 1; 2, Z t 2 kðuj ðtÞ; vj ðtÞ; hj ðtÞÞkH 1 þ ðkujx k2 þ kvj k2H 2 þ khjx k2H 1 þ khjt ðtÞk2 þ kvjt k2 ÞðsÞds  CR;d ; 0

ð5:5:11Þ where constant CR;d [ 0 depends only on R and d. Multiplying (5.5.7), (5.5.8) and (5.5.9) by u; v and h respectively, adding the resulting equalities up and integrating the final resulting equality over ½0; 1, and using (5.1.24), (5.5.11), the Cauchy’s inequality, the embedding theorem, the mean value theorem and noting that inequalities khk2L1  C ðkhkkhx k þ khk2 Þ; kvkL1  kvx k, we conclude that for any small  [ 0,  Z 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1d l0 vx 2 2 2 2 ðkuðtÞk þ kvðtÞk þ k eh ðu1 ; h1 ÞhðtÞk Þ þ þ kðu1 ; h1 Þhx dx 2 dt u1 0 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðkvx ðtÞk2 þ khx ðtÞk2 Þ þ CR;d ðÞH ðtÞðkuðtÞk2 þ k eh ðu1 ; h1 ÞhðtÞk2 þ kvðtÞk2 Þ; which, together with lemmas 5.4.1, 5.4.2, theorem 5.4.3 and (5.1.24), gives to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ðkuðtÞk2 þ kvðtÞk2 þ k eh ðu1 ; h1 ÞhðtÞk2 Þ þ Cd1 ðkvx ðtÞk2 þ khx ðtÞk2 Þ dt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  CB;d H ðtÞðkuðtÞk2 þ k eh ðu1 ; h1 ÞhðtÞk2 þ kvðtÞk2 Þ;

ð5:5:12Þ

where, by (5.6.11), H ðtÞ ¼ kh1t ðtÞk2 þ kh2t ðtÞk2 þ kv1xx ðtÞk2 þ kv2xx ðtÞk2 þ kh1xx ðtÞk2 þ kh2xx ðtÞk2 þ 1 satisfies for any t [ 0, Z t H ðsÞds  CR;d ð1 þ tÞ: ð5:5:13Þ 0

Therefore, we can derive the conclusion of this lemma using the Gronwall’s inequality, (5.5.12), (5.5.13) and (5.1.20). □ Now following agian the same arguments as in [53] (see also theorem 4.5.4) to obtain the following result on existence of an universal attractor in Hd1 . Theorem 5.5.4. The set A1;d ¼

\ [

SðtÞB1 ;

ð5:5:14Þ

s0 ts 1 where the closures are taken in the weak topology of H þ , is the (maximal) universal 1 attractor in Hd . 2 1 Remark 5.5.2. Noting that A2;d is bounded in H þ , and also is bounded in H þ , by the invariance property (5.5.5), we conclude

A2;d A1;d :

ð5:5:15Þ

Universal Attractors for a Nonlinear Compressible System

111

2 On the contrary, if we knew that A1;d is bounded in H þ , then the opposite inclusion would hold.

5.6

Bibliographic Comments

Recall that the global existence and asymptotic behaviour of smooth (generalized) solutions to the system (5.1.1)–(5.1.3) have been studied by many authors, e.g., see [94, 99, 105] on the initial boundary value problems and the Cauchy problem. Zheng and Qin [159] obtained the existence of maximal attractor for the problem (5.1.1)– (5.1.3) and (5.1.5)–(5.1.6) for (5.1.11). Qin [107] established the existence and exponential stability of a C0 -semigroup in the subspace of H i  H i  H i ði ¼ 1; 2Þ for a viscous ideal gas (5.1.11) in a bounded domain in R (see chapter 4) and in bounded annular domains Gn ¼ fx 2 Rn j0\a\jxj\bg ðn ¼ 2; 3Þ in Rn (see chapter 4) for a viscous spherically symmetric ideal gas. We point out here that these results have improved those in [99] for (5.1.11). Moreover, Jiang [62] proved the global existence for the system (5.1.1)–(5.1.3) with the boundary conditions (5.1.6) or (5.1.7) or Qð0; tÞ ¼ Qð1; tÞ ¼ 0;

rð0; tÞ ¼ vð0; tÞ;

rð1; tÞ ¼ vð1; tÞ

ð5:6:1Þ

hð0; tÞ ¼ hð1; tÞ ¼ T0 ;

rð0; tÞ ¼ vð0; tÞ;

rð1; tÞ ¼ vð1; tÞ

ð5:6:2Þ

or

under the assumptions (5.1.8), (5.1.12)–(5.1.14), (5.1.20), (5.6.8) (below) (for (5.6.1)–(5.6.2)), (5.1.24), (5.1.25), (5.6.12) (below) and jph ðu; hÞj  p3 ðuÞu 1 ð1 þ hr Þ; upðu; hÞ  p4 ð1 þ hr þ 1 Þ;

pu ðu; T0 Þ  0;

ð5:6:3Þ for ð5:1:7Þ;

ð5:6:4Þ

jph ðu; hÞj  N ðuÞð1 þ hr Þ; 0\l0  lðuÞ  l1 ; lðuÞ ¼ l0 ;

for

for

ð5:6:5Þ

ð5:6:1Þ  ð5:6:2Þ;

ð5:6:6Þ

ð5:1:6Þ  ð5:1:7Þ;

ð5:6:7Þ

with the exponents r 2 ½0; 1; q  r þ 1. Under the assumptions (5.1.8), (5.1.12)– (5.1.14), (5.1.20), (5.1.23)–(5.1.25) and 

p2 ½l þ ð1  lÞh þ h1 þ r  p1 ½l þ ð1  lÞh þ h1 þ r   p ðu; hÞ   ; u u2 u2

l¼0

or

l ¼ 1; ð5:6:8Þ

0  pðu; hÞ;

pðu; hÞ ! 0;

as

u ! þ 1;

r 2 ½0; 1;

q  r þ 1;

ð5:6:9Þ

Attractors for Nonlinear Autonomous Dynamical Systems

112

Under the assumptions (5.1.8), (5.1.12)–(5.1.14), (5.1.20), and p2 ð1 þ h1 þ r Þu 2  pu ðu; hÞ   p1 ð1 þ h1 þ r Þu 2 ; jph ðu; hÞj  p3 ðuÞu 1=2 ð1 þ hr Þ;

upðu; hÞ  p4 ð1 þ h1 þ r Þ;

ð5:6:10Þ ð5:6:11Þ

0\pðu; hÞ  N ðuÞð1 þ hr þ 1 Þ;

ð5:6:12Þ

k0 ð1 þ hq Þ  kðu; hÞ  k2 ð1 þ hq Þ;

ð5:6:13Þ

jku ðu; hÞj þ jkuu ðu; hÞj  k2 ð1 þ hq Þ;

ð5:6:14Þ

with r 2 ½0; 1; q  2r þ 2 and for some constants c0 \2;

g0 [ 0,

c0

gðu; hÞ  ððMuÞ þ g0 Þeðu; hÞ

ð5:6:15Þ

R u lðnÞ

with Mu :¼ 1 n dn; Kawhol [70] obtained the classical solutions to the system (5.1.1)–(5.1.3) with the boundary conditions (5.1.6) or Qð0; tÞ ¼ Qð1; tÞ ¼ 0;

rð0; tÞ ¼ rð1; tÞ ¼ 0:

ð5:6:16Þ

Note that our assumption (5.1.22) is weaker than (5.6.8) and (5.6.10). Moreover, our assumption (5.1.23) is weaker than (5.6.3) and (5.6.11), while (5.1.24) and (5.1.25) are more general than (5.6.13) and (5.6.14), respectively. Under the assumptions (5.1.8), (5.1.12)–(5.1.26), Qin [105] improved the results in [62, 70] to obtain global existence and asymptotic behaviour of smooth solutions (generalized solutions in H 1  H 1  H 1 ) to problems (5.1.1)–(5.1.3), (5.1.5)–(5.1.6) and (5.1.1)– (5.1.3), (5.1.5), (5.1.7). Qin [108] proved the existence and exponential stability of a i nonlinear C0 -semigroup SðtÞ in the subspace H þ of H i  H i  H i ði ¼ 1; 2Þ for the equations (5.1.1)–(5.1.3) subject to the boundary conditions (5.1.6) or (5.1.7) with more general constitutive relations (5.1.8) and (5.1.12)–(5.1.26). We should point out here that assumptions (5.1.15)–(5.1.19) on the exponents q; r do not cover the case of q ¼ r ¼ 0. The case of a one-dimensional polytropic viscous ideal gas (see (5.1.11)) is a special case of q ¼ r ¼ 0, for which many authors proved the global existence and asymptotic behaviour of smooth (generalized) solutions, see [71, 72, 94, 106, 112], etc. For the infinite-dimensional dynamics, we refer to [146]. Concerning compressible fluid, we have the recent works of Zheng and Qin [158, 159] where the existence of maximal universal attractors for the one-dimensional viscous heat-conductive polytropic ideal gas in a bounded domain with (5.1.11)) (see chapter 4) and in bounded annular domains in Rn ; n ¼ 2; 3 (see the next chapter) was proved. In this chapter, we use the abstract framework established in [53] and the ideas in [108, 158, 159] to establish the existence of universal (maximal, global) attractors for problems (5.1.1)–(5.1.3), (5.1.5)–(5.1.6) and (5.1.1)–(5.1.3), (5.1.5), (5.1.7) with more general constitutive relations (5.1.8) and (5.1.12)–(5.1.26).

Universal Attractors for a Nonlinear Compressible System

113

Since some of present mathematical difficulties are similar to the four points (1)–(4) in section 4.6, we omit these four points here. Besides, we have the next mathematical difficulty. (5) Unlike the one-dimensional polytropic viscous idealgas (5.1.11) (the special case of q ¼ r ¼ 0), see also chapter 4. Since equations (5.1.1)– (5.1.3) look more complicated than those of the special case of q ¼ r ¼ 0 and those in chapter 4. Thus much more delicate estimates are needed here.

Chapter 6 Global Attractors for the Compressible Navier–Stokes Equations in Bounded Annular Domains This chapter will prove the existence of universal (global) attractor for Navier–Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains Xn in Rn ðn ¼ 2; 3Þ. One important feature of this problem is that our working metric spaces H ð1Þ , H ð2Þ and H ð4Þ are three incomplete metric spaces, as can be known from the constraints h [ 0 and u [ 0 with h and u being absolute temperature and specific volume, respectively. For any constants d1 ; d2 ; . . .; d5 satisfying certain conðiÞ ditions, three sequences of closed subspaces Hd  H ðiÞ ði ¼ 1; 2; 4Þ have ð1Þ

been constructed, and the existence of three (maximal) universal attractors in Hd , ð2Þ

ð4Þ

Hd and Hd is established. This chapter is picked from Zheng and Qin [158] and the weakly compact semigroup method (see theorem 1.2.24) will be used here.

6.1

Main Result

This chapter shall establish the existence of global attractors of the following three-dimensional spherically symmetric compressible Navier–Stokes equations for a polytropic viscous and heat-conductive ideal gas in the Eulerian coordinates in bounded annular domains Gn ¼ fx 2 Rn ; 0\a\jxj\b\ þ 1g ðn ¼ 2; 3Þ in Rn (see [46, 47, 63, 64, 96]): for r 2 Gn ; t [ 0, @t q þ @r ðqvÞ þ

ðn  1Þ qv ¼ 0; r


 ðn  1Þ ðn  1Þ qð@t v þ v@r vÞ  b @r2 v þ @r v  v þ R@r ðqhÞ ¼ 0; r r2

ð6:1:1Þ

ð6:1:2Þ

DOI: 10.1051/978-2-7598-2702-2.c006 © Science Press, EDP Sciences, 2022

Attractors for Nonlinear Autonomous Dynamical Systems

116


 ðn  1Þ ðn  1Þv @r h þ Rqh @r v þ CV qð@t h þ v@r hÞ  j r r
2 2 ðn  1Þv 2lðn  1Þv  k @r v þ 2lð@r vÞ2  ¼ 0; r r2 j@r2 h

ð6:1:3Þ

where q; v; h are the density, the velocity, and the absolute temperature, respectively; k and l are the constant viscosity coefficients, R; CV ; and j are the gas constant, the specific heat capacity, and the thermal conductivity, respectively, satisfying that R; CV ; j; l [ 0; k þ 2l=n  0: In (6.1.2), we denote b ¼ k þ 2l. Consider problem (6.1.1)–(6.1.3) in the region fr 2 Gn ; t  0g subject to the following initial and boundary conditions qðr; 0Þ ¼ q0 ðrÞ; vðr; 0Þ ¼ v0 ðrÞ; hðr; 0Þ ¼ h0 ðrÞ; r 2 Gn ;

ð6:1:4Þ

vða; tÞ ¼ vðb; tÞ ¼ 0; hr ða; tÞ ¼ hr ðb; tÞ ¼ 0; t  0:

ð6:1:5Þ

In the sequel, we first turn problem (6.1.1)–(6.1.5) into that in the Lagrangian coordinates and obtain the results on existence of maximal universal attractors. Then we go back to the Eulerian coordinates to conclude corresponding results. We know that the Eulerian coordinates ðr; tÞ and the Lagrangian coordinates ðn; tÞ have the following relationship Z t ~v ðn; sÞds; rðn; tÞ ¼ r0 ðnÞ þ ð6:1:6Þ 0

where ~ v ðn; tÞ ¼ vðrðn; tÞ; tÞ and r0 ðnÞ ¼ g1 ðnÞ; gðrÞ ¼

Z

r

s n1 q0 ðsÞds; r 2 Gn :

ð6:1:7Þ

a

Obviously, from equation (6.1.1) and boundary condition (6.1.5), we derive that ! Z rðn;tÞ @ n1 s qðs; tÞds @t a Z r ð6:1:8Þ ¼ vr n1 q þ s n1 q ds ¼ vr n1 q  r n1 qv ¼ 0: t

a

Thus,

Z

r

Z s n1 qðs; tÞds ¼

r0

s n1 q0 ðsÞds ¼ n;

ð6:1:9Þ

a

a

and Gn is changed into Xn ¼ ð0; LÞ with Z Z b s n1 qðs; tÞds ¼ L¼ a

a

b

s n1 q0 ðsÞds:

ð6:1:10Þ

Global Attractors for the Compressible Navier–Stokes Equations

117

Moreover, differentiating (6.1.9) yields @n rðn; tÞ ¼ ½rðn; tÞn1 qðrðn; tÞ; tÞ1 : e tÞ ¼ /ðrðn; tÞ; tÞ, then using Indeed, generally for a function /ðr; tÞ, setting /ðn; the chain rule, we can deduce e tÞ ¼ @t /ðr; tÞ þ v@r /ðr; tÞ; @t /ðn; ð6:1:11Þ

e tÞ ¼ @r /ðr; tÞ@n rðn; tÞ ¼ @r /ðr; tÞ : @n /ðn; r n1 qðr; tÞ

In the sequel, without danger of confusion, denote ð~ q; v~; e hÞ still by ðq; v; hÞ and ðn; tÞ by ðx; tÞ, and using u ¼ 1=q to denote the specific volume, then we can transform the equations (6.1.1)–(6.1.3) in the Eulerian coordinates to the following equations in the Lagrangian coordinates in the new variables ðx; tÞ, x 2 Xn ; t  0: ut ¼ ðr n1 vÞx ; vt ¼ r

n1

ð6:1:12Þ


n1  bðr vÞx h R ; u x u

ð6:1:13Þ


2n2   r hx 1  n1 bðr vÞx  Rh ðr n1 vÞx  2lðn  1Þðr n2 v 2 Þx ð6:1:14Þ CV ht ¼ j þ u u x subject to the following initial and boundary conditions uðx; 0Þ ¼ u0 ðxÞ; vðx; 0Þ ¼ v0 ðxÞ; hðx; 0Þ ¼ h0 ðxÞ; x 2 Xn ;

ð6:1:15Þ

vð0; tÞ ¼ vðL; tÞ ¼ 0; hx ð0; tÞ ¼ hx ðL; tÞ ¼ 0; t  0:

ð6:1:16Þ

By (6.1.6), rðx; tÞ is determined by rt ðx; tÞ ¼ vðx; tÞ; rjt¼0


Z n ¼ r0 ðxÞ ¼ a þ n

x

1=n u0 ðyÞdy

:

ð6:1:17Þ

0

From (6.1.9), that there holds that: r n1 ðx; tÞrx ðx; tÞ ¼ uðx; tÞ:

ð6:1:18Þ

For the simplicity, we always assume that k and l satisfy (see [64]) nk þ 2l [ 0:

ð6:1:19Þ

Thus, we need first to study the initial boundary value problem (6.1.12)–(6.1.16) where L is fixed.

Attractors for Nonlinear Autonomous Dynamical Systems

118

Introduce three spaces H ð1Þ ¼ fðu; v; hÞ 2 H 1 ½0; L  H 1 ½0; L  H 1 ½0; L : uðxÞ [ 0; hðxÞ [ 0; x 2 ½0; L; vjx¼0 ¼ vjx¼L ¼ 0g; H ð2Þ ¼ fðu; v; hÞ 2 H 2 ½0; L  H 2 ½0; L  H 2 ½0; L : uðxÞ [ 0; hðxÞ [ 0; x 2 ½0; L; vjx¼0 ¼ vjx¼L ¼ hx jx¼0 ¼ hx jx¼L ¼ 0g and H ð4Þ ¼ fðu; v; hÞ 2 H 4 ½0; L  H 4 ½0; L  H 4 ½0; L : uðxÞ [ 0; hðxÞ [ 0; x 2 ½0; L; vjx¼0 ¼ vjx¼L ¼ hx jx¼0 ¼ hx jx¼L ¼ 0g; 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. Assume di ði ¼ 1; 2; :::; 8Þ are any given constants suchi that d1 2 R; h

d2 [ 0; d6 [ 0; d7 [ 0; d8 [ 0; 0\d5 \d2 ; d4  max ( ðiÞ Hd

ðu; v; hÞ 2 H ðiÞ :

:¼

Z

L

ed1 =LR 2ð2d2 =CV LÞCV =R

Z ðCV logðhÞ þ R logðuÞÞdx  d1 ; d5 

0

Z d3  0

L

; d3 [ 0 and let

L

ðCV h þ v 2 =2Þdx  d2 ;

0

) d5 2d2 d3 2d4 udx  d4 ; h ; u ; d6  d7 ; i ¼ 1; 2; 4: 2LCV CV L 2L L ðiÞ

Clearly, Hd is a sequence of closed subspaces of H ðiÞ ði ¼ 1; 2; 4Þ: Similarly as chapters 4 and 5, the first three constraints are invariant, while the last two constraints are not invariant, which are just introduced to overcome the difficulty that the original spaces H ðiÞ are incomplete. It is very crucial to prove that ðiÞ ðiÞ the orbit starting from any bounded set of Hd will re-enter Hd after a finite time. The notation in this chapter will be as follows: Lp ; 1  p  þ 1; W m;p ; m 2 N; H 1 ¼ W 1;2 ; H01 ¼ W01;2 denote the usual (Sobolev) spaces on ð0; LÞ: In addition, k  kB denotes the norm in the space B; we also put k  k ¼ k  kL2 : We denote by C k ðI ; BÞ; k 2 N0 ; the space of k-times continuously differentiable functions from I R into a Banach space B, and likewise by Lp ðI ; BÞ; 1  p  þ 1 the corresponding Lebesgue spaces. Subscripts t and x denote ðiÞ the (partial) derivatives with respect to t and x, respectively. We use C0 ; i ¼ 1; 2; 4 to denote the universal constant depending only on the H ðiÞ norm of initial data, minx2½0;L u0 ðxÞ and minx2½0;L h0 ðxÞ. Cd denotes the universal constant depending ðiÞ

only on di ði ¼ 1; 2; :::; 8Þ, but independent of initial data. Cd denotes the universal constant depending on both dj ðj ¼ 1; 2; :::; 8Þ, H ðiÞ norm of initial data, minx2½0;L h0 ðxÞ and minx2½0;L u0 ðxÞ. Our main theorem reads as follows.

Global Attractors for the Compressible Navier–Stokes Equations

119

Theorem 6.1.1. The nonlinear semigroup SðtÞ defined by the solution to problem (6.1.12)–(6.1.17) maps H ðiÞ ði ¼ 1; 2; 4Þ into itself. Moreover, for any di ði ¼ 1; 2; :::; 8Þ with " # ed1 =LR d1 \0; d2 [ 0; d4  max ; d3 [ 0; 0\d5 \d2 ; 0\d6 \d7 ; 0\d8 ; 2ð2d2 =CV LÞCV =R ðiÞ

it possesses in Hd a maximal universal attractor Ai;d ði ¼ 1; 2; 4Þ: Remark 6.1.1. See section 6.5 in this chapter for more precise definition of (maximal) universal attractor. S Remark 6.1.2. The set Ai ¼ d1 ;d2 ;...;d8 Ai;d ði ¼ 1; 2Þ is a global noncompact

attractor in the metric space H ðiÞ in the following sense that it attracts any bounded sets of H ðiÞ with constraints u  g1 ; h  g2 with g1 ; g2 being any given positive constants. This chapter is organized as follows. Section 6.2 is concerned with nonlinear semiðiÞ groups on H ðiÞ and on HL;G ði ¼ 1; 2; 4Þ. To prove theorem 6.1.1, sections 6.3 and 6.4 ðiÞ

are devoted to proofs of the existence of absorbing sets in Hd ði ¼ 1; 2Þ, then we complete the proof of theorem 6.1.1 for i ¼ 1; 2 in section 6.5.

6.2

Nonlinear Semigroup on H ð2Þ

As known in section 6.1, for any initial data ðu0 ; v0 ; h0 Þ 2 H ð1Þ , the results on global existence, uniqueness and asymptotic behaviour of solutions to problem (6.1.12)– (6.1.17) have been established in [64] and [108], respectively. It has been proved in [108] that the operators SðtÞ defined by the solutions are C0 -semigroups on H ðiÞ ði ¼ 1; 2Þ, respectively. The following lemma has been in [64, 108]. Lemma 6.2.1. The generalized global solution ðuðtÞ; vðtÞ; hðtÞÞ to problem (6.1.12)– (6.1.17) defines a nonlinear C0 -semigroup SðtÞ on H ð1Þ such that for any ðu0 ; v0 ; h0 Þ 2 H ð1Þ ; SðtÞðu0 ; v0 ; h0 Þ ¼ ðuðtÞ; vðtÞ; hðtÞÞ 2 C ð½0; þ 1Þ; H ð1Þ Þ, ut ; vt ; ht ; hx ; vx ; ux ; vxx ; hxx ; ðr  rÞx ; ðr  rÞxx ; rtxx 2 L2 ð½0; þ 1Þ; L2 Þ. Moreover, ð1Þ

ð1Þ

on ½0; L  ½0; þ 1Þ;

ð6:2:1Þ

ð1Þ

ð1Þ

on ½0; L  ½0; þ 1Þ;

ð6:2:2Þ

0\1=C0  hðx; tÞ  C0

0\1=C0  uðx; tÞ  C0 ð1Þ

ð1Þ

0\a  rðx; tÞ  b; 0\1=C0  rx ðx; tÞ  C0 krt ðtÞk2H 1

þ krðtÞk2H 2

þ khx k2H 1

2

þ kuðtÞk2H 1 2

þ khðtÞk2H 1

þ kvt k þ kht k þ kr 

rk2H 2

on ½0; L  ½0; þ 1Þ;

þ kvðtÞk2H 1

þ krt k2H 2

ð6:2:3Þ

Z th þ kux k2 þ kvk2H 2

0 i ð1Þ ðsÞds  C0 ; 8t [ 0;

ð6:2:4Þ

Attractors for Nonlinear Autonomous Dynamical Systems

120

ð1Þ ð1Þ and there exist constants ~c1 ¼ ~ci ðC0 Þ; C0 [ 0 ði ¼ 1; 2Þ such that for any fixed ~c 2 ð0; ~ci  and for any t [ 0, it holds that

kSðtÞðu0 ; v0 ; h0 Þ  ðu; 0; hÞk2H 1 þ krðtÞ  rðxÞk2H 2  C1 e~ct ;

ð6:2:5Þ

where u¼

1 L

Z

1

u0 ðxÞdx; h ¼

0

1

Z 1


CV L

0

rðxÞ ¼ ða n þ nuxÞ1=n ;

CV h0 þ

 v02 ðxÞdx; 2

ð6:2:6Þ

8x 2 ½0; L:

ð6:2:7Þ

The next lemma is also proved in [108]. Lemma 6.2.2. For any ðu0 ; v0 ; h0 Þ 2 H ð2Þ , the global solution ðuðtÞ; vðtÞ; hðtÞÞ to problem (6.1.12)–(6.1.17) defines a nonlinear C0 -semigroup SðtÞ on H ð2Þ such that SðtÞðu0 ; v0 ; h0 Þ ¼ ðuðtÞ; vðtÞ; hðtÞÞ 2 C ð½0; þ 1Þ; H ð2Þ Þ. In addition to lemma 6.2.1, we have hxt ; vxt 2 L2 ð½0; þ 1Þ; L2 Þ and krt ðtÞk2H 2 þ krðtÞk2H 3 þ kuðtÞk2H 2 þ kvðtÞk2H 2 þ khðtÞk2H 2 Z t ð2Þ þ ðkhxt k2 þ kvxt k2 ÞðsÞds  C0 ; 8t [ 0:

ð6:2:8Þ

0

ð1Þ

6.3

Existence of an Absorbing Set in Hd

ð1Þ

This section is devoted to the existence of an absorbing set in Hd . We always ð1Þ

assume in this section that initial data belong to a bounded set of Hd . First, we ð1Þ

ð1Þ

need to prove that the orbit starting from any bounded set in Hd will re-enter Hd after a finite time which should be uniform with respect to all orbits starting from that bounded set. ð1Þ

Lemma 6.3.1. If ðu0 ; v0 ; h0 Þ 2 Hd , then the following estimates hold, Z L Z L uðx; tÞdx ¼ u0 ðxÞdx  d4 ; 8t [ 0; d3  0

d5 

Z L
0

Z 

L

  Z L
v2 v02 CV h þ C V h0 þ ðx; tÞdx ¼ ðxÞdx  d2 ; 8t [ 0; 2 2 0

ðCV log h þ R log uÞðx; tÞdx þ

0

Z

 0

L

ð6:3:1Þ

0

ð6:3:2Þ

 Z t Z L
2n2 2 jr hx 2lð2l þ nkÞr 2n2 vx2 dxds þ ð2l þ ðn  1ÞkÞuh uh2 0 0

ðCV log h0 þ R log u0 Þdx   d1 ;

8t [ 0:

ð6:3:3Þ

Global Attractors for the Compressible Navier–Stokes Equations

121

Proof. Indeed, if we integrate the equation (6.1.12) with respect to x and t and exploit the boundary conditions, we will end up with Z L Z L uðx; tÞdx ¼ u0 ðxÞdx; 8t [ 0: ð6:3:4Þ 0

0

Next, if we multiply (6.1.13) by v, integrate the resultant and also integrate the equation (6.1.14) with respect to x and t, then add together, we finally get   Z L
Z L
v2 v2 ð6:3:5Þ CV h þ CV h0 ðxÞ þ 0 ðxÞ dx: dx ¼ 2 2 0 0 A straightforward calculation, using (6.1.18), (6.1.19) and (6.1.12), yields b 2lðn  1Þðr n2 v 2 Þx ððr n1 vÞx Þ2  uh h " #
2 1 kr n1 vx 2lð2l þ nkÞ 2n2 2 1 ðn  1Þð2l þ ðn  1ÞkÞ r uv þ r ¼ þ vx uh 2l þ ðn  1Þk 2l þ ðn  1Þk 

2lð2l þ nkÞr 2n2 vx2 ½2l þ ðn  1Þkuh ð6:3:6Þ

with b ¼ 2l þ k. Multiplying equation (6.1.14) by h1 and exploiting (6.1.12), we arrive at
 j r 2n2 hx b 2lðn  1Þðr n2 v 2 Þx ððr n1 vÞx Þ2  ; þ ðCV log h þ R log uÞt ¼ h uh u h x which, together with (6.3.6), yields the estimate (6.3.3). The proof is now complete. h ð1Þ

Lemma 6.3.2. If ðu0 ; v0 ; h0 Þ 2 Hd , then Z t Z L
0

 h2x vx2 þ ðx; sÞdxds  Cd ; 8t [ 0; uh2 uh

0

0\Cd1 

Z

L

ð6:3:7Þ

kvðtÞk  Cd ; 8t [ 0;

ð6:3:8Þ

hl0 ðx; tÞdx  Cd ; 8l0 2 ½0; 1; 8t [ 0:

ð6:3:9Þ

0

Proof. It follows from (6.3.2) that (6.3.8) and khðtÞkL1  Cd :

Attractors for Nonlinear Autonomous Dynamical Systems

122

We can derive from (6.3.3) and a  r  b in (6.2.3) that Z L log hdx  Cd  0

holds. Applying the Jensen’s inequality to the convex function  log y yields Z L hl0 dx: 0\Cd  0

Combining (6.3.3) with (6.3.9), (6.3.1) and (6.1.19) yields (6.3.7). h For estimating of u, we need the next expression of u which is similar to that in the case n ¼ 1. Lemma 6.3.3. For each t  0, there exists a point x0 ¼ x0 ðtÞ 2 ½0; L such that the specific volume uðx; tÞ has the following expression:   Z Dðx; tÞ R t hðx; sÞBðx; sÞ 1þ ds ; uðx; tÞ ¼ Bðx; tÞ b 0 Dðx; sÞ

8x 2 ½0; L;

ð6:3:10Þ

Z x n1h 1 Z 1 Dðx; tÞ ¼ u0 ðxÞ exp u0 ðxÞ r01n ðyÞv0 ðyÞdydx b u 0 0 Z x Z x io þ r 1n ðy; tÞvðy; tÞdy  r01n ðyÞv0 ðyÞdy ;

ð6:3:11Þ

with D; B being given as follows

x0 ðtÞ

0

( " Z Z
 1 1 t 1 v2 þ Rh ðx; sÞdxds Bðx; tÞ ¼ exp b u 0 0 n Z tZ 1 ðn  1Þa n þ  r n ðx; sÞv 2 ðx; sÞdxds nu

0 0 #) Z Z t

þ ðn  1Þ

L

0

r n ðy; sÞv 2 ðy; sÞdyds

;

ð6:3:12Þ

x

where Z u ¼

1

u0 ðxÞdx:

ð6:3:13Þ

0

h ð1Þ

Lemma 6.3.4. If ðu0 ; v0 ; h0 Þ 2 Hd , then 0\Cd1  uðx; tÞ  Cd ;

0\Cd1  rx ðx; tÞ  Cd ;

8ðx; tÞ 2 ½0; L  ½0; þ 1Þ: ð6:3:14Þ

Global Attractors for the Compressible Navier–Stokes Equations

123

Proof. The proof borrowed some ideas from that in [159] for the case of n ¼ 1. Noting that Bðx; tÞ depends on the variables x and t for the case of n ¼ 2 or 3, the present situation is more complicated than that for the case of n ¼ 1. To this end, let Bðx; tÞ ¼ Z1 ðtÞZ2 ðx; tÞ where  n 1 h 1 Z t Z 1
v 2 þ Rh ðx; sÞdxds Z1 ðtÞ ¼ exp b u 0 0 n Z Z io ðn  1Þa n t 1 n þ r ðx; sÞv 2 ðx; sÞdxds ; nu

0 0 

ðn  1Þ Z2 ðx; tÞ ¼ exp b

Z tZ

L

r 0

n

ðy; sÞv ðy; sÞdyds : 2

x

Thus, from lemmas 6.3.1–6.3.3 and the Cauchy’s inequality, it follows that for any t  s  0; x 2 ½0; L, 0\Cd1  Dðx; tÞ  Cd ; 0\Cd1 

1 bu

0\Cd1 

ð6:3:15Þ

  Z 1 
2 v ðn  1Þa n n 2 þ Rh þ r v ðx; sÞdx  Cd ; n n 0

  v2 ðn  1Þa n n 2 þ Rh þ r v ðx; sÞdx n n 0 Z L

n  r ðy; sÞv 2 ðy; sÞdy  Cd ; þ ðn  1Þ 1 bu

ð6:3:16Þ

Z 1 


ð6:3:17Þ

0 0

1 ¼ Z2 ðL; tÞ  Z2 ðx; tÞ  Z2 ð0; tÞ; eCd t  Bðx; tÞ  eCd t ; 1

eCd ðtsÞ  Z11 ðtÞZ1 ðsÞ  eCd

ðtsÞ

ð6:3:18Þ

;

ð6:3:19Þ

eCd ðtsÞ  Z2 ðx; sÞ=Z2 ðx; tÞ  1;

ð6:3:20Þ 1

eCd ðtsÞ  Bðx; sÞ=Bðx; tÞ  Z1 ðsÞ=Z1 ðtÞ  eCd

ðtsÞ

:

ð6:3:21Þ

Hence, similarly as in [159], we may show uðx; tÞ  Cd ;

8ðx; tÞ 2 ½0; L  ½0; þ 1Þ:

ð6:3:22Þ

By lemma 6.3.3 and (6.3.18), we have uðx; tÞ  Dðx; tÞ=Bðx; tÞ  Cd1 eCd t ;

8ðx; tÞ 2 ½0; L  ½0; þ 1Þ:

ð6:3:23Þ

Attractors for Nonlinear Autonomous Dynamical Systems

124

Then using the asymptotic behaviour (6.2.5) in lemma 6.2.1, (6.3.23) and the similar contradiction argument as in [159], we show uðx; tÞ  Cd1 ; 8ðx; tÞ 2 ½0; L  ½0; þ 1Þ;

ð6:3:24Þ

which, with (6.3.23), (6.1.18) and (6.2.3), implies the estimates (6.3.12). Thus the proof is complete. h ð1Þ

Corollary 6.3.1. If ðu0 ; v0 ; h0 Þ 2 Hd , then Z t kvðsÞk2L1 ds  Cd ;

8t [ 0:

ð6:3:25Þ

0

Proof. The estimate (6.3.25) follows from lemmas 6.3.2 and 6.3.4, the Cauchy’s inequality, and the boundary condition (6.1.16) readily. This corollary is proved.h The next lemma will proves the boundedness of h from below. For this purpose, we need more delicate estimates to deal with the cases of n ¼ 2 and n ¼ 3. ð1Þ

Lemma 6.3.5. If ðu0 ; v0 ; h0 Þ 2 Hd , then Cd1  hðx; tÞ; 8ðx; tÞ 2 ½0; L  ½0; þ 1Þ:

ð6:3:26Þ

Proof. Let w ¼ 1h : From (6.1.19), we derive that 0

2ðn  2Þl 2ðn  1Þl \ \1 ðn  1Þb nb

ð6:3:27Þ

with b ¼ k þ 2l. Let e 2 ð0; 1Þ such that 2ðn  1Þl \e\1: nb

ð6:3:28Þ

0\ðn  1Þbe  2ðn  2Þl\nbe  2ðn  1Þl:

ð6:3:29Þ

Then

By a straightforward calculation, equation (6.1.14) can be rewritten as ( CV wt ¼ jðqr 2n2 wx Þx 

2jqr 2n2 hwx2 þ qw 2 ½ðn  1Þeb  2ðn  2Þl


 r 1 uv þ

2 ðbe  2lÞr n1 vx 2l½neb  2ðn  1Þlqw 2 r 2n2 vx2 þ ðn  1Þeb  2ðn  2Þl ðn  1Þeb  2ðn  2Þl  2 ) Rh R2 q : þ bð1  eÞqw 2 ðr n1 vÞx  þ 2bð1  eÞ 4ð1  eÞb ð6:3:30Þ

Global Attractors for the Compressible Navier–Stokes Equations

125

Multiplying (6.3.30) by 2mw 2m1 where m is an arbitrary natural number, then integrating the resulting equation over Xn ¼ ð0; LÞ; and noting that the expression in the bracket fg is nonnegative, using Hölder inequality and lemma 6.3.4, we conclude Z L d R2 kwðtÞk  qw 2m1 dx CV kwðtÞk2m1 L2m L2m dt 4ð1  eÞb 0 2m1  Cd kqkL2m kwðtÞk2m1 L2m  Cd kwðtÞkL2m ;

which result in that, when m ! þ 1; kwðtÞkL1  k1=h0 kL1 þ Cd t  Cd ð1 þ tÞ: Thus, for all x 2 ½0; L; t  0, hðx; tÞ 

1 : Cd ð1 þ tÞ

ð6:3:31Þ

Using the asymptotic behaviour in (6.2.5), the estimate (6.3.26) follows immediately, this proves the lemma. h ð1Þ

Lemma 6.3.6. For initial data belonging to a bounded set of Hd , there is t0 [ 0 depending only on boundedness of this bounded set such that for all t  t0 , x 2 ½0; L, d5 2d2 d3 2d4  uðx; tÞ  :  hðx; tÞ  ; 2LCV LCV 2L L

ð6:3:32Þ

Proof. Since the proof is similar to that of lemma 5.3.6 of chapter 5, see also [159] for the case n ¼ 1, we can omit the proof here. h Remark 6.3.1. From lemmas 6.3.1 and 6.3.6, we know that for initial data belonging ð1Þ ð1Þ to a given bounded set of Hd , the orbit will re-enter Hd after a finite time. ð1Þ

In the sequel, we shall prove the existence of an absorbing set in Hd . To this end, similarly to the discussions in [159] for the case of n ¼ 1, we introduce Eðu; v; SÞ ¼

v2 @e @e ðu; SÞðu  uÞ  ðu; SÞðS  SÞ; ð6:3:33Þ þ eðu; SÞ  eðu; SÞ  @u @S 2 S ¼ CV log h þ R log u;

eðu; SÞ ¼ CV h ¼ CV hðu; SÞ ¼ CV

(entropyÞ

expðS=CV Þ ; u R=CV

(internal energyÞ

ð6:3:34Þ ð6:3:35Þ

where S ¼ CV log h þ R log u

ð6:3:36Þ

Attractors for Nonlinear Autonomous Dynamical Systems

126

and u; h are defined in (6.2.6) in lemma 6.2.1. Since we have assumed that initial data ðu0 ; v0 ; h0 Þ belong to an arbitrarily ð1Þ bounded set of Hd , there is a positive constant B such that kðu0 ; v0 ; h0 ÞkH 1  B. We 0 00 use Cd;B or Cd;B ; Cd;B to denote universal positive constants depending on B and di ; ði ¼ 1; 2; . . .; 5Þ. Then, similarly to the proof of lemma 3.6 in [159], we have the following lemma. Lemma 6.3.7. There holds that v2 1 v2 þ þ Cd;B ðju  uj2 þ jS  Sj2 Þ: ðju  uj2 þ jS  Sj2 Þ  Eðu; v; SÞ  Cd 2 2

Proof. For the proof, see [159] for the case n ¼ 1.

ð6:3:37Þ h

Lemma 6.3.8. There exists a positive constant c1 ¼ c1 ðCd;B Þ [ 0 such that for any fixed c 2 ð0; c1 , there holds that Z t 2 2 2 2 ct e ðkvðtÞkH 1 þ kuðtÞ  ukH 1 þ khðtÞ  hkH 1 þ kSðtÞ  Sk Þ þ ecs ðkux k2 0

þ kqx k2 þ khx k2 þ kvx k2 þ kvxx k2 þ khxx k2 ÞðsÞds  Cd;B ; 8t [ 0;

ð6:3:38Þ

which leads to that for any fixed c 2 ð0; c1 , we have kðuðtÞ  u; vðtÞ; hðtÞ  hÞk2H 1  Cd;B ect ; 8t [ 0:

ð6:3:39Þ

Proof. Exploiting equations (6.1.12)–(6.1.14) and noting that u t ¼ 0; ht ¼ 0, we readily verify that ðq; v; SÞ fulfills " # qh jðr n1 hx Þ2 n1 2 bðr vÞx þ Et ðq ; v; SÞ þ h h 1

¼ ½bqðr n1 vÞðr n1 vÞx þ jð1  h=hÞqr 2n2 hx  Rðqh  qhÞr n1 vx  2ðn  1Þlð1  h=hÞðr n2 v 2 Þx ;

ð6:3:40Þ

½bðqx =qÞ2 =2 þ qx r 1n v=qt þ ðn  1Þr n v 2 qx =q þ Rhq2x =q ¼ qðr 1n vÞx ðr n1 vÞx  Rqx hx  ½qr 1n vðr n1 vÞx x

ð6:3:41Þ

with q ¼ 1=u. Multiplying (6.3.40), (6.3.41) by ect ; gect respectively, and adding the resulting equation up, we conclude

Global Attractors for the Compressible Navier–Stokes Equations

127

@ M ðtÞ þ ect hq½bððr n1 vÞx Þ2 þ jðr n1 hx Þ2 =h=h @t þ gect ½Rhq2x =q þ Rqx hx  qðr 1n vÞx ðr n1 vÞx þ ðn  1Þr n v 2 qx =q ¼ cect ½E þ gbðqx =qÞ2 =2 þ gqx r 1n v=q þ ect ½qðbr n1 v  gr 1n vÞðr n1 vÞx þ jð1  h=hÞqr 2n2 hx  Rðqh  qhÞr n1 vx  2ðn  1Þlð1  h=hÞðr n2 v 2 Þx ect

ð6:3:42Þ

with M ðtÞ ¼ ect ½E þ gbðqx =qÞ2 =2 þ gr 1n vqx =q. Integrating (6.3.42) over ½0; L  ½0; t, integrating the last term by parts, using lemmas 6.3.1–6.3.6, Young’s and Poincare’s inequalities, we infer that Z

L

M1 ðtÞ  ect

M ðtÞdx þ

0

þg

0

Z tZ 0

Z tZ

L

0

L 0

ecs hq½bððr n1 vÞx Þ2 þ jðr n1 hx Þ2 =hÞ=hðx; sÞdxds

ecs ðRhq2x =q þ Rqx hx  qðr 1n vÞx ðr n1 vÞx

þ ðn  1Þr n v 2 qx =qÞðx; sÞdxds Z tZ L  Cd;B þ c ecs ½ð1=2 þ gr 22n =2bÞv 2 þ gbðqx =qÞ2 þ Cd;B ðju  uj2 0 0 Z t þ jS  Sj2 Þdxds þ g ecs kvx k2 ðsÞds 0 Z t þ Cd g1 ecs kvk2L1 khx k2 ðsÞds:

ð6:3:43Þ

0

On the other hand, it follows from (6.1.18), (6.3.14), Poincaré’s inequality and lemma 6.3.1 that kðr 1n vÞx k  Cd kðr n1 vÞx k; kvk4  Cd kðr n1 vÞx k2 ;

ð6:3:44Þ

kvx k  Cd kðr n1 vÞx k;

ð6:3:45Þ

2 bqr n1 vxx ðr n1 vÞxx  Cd1 vxx  Cd;B ðv 2 þ vx2 þ ux2 Þ;

ð6:3:46Þ

qðr 2n2 hx Þx hxx  Cd1 h2xx  Cd h2x ;

ð6:3:47Þ


2n2  r hx htx  Cd1 h2tx  Cd ½v 2 h2x þ h2x ððr n1 vÞx Þ2 : u t

ð6:3:48Þ

Attractors for Nonlinear Autonomous Dynamical Systems

128

Thus, using lemmas 6.3.1–6.3.6 and Young’s inequality to get M1 ðtÞ  ect ½ð1=2  ga 22n =bÞkvk2 þ Cd1 ðku  uk2 þ kS  Sk2 Þ þ gbkqx =qk2 =4 Z tZ L þ ecs ½Cd1 ðh=h  Cd gÞððr n1 vÞx Þ2 0

0

þ ðjh=h  Rg=a2n2 Þqðr n1 hx Þ2 =h þ ghq2x =2qdxds  Cd1 ect ðkvðtÞk2 þ kuðtÞ  uk2 þ kSðtÞ  Sk2 þ kqx k2 þ kux k2 Þ Z t 1 þ Cd;B ecs ðkvx k2 þ khx k2 þ kqx k2 þ kux k2 Þds

ð6:3:49Þ

0

1 0 with having chosen g such that 0\1=2  ga 22n =b; Cd;B  g [ 0; h=h  Cd g  Cd;B  2n2 00 Cd g [ 0 and jh  Cd;B  Rg=a 2n2 [ 0. Thus exploiting the mean value h  Rg=a theorem, (6.3.33)–(6.3.37), lemmas 6.3.1–6.3.6 and Poincaŕe’s inequality, we can derive

kS  Sk  Cd ðku  uk þ kh  hkÞ  Cd ðkux k þ khx k þ kvx kÞ;

ð6:3:50Þ

kh  hk  Cd;B ðkS  Sk þ ku  ukÞ:

ð6:3:51Þ

and

Thus, by use of (6.3.45), (6.3.49)–(6.3.51), lemma 6.3.4 and the Poincaré’s inequality, we arrive at ect ðkvðtÞk2 þ kuðtÞ  uk2 þ kSðtÞ  Sk2 þ kqx ðtÞk2 þ kux ðtÞk2 Þ Z t þ ecs ðkvx k2 þ khx k2 þ kqx k2 þ kux k2 ÞðsÞds 0 Z t  Cd;B þ Cd;B c ecs ðkvx k2 þ khx k2 þ kux k2 þ kqx k2 ÞðsÞds 0 Z t þ Cd;B ecs kvk2L1 khx k2 ðsÞds; 0

which, with choosing c1 ¼ 2C1d;B , implies that for any fixed c 2 ð0; c1 , ect ðkvðtÞk2 þ kuðtÞ  uk2 þ kSðtÞ  Sk2 þ kqx ðtÞk2 þ kux ðtÞk2 Þ Z t þ ecs ðkvx k2 þ khx k2 þ kqx k2 þ kux k2 ÞðsÞds 0 Z t ecs kvk2L1 khx k2 ðsÞds:  Cd;B þ Cd;B

ð6:3:52Þ

0

On the other hand, equations (6.1.12)–(6.1.14) can be rewritten as follows qt þ q2 ðr n1 vÞx ¼ 0;

ð6:3:53Þ

Global Attractors for the Compressible Navier–Stokes Equations

129

vt  br n1 ½ðqðr n1 vÞx x þ Rr n1 ðqhÞx ¼ 0;

ð6:3:54Þ

CV ht  jðqr 2n2 hx Þx þ Rqhðr n1 vÞx  bqðr n1 vÞ2x þ 2lðn  1Þðr n2 v 2 Þx ¼ 0:

ð6:3:55Þ

Multiplying (6.3.53), (6.3.54), respectively, by ect vxx ; ect hxx , integrating the resulting equations over ½0; L  ½0; t, and adding up the final resultants, using Young’s inequality, the imbedding theorem, (6.3.52), (6.3.46) and (6.3.47), we finally deduce that Z t 1 ct 2 2 1 e ðkvx ðtÞk þ CV khx ðtÞk Þ þ Cd ecs ðkvxx k2 þ khxx k2 ÞðsÞds 2 0 Z t  Cd;B þ Cd;B ðcÞ ecs ðkvx k2 þ CV khx k2 ÞðsÞds 0 ( Z Z t

þ 0

L

0

 Rqhðr

ecs r n1 ½RðqhÞx  bqx ðr n1 vÞx vxx þ ½2lðn  1Þðr n2 v 2 Þx )

n1

vÞx  bqððr

Z  Cd;B þ Cd;B

t

n1

2

vÞx Þ  jqx r

2n2

hx hxx dxds

ecs ½ðkqx k þ khx k þ kqx kkvx k1=2 kvxx k1=2 Þkvxx k

0

þ ðkvx k þ kvx k3=2 kvxx k þ kqx kkhx k1=2 khxx k1=2 þ kqx kkhx kÞkhxx kds Z t þ Cd;B ðcÞ ecs ðkvx k2 þ kux k2 þ khx k2 ÞðsÞds 0 Z t Z t 1  Cd;B þ ecs ðkvxx k2 þ khxx k2 ÞðsÞds þ Cd;B ecs kvk2L1 khx k2 ðsÞds 2Cd 0 0 which, with Gronwall’s inequality, (6.3.52) and equations (6.1.13), (6.1.14), gives (6.3.38). h Thus, combining lemmas 6.3.1–6.3.7, we have the next theorem on existence of ð1Þ an absorbing ball in Hd . Theorem 6.3.9. B1 is an absorbing ball in t1 ðCd;B Þ  t0 such that when t þ CV2 d24 Þ CV2 L2

4ðd22

6.4

ð1Þ

Hd , i.e., there exists some

 t1 ; kðuðtÞ; vðtÞ; hðtÞÞk2H ð1Þ ð1Þ

 R21 .

t1 ¼

Here R1 ¼ R1 ðdÞ ¼

ð1Þ

and B1 ¼ fðu; v; hÞ 2 Hd ; kðu; v; hÞkH  R1 g.

ð2Þ

Existence of an Absorbing Set in Hd

ð2Þ

This section shall establishes the existence of an absorbing set in Hd . Similarly as in ð2Þ

chapters 4 and 5, we always assume that initial data belong to a bounded set in Hd ,

Attractors for Nonlinear Autonomous Dynamical Systems

130

i.e., kðu0 ; v0 ; h0 ÞkH ð2Þ  B where B is any given positive constant. Next, we first need to obtain the uniform estimates on H 2 norms of v and h. Lemma 6.4.1. There exists a positive constant c2 ¼ c2 ðCd;B Þ  c1 ðCd;B Þ satisfying that for any fixed c 2 ð0; c2 ; and for all t [ 0, khðtÞk2H 2 þ kvðtÞk2H 2 

2d22 þ Cd;B ect : CV2 L2

ð6:4:1Þ

Proof. First, a straight forward calculation shows ðr n1 vt Þx ¼ ðr n1 vÞtx  ðn  1Þðr n2 v 2 Þx :

ð6:4:2Þ

Clearly, from the Poincaré’s inequality and (6.1.17), (6.1.18) and (6.2.3), (6.3.14), it follows that kvt k  C kvtx k  Cd;B ðkðr n1 vÞx k þ kðr n1 vÞtx kÞ  Cd;B ðkvx k þ kðr n1 vÞtx kÞ:

ð6:4:3Þ

Differentiating equation (6.1.13) in t, multiplying the resulting equation by vt ect, integrating the resultant over ½0; L  ½0; t, and using lemmas 6.3.1 and 6.3.2, Young’s inequality to obtain Z t 1 ct 1 e kvt ðtÞk2 þ ecs kðr n1 vÞxt k2 ðsÞds 2 Cd 0 Z t Z t  Cd;B þ c=2 ecs kvt k2 ðsÞds þ Cd;B ecs ðkvx k2 þ kht k2 þ kvxx k2 Þds 0 0 Z t  Cd;B þ Cd;B c ecs kðr n1 vÞxt k2 ðsÞds 0 Z t ecs ðkht k2 þ kvx k2 þ kvxx k2 ÞðsÞds; þ Cd;B ðcÞ 0

which yields the existence of a positive constant c2 ¼ c2 ðCd;B Þ  c1 satisfying that for any fixed c 2 ð0; c2 ; Z t ect ðkvt ðtÞk2 þ kvxx ðtÞk2 Þ þ ecs kvxt k2 ðsÞds  Cd;B : ð6:4:4Þ 0

Similarly, using equation (6.1.14), lemma 6.3.8 and (6.3.48) to be able to get Z t ect ðkht ðtÞk2 þ khxx ðtÞk2 Þ þ ecs khxt k2 ðsÞds  Cd;B ; 0

which, together with (6.4.4) and lemma 6.3.8, gives (6.4.1). Thus the proof is complete. h

Global Attractors for the Compressible Navier–Stokes Equations

131

2 2 2 Corollary 6.4.1. Let t2 ¼ t2 ðCd;B Þ  maxðt1 ðCd;B Þ; c1 2 lnð2d2 =ðCV L Cd;B ÞÞÞ. Then estimate (6.4.1) implies that for any t  t2 ðCd;B Þ;

khðtÞk2H 2 þ kvðtÞk2H 2 

4d22 : CV2 L2

ð6:4:5Þ

The next lemma gives the uniform estimate of uðtÞ on H 2 . Lemma 6.4.2. There exists a positive constant c3 ¼ c3 ðCd;B Þ  c2 satisfying that for any fixed c 2 ð0; c3  and for all t [ 0, kuðtÞ  uk2H 2  Cd;B ect :

ð6:4:6Þ

Proof. Differentiating equation (6.1.13) in x, and by equation (6.1.12), we can deduce @ uxx huxx b þ 2 @t u u(   Rhx  bðr n1 vÞxx bðr n1 vÞx ux  Rhux 1n n þ ¼ r vtx þ ðn  1Þr u u u2 ) Rhxx 2bðr n1 vÞxx ux  2Rhx ux 2Rhux2  2bðr n1 vÞx ux2 ð6:4:7Þ þ þ þ : u u2 u3

Multiplying (6.4.7) by uxx =u, integrating the resulting equation over ½0; L, using Young’s inequality and lemmas 6.3.1–6.3.8 and 6.4.1, we can conclude d uxx 2 uxx 2 k k þ Cd1 k k dt u u 1 uxx 2 k þ þ Cd;B ðkhx k2 þ kux k2 þ kðr n1 vÞxx k2 þ khxx k2 þ kvtx k2 Þ: ð6:4:8Þ  k 2Cd u Multiplying (6.4.8) by et=2Cd and taking c such that c  c3  minðc2 ; 1=4Cd Þ and exploiting lemmas 6.3.4–6.3.8 and 6.4.1, we obtain      u ðtÞ 2  u ð0Þ 2  t=2Cd  xx   xx þ Cd;B ect  Cd;B ect ;    e   uðtÞ   uð0Þ  which, with lemmas 6.3.1–6.3.8, leads to (6.4.6). Thus this completes the proof. h h 2ð2d22 þ CV2 d24 Þ1=2 Now, letting R2 ¼ and t ¼ t ðC Þ  max t2 ðCd;B Þ; 3 3 d;B CV L 2 2 2 i 2ð2d þ C d Þ 2 V 4 c1 , then we immediately infer from corollary 6.4.1 and lemma 3 ln C 2 L2 V

6.4.2 the following theorem.

Attractors for Nonlinear Autonomous Dynamical Systems

132

Theorem 6.4.3. When t  t3 ; kðuðtÞ; vðtÞ; hðtÞÞk2H 2  R22 , which implies that the ball ð2Þ

ð2Þ

B2 ¼ fðu; v; hÞ 2 Hd ; kðuðtÞ; vðtÞ; hðtÞÞk2H 2  R22 g is an absorbing ball in Hd . ð2Þ

ð1Þ

Noticing that the existence of absorbing balls in Hd and Hd has been established, similarly as in chapters 4 and 5, we can exactly follow the abstract framework by Ghidaglia [51] to conclude the next two results. Lemma 6.4.4. The set xðB2 Þ ¼

\ [

ð6:4:9Þ

SðtÞB2

s0 t s

is included in B2 and nonempty, where the closures are taken in the weak topology of H ð2Þ . It is invariant by SðtÞ, i.e., SðtÞxðB2 Þ ¼ xðB2 Þ;

8t [ 0:

ð6:4:10Þ

Theorem 6.4.5. The set A2;d ¼ xðB2 Þ

ð6:4:11Þ

satisfies A2;d

is bounded and weakly closed in SðtÞA2;d ¼ A2;d ;

ð2Þ

Hd ;

ð6:4:12Þ

8t  0;

ð6:4:13Þ

lim d w ðSðtÞB; A2;d Þ ¼ 0:

ð6:4:14Þ

ð2Þ

for every bounded set B in Hd , t! þ 1

Moreover, it is the maximal set in the sense of inclusion that satisfies (6.4.12)– (6.4.14). ð2Þ

The set A2;d can be also called the universal attractor of SðtÞ in Hd . In order to ð1Þ

get an universal attractor in Hd , we plan to prove the next lemma. Lemma 6.4.6. For every t  0, the mapping SðtÞ is continuous on bounded sets of ð1Þ Hd for the topology induced by the norm in L2  L2  L2 . ð1Þ

Proof. Let ðu0i ; v0i ; h0i Þ 2 Hd ; ði ¼ 1; 2Þ, kðu0i ; v0i ; h0i ÞkH 1  R; ði ¼ 1; 2Þ, ðui ; vi ; hi Þ ¼ SðtÞðu0i ; v0i ; h0i Þ, and ðu; v; hÞ ¼ ðu1 ; v1 ; h1 Þ  ðu2 ; v2 ; h2 Þ. Subtracting the equations (6.1.12)–(6.1.14) satisfied by ðu1 ; v1 ; h1 Þ and ðu2 ; v2 ; h2 Þ, multiplying the resulting equations by u; v; h, respectively, adding them up and integrating over ½0; L, we finally arrive at

Global Attractors for the Compressible Navier–Stokes Equations

133

Z L 1d bðr1n1 vÞ2x þ jr12n2 h2x ðkuk2 þ kvk2 þ CV khk2 Þ þ dx 2 dt u1 0  Z L h
n1 bðr1 v2 Þx  Rh2 ¼ þ 1 uðr1n1 vÞx u1 u2 0 i R þ bðr1n1 v1 Þx þ bðr2n1 v2 Þx  Rh2 n1 þ ðr1 vÞx h dx u1 Z 1 jh2x r22n2 ½Rh2  bðr2n1 v2 Þx ðr2n1 v2 Þx Rðr1n1 v1 Þx 2 þ uhx  uh  h dx u1 u2 u1 u2 u1 0 Z 1 Z 1 n1 Rh  bðr v Þ 2 2 2 x þ 2lðn  1Þ r1n2 ðv1 þ v2 Þvhx dx þ ½ðr1n1  r2n1 Þvx dx u2 0 0   Z 1 
n1 bðr1 v1 Þx bðr2n1 v2 Þx  Rh2 bðr1n1 vÞx h þ þ u ½ðr1n1  r2n1 Þv2 x dx þ u1 u2 u2 0  Z 1 2n2 2n2 jh2x ðr1  r2 Þhx   2lðn  1Þv22 ðr1n2  r2n2 Þhx dx; ð6:4:15Þ u1 0

where

Z

ri ðx; tÞ ¼ r0i ðxÞ þ

t

Z vi ðx; sÞds;

x

u0i ðyÞdyÞ1=n

ð6:4:16Þ

8ðx; tÞ 2 ½0; L  ½0; þ 1Þ:

ð6:4:17Þ

r0i ðxÞ ¼ ða n þ n

0

0

and rin1 ðx; tÞrix ðx; tÞ ¼ ui ðx; tÞ;

i ¼ 1; 2;

Therefore, we can derive from lemma 6.2.1 that for any t [ 0, Z t ðkvi k2H 2 þ khix k2 þ khixx k2 ÞðsÞds  Cd ðRÞ; i ¼ 1; 2 kðui ðtÞ; vi ðtÞ; hi ðtÞÞk2H 1 þ 0

ð6:4:18Þ where the constant Cd ðRÞ [ 0 depends only on R and d: Obviously, it follows from (6.1.17), (6.1.18) and (6.2.7) that Z x rin ðx; tÞ ¼ r n ðxÞ þ n ðui ðyÞ  uÞdy; i ¼ 1; 2; 0

which yields that for any ðx; tÞ 2 ½0; L  ½0; þ 1Þ; Z x r1n ðx; tÞ  r2n ðx; tÞ ¼ n uðy; tÞdy; ðr1n ðx; tÞ  r2n ðx; tÞÞx ¼ nuðx; tÞ:

ð6:4:19Þ

0

Applying the mean value theorem to the function gðzÞ ¼ z k=n over where k [ 0 is any given constant, and using (6.4.19) and lemma 6.2.1, we can conclude

½r1n ; r2n  ½a n ; bn 

Attractors for Nonlinear Autonomous Dynamical Systems

134

jr1k ðx; tÞ  r2k ðx; tÞj  jg 0 ðz0 Þjjr1n ðx; tÞ  r2n ðx; tÞj  Cd kuðtÞkL1 ;

ð6:4:20Þ

where z0 is a point between r1n and r2n (therefore, z0 2 ½a n ; bn ), and by lemma 6.2.1, jg 0 ðz0 Þj  Cd . In the same manner, again applying the mean value theorem to the function hðzÞ ¼ z n=k over ½r1k ; r2k  ½a k ; bk , we can derive from lemma 6.2.1, (6.4.19) and (6.4.20) that   nuðx; tÞ ¼ ½hðr1k Þ  hðr2k Þx ¼ h 0 ðz1 Þ r1k ðx; tÞ  r2k ðx; tÞ r2k ðx; tÞx þ h 0 ðr1k Þðr1k  r2k Þx ; where z1 is a point between r1k and r2k , hence, z1 2 ½a k ; bk . By lemmas 6.2.1 and 6.3.4, 0\Cd1  jh 0 ðz1 Þj  Cd ; j½h 0 ðr1 Þx j  Cd . Therefore,    nuðx; tÞ  h 0 ðz Þðr k ðx; tÞ  r k ðx; tÞÞr k ðx; tÞ   1 k k 1 2 2 x jðr1  r2 Þx j ¼     h 0 ðz1 Þ ð6:4:21Þ  Cd ðjuðx; tÞj þ kuðtÞkL1 Þ: By exploiting Young’s inequality and the embedding theorem, from (6.3.45), (6.4.20) and (6.4.21) it thus follows that d ðkuðtÞk2 þ kvðtÞk2 þ CV khðtÞk2 Þ þ Cd1 ðkvx ðtÞk2 þ kðr1n1 vðtÞÞx k2 þ khx ðtÞk2 Þ dt 1  ðkvx ðtÞk2 þ kðr1n1 vðtÞÞx k2 þ khx ðtÞk2 Þ 2Cd þ Cd ðRÞH ðtÞðkuðtÞk2 þ kvðtÞk2 þ khðtÞk2 Þ; ð6:4:22Þ

where H ðtÞ ¼ kðr1n1 v2 Þx k2L1 þ kðr2n1 v2 Þx k2L1 þ kðr1n1 v1 Þx k2 þ kv2x k2L1 ð6:4:23Þ

þ kh2 k2L1 þ kv2 k4L4 þ kv1 k2L1 þ 1

Rt fulfilling 0 H ðsÞds  Cd ðRÞð1 þ tÞ for any t [ 0. Now employing the Gronwall’s inequality and (6.4.23), we therefore proves this lemma. h Now in the manner as in chapters 4 and 5, we can also prove the existence of an ð1Þ universal attractor in Hd in the next theorem. Theorem 6.4.7. The set A1;d ¼

\ [

SðtÞB1

ð6:4:24Þ

s0 ts ð1Þ

is the (maximal) universal attractor in Hd . Here the closures are taken in the weak topology of H ð1Þ . Proof. Having proved lemmas and theorems in this section, we can follow the line of the proof in [158] to prove this theorem. h

Global Attractors for the Compressible Navier–Stokes Equations

135

Remark 6.4.1. Since A2;d is bounded in H ð2Þ , it is bounded in H ð1Þ and by the invariance property (6.4.13), we have A2;d A1;d :

ð6:4:25Þ

On the contrary, if we knew that A1;d is bounded in H ð2Þ , then the opposite inclusion would hold.

6.5

ð4Þ

Existence of an Absorbing Set in Hd ð4Þ

This section will establish an absorbing set in Hd . For this purpose, we assume all the time in this section that initial data belong to an any given bounded set B4 in ð4Þ Hd , and thus there exists a positive constant R4  R2  R1 such that kðu0 ; v0 ; h0 ÞkH ð4Þ  R4 . We start the next lemma. Lemma 6.5.1. If ðu0 ; v0 ; h0 Þ 2 H ð4Þ , then there exists a unique global solution ðuðtÞ; vðtÞ; hðtÞÞ 2 C ð½0; þ 1Þ; H ð4Þ Þ to problem (6.1.12)–(6.1.17), which defines a nonlinear C0 -semigroup SðtÞ on H ð4Þ with SðtÞðu0 ; v0 ; h0 Þ ¼ ðuðtÞ; vðtÞ; hðtÞÞ satisfying kvtx ðx; 0Þk þ khtx ðx; 0Þk þ kvtt ðx; 0Þk þ khtt ðx; 0Þk þ kvtxx ðx; 0Þk þ khtxx ðx; 0Þk  C4 ; 8x 2 ½0; L;

ð6:5:1Þ

kuðtÞ  uk2H 4 þ kvðtÞk2H 4 þ khðtÞ  hk2H 4  C4 ; 8t [ 0;

ð6:5:2Þ

Z

t

0

ðku  uk2H 4 þ kvk2H 5 þ kh  hk2H 5 ÞðsÞds  C4 ; 8t [ 0:

ð6:5:3Þ h

ð4Þ

ðu0 ; v0 ; h0 Þ 2 B4 Hd , there exists a positive constant

Lemma 6.5.2. If ð1Þ

ð1Þ

ð1Þ

c4 ¼

c4 ðCd;B4 Þ  ^c3 ðCd;B2 Þ such that for any fixed c 2 ð0; c4 , the following inequalities hold for any t [ 0, Z t 2 ct e kvtt ðtÞk þ ecs kvttx k2 ðsÞds 0 Z t ð6:5:4Þ  Cd;B4 þ Cd;B4 ecs ðkhtxx k2 þ kvtxx k2 ÞðsÞds; 0

Z 2

t

1

cs

2

Z

t

e khttx k ðsÞds  Cd;B4 ðÞ þ Cd;B4  ecs khtxx k2 ðsÞds e khtt ðtÞk þ 0 0 Z t þ Cd;B4  ecs kvttx k2 ðsÞds þ Cd;B4 ect sup kvtx ðsÞk2 ct

0

with  2 ð0; 1Þ small enough.

0st

ð6:5:5Þ

Attractors for Nonlinear Autonomous Dynamical Systems

136

Proof. Differentiating (6.1.13) in t twice, multiplying the resulting equation by vtt in L2 ð0; LÞ, integrating by parts, and exploiting Poincaré’s inequality, we obtain that for any  [ 0, d 1 kvtt ðtÞk2   ðCd;B  Þkvttx ðtÞk2 þ Cd;B4 ðÞðkhx ðtÞk2 þ kvxx ðtÞk2 þ kux ðtÞk2 4 dt ð6:5:6Þ þ kvt ðtÞk2 þ kvxt ðtÞk2 þ kvtt ðtÞk2 þ kht ðtÞk2 þ khtt ðtÞk2 Þ:

Multiplying (6.5.6) by ect, we easily arrive at 2

e kvtt ðtÞk  Cd;B4  ct

1 ðCd;B 4

Z þ Cd;B4 ðÞ

t

Z

t

   Cd;B4 cÞ

ecs kvttx k2 ðsÞds

0

ecs ðkux k2 þ kvt k2 þ kvxx k2 þ kvtx k2 þ kvtt k2

0

ð6:5:7Þ

þ khx k2 þ kht k2 þ khtt k2 ÞðsÞds:

  h i Thus choosing c and  such that   min 1; 4C1d;B and 0\c  min 4C12 ; ^c3 ðCd;B4 Þ , 4

d;B4

we can derive estimate (6.5.4) from (6.5.7). Similarly, differentiating (6.1.14) in t twice, multiplying the resulting equation by htt in L2 ð0; LÞ, integrating by parts, using Poincaré’s inequality and the embedding theorem and noting that kðr n1 vÞxt ðtÞk  Cd;B4 ðkvx ðtÞk þ kvtx ðtÞkÞ; kðr n1 vÞxtt ðtÞk  Cd;B4 ðkvx ðtÞk þ kvtx ðtÞk þ kvttx ðtÞkÞ; k½ðn  2Þr n3 rx v 2 þ 2r n2 vvx tt ðtÞk  Cd;B4 ðkvx ðtÞk þ kvtx ðtÞk þ kvttx ðtÞk þ kvtx ðtÞk2 Þ; we conclude that for any  2 ð0; 1Þ, d 1  Þkhttx ðtÞk2 þ kvttx ðtÞk2 þ Cd;B4 1 ðkhx ðtÞk2 þ khtx ðtÞk2 khtt ðtÞk2   ðCd;B 4 dt þ kvx ðtÞk2 þ kvtx ðtÞk2 þ kht ðtÞk2 þ khtt ðtÞk2 þ khtxx ðtÞk2 Þ þ Cd;B4 khtt ðtÞkkvtx ðtÞk2 :

ð6:5:8Þ

Global Attractors for the Compressible Navier–Stokes Equations

137

Multiplying (6.5.8) by ect, we have Z t ect khtt ðtÞk2  Cd;B4 þ Cd;B4 c ecs ðkvx k2 þ kvtx k2 þ kht k2 þ khx k2 þ khtx k2 0 Z t 1 þ khxx k2 þ khtxx k2 ÞðsÞds  ðCd;B  Þ ecs khttx k2 ðsÞds 4 0 Z t 2 2 2 1 cs e ðkhx k þ khtx k þ kht k þ khtt k2 þ Cd;B4  0 Z t 2 þ kvx k þ kvtx k2 ÞðsÞds þ Cd;B4  ecs kvttx k2 ðsÞds
Z

t

þ Cd;B4

0

1=2 ecs khtt k2 ðsÞds

0

Z

e

c 2t


Z

t

sup kvtx ðsÞk

0st

1=2 kvtx k2 ðsÞds

0

t

 Cd;B4 ðÞ þ Cd;B4 ð1 þ cÞ ecs khtxx k2 ðsÞds 0 Z t Z t 2 1 cs  ðCd;B4  Þ e khttx k ðsÞds þ Cd;B4  ecs kvttx k2 ðsÞds 0

0

þ ect sup kvtx ðsÞk2 0st


 ð1Þ 1 implying estimate (6.5.5) after taking 0\c  min 1; min 4C 2 ; ^c3 ðCd;B2 Þ  c4 d;B4 h i and  [ 0 small enough for example; 0\  min 2C1d;B ; 1 . This proves the 4

h

lemma. ð4Þ

ð1Þ

Lemma 6.5.3. For any ðu0 ; v0 ; h0 Þ 2 B4 Hd and for any fixed c 2 ð0; c4 , there holds that for any t [ 0, Z t ect kvtx ðtÞk2 þ ecs kvtxx k2 ðsÞds 0 Z t 6 ð6:5:9Þ  Cd;B4  þ Cd;B4 2 ecs ðkhtxx k2 þ kvtxxx k2 ÞðsÞds; 0

Z ect khtx ðtÞk2 þ

t

ecs khtxx k2 ðsÞds Z t  Cd;B4 6 þ Cd;B4 2 ecs ðkvtxx k2 þ khtxxx k2 ÞðsÞds 0

ð6:5:10Þ

0

with  2 ð0; 1Þ small enough. Proof. Differentiating (6.1.13) in x and t, multiplying the resultant by vtx in L2 ð0; LÞ, and integrating by parts, we have

Attractors for Nonlinear Autonomous Dynamical Systems

138

1d kvtx ðtÞk2 ¼ I0 ðtÞ þ I1 ðtÞ 2 dt with

ð6:5:11Þ



 bðr n1 vÞx  Rh I0 ðtÞ ¼ r vtx jx¼L x¼0 ; u x t  n1 Z L bðr vÞx  Rh I1 ðtÞ ¼  r n1 vtxx dx: u 0 x t 

n1

Using Sobolev’s interpolation inequality, and using the fact that for any ðx; tÞ 2 ½0; L  ½0; þ 1Þ, jðr n1 vÞtxx ðx; tÞj  Cd;B4 ðjvðx; tÞj þ jvx ðx; tÞj þ jvt ðx; tÞj þ jux ðx; tÞj þ jvxx ðx; tÞj þ jvtx ðx; tÞj þ jvtxx ðx; tÞjÞ; we derive that I0  Cd;B4 ðkvxx ðtÞkL1 þ kux ðtÞkL1 þ khx ðtÞkL1 þ khtx ðtÞkL1 þ kht ðtÞkL1 þ kvtxx ðtÞkÞkvtx ðtÞkL1  Cd;B4 ðkvxx ðtÞk1=2 kvxxx ðtÞk1=2 þ kvxx ðtÞk þ kux ðtÞk1=2 kuxx ðtÞk1=2 þ kux ðtÞk þ khx ðtÞk1=2 khxx ðtÞk1=2 þ khx ðtÞk þ kht ðtÞk1=2 khtx ðtÞk1=2 þ kht ðtÞk þ khtx ðtÞk þ khtx ðtÞk1=2 khtxx ðtÞk1=2 þ kvtxx ðtÞk1=2 kvtxxx ðtÞk1=2 þ kvtxx ðtÞkÞ  ðkvtx ðtÞk1=2 kvtxx ðtÞk1=2 þ kvtx ðtÞkÞ  Cd;B4 ðI01 þ I02 Þðkvtx ðtÞk1=2 kvtxx ðtÞk1=2 þ kvtx ðtÞkÞ;

ð6:5:12Þ

where I01 ¼ kvxx ðtÞk1=2 kvxxx ðtÞk1=2 þ kvxx ðtÞk þ kux ðtÞk1=2 kuxx ðtÞk1=2 þ kux ðtÞk þ khx ðtÞk1=2 khxx ðtÞk1=2 þ khx ðtÞk þ kht ðtÞk1=2 khtx ðtÞk1=2 þ kht ðtÞk þ khtx ðtÞk and I02 ¼ khtx ðtÞk1=2 khtxx ðtÞk1=2 þ kvtxx ðtÞk1=2 kvtxxx ðtÞk1=2 þ kvtxx ðtÞk: Using Young’s inequality several times, we know that for any  2 ð0; 1Þ, Cd;B4 I01 ðkvtx ðtÞk1=2 kvtxx ðtÞk1=2 þ kvtx ðtÞkÞ 2 2 kvtxx ðtÞk2 þ Cd;B4 2 ðI01 Þ4=3 kvtx ðtÞk2=3 þ Cd ðI01 þ kvtx ðtÞk2 Þ 2 2  kvtxx ðtÞk2 þ Cd;B4 2 ðkvxx ðtÞk2H 1 þ kux ðtÞk2H 1 þ khx ðtÞk2H 1 þ khtx ðtÞk2 2 ð6:5:13Þ þ kht ðtÞk2 þ kvtx ðtÞk2 Þ 

Global Attractors for the Compressible Navier–Stokes Equations

139

and Cd;B4 I02 ðkvtx ðtÞk1=2 kvtxx ðtÞk1=2 þ kvtx ðtÞkÞ 



2 kvtxx ðtÞk2 þ Cd;B4 2=3 khtx ðtÞk2=3 khtxx ðtÞk2=3 kvtx ðtÞk2=3 2 2 þ Cd;B4 2 ðkvtx ðtÞkkvtxxx ðtÞk þ kvtx ðtÞk2 Þ þ khtxx ðtÞk2 2 þ Cd;B4 2=3 ðkhtx ðtÞk2=3 kvtx ðtÞk4=3 þ kvtx ðtÞk4=3 kvtxxx ðtÞk2=3 Þ þ Cd;B4 6 kvtx ðtÞk2 2 kvtxx ðtÞk2 þ 2 ðkhtxx ðtÞk2 þ kvtxxx ðtÞk2 Þ 2 þ Cd;B4 6 ðkhtx ðtÞk2 þ kvtx ðtÞk2 Þ:

ð6:5:14Þ

Thus combining (6.5.11)–(6.5.14) gives I0  2 ðkvtxx ðtÞk2 þ kvtxxx ðtÞk2 þ khtxx ðtÞk2 Þ þ Cd;B4 6 ðkux ðtÞk2H 1 þ kvxx ðtÞk2H 1

ð6:5:15Þ

þ khx ðtÞk2H 1 þ khtx ðtÞk2 þ kht ðtÞk2 þ kvtx ðtÞk2 Þ:

Similarly, we get that for any  2 ð0; 1Þ, Z I1   0

L

br n1 ðr n1 vÞtxx vtxx dx þ 2 kvtxx ðtÞk2 þ Cd;B4 2 ðkðr n1 vÞxx ðtÞk2 þ khx ðtÞk2 u

þ kux ðtÞk2 þ khtx ðtÞk2 þ kðr n1 vÞxt ðtÞk2 þ kht ðtÞk2 þ kðr n1 vÞx ðtÞk2 Þ Z L 2n2 2 r vtxx dx þ 2 kvtxx ðtÞk2 þ Cd;B4 2 ðkux ðtÞk2 þ kvxx ðtÞk2 þ kvxt ðtÞk2  b u 0 þ khx ðtÞk2 þ kht ðtÞk2 þ khtx ðtÞk2 Þ:

ð6:5:16Þ

Multiplying (6.5.11) by ect and exploiting (6.5.15) and (6.5.16), we obtain Z tZ L Z 2 1 ct r 2n2 vtxx c t cs e kvtx ðtÞk2 þ b dxds  Cd;B4 þ ecs e kvtx k2 ðsÞds 2 2 u 0 0 0 Z t Z t Z t 2 cs 2 cs 2 jI0 je ds þ  e kvtxx k ðsÞds þ Cd;B4  ecs ðkux k2 þ kvxx k2 þ 0

0

0

þ khx k2 þ kvtx k2 þ kht k2 þ khtx k2 ÞðsÞds Z t Z t 2 2 2 2 cs 6  Cd;B4 þ Cd;B4  e ðkvtxx k þ khtxx k þ kvtxxx k ÞðsÞds þ Cd;B4  ecs ðkux k2H 1 0

0

þ kvtx k2 þ kvxx k2H 1 þ kht k2 þ khx k2H 1 þ khtx k2 ÞðsÞds Z t  Cd;B4 6 þ Cd;B4 2 ecs ðkvtxx k2 þ khtxx k2 þ kvtxxx k2 ÞðsÞds; 0 ð1Þ

which gives estimate (6.5.9) for any fixed c 2 ð0; c4  and  2 ð0; 1Þ small enough. Similarly, we can show (6.5.10), which completes the lemma. h ð1Þ

Lemma 6.5.4. For any ðu0 ; v0 ; h0 Þ 2 B4 H ð4Þ and for any fixed c 2 ð0; c4 , there holds that for any t [ 0,

Attractors for Nonlinear Autonomous Dynamical Systems

140

Z 2

2

t

e ðkvtx ðtÞk þ khtx ðtÞk Þ þ ecs ðkvtxx k2 þ khtxx k2 ÞðsÞds 0 Z t 6 2 ecs ðkhttx k2 þ kvttx k2 ÞðsÞds  Cd;B4  þ Cd;B4  ct

ð6:5:17Þ

0

with  2 ð0; 1Þ small enough. Proof. Differentiating (6.1.13) in t and x, we arrive at  n1   bðr vÞx  Rh br n1 ðr n1 vÞtxxx þ DðtÞ; ¼ vttx ¼ r n1 u u x tx

ð6:5:18Þ

where kDk  Cd;B4 ðkvxx ðtÞkH 1 þ khx ðtÞkH 1 þ kux ðtÞkH 1 þ khtx ðtÞk þ kht ðtÞk þ khtxx ðtÞk þ kvtxx ðtÞkÞ:

ð6:5:19Þ

But by a simple calculation with (6.5.18) and (6.5.19), we get kvtxxx ðtÞk  Cd;B4 kðr n1 vÞtxxx ðtÞk þ Cd;B4 ðkvxx ðtÞk þ kvtxx ðtÞk þ kvxxx ðtÞkÞ  Cd;B4 ðkvttx ðtÞk þ kDðtÞkÞ þ Cd;B4 ðkvxx ðtÞk þ kvtxx ðtÞk þ kvxxx ðtÞkÞ  Cd;B4 kvttx ðtÞk þ Cd;B4 ðkvxx ðtÞkH 1 þ khx ðtÞkH 1 þ kux ðtÞkH 1 þ khtx ðtÞk

ð6:5:20Þ

þ kht ðtÞk þ kvtxx ðtÞk þ khtxx ðtÞkÞ:

Analogously, we can derive khtxxx ðtÞk  Cd;B4 khttx ðtÞk þ Cd;B4 ðkux ðtÞk þ kvxx ðtÞkH 1 þ kvtxx ðtÞk þ khx ðtÞkH 2 þ khtx ðtÞk þ kht ðtÞk þ khtxx ðtÞkÞ:

ð6:5:21Þ

Adding (6.5.9) to (6.5.10) and choosing  2 ð0; 1Þ small enough, we obtain Z t ect ðkvtx ðtÞk2 þ khtx ðtÞk2 Þ þ ecs ðkvtxx k2 þ khtxx k2 ÞðsÞds 0 Z t  Cd;B4 6 þ Cd;B4 2 ecs ðkhtxxx k2 þ kvtxxx k2 ÞðsÞds; 0

which, combined with (6.5.20) and (6.5.21) and choosing  2 ð0; 1Þ suitably small, gives the estimate (6.5.17). This proves the lemma. h ð4Þ

ð2Þ

Lemma 6.5.5. For any ðu0 ; v0 ; h0 Þ 2 B4 Hd , there is a positive constant c4 ¼ ð2Þ

ð1Þ

ð2Þ

c4 ðCd;B4 Þ  c4 such that for any fixed c 2 ð0; c4 , there holds that for any t [ 0, Z t 2 2 2 2 ct e ðkvtt ðtÞk þ kvtx ðtÞk þ khtt ðtÞk þ khtx ðtÞk Þ þ ecs ðkvttx k2 þ kvtxx k2 0

þ khttx k2 þ khtxx k2 ÞðsÞds  Cd;B4 ;

ð6:5:22Þ

Global Attractors for the Compressible Navier–Stokes Equations Z e

ct

kuxxx ðtÞk2H 1

þ

t

0

141

ecs kuxxx k2H 1 ÞðsÞds  Cd;B4 ;

ð6:5:23Þ

ect ðkvxxx ðtÞk2H 1 þ khxxx ðtÞk2H 1 þ kutxxx ðtÞk2 þ kvtxx ðtÞk2 þ khtxx ðtÞk2 Þ Z t þ ecs ðkvxxxx k2H 1 þ kvtxx k2H 1 þ khxxxx k2H 1 0

ð6:5:24Þ

þ khtxx k2H 1 þ kutxxx k2H 1 ÞðsÞds  Cd;B4 :

Proof. Multiplying (6.5.4) and (6.5.5) by  and 3=2 respectively, adding the resulting inequality to (6.5.17), and then choosing e [ 0 suitably small, we can derive the estimate (6.5.22). Differentiating (6.1.13) in x, and using (6.1.12), we infer ( " Rhx  bðr n1 vÞxx @ uxx Rhuxx 1n n b ¼ r vtx þ ðn  1Þr u þ 2 u u @t u # bðr n1 vÞx ux  Rhux Rhxx þ þ u2 u 2bðr n1 vÞxx ux  2Rhx ux 2Rhux2  2bðr n1 vÞx ux2 þ þ 2 u u3

ð6:5:25Þ

 r 1n vtx þ Eðx; tÞ:

Differentiating (6.5.25) in x, we have @ uxxx Rhuxxx ¼ E1 ðx; tÞ b þ @t u u2 with

)

ð6:5:26Þ

 n1  ðr vÞxxx ux þ uxx ðr n1 vÞxx 2ux uxx ðr n1 vÞx  u2 u3 hx uxx 2Rhux uxx  2 þ þ ð1  nÞr 12n uvtx þ r 1n vtxx þ Ex ðx; tÞ: u u3

E1 ðx; tÞ ¼ b

Obviously, it follows from (6.5.22) that kE1 k  Cd;B4 ðkux ðtÞkH 1 þ kvxx ðtÞkH 1 þ khx ðtÞkH 2 þ kvtx ðtÞkH 1Þ;

ð6:5:27Þ

which yields Z 0

t

kE1 k2 ðsÞds  Cd;B4 ;

8t [ 0:

ð6:5:28Þ

Attractors for Nonlinear Autonomous Dynamical Systems

142

uxxx u

Now multiplying (6.5.26) by

in L2 ð0; LÞ, we obtain

d uxxx 2 1 uxxx 2 k k þ Cd;B k  Cd;B4 kE1 k2 : k 4 dt u u

ð6:5:29Þ

Multiplying (6.5.29) byi ect, using (6.5.27), (6.5.22), and taking c [ 0 such that h ð2Þ ð1Þ 0\c  c4  min 2C1d;B1 ; c4 , we derive that for any t [ 0, Z t Z t uxxx 1 uxxx 2 ðtÞk2 þ k ðsÞds  Cd;B4 þ Cd;B4 ecs k ecs kE1 ðsÞk2 ds 2Cd;B4 0 u u 0 Z t  Cd;B4 þ Cd;B4 ecs ðkux k2H 1 þ kvxx k2H 1 þ khx k2H 2 þ kvtx k2H 1 ÞðsÞds  Cd;B4 :

ect k

0

Thus, Z ect kuxxx ðtÞk2 þ

t

ecs kuxxx k2 ðsÞds  Cd;B4 ;

8t [ 0:

ð6:5:30Þ

kvxxx ðtÞk  Cd;B4 ðkvx ðtÞk þ kux ðtÞkH 1 þ khx ðtÞkH 1 þ kvtx ðtÞkÞ:

ð6:5:31Þ

0

Differentiating (6.1.13) in x, we obtain

Differentiating (6.1.13) in x twice, employing the embedding theorem, we conclude kvxxxx ðtÞk  Cd;B4 ðkux ðtÞkH 2 þ kvx ðtÞkH 2 þ khx ðtÞkH 2 þ kvtxx ðtÞkÞ:

ð6:5:32Þ

Similarly, it follows from (6.1.14) that khxxx ðtÞk  Cd;B4 ðkhx ðtÞkH 1 þ kvx ðtÞkH 1 þ kuxx ðtÞk þ khtx ðtÞkÞ;

ð6:5:33Þ

khtxx ðtÞk  Cd;B4 ðkux ðtÞkH 2 þ kvx ðtÞkH 2 þ khx ðtÞkH 3 Þ;

ð6:5:34Þ

khxxxx ðtÞk  Cd;B4 ðkux ðtÞkH 2 þ kvx ðtÞkH 2 þ khx ðtÞkH 2 þ khtxx ðtÞkÞ:

ð6:5:35Þ

Thus it follows from (6.5.30)–(6.5.33), (6.5.35) and (6.5.22) that Z t 2 2 ct e ðkvxxx ðtÞk þ khxxx ðtÞk Þ þ ecs ðkvxxx k2H 1 þ khxxx k2H 1 ÞðsÞds  Cd;B4 ; 8t [ 0: 0

ð6:5:36Þ Differentiating (6.1.13) in t and using (6.5.22), we obtain kvtxx ðtÞk  Cd;B4 kvtt ðtÞk þ Cd;B4 ðkux ðtÞk þ kvxx ðtÞk þ kvtx ðtÞk þ khx ðtÞk þ kht ðtÞk þ khtx ðtÞkÞ  Cd;B4 ; 8t [ 0;

ð6:5:37Þ

Global Attractors for the Compressible Navier–Stokes Equations

which, together with (6.5.32), implies Z t 2 ct e kvxxxx ðtÞk þ ecs ðkvtxx k2 þ kvxxxx k2 ÞðsÞds  Cd;B4 ; 8t [ 0:

143

ð6:5:38Þ

0

Similarly, we can derive from (6.5.30), (6.5.33), (6.5.34) and (6.5.36) that Z t ect ðkhtxx ðtÞk2 þ khxxxx ðtÞk2 Þ þ ecs ðkhtxx k2 þ khxxxx k2 ÞðsÞds  Cd;B4 ; 8t [ 0: ð6:5:39Þ 0

Now differentiating (6.5.26) in x gives @ uxxxx Rhuxxxx b ¼ E2 ðx; tÞ; þ @t u u2 where

ð6:5:40Þ

 n1  ðr vÞxx uxxx þ ux ðr n1 vÞxxxx 2ux ðr n1 vÞx uxxx E2 ðx; tÞ ¼ b  u2 u3 2Rhux uxxx Rhx uxxx þ  þ E1x ðx; tÞ: u3 u2

Appealing to the embedding theorem, (6.5.25) and (6.5.30), (6.5.36), (6.5.38) and (6.5.39), we can conclude kExx ðtÞk  Cd;B4 ðkhx ðtÞkH 3 þ kux ðtÞkH 2 þ kvx ðtÞkH 3 Þ; which, combined with the expressions of E1 and E2 , yields kE1x ðtÞk  Cd;B4 ðkvx ðtÞkH 3 þ kux ðtÞkH 2 þ kvtx ðtÞkH 2 þ kExx ðtÞkÞ  Cd;B4 ðkvx ðtÞkH 3 þ kux ðtÞkH 2 þ kvtx ðtÞkH 2 þ khx ðtÞkH 3Þ;

ð6:5:41Þ

kE2 ðtÞk  Cd;B4 ðkvx ðtÞkH 3 þ kux ðtÞkH 2 þ kE1x ðtÞkÞ  Cd;B4 ðkvx ðtÞkH 3 þ kux ðtÞkH 2 þ kvtx ðtÞkH 2 þ khx ðtÞkH 3Þ:

ð6:5:42Þ

On the other hand, it follows from (6.5.21) and (6.5.22) that Z t ecs ðkvtxxx k2 þ khtxxx k2 ÞðsÞds  Cd;B4 ; 8t [ 0:

ð6:5:43Þ

0

Thus combining (6.5.20), (6.5.38), (6.5.39), (6.5.42) and (6.5.43) yields Z t ecs kE2 k2 ðsÞds  Cd;B4 ; 8t [ 0: ð6:5:44Þ 0

Multiplying (6.5.40) by

uxxxx u

in L2 ð0; LÞ to get

d uxxxx 1 uxxxx k ðtÞk2 þ Cd;B ðtÞk2  Cd;B4 kE2 ðtÞk2 : k 4 dt u u

ð6:5:45Þ

Attractors for Nonlinear Autonomous Dynamical Systems

144

Multiplying (6.5.45) by ect and using (6.5.22), (6.5.30), (6.5.36)–(6.5.39) and ð2Þ (6.5.42), we get that for any fixed c 2 ð0; c4 ; Z t uxxxx uxxxx 2 ðtÞk2 þ k ðsÞds ecs k ect k u u 0 Z t  Cd;B4 þ Cd;B4 ecs kE1 ðsÞk2 ds 0 Z t ecs ðkux k2H 2 þ kvx k2H 3 þ kvtx k2H 2 þ khx k2H 3 ÞðsÞds  Cd;B4 þ Cd;B4 0

 Cd;B4 ; whence

Z 2

e kuxxxx ðtÞk þ ct

t

ect kuxxxx k2 ðsÞds  Cd;B4 ; 8t [ 0:

ð6:5:46Þ

0

Let rðx; tÞ ¼

bðr n1 vÞx  Rh : u

Differentiating (6.1.13) in x three times, employing Poincaré’s inequality, and noting that krx ðtÞk  Cd;B4 ðkvxx ðtÞk þ khx ðtÞk þ kux ðtÞkÞ; krxx ðtÞk  Cd;B4 ðkvx ðtÞkH 2 þ khx ðtÞkH 1 þ kux ðtÞkH 1 Þ; krxxx ðtÞk  Cd;B4 ðkvx ðtÞkH 3 þ khx ðtÞkH 2 þ kux ðtÞkH 2 Þ; we deduce kvxxxxx ðtÞk  Cd;B4 kvtxxx ðtÞk þ Cd;B4 ðkux ðtÞkH 3 þ kvx ðtÞkH 3 þ khx ðtÞkH 3 Þ:

ð6:5:47Þ

Thus we conclude from (6.1.12), (6.5.38), (6.5.39), (6.5.43), (6.5.46) and (6.5.47) ð2Þ that for any fixed c 2 ð0; c4 ; Z t ecs ðkvxxxxx k2 þ kutxxx k2H 1ÞðsÞds  Cd;B4 ; 8t [ 0: ð6:5:48Þ 0

Similarly, we can deduce Z t ecs khxxxxx k2 ðsÞds  Cd;B4 ; 8t [ 0:

ð6:5:49Þ

0

h

Thus estimates (6.5.23) and (6.5.24) readily follow. ð3Þ

ð3Þ

ð2Þ

Lemma 6.5.6. There exists a positive constant c4 ¼ c4 ðCd;B4 Þ  c4 ðCd;B4 Þ such ð3Þ

that for any fixed c 2 ð0; c4 , we have that for any t [ 0;

Global Attractors for the Compressible Navier–Stokes Equations Z e ðku  ct

uk2H 4

þ kvk2H 4

þ kh 

hk2H 4 Þ þ

t 0

145

ecs ðku  uk2H 4 þ kvk2H 5 þ kh  hk2H 5 Þds

 Cd;B4 ; which implies 2

kuðtÞk2H 4 þ kvðtÞk2H 4 þ khðtÞk2H 4  2ðu 2 þ h Þ þ Cd;B4 ect ^ 1 ðdÞ þ Cd;B ect C ^ 1 ðdÞ ¼ 2 with C



d24 L2

þ

d22 CV2 L2

ð6:5:50Þ

4

.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi h

i ^ 1 ðdÞ and ^t4 ¼ ^t4 ðCd;B Þ ¼ max ^t3 ðCd;B Þ; ^c1 ln C^ 1 ðdÞ , Now taking R4 ¼ 2C 4 2 4 Cd;B 4

we hence prove the next lemma. ^ 4 ¼ fðu; v; hÞ 2 H ð4Þ ; kðu; v; hÞk ð4Þ  R4 g is an absorbing Lemma 6.5.7. The ball B H d ð4Þ set in H , i.e., when t  ^t4 ðCd;B4 Þ, d

kðuðtÞ; vðtÞ; hðtÞÞk2H ð4Þ  R24 : ð1Þ

ð2Þ

ð4Þ

Since having obtained absorbing balls in Hd ; Hd and Hd , same as in chapters 4 and 5 and also following the abstract framework in [53], we can conclude the next lemma. Lemma 6.5.8. The set ^ 4Þ ¼ xðB

\ [

^4 SðtÞB

ð6:5:51Þ

s0 t s

^ 4 and is nonempty, invariant by SðtÞ. Here the closures are taken in is included in B the weak topology of H ð4Þ , ^ 4 Þ ¼ xðB ^ 4 Þ; SðtÞxðB

8t [ 0:

Remark 6.5.1. As in chapters 4 and 5, the set inclusion.

ð6:5:52Þ

is maximal in the sense of

Similarly as establishing attractors A1;d and A2;d , we can also get the next theorem of the existence of attractor A4;d . Theorem 6.5.9. The set ^ 4Þ A4;d ¼ xðB

ð6:5:53Þ

satisfies A4;d

is bounded and weakly closed in

ð4Þ

Hd ;

ð6:5:54Þ

Attractors for Nonlinear Autonomous Dynamical Systems

146

SðtÞA4;d ¼ A4;d ;

8t  0;

ð6:5:55Þ

lim d w ðSðtÞB; A4;d Þ ¼ 0:

ð6:5:56Þ

ð4Þ

for every bounded set B in Hd , t! þ 1

Moreover, it is the maximal set in the sense of inclusion that satisfies (6.5.54)– (6.5.56). Remark 6.5.2. Since having obtained three attractors A4;d , A2;d and A1;d satisfying ð4Þ

ð2Þ

ð1Þ

ð2Þ

ð1Þ

that A4;d is bounded in Hd ð Hd Hd Þ and A2;d is bounded in Hd ð Hd Þ, so ð1Þ Hd

ð2Þ Hd

A4;d is bounded in both and and A2;d is bounded in invariance property (6.5.55), we can conclude A4;d A2;d A1;d : ð2Þ

in

thus by the ð6:5:57Þ

On the contrary, if we knew that A1;d is bounded in Hd ð4Þ Hd ,

ð1Þ Hd ,

or/and A2;d is bounded

then we obtain that A1;d ¼ A2;d or/and A2;d ¼ A4;d .

Till now we have proved the conclusion for the case of i ¼ 4 in theorem 6.1.1.h

6.6

Bibliographic Comments

In this section, we shall recall some related results from the literature. For two or three dimensions, the global existence and asymptotic behavior in time of smooth solutions to the equations of a viscous and heat-conductive polytropic ideal gas in general domains have been studied only for sufficiently small smooth initial data (see [85–88, 149]). For the spherically symmetric motion of ideal gas in a bounded annular domain and exterior domain, the global existence and uniqueness of generalized solutions with arbitrary large initial data have been shown in [46, 96] and [47, 63] for various boundary conditions. The corresponding study on asymptotic behaviour of solutions for any given initial datum has been done in [64, 108]. In one-dimensional case, for the initial boundary value problems in bounded domains, we know that for arbitrarily large initial datum, there is a unique global (generalized or smooth) solution that converges (exponentially) to a steady state as time goes to infinity, see [2, 60, 61], and [106–109]; see also [107, 123, 131] and references therein for the counterpart in nonlinear thermoviscoelasticity). For the one-dimensional Cauchy problem, the results on global existence and asymptotic behavior were still established with assumption of small initial data, see [64, 69, 71, 122, 156, 160] and the references therein. For the global attractors, we may refer to [44] and the references therein for the Navier–Stokes equations for incompressible fluid. We also recall two references [33, 59]. In [59], the authors briefly described their result on the existence of a

Global Attractors for the Compressible Navier–Stokes Equations

147

compact attractor for the one-dimensional isentropic compressible viscous flow in a finite interval, and worked on the following incomplete metric space:  Z 1 qdx ¼ 1; q [ 0; q1 2 L1 : X ¼ ðq; uÞ 2 H 1  L2 ; 0

In [33], the author considered the isentropic compressible viscous flow in a bounded domain in R3 where the uniqueness of weak solutions is not known since it was based on the fundamental result on global existence of weak solutions by P. L. Lions [79]. Thus it is impossible to adopt the usual solution semigroup approach. Consequently, the author [33] used 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, this chapter is quite different from [33] in the following aspects: non-isentropic via isentropic; spherically symmetric motion via non-spherically symmetric motion; solution semigroup approach via simple time shift. Since some present mathematical difficulties in studying global attractors are similar to those four points (1)–(4) in section 4.6 of chapter 4, we omit the details here.

Chapter 7 Global Attractor for a Nonlinear Thermoviscoelastic System in Shape Memory Alloys This chapter shall prove the existence of a global attractor for a semiflow governed by the weak solutions to a nonlinear one-dimensional thermoviscoelastic system with clamped boundary conditions in shape memory alloys. In order to describe physically phase transitions between different configurations of crystal lattices, we shall work in a framework where the strain u belongs to L1 and introduce more delicate estimates to derive the crucial L1 -estimate of strain u in establishing the uniform compactness of the orbit of the semiflow and existence of an absorbing set. We shall devise a new method in this chapter to prove uniform compactness of semiflow SðtÞ, see lemmas 7.3.2–7.3.4. The content of this chapter is adapted from Qin [119]. The nonlinear thermoviscoelastic system studied in this chapter is quite different from that in chapter 5, which is the one-dimensional compressible Navier–Stokes equations of real gas with different constitutive relations from those in chapter 5. The method used here is the uniformly compact semigroup method, see theorem 1.2.23.

7.1

Main Result

This chapter will study the existence of global attractors for a semiflow generated by the global weak solutions to the following nonlinear one-dimensional thermoviscoelastic system in the Lagrangian coordinates (see [115, 123, 131, 152, 158]) ut  vx ¼ 0;

ð7:1:1Þ

vt  rx ¼ 0;

ð7:1:2Þ

ðe þ v 2 =2Þt  ðrvÞx þ qx ¼ 0;

ð7:1:3Þ

where u; v; r; e; Q and h stand for strain, velocity, stress, internal energy, heat flux and absolute temperature, respectively. DOI: 10.1051/978-2-7598-2702-2.c007 © Science Press, EDP Sciences, 2022

150

Attractors for Nonlinear Autonomous Dynamical Systems

Consider problem (7.1.1)–(7.1.3) in the region f0  x  1; t  0g. It is well-known that for one-dimensional homogeneous, thermoviscoelastic materials, the constitutive relations of e; r; g (specific entropy) and Q satisfy the following relations (see [115, 123, 131, 152, 158]) ^ðu; h; vx Þ; g ¼ ^ e ¼ ^eðu; hÞ; r ¼ r gðu; hÞ; Q ¼ q^ðu; h; hx Þ:

ð7:1:4Þ

We know that in order to comply with the second law of thermodynamics expressed by the Clausius–Duhem inequality, (7.1.4) must fulfill ^ u ðu; hÞ; ^ ^ h ðu; hÞ; ^ðu; h; 0Þ ¼ W r gðu; hÞ ¼ W ^ðu; h; 0ÞÞw  0; ð^ rðu; h; wÞ  r

^q ðu; h; gÞg  0:

ð7:1:5Þ ð7:1:6Þ

^ ¼ W ¼ e  hg is the Helmholtz free energy function, and g is the specific Here W entropy. The following assumptions are supposed to hold for problem (7.1.1)–(7.1.5) (see [131]): (i) The material model considered here has viscoelastic damping of rate type, i.e., r¼

@W þ cvx @u

ð7:1:7Þ

with a constant c [ 0. (ii) The heat flux Q is proportional to hx : Q ¼ jhx

ð7:1:8Þ

with a constant j [ 0. (iii) The Helmholtz free energy W takes the following form: Wðu; hÞ ¼ cv h ln h þ c1 h þ F1 ðuÞh þ F2 ðuÞ

ð7:1:9Þ

with positive constants cv ; c1 and smooth functions F1 ; F2 . (iv) In order to include the model to study of phase transitions problems in shape memory alloys (see [31, 32]), F1 and F2 take the following forms: F1 ðuÞ ¼ u 2 ;

F2 ðuÞ ¼ u 6  c2 u 4  c3 u 2 ;

ð7:1:10Þ

where c2 and c3 are positive constant. It therefore turns out that (7.1.1)–(7.1.3) reduces to the following problem ut ¼ vx ;

ð7:1:11Þ

vt  ðf1 ðuÞh þ f2 ðuÞ þ cvx Þx ¼ 0;

ð7:1:12Þ

cv ht  hf1 ðuÞvx  cvx2  jhxx ¼ 0;

ð7:1:13Þ

Global Attractor for a Nonlinear Thermoviscoelastic System

151

where fi ¼ Fi0 ði ¼ 1; 2Þ and r is given by r ¼ r1 þ cvx ; r1 ¼ r1 ðu; hÞ ¼ f1 ðuÞh þ f2 ðuÞ:

ð7:1:14Þ

Now we assume that equations (7.1.11)–(7.1.13) are subject to the following boundary conditions and initial conditions: v ¼ 0; hx ¼ 0; for x ¼ 0; 1;

ð7:1:15Þ

t ¼ 0 : u ¼ u0 ðxÞ; v ¼ v0 ðxÞ; h ¼ h0 ðxÞ:

ð7:1:16Þ

The boundary conditions implies physically that there is no heat flux through the boundary and the rod is clamped at the both ends. In order to describe physically martensitic phase transitions between different configurations of crystal lattices, we have to work in a framework in which the strain u belongs to L1 ð0; 1Þ, hence we assume that (v) u0 ðxÞ 2 L1 ð0; 1Þ; v0 ðxÞ 2 W 1;1 ð0; 1Þ; h0 ðxÞ 2 H 1 ð0; 1Þ with h0 ðxÞ [ 0 for any x 2 ½0; 1. Moreover, assume that the compatibility conditions v0 ð0Þ ¼ v0 ð1Þ ¼ 0 hold. Now define two spaces as H ¼ fðu; v; hÞ 2 L1  W 1;1  H 1 : hðxÞ [ 0; x 2 ½0; 1; vjx¼0 ¼ vjx¼1 ¼ 0g and

( Hd ¼

Z ðu; v; hÞ 2 H : 0\d1  h;

1

ðcv h þ F2 ðuÞ þ v 2 =2Þdx  d2 ;

0

Z

1

Z

1

udx  d3 ;

0

) ðF1 ðuÞ  log hÞdx  d4 ;

0

where di 2 R; i ¼ 1; 2; 3; 4 verify 0\d1 \ed4 :

ð7:1:17Þ

Clearly, Hd is a closed subspace of H. The next is the main result in this chapter. Theorem 7.1.1. The nonlinear semiflow fSðtÞg defined by the global weak solutions to problem (7.1.11)–(7.1.16) maps H into itself. Furthermore, for any given constants di 2 R ði ¼ 1; 2; 3; 4Þ verifying (7.1.17), the semiflow fSðtÞg possesses in Hd a global attractor Ad . Furthermore, the following conclusions hold: (1) Ad attracts all bounded sets in Hd ; (2) Ad is maximal in the sense that every compact invariant set in Hd lies in Ad ;

152

Attractors for Nonlinear Autonomous Dynamical Systems

(3) Ad is minimal in the sense that if B is any closed set in Hd that attracts each compact set in Hd , then one has Ad  B; (4) For each bounded set B in Hd , the omega-limit set xðBÞ satisfies xðBÞAd ; (5) Ad is a connected set in Hd ; (6) Ad is Lyapunov stable, i.e., for every neighborhood V of Ad and every s [ 0, there is a neighborhood U of Ad with the property that SðtÞU V , for all t  s. (7) Ad ¼ \ s  0 [ t  s SðtÞBd is invariant under SðtÞ, i.e., SðtÞAd ¼ Ad ; t  0. (8) Ad is compact. S Remark 7.1.1. We should point out here that the set A ¼ d1 ;...;d4 Ad is a noncompact global attractor in the metric space H in the sense that it attracts any bounded sets of H with constraint h  d1 [ 0 where d1 [ 0 is any given positive constant. Remark 7.1.2. For some polynomial functions, theorem 7.1.1 is also valid, for example, the case F1 ¼ F1 ðsÞ ¼ b0 s2n þ 2 þ b1 s 2n þ 1 þ þ b2n þ 2 and F2 ¼ F2 ðsÞ ¼ a0 s 2m þ 2 þ a1 s 2m þ 1 þ þ a2m þ 2 with a0 [ 0; b0 [ 0; aj ; bk 2 R ðj ¼ 1; 2; . . .; 2m þ 2; bk ¼ 1; 2; . . .; 2n þ 2Þ being constants and m  n  0 being integers. The usual notation in this chapter will follow that in chapters 4–6. Specially, by C0 [ 0 we denote the generic positive constant depending only on the L1  W01;1  H 1 -norm of the initial datum ðu0 ; v0 ; h0 Þ. Cd (sometimes Cid ði ¼ 1; 2; . . .Þ) stands for the generic positive constant depending only on di ði ¼ 1; . . .; 4Þ, but is independent of initial data. CB denotes the generic positive constant depending on the parameters di ði ¼ 1; . . .; 4Þ and the L1  W01;1  H 1 -norm of initial datum ~ where ðu0 ; v0 ; h0 Þ 2 BL1  W 1;1  H 1 and B ðu0 ; v0 ; h0 Þ with kðu0 ; v0 ; h0 ÞkH  B, 0 ~ is a positive constant depending only on is a bounded set in L1  W01;1  H 1 and B the bounded set B. Similarly as in chapter 5, we use the same symbol to denote the state functions as well as their values along a thermodynamic process, e.g., r1 ðu; hÞ, and r1 ðuðx; tÞ; hðx; tÞÞ and r1 ðx; tÞ. We organize this chapter as follows. Section 7.2 is devoted to the existence of an absorbing set in Hd or equivalently the semiflow fSðtÞg is point dissipative. Section 7.3 focuses on the compactness of the orbit which completes the proof of theorem 7.1.1.

7.2

An Absorbing Set Bd in Hd

This section will establish the existence of an absorbing set Bd in Hd for the semiflow fSðtÞg. The proof of theorem 7.1.1 consists of a series of the following lemmas. For simplicity, we assume that cv ¼ c ¼ 1 in (7.1.11)–(7.1.13). The next lemma has been given in [119].

Global Attractor for a Nonlinear Thermoviscoelastic System

153

Lemma 7.2.1. Assume that above assumptions (i)–(v) hold, then there exists a unique global weak solution ðuðtÞ; vðtÞ; hðtÞÞ 2 C ð½0; þ 1Þ; L1 Þ  C ðð0; þ 1Þ; W01;1 Þ \ L1 ð½0; 1Þ; W 1;1 Þ  C ð½0; þ 1Þ; H 1 Þ to problem (7.1.11)–(7.1.13) such that the following estimates hold: hðx; tÞ [ 0; in ½0; 1  ½0; 1Þ;

ð7:2:1Þ

kuðtÞkL1 þ kvðtÞkW 1;1 þ khðtÞkH 1 þ knðtÞkH 1 þ kr 1 ðtÞk  C0 ;  Z t Z 1 2 hx vx2 þ ðx; sÞdxds  C0 ; 8t [ 0; h h2 0 0 Z 0

t

ðkvk2L1 þ kvk2H 2 þ kvt k2 þ kht k2 þ khx k2H 1 þ knt k2 ÞðsÞds  C0 :

ð7:2:2Þ

ð7:2:3Þ

ð7:2:4Þ

Moreover, as t ! þ 1, kvðtÞkH 1 ! 0; khx ðtÞk ! 0; khðtÞ  hkL1 ! 0;

ð7:2:5Þ

kr 1 ðtÞk ! 0; kr ðtÞk ! 0;

ð7:2:6Þ

where Z h¼

1

hðx; tÞdx; r1 ¼ f1 ðuÞh þ f2 ðuÞ; r ¼ r1 

0

r ¼ r 

Z

1

rdx; n ¼

0

Z tZ 0

0

Z

1

r1 dx; 0

1

H ðx; y; t  sÞr 1s ðy; sÞdyds

ð7:2:7Þ

P ðnpÞ2 t cos npx cos npy stands for the fundamental solution to and H ðx; y; tÞ ¼ 1 n¼0 e the heat equation subject to the homogeneous Neumann boundary condition. We thus derive from lemma 7.2.2 that the mapping SðtÞ : ðu0 ; v0 ; h0 Þ 2 H ! ðuðtÞ; vðtÞ; hðtÞÞ ¼ SðtÞðu0 ; v0 ; h0 Þ exists for any t [ 0, and satisfies SðtÞðu0 ; v0 ; h0 Þ 2 C ðð0; 1Þ; HÞ. On the other hand, it follows from the proof in [131] that the family operators fSðtÞg, defined by the global weak solution ðuðtÞ; vðtÞ; hðtÞÞ 2 H above are continuous operators from H into itself and enjoy the usual semiflow properties. Now we always assume that the initial datum ðu0 ; v0 ; h0 Þ 2 B with ~ where B is an arbitrarily bounded set in Hd ; B ~ [ 0 is a constant kðu0 ; v0 ; h0 ÞkH  B depending on B. Then the generic constant C0 [ 0 in lemma 7.2.2 can be replaced by CB under the above assumption. Lemma 7.2.2. There holds that for any t [ 0, Z 1 Z 1 uðx; tÞdx ¼ u0 ðxÞdx  d3 ; 0

0

ð7:2:8Þ

Attractors for Nonlinear Autonomous Dynamical Systems

154

Z

1

Z

1

ðcv h þ F2 ðuÞ þ v =2Þðx; tÞdx ¼ 2

0

0

ðcv h0 þ F2 ðu0 Þ þ v02 =2ÞðxÞdx  d2 ;

 Z t Z 1 2 hx vx2 ðF1 ðuÞ  log hÞðx; tÞdx þ þ ðx; sÞdxds h h2 0 0 0 Z 1 ðF1 ðu0 Þ  log h0 Þdx  d4 : ¼

Z

ð7:2:9Þ

1

ð7:2:10Þ

0

Proof. We easily first integrate (7.1.11) in x to get (7.2.8). Second, multiplying (7.1.12) by v, adding the result to (7.1.13), and integrating the resulting equation in x in ½0; 1 to obtain  Z  d 1 1 cv h þ F2 ðuÞ þ v 2 dx ¼ 0; ð7:2:11Þ dt 0 2 which yields (7.2.9). Third, multiplying (7.1.13) by h1 and integrating in x in ½0; 1 to yield  Z Z 1 2 d 1 hx vx2 ðF1 ðuÞ  log hÞdx þ þ dx ¼ 0; ð7:2:12Þ dt 0 h h2 0 which, with the definition of Hd , gives (7.2.10). This thus completes the proof.

h

Lemma 7.2.3. The following estimates hold that for any t [ 0,

Z 0

t

khðtÞkL1 þ kuðtÞkL6 þ kvðtÞk  Cd ;

ð7:2:13Þ

 Z t Z 1 2 hx vx2 þ ðx; sÞdxds  Cd ; h h2 0 0

ð7:2:14Þ

kvðsÞk2L1 ds  Cd ;

e

ad4

Z

1



ha ðx; tÞd  Cd ðaÞ; 8a 2 ð0; 1;

ð7:2:15Þ

0

where Cd ðaÞ is a generic positive depending on di ði ¼ 1; 2; 3; 4Þ.

a 2 ð0; 1 and parameters

Proof. Employing the Young’s inequality, we can deduce F2 ðuÞ  C 1 u 6  C ;

ð7:2:16Þ

where C [ 0 is a generic absolute constant. Thus (7.2.13) easily follows from (7.2.11) and (7.2.16). We can derive from (7.2.10) and (7.2.13) that  Z t Z 1 2 Z 1 hx vx2 þ ðh  1Þdx  Cd ; ðx; sÞdxds  d4 þ h h2 0 0 0

Global Attractor for a Nonlinear Thermoviscoelastic System

155

which gives (7.2.14). From (7.2.10), we have Z 1 Z 1 log hdx   d4 þ F1 ðuÞdx   d4 : 0

0

Thus Z

1

a

log hdx   ad4 ; 8a 2 ð0; 1:

0

Exploiting Jensen’s inequality, we conclude Z 1 Z 1 log ha dx  log ha dx   ad4 ; 0

0

which, along with (7.2.13), gives (7.2.15). This completes the proof.

h

Lemma 7.2.4. For any a 2 ð0; 1 and s  0, there holds that for any t  s  0,  Z t Z 1 2 hx vx2 þ a ðx; sÞdxds  Cd ðaÞ; ð7:2:17Þ h1 þ a h s 0 Z

t

Z

1

kha 

s

0

Z ha dxk2L1  Cd ðaÞ;

t s

kvx ðsÞk2 ds  Cd ðaÞ sup khðsÞkaL1 : sst

ð7:2:18Þ

Proof. We apply lemmas 7.2.1–7.2.3 and the method of the proof of lemma 2.2 in [131] to derive (7.2.17) and (7.2.18). The difference here is that the constants Cd ðaÞ in (7.2.17) and (7.2.18) depends only on parameters di ði ¼ 1; 2; 3; 4Þ. Thus this proves the proof. h The next lemma focuses on the uniform estimate on the L1 -norm of the strain u, which is very important to prove our main theorem. Lemma 7.2.5. We have that for any t  1, kuðtÞkL1  C1d þ CB ec1 t ;

ð7:2:19Þ

where an absolute constant c1 [ 0 is independent of parameters di ði ¼ 1; 2; 3; 4Þ and kðu0 ; v0 ; h0 ÞkH . Proof. Set Gðx; y; tÞ ¼

1 X

2

eðnpÞ t sin npx sin npy;

n¼1

which is the fundamental solution to the heat equation subject to the homogeneous Dirichlet boundary condition with the property Gy ¼ Hx : Thus (7.1.12) can be reduced as vt  vxx  r 1x ¼ 0

ð7:2:20Þ

Attractors for Nonlinear Autonomous Dynamical Systems

156

with Z

r 1

1

¼ r1 

r1 dx:

ð7:2:21Þ

0

Hence, v satisfies the next integral equation Z 1 Z tZ 1 vðx; tÞ ¼ Gðx; y; tÞv0 ðyÞdy  Gy ðx; y; t  sÞr 1 ðy; sÞdyds: 0

0

ð7:2:22Þ

0

Thus, by integrating by parts, we derive from (7.1.11), Z tZ 1 Z 1 @ ut ¼ vx ¼ Gx ðx; y; tÞv0 ðyÞdy  Gy ðx; y; t  sÞr 1 ðy; sÞdyds @x 0 0 0 Z 1 0 ¼ H ðx; y; tÞv0 ðyÞdy þ wxx 0

Z ¼

0

1

H ðx; y; tÞv00 ðyÞdy þ wt  r 1 ðx; tÞ:

ð7:2:23Þ

Rt R1 Here w ¼ 0 0 H ðx; y; t  sÞr 1 ðy; sÞdyds: In the same manner as the derivation of (7.2.44) in [37], from lemma 1.2.22 and lemmas 7.3.1–7.2.3, it follows that for all ðx; tÞ 2 ½0; 1  ½0; 1Þ, jwðx; tÞj  Cd :

ð7:2:24Þ

qðx; tÞ ¼ uðx; tÞ  wðx; tÞ:

ð7:2:25Þ

Set

Then we can derive from (7.2.23) that q fulfills qt ¼ hðx; tÞ  r 1 ðx; tÞ; where h ¼

R1 0

ð7:2:26Þ

H ðx; y; tÞv00 ðyÞdy satisfies ht  hxx ¼ 0; hx jx¼0;1 ¼ 0; hjt¼0 ¼ v00 ðxÞ:

ð7:2:27Þ

Obviously, Z

1

hðx; tÞdx ¼ 0:

ð7:2:28Þ

0

We easily deduce from (7.2.27), 1 khðtÞk2 þ 2

Z

t 0

1 khx ðsÞk2 ds ¼ kv00 k2 ; 2

1d khx k2 þ khxx k2 ¼ 0: 2 dt

ð7:2:29Þ

ð7:2:30Þ

Global Attractor for a Nonlinear Thermoviscoelastic System

157

Multiplying (7.2.30) by t [ 0, we have 1d 1 ðtkhx k2 Þ þ tkhxx k2 ¼ khx k2 : 2 dt 2

ð7:2:31Þ

Integrating (7.2.31) in t and exploiting (7.2.29), we conclude that for all t [ 0, 1 khx ðtÞk2  kv00 k2 : ð7:2:32Þ 4t By virtue of (7.2.27) and (7.2.28), and using the Poincaré’s inequality, khx k  khxx k:

ð7:2:33Þ

Thus from (7.2.30) and (7.2.33), it follows that for any c2 2 ð0; 1, 1d khx k2 þ c2 khx k2  0; 2 dt which, with (7.2.32), leads to khx ðtÞk2  e2c2 kv00 k2 e2c2 t =4; 8t  1:

ð7:2:34Þ

Using (7.2.29), (7.2.34) and the interpolation inequality, we conclude that for all t  1, khðtÞkL1  C khðtÞk1=2 khx ðtÞk1=2  CB ðc2 Þec2 t=2 : ð7:2:35Þ Put GðtÞ ¼ q 2 ðtÞ. Then from (7.2.26), we know that GðtÞ fulfills G 0 ðtÞ  2hq þ 2r1 q  2qr1 ¼ 0 where r1 ¼

ð7:2:36Þ

R1

r1 dx: Noting that from (7.2.25), we can write 4 3 X X ai q i þ 1 w 5i  8c2 bj q j þ 1 w 3j  4c3 ðq þ wÞq 2r1 q ¼ 4hq 2 þ 12q 6 þ 4whq þ 12 0

i¼0

i¼0

ð7:2:37Þ where coefficients ai ; bj ði ¼ 0; 1; 2; 3; 4; j ¼ 0; 1; 2; 3Þ are absolute positive constants. Exploiting lemmas 7.2.1–7.2.3 and the Young’s inequality, it follows from (7.2.24) that Z 1 2jr1 qj  Cd jqj þ Cd jqj jujhdx 0 ( " Z 4=3 # ) Z 1

 Cd 1 þ (

Z

0 1

 Z 3=4 juj h þ

0

 Cd 1 þ kh

dx jqj

h3=4 dx

0

 Cd 1 þ (

1

juj h 

h3=4 dx

Z jh3=4 

0

Z 3=4

1=3

1



h 0

3=4

dxkL1 kukL6

 Z  3q 6 þ Cd 1 þ kh3=4  0

1

) h3=4 dxjdx jqj

0

Z

1

1

 h3=4 dxk2L1 ;

1

5=6 ) 3=10

ðh þ 1Þ

dx

jqj

0

ð7:2:38Þ

Attractors for Nonlinear Autonomous Dynamical Systems

158

Z 4jwhqj  Cd hjqj  Cd ðh

1=4

þ 1Þkh

3=4

1

 0

Z

1

 Cd þ 3q 6 þ Cd kh3=4  0

h3=4 dxkL1 jqj þ Cd jqj ð7:2:39Þ

h3=4 dxk2L1 þ hq 2 ; 6=5

2jhqj  Cd þ 2q 6 þ Cd jhjL1 :

ð7:2:40Þ

Inserting (7.2.37)–(7.2.40) into (7.2.36) gives G 0 ðtÞ þ 3hq 2 þ 2q 6  Cd þ Cd jhjL1 þ Cd kh3=4 

Z

1

6=5

0

h3=4 dxk2L1 :

ð7:2:41Þ

Using the Young’s inequality to get GðtÞ ¼ q 2  2q 6 þ C :

ð7:2:42Þ

Thus it follows from (7.2.41) and (7.2.42) that for any t  1, G 0 ðtÞ þ GðtÞ  Cd þ Cd jhjL1 þ Cd kh3=4 

Z

1

6=5

0

h3=4 dxk2L1 :

ð7:2:43Þ

Using (7.2.35) and lemma 7.2.4, we can derive from (7.2.43) that for any t  1, GðtÞ  Cd þ CB et þ CB e3c2 t=5  Cd þ CB e3c2 t=5 ; c2 2 ð0; 1

ð7:2:44Þ

where we have used the estimate Gð1Þ  CB , using lemma 7.2.2 and (7.2.24), (7.2.25). Thus (7.2.44) and (7.2.24) yields (7.2.19) with c1 ¼ 3c2 =10 [ 0. This completes the proof. h Lemma 7.2.6. For any a 2 ð0; 1, we have that for any t  1, Z 3 t ts knx ðtÞk2 þ ðknt k2 þ knxx k2 ÞðsÞe 4 ds 4 1 ( Z  CB e2c1 t þ Cd 1 þ sup

s2½1;t

khðsÞk2L1þ a

t

þ

2 ts 4

kht ðsÞk e

) ds ;

ð7:2:45Þ

1

where n is defined in (7.2.7). Proof. Since n satisfies nt  nxx ¼ r 1t ¼ ðr 1 Þt ;

ð7:2:46Þ

nx jx¼0;1 ¼ 0; njt¼0 ¼ 0;

ð7:2:47Þ

Global Attractor for a Nonlinear Thermoviscoelastic System

159

we obtain Z

1

ndx ¼ 0;

ð7:2:48Þ

0

which, using the Poincaré’s inequality, leads to knðtÞkL1  knx ðtÞk  knxx ðtÞk:

ð7:2:49Þ

Multiplying (7.2.46) by nxx in L2 ð0; 1Þ and employing (7.2.49), we get d 1 3 knx ðtÞk2 þ knx ðtÞk2 þ knxx ðtÞk2  kr 1t k2 dt n 4 4 o 2  Cd ðkhkL1 þ 1Þ þ CB e2c1 t kvx ðtÞk2 þ ðCd þ CB e2c1 t Þkht ðtÞk2 :

ð7:2:50Þ

Thus using lemmas 7.2.1, 7.2.4, and (7.2.50), we can derive that for any c1 2 ð0; 1=8Þ, knx ð1Þk  CB ; knx ðtÞk2 et=4 þ

3 4

Z

t

1

ð7:2:51Þ

  knxx ðsÞk2 es=4 ds  CB þ Cd 1 þ sup khðsÞk2L1þ a

þ CB eð1=42c1 Þt þ Cd

Z

1st

t

kht k2 es=4 ds:

ð7:2:52Þ

1

Now using lemmas 7.2.1–7.2.5, it follows from (7.2.46) that knt ðtÞk  knxx ðtÞk þ kr 1t ðtÞk  knxx ðtÞk  þ Cd ðkhkL1 þ 1Þ þ CB ec1 t kvx ðtÞk þ ðCd þ CB ec1 t Þkht ðtÞk; h

which, together with (7.2.52), yields (7.2.45). This proves the lemma. Lemma 7.2.7. For any a 2 ð0; 1, we have that for any t  1, Z t ts kr 1 ðsÞk2 e 4 ds 1

 Cd þ CB e

c1 t

þ Cd sup

s2½1;t

1 þ a=2 khðsÞkL1

Z

t

þ Cd

2 ts 4

kht ðsÞk e

1=2 ds

:

ð7:2:53Þ

1

Proof. Now we rewrite (7.1.12) as vt  vxx ¼ r 1x ;

ð7:2:54Þ

Attractors for Nonlinear Autonomous Dynamical Systems

160

Thus it follows from (7.2.54) that       Z x Z x Z x r 1 dy ¼ vt ;  r 1 dy þ vxx ; r 1 dy kr 1 ðtÞk2 ¼ r1x ;  0  Z x 0  0 Z x @ ¼ vt dy;  r 1 dy  ðvx ; r 1 Þ @x 0 0 Z x  Z x  @ vdy; r 1  vdy; r 1t  ðvx ; r 1 Þ ¼ @t 0 Z x  Z 0 x  @ 1 1 1



vdy; r1  vdy; r1t þ kvx k2 þ kvx k2 þ kr 1 k2 ;  @t 2 2 2 0 0 i.e., kr 1 ðtÞk2 et=4

 Z x  Z x  @ 1 t=4



e 2 vdy; r1 vdy; r1 et=4  @t 2 0 0 Z x 

2 vdy; r1t et=4 þ kvx k2 et=4 ; 0

which, combined with lemmas 7.2.1–7.2.5 and (7.2.50), results in that for any t  1, Z

t 1

kr 1 ðsÞk2 es=4 ds  CB þ 2et=4 kvkL1 kr 1 kL1 þ Cd Z

t

þ Cd 1

Z 1

1=2 Z kvk2L1 es=4 ds

t

t 1

1=2 Z kvk2L1 es=4 ds

kr 1t k2 es=4 ds

t

kr 1 k2 es=4 ds

1

1=2

Z

t

þ 1

1=2

kvx k2 es=4 ds (

 1Z t 1=6  CB þ Cd et=4 1 þ khkL1 þ kr 1 ðsÞk2 es=4 ds þ Cd et=8 CB eð1=8c1 Þt 2 1 1=2 )   Z t 1 þ a=2 2 s=4 t=8 þe 1 þ sup khkL1 kht k e ds : þ 1st

1

Here using lemma 7.2.1, we have used the estimate: je1=4 Thus estimate (7.2.53) is obtained. This completes the proof.

R x 0

vdy; r 1 jt¼1 j  CB . h

Lemma 7.2.8. For any a 2 ð0; 1, we know that for any t  1,  7=8 Z t knðtÞk2L1  Cd þ CB ec1 t=4 þ Cd sup khðsÞk2L1þ a þ kht k2 eðtsÞ=4 ds ; ð7:2:55Þ 1st

1

where n is defined in (7.2.7). Proof. Multiplying (7.2.46) by n to give immediately Z Z 1 Z 1 1d d 1

knk2 þ knx k2 ¼ nðr 1 Þt dx ¼ nr1 dx  nt r 1 dx: 2 dt dt 0 0 0

Global Attractor for a Nonlinear Thermoviscoelastic System

161

Thus, with (7.2.48), implies  7 d

knðtÞk2 et=4 þ et=4 knx ðtÞk2 dt 4   Z 1 Z 1 Z 1 d 1 et=4 2 nr 1 dx  et=4 nr 1 dx  2et=4 nt r 1 dx: dt 2 0 0 0

ð7:2:56Þ

Now we integrate in t to get that for any t  1, Z Z 7 t s=4 1 t=4 5 t s=4 2 t=4 2 2 knðtÞk e þ e knx ðsÞk ds  CB þ e knðtÞk þ e knx ðsÞk2 ds 4 1 2 4 1 Z t kr 1 k2 es=4 ds þ Cd et=4 kr 1 ðtÞk2 þ Cd 1

Z þ Cd

1=2 Z

t

e

s=4

t

2

knt ðsÞk ds

e

1

1

s=4

kr 1 ðsÞk2 ds

1=2 :

ð7:2:57Þ

Thus it follows from lemmas 7.2.6, 7.2.7 and (7.2.57) that Z t 2 eðtsÞ=4 knx ðsÞk2 ds knðtÞk þ 1 Z t 4=3  CB et=4 þ Cd þ CB e4c1 t þ Cd khðtÞkL1 þ Cd eðtsÞ=4 kr 1 k2 ds Z þ Cd

t

2 ðtsÞ=4

knt k e

1=2 Z

1

t

ds

1

kr 1 ðsÞk2 eðtsÞ=4 ds

1=2

1 3ð1 þ a=2Þ=2

 Cd þ CB ec1 t=2 þ Cd sup khðsÞkL1 Z þ Cd

1st

t

eðtsÞ=4 kht ðsÞk2 ds

3=4

ð7:2:58Þ

:

1

Therefore from (7.2.45) and (7.2.58) and the interpolation inequality, it follows knðtÞk2L1  C knðtÞkknx ðtÞk  Cd þ CB ec1 t=4 Z t 7=8 7ð2 þ aÞ=8 2 ðtsÞ=4 þ Cd sup khðsÞkL1 þ Cd kht k e ds : 1st

1

h

We thus complete the proof. Lemma 7.2.9. For any a 2 ð0; 2=7Þ, we have that for any t  1, Z t 7=16 2 ðtsÞ=4 c1 t=8 þ Cd sup khðsÞkL1 þ Cd kht k e ds : kvx ðtÞkL1  Cd þ CB e 1st

1

ð7:2:59Þ

Attractors for Nonlinear Autonomous Dynamical Systems

162

Proof. By (7.2.23), we can write Z 1 Z H ðx; y; tÞv00 ðyÞdy  r 1 þ vx ¼ 0

þ

0

Z tZ 0

h

r 1

1

1

0

H ðx; y; tÞr 1 ðy; 0Þdy

H ðx; y; t  sÞr 1s ðy; sÞdyds Z

1

þ 0

ð7:2:60Þ

H ðx; y; tÞr 1 ðy; 0Þdy þ n:

Let Z fðx; tÞ ¼ 0

1

H ðx; y; tÞr 1 ðy; 0Þdy:

Then f fulfills ft  fxx ¼ 0;

ð7:2:61Þ

fx jx¼0;1 ¼ 0; fjt¼0 ¼ r 1 ðx; 0Þ:

ð7:2:62Þ

Similarly to the derivation of (7.2.32)–(7.2.35), we know kfx ðtÞk 

1

kr ðx; 0Þk2 ; 8t  1; 4t 1

ð7:2:63Þ

kfðtÞkL1  C kfðtÞk1=2 kfx ðtÞk1=2  CB ðc2 Þec2 t=2 ; 8t  1;

ð7:2:64Þ

where c2 2 ð0; 1 is an arbitrary constant. Thus using (7.2.35), (7.2.64) and lemmas 7.2.1–7.2.3, 7.2.5 and 7.2.8, we conclude from (7.2.60) that for a 2 ð0; 2=7Þ, kvx ðtÞkL1  khkL1 þ kr 1 kL1 þ kfkL1 þ knkL1  Cd þ CB ec1 t=8 þ Cd sup khðsÞkL1 7ð2 þ aÞ=16

þ Cd sup khðsÞkL1 1st

 Cd þ CB e

c1 t=8

1st

Z

t

þ Cd

kht ðsÞk2 eðtsÞ=4 ds

1

Z

þ Cd sup khðsÞkL1 þ Cd 1st

t

7=16

2 ðtsÞ=4

kht ðsÞk e

7=16 :

ds

1

h

Therefore we complete the proof. Lemma 7.2.10. For any a 2 ð0; 2=7Þ, we have that for any t  1, Z t ðkhxx k2 þ kht k2 ÞðsÞeðtsÞ=4 ds khx ðtÞk2 þ 1

 Cd þ CB e

c1 t=4

þ Cd sup

1st

khðsÞk2L1þ a

Z þ Cd

t

2 ðtsÞ=4

kht k e

7ð2 þ aÞ=16 ds

;

1

ð7:2:65Þ

Global Attractor for a Nonlinear Thermoviscoelastic System

Z

t

2

khx ðtÞk þ

163

ðkhxx k2 þ kht k2 ÞeðtsÞ=4 ds  Cd þ CB ec1 t=4 ;

ð7:2:66Þ

1

khðtÞkL1  Cd þ CB ec1 t=12 :

ð7:2:67Þ

Proof. Multiplying (7.1.13) by hxx , integrating the resulting equation in ½0; 1, and using the estimate khxx k  khx k, we can obtain that  3 d

1 1 khx ðtÞk2 et=4 þ et=4 khxx ðtÞk2  khxx ðtÞk2 et=4 þ et=4 khf1 ðuÞvx þ vx2 k2 : dt 4 4 2 Thus khx ðtÞk2 þ

1 2

Z

t

khxx ðsÞk2 eðtsÞ=4 ds

1

 CB et=4 þ Cd sup khðsÞk2L1þ a þ CB sup khðsÞk2L1 1st

Z

t

þ Cd

2 ðtsÞ=4

kht ðsÞk e

1

Z

1st

7=8 Z

t

ds

t

et=4 þ ð1=42c1 Þs kvx ðsÞk2 ds

1

kvx ðsÞk2 eðtsÞ=4 ds

1

 CB et=4 þ Cd þ Cd sup khðsÞk2L1þ a þ CB sup khðsÞk2L1 e2c1 t 1st

1st

þ CB sup khðsÞkL1 ec1 t þ Cd sup khðsÞkaL1 1st

1st

 CB ec1 t=4 þ Cd þ Cd sup khðsÞk2L1þ a þ Cd

Z

1st

Z

t

t

kht ðsÞk2 eðtsÞ=4 ds

1

kht k2 eðtsÞ=4 ds

7=8

7ð2 þ aÞ=16 :

1

ð7:2:68Þ

On the other hand, noting from (7.1.13), kht ðtÞk  khxx ðtÞk þ Cd ðkhðtÞkL1 kuðtÞkL1 þ kvx ðtÞkL1 Þkvx ðtÞk; we infer from lemmas 7.2.1–7.2.9 and (7.2.68) that Z t kht ðsÞk2 eðtsÞ=4 ds  Cd þ Cd sup khðsÞk2L1þ a þ CB ec1 t=4 1st

1

Z þ Cd

t

2 ðtsÞ=4

kht k e

3ð2 þ aÞ 16ð1 þ aÞ

ds

;

1 3ð2 þ aÞ which, by noting that 16ð1 þ aÞ \1 and applying the Young’s inequality, implies Z t kht ðsÞk2 eðtsÞ=4 ds  Cd þ Cd sup khðsÞk2L1þ a þ CB ec1 t=4 : ð7:2:69Þ 1

1st

Attractors for Nonlinear Autonomous Dynamical Systems

164

Thus (7.2.65) follows from (7.2.68) and (7.2.69). By lemmas 7.2.1 and the interpolation inequality, it follows that 1=3

khðtÞkL1  C khðtÞkL1 khx ðtÞk2=3 þ C khðtÞkL1  Cd khx ðtÞk2=3 þ Cd :

ð7:2:70Þ

Thus from (7.2.68) and (7.2.70), we can derive that for any t  1, Z t 2 khx ðtÞk þ ðkht k2 þ khxx k2 ÞðsÞeðtsÞ=4 ds  Cd þ CB ec1 t=4 þ sup khx ðsÞk2ð2 þ aÞ=3 1st

1

which, combined with the Young’s inequality, yields Z t sup khx ðsÞk2 þ ðkht k2 þ khxx k2 ÞeðtsÞ=4 ds  Cd þ CB ec1 t=4 : 1st

1

Thus (7.2.66) is obtained and estimate (7.2.67) follows from (7.2.66) and (7.2.70). This proves the lemma. h Lemma 7.2.11. For any a 2 ð0; 2=7Þ, we know that for any t  1, Z t ðknxx k2 þ knt k2 ÞðsÞeðtsÞ=4  Cd þ CB ec1 ð2 þ aÞt=12 ; knx ðtÞk2 þ

ð7:2:71Þ

1

Z 1

t

kr 1 ðsÞk2 eðtsÞ=4 ds  Cd þ CB ec1 ð2 þ aÞt=24 ; knðtÞkL1  Cd þ C 7c1 ð2 þ aÞt=192 ;

Z

t

khx ðsÞk2 eðtsÞ=4 ds 

1

Z

t

khxx ðsÞk2 eðtsÞ=4 ds  Cd þ CB ec1 t=4 ;

ð7:2:72Þ

ð7:2:73Þ ð7:2:74Þ

1

kvðtÞkW 1;1  C kvx ðtÞkL1  C2d þ CB ec1 t=12 ;

ð7:2:75Þ

khðtÞkH 1  C2d þ CB ec1 t=24 :

ð7:2:76Þ

0

Proof. Inserting (7.2.65)–(7.2.67) into (7.2.45), (7.2.53), (7.2.55) and (7.2.59) 1=2 1=2 gives estimates (7.2.71)–(7.2.75). Using the estimate khðtÞk  khðtÞkL1 khðtÞkL1  1=2

Cd khðtÞkL1 and (7.2.66), (7.2.67), we easily derive (7.2.76). This completes the proof. h The next lemma shows that the orbit starting from any bounded set B in Hd will re-enter Hd and stay there forever after a large time t1 ðBÞ.

Global Attractor for a Nonlinear Thermoviscoelastic System

165

Lemma 7.2.12. For any ðu0 ; v0 ; h0 Þ 2 B, there exists some time t1 ¼ t1 ðBÞ [ 0 ~ of B, such that for all t  t1 ðBÞ; x 2 ½0; 1, depending only on the boundedness B hðx; tÞ  d1 [ 0: Proof. Since the proof is the same as that of lemma 5.3.6 of chapter 5, we omit the details of the proof. h The next lemma is concerned with the existence of an absorbing set Bd in Hd . Lemma 7.2.13. There exists some time t2 ¼ t2 ðBÞ  maxft1 ðBÞ; 1g such that as t  t2 ðBÞ, ðuðtÞ; vðtÞ; hðtÞÞ 2 Bd ;

ð7:2:77Þ

where Bd ¼ fðu; v; hÞ 2 Hd : kðuðtÞ; vðtÞ; hðtÞÞkHd  R0 g, R0 ¼ R0 ðCd Þ [ 0 is a constant depending only on the parameters di ði ¼ 1; 2; 3; 4Þ. Proof. Fix a 2 ð0; 2=7Þ and c2 2 ð0; 1 with c1 ¼ 3c2 =10. It follows from (7.2.19) and (7.2.75), (7.2.76) that kuðtÞkL1  2C1d ; kvðtÞkW 1;1  2C2d ; khðtÞkH 1  2C2d : ð7:2:78Þ  1 1 Therefore choosing t2 ¼ t2 ðBÞ ¼ max 1; t1 ðBÞ; c1 1 logðC1d CB Þ; 24c1 1 logðC2d CB Þg, we derive (7.2.77) from (7.2.78) where R0 ¼ 2 maxfC1d ; C2d g. This completes the proof. h

7.3

Compactness of the Orbit in Hd

This section is devoted to verification that the semiflow fSðtÞg is compact, i.e., we S need to prove that there exists some time t0 ¼ t0 ðBÞ [ 0 such that t  t0 SðtÞB is relatively compact in Hd , where B is an arbitrarily bounded set in Hd . For this purpose, we prove that for any ðu0 ; v0 ; h0 Þ 2 B, the orbit ðuðtÞ; vðtÞ; hðtÞÞ 2 Hd is uniformly bounded, i.e., the bound depends only on the parameters, but is independent of initial data in the space H1d . This will be carried out in a series of lemmas below and lemma 1.2.17. @2 2 2 We consider a closed linear operator A ¼ @x 2 : X0 ¼ L ð0; 1Þ ! X0 ¼ L ð0; 1Þ 2 1 with the domain DðAÞ ¼ H ð0; 1Þ \ H0 ð0; 1Þ verifying that AuðxÞ ¼ uxx ðxÞ 2 L2 ð0; 1Þ; 8u 2 DðAÞ:

ð7:3:1Þ

It is well-known (see [104, 156]) that A generates an analytic semigroup of  ^ ðtÞ on X0 ¼ L2 ð0; 1Þ satisfying bounded linear operators T ^ ðtÞuk  kuk; kT

8u 2 X0

where kuk ¼ kukX0 and A has a bounded inverse operator A1 given by

Attractors for Nonlinear Autonomous Dynamical Systems

166

1

Z

x

ðA uÞðxÞ ¼

Z ðx  nÞuðnÞdn þ x

0

Set

1

ðn  1ÞuðnÞdn;

8u 2 X0 :

ð7:3:2Þ

ðvðy; tÞ  vÞdy

ð7:3:3Þ

0

Z

x

u^ ðx; tÞ ¼

Z

x

ðuðy; tÞ  uÞdy; ^v ðx; tÞ ¼

0

0

R1 R1 where, by (7.2.8), u ¼ 0 udx ¼ u0 ; v ¼ 0 vdx. Therefore, from (7.3.3), (7.1.11)– (7.1.13) we can derive that ð^ u ; ^v Þ that

Z u^ jt¼0 ¼

x

0

u^t ¼ ^vx þ v;

ð7:3:4Þ

^ vt ¼ ^ vxx þ g1 ðx; tÞ;

ð7:3:5Þ

x ¼ 0; 1 : u^ ¼ 0; ^ v ¼ 0;

ð7:3:6Þ

Z ðu0  u 0 Þdy; ^v jt¼0 ¼

x

ðv0  v 0 Þdy

ð7:3:7Þ

0

with g1 ðx; tÞ ¼ r1 ðx; tÞ  r1 ð0; tÞ  vx ð0; tÞ:

ð7:3:8Þ

Letting zðx; tÞ ¼

Z x

Z

1

u^0 

0

 Z t u^0 dx dy þ v^ðx; sÞds;

0

then it folllows from (7.3.3) and (7.3.9) that Z t Z 1 v ; zx ¼ u^  vds  u^0 dx; Az ¼ zxx ¼ u^x ¼ u  u: zt ¼ ^ 0

ð7:3:9Þ

0

ð7:3:10Þ

0

Thus we derive from (7.3.3), (7.3.5) and (7.3.10) that z fulfills ztt  Azt  Az ¼ g2 ðx; tÞ; zjx¼0;1 ¼ 0; zt jx¼0;1 ¼ 0; zjt¼0

Z x Z ¼ z0  u^0  0

1 0

ð7:3:11Þ 

u^0 dx dy; zt jt¼0 ¼ ^v0 ð7:3:12Þ

with g2 ðx; tÞ ¼ g1 ðx; tÞ  ðu  uÞ:

ð7:3:13Þ

V ðtÞ ¼ ðv1 ðtÞ; v2 ðtÞÞT ¼ ðzt ; AzÞT ¼ ð^v ; u^x ÞT :

ð7:3:14Þ

Put

Global Attractor for a Nonlinear Thermoviscoelastic System

167

Hence, V ðtÞ verifies that for any t  s  0, d V ðtÞ ¼ AV ðtÞ þ FðV ðtÞÞ; t  s  0; dt

ð7:3:15Þ

V jt¼s ¼ V0s ¼ ð^ v jt¼s ; u^x jt¼s ÞT ;

ð7:3:16Þ

where

 A¼

 A 1 ; A 0

as shown in [156], can generate an analytic semigroup of bounded linear operators  fT ðtÞg ¼ etA on X0  X0 with the domain DðAÞ ¼ DðAÞ  X0 and FðV ðtÞÞ ¼ ðg2 ; 0ÞT : Now consider the following problem ^ztt  A^zt  A^z ¼ 0;

ð7:3:17Þ

^z ðsÞ ¼ ^z0s ; ^zt ðsÞ ¼ ^z1s ;

ð7:3:18Þ

where ð^z1s ; A^z0s Þ 2 X0  X0 and ^z ¼ ^z s ðtÞ. Then problem (4.3.17) and (4.3.18) reduces to Wt ¼ AW ; t  s  0; W ðsÞ ¼ W0s ¼ ð^z1s ; A^z0s ÞT 2 X0  X0 ;

ð7:3:19Þ

W ðtÞ ¼ ðw1s ; w2s ÞT ¼ ð^zts ; A^z s ÞT ¼ T ðt  sÞW0s ; t  s  0:

ð7:3:20Þ

Thus from (4.3.15) to (4.3.20) we can verify that V ðtÞ fulfills Z t T ðt  sÞFðV ðsÞÞds; t  s  0: V ðtÞ ¼ T ðt  sÞV0s þ

ð7:3:21Þ

where

s

The following lemma shows the properties on the semigroup fT ðtÞg. Lemma 7.3.1. For any t  s þ 1; s  0, there exist an absolute constant q [ 0 and a constant K1 ¼ K1 ðsÞ [ 0 depending only on s such that for any W0s 2 X0  X0 ; s 2 R þ ; we have kT ðt  sÞW0s kH 2 H 1  K1 eqt=2 kW0s kX0 X0 :

ð7:3:22Þ

Proof. As in (4.3.19) and (4.3.20), we can write W ðsÞ ¼ W0s ¼ ð^z1s ; A^z0s ÞT 2 X0  X0 ; W ðtÞ ¼ ð^zts ; A^z s ÞT ¼ ðw1s ðtÞ; w2s ðtÞÞT ¼ T ðt  sÞW0s :

ð7:3:23Þ

Attractors for Nonlinear Autonomous Dynamical Systems

168

Set u~ ¼ ^zxs ; ~v ¼ ^zts :

ð7:3:24Þ

Then (7.3.17) and (7.3.18) or (7.3.19) and (7.3.20) is now transformed into vx ¼ 0; t  s  0; u~t  ~

ð7:3:25Þ

~ vt  ~ vxx  u~x ¼ 0; t  s  0;

ð7:3:26Þ

s ~ v jx¼0;1 ¼ 0; u~ jt¼s ¼ ^z0;x ðxÞ; ~v jt¼s ¼ ^z1s ðxÞ:

ð7:3:27Þ

Here W0s ¼ ð^z1s ; A^z0s ÞT 2 X0  X0 : From (4.3.25) and (4.3.27), we have Z 1 Z 1 ^zx dx ¼ 0; k~ u k  k~ ux k; k~v k  k~vx k: u~ dx ¼ 0

ð7:3:28Þ

0

Multiplying (7.3.26) by ~vt eqt in L2 ð0; 1Þ gives o 1dn ðk~ v ðtÞk2 þ k~ u ðtÞk2 Þeqt þ ð1  q=2Þk~ u ðtÞk2 eqt : vx ðtÞk2 eqt  ðq=2Þk~ 2 dt

ð7:3:29Þ

Multiplying (7.3.26) by u~x eqt in L2 ð0; 1Þ and using (7.3.25), we obtain    d 1 2 k~ ux ðtÞk  ð~ v ; u~x Þ eqt þ ð1  qÞk~ ux ðtÞk2 eqt  ½ðq=2Þk~v ðtÞk2 þ k~vx ðtÞk2 eqt ; dt 2 ð7:3:30Þ R1 where ð~ v ; u~x Þ ¼ 0 ~ v u~x dx: Multiplying (7.3.30) by a parameter k 2 ð0; 1Þ, then adding the resulting inequality to (7.3.29), we conclude d G1 ðtÞ þ ð1  q=2  kq=2  kÞk~ vx ðtÞk2 eqt þ ½kð1  qÞ  q=2k~ ux ðtÞk2 eqt  0 dt ð7:3:31Þ with

 G1 ðtÞ ¼

 1 1 k 2 2 2 k~ u ðtÞk þ k~ v ðtÞk þ k~ ux ðtÞk  kð~v ; u~x Þ eqt : 2 2 2

Now fix k 2 ð0; 1Þ and choose q [ 0 so small that   2ð1  kÞ 2k ;  q0 : 0\q\ min 1 þ k 1 þ 2k

ð7:3:32Þ

ð7:3:33Þ

Global Attractor for a Nonlinear Thermoviscoelastic System

169

Thus using the estimate 1 q k~ v ðtÞk2 þ k~ ux ðtÞk2  kð~ v ; u~x Þ  C0 ðkÞðk~v ðtÞk2 þ k~ ux ðtÞk2 Þ 2 2

ð7:3:34Þ

with C0 ðkÞ ¼ kð1kÞ 2 , it follows from (7.3.31)–(7.3.34) that for any q 2 ð0; q0 Þ and for any t  s þ 1; Z t 2 2 qt e ðk~ u ðtÞkH 1 þ k~v ðtÞk Þ þ ðk~vx k2 þ k~ ux k2 ÞðsÞeqs ds  Cs ðk~ u ðs þ 1Þk2H 1 þ k~ v ðs þ 1Þk2 Þ; sþ1

ð7:3:35Þ here and hereafter Cs [ 0 is a generic constant depending only on q; s and k. In the same manner, we can derive from (7.3.26) for q 2 ð0; q0 Þ, d ½k~ vx ðtÞk2 eqt  þ k~ vt ðtÞk2 eqt  ðq=2Þk~ vx ðtÞk2 eqt þ ð1=2Þk~ ux ðtÞk2 eqt ; dt which and (7.3.35) imply that for any q 2 ð0; q0 Þ; t  s þ 1, Z

eqt k~vx ðtÞk2 þ

t

sþ1

Z

k~vt ðsÞk2 eqs ds  Cs k~vx ðs þ 1Þk2 þ Cs

t

sþ1

ðk~ vx k2 þ k~ ux k2 ÞðsÞeqs ds

u ðs þ 1Þk2H 1 þ k~v ðs þ 1Þk2H 1 Þ:  Cs ðk~

ð7:3:36Þ

Thus for any q 2 ð0; q0 Þ and for any t  s þ 1, it follows from (7.3.35) and (7.3.36) that Z t 2 2 qt e ðk~ u ðtÞkH 1 þ k~ v ðtÞkH 1 Þ þ ðk~ v k2H 2 þ k~ u k2H 1 ÞðsÞeqs ds sþ1

 Cs ðk~ u ðs þ 1Þk2H 1 þ k~ v ðs þ 1Þk2H 1 Þ:

ð7:3:37Þ

Differentiating (7.3.26) in t, multiplying the resultant by eqt ~vt in L2 ð0; 1Þ, and using (7.3.36) and (7.3.37), we readily conclude that for any q 2 ð0; q0 Þ and for any t  s þ 1, n o Z t eqt k~vxx ðtÞk2 þ k~ vt ðtÞk2 þ k~ vtx ðsÞk2 eqs ðsÞds Z 2

 Cs k~ vt ðs þ 1Þk þ Cs

sþ1

t sþ1

ðk~ vt k2 þ k~vxx k2 ÞðsÞeqs ds

u ðs þ 1Þk2H 1 þ k~ v ðs þ 1Þk2H 1 þ k~vt ðs þ 1Þk2 Þ:  Cs ðk~

ð7:3:38Þ

Differentiating (7.3.26) in x and using (7.3.25), we obtain vtx : u~txx þ u~xx ¼ ~ Multiplying (7.3.39) by eqt u~xx in L2 ð0; 1Þ to obtain o 1dn k~ uxx ðtÞk2 eqt þ ð1  qÞk~ uxx ðtÞk2 eqt  ð2qÞ1 k~vtx ðtÞk2 eqt ; 2 dt

ð7:3:39Þ

ð7:3:40Þ

170

Attractors for Nonlinear Autonomous Dynamical Systems

which, combined with (7.3.36) and (7.3.37), gives that for any q 2 ð0; q0 Þ and for any t  s þ 1; Z t 2 2 2 qt e ðk~ u ðtÞkH 2 þ k~v ðtÞkH 2 þ k~vt ðtÞk Þ þ ðk~v k2H 3 þ k~ u k2H 2 þ k~ vt k2H 1 ÞðsÞeqs ds sþ1

u ðs þ 1Þk2H 2 þ k~v ðs þ 1Þk2H 1 þ k~vt ðs þ 1Þk2 Þ:  Cs ðk~

ð7:3:41Þ

Noting that semigroup fT ðtÞg is an analytic semigroup on X0  X0 with its infinitesimal generator A, we easily prove by the regularity of elliptic equations that 0 2 qðAÞ;

ð7:3:42Þ

where qðAÞ is the resolvent set of A. Thus from lemma 1.2.17 and (7.3.42) we can derive that for any t [ 0; m  0; k Am T ðtÞkLðX0 Þ  C ðmÞt m edt ;

ð7:3:43Þ

where d [ 0 and C ðmÞ are positive constants depending only on the operator A and m, but independent of t. Using u~x ðtÞ ¼ w2s ðtÞ; ~ v ¼ w1s ðtÞ;

ð7:3:44Þ

it follows from (7.3.28), (7.3.41) and (7.3.43) that k~ u ðs þ 1ÞkH 2  C k~ ux ðs þ 1ÞkH 1  C kw2s ðs þ 1ÞkH 1  s  C kw2 ðs þ 1Þk þ kAw2s ðs þ 1Þk  C fkW ðs þ 1Þk þ k AW ðs þ 1Þkg n o  C kT ðs þ 1ÞkLðX0 Þ þ k AT ðs þ 1ÞkLðX0 Þ kW0s k  C kW0s k;

ð7:3:45Þ

 k^ v ðs þ 1ÞkH 1 ¼ kw1s ðs þ 1ÞkH 1  C kw1s ðs þ 1Þk þ kAw1s ðs þ 1Þk n o  C kT ðs þ 1ÞkLðX0 Þ þ k AT ðs þ 1ÞkLðX0 Þ kW0s k  C kW0s k;

ð7:3:46Þ

s ðs þ 1Þk  kWt ðs þ 1Þk k^vt ðs þ 1Þk ¼ kw1t

 k AT ðs þ 1ÞkLðX0 Þ kW0s k  C kW0s k;

ð7:3:47Þ

where C [ 0 is a generic constant depending only on operators A and fT ðtÞg, but independent of t [ 0. Thus we finally conclude from (7.3.41), (7.3.45)–(7.3.47) that k~ u ðs þ 1ÞkH 2 þ k~ v ðs þ 1ÞkH 1 þ k~vt ðs þ 1Þk  C kW0s k;

ð7:3:48Þ

which, together with (7.3.41), gives for any q 2 ð0; q1 Þ and for any t  s þ 1; u ; ~v ÞkH 2 H 2  K1 ðsÞkW0s keqt=2 : kT ðtÞW0 kH 2 H 1 ¼ kW ðtÞkH 2 H 1  kð~

Global Attractor for a Nonlinear Thermoviscoelastic System

Thus this proves the lemma. The following lemma is very important to the compactness of the orbit.

171

h

Lemma 7.3.2. (i) For any t  1; kuðtÞkH 1  C3d þ CB ec3 t :

ð7:3:49Þ

(ii) There exists some time t3 ¼ t3 ðBÞ  t2 ðBÞ [ 0 such that as t [ t3 ðBÞ, kuðtÞkH 1  2C3d

ð7:3:50Þ

h i h i þ aÞc1 c1 1 ; 12 and 0\c1 \ min 32 ; 7ð296q with c3 2 0; min q=2; 7ð2192 þ aÞ : Proof. Employing lemmas 7.2.6, 7.2.11, 7.2.12 and (7.3.21), we can derive kFðV ðtÞÞk  Cd ðkr1 kL1 þ kukL1 þ kvx kL1 Þ  Cd þ CB ec1 t=12 :

ð7:3:51Þ

Now applying lemma 7.3.1 with s ¼ 1, it follows from (7.3.22) and (4.3.51) that for any t  1, Z t eqðtsÞ=2 kFðV ðsÞÞkds kV ðtÞkH 2 H 1  K1 eqt=2 kV01 k þ K1 eq=2 1 Z t  Ceqt=2 kV01 k þ C eqðtsÞ=2 ðCd þ CB ec1 s=12 Þds 1

ð7:3:52Þ  Cd þ CB ec3 t

h i þ aÞc1 c1 with c3 2 0; min q=2; 7ð2192 ; 12 and, by lemmas 7.2.2 and 7.2.3, kV01 k ¼ kð^ v ; u^x Þjt¼1 k  CB . Thus uxx k  kv2 ðtÞkH 1  kV ðtÞkH 2 H 1  Cd þ CB ec3 t ; kux ðtÞk ¼ k^ which, along with (7.2.19), gives (4.3.49). Choosing  1 t3 ¼ t3 ðBÞ ¼ max t2 ; c1 3 logðC3d CB Þ ; (7.3.50) readily follows. This completes the proof. Lemma 7.3.3. For t [ t3 ðBÞ, we have Z t s  t3 ðtsÞ=4 2 kvt ðtÞk þ e kvtx ðsÞk2 ds  C4d þ CB ec1 at=12 ; t  t 3 t3 Z

t

kht ðtÞk2 þ t3

with a 2 ð0; 2=7Þ.

s  t3 ðtsÞ=4 e khtx ðsÞk2 ds  C4d þ CB ec1 at=12 t  t3

h

ð7:3:53Þ

ð7:3:54Þ

Attractors for Nonlinear Autonomous Dynamical Systems

172

Proof. Differentiating (7.1.12) in t, and using lemma 7.2.13 and the Poincaré’s inequality, we derive that for t [ t3 , d 1 3 kvt ðtÞk2 þ kvt ðtÞk2 þ kvtx ðtÞk2  kr 1t k2  Cd ðkvx ðtÞk2 þ kht ðtÞk2 Þ; dt 4 4 which implies that for all for t [ t3 , o 3 dn ðt  t3 Þet=4 kvt ðtÞk2 þ ðt  t3 Þkvtx ðtÞk2 et=4 dt 4 n o tt 3 2 t=4 kvt ðtÞk2 et=4 :  Cd ðt  t3 Þe kvx ðtÞk þ kht ðtÞk2 þ 4

ð7:3:55Þ

Using lemmas 7.2.5, 7.2.11, (7.2.18), (7.2.66) and (7.2.67), integrating (7.3.55) over ½t3 ; t, we obtain that for all t [ t3 ðBÞ, Z 1 t s  t3 2 kvt ðtÞk þ kvtx ðsÞk2 eðtsÞ=4 ds 2 t3 t  t3 Z t s  t3 ðtsÞ=4  Cd e ðkvx ðsÞk2 þ kht ðsÞk2 Þds t3 t  t3 Z t Z t  Cd kvx ðsÞk2 ds þ Cd eðtsÞ=4 kht ðsÞk2 ds t3

 Cd þ CB e

t3 c1 at=12

;

ð7:3:56Þ

which gives (7.3.53). Similarly, we infer from (7.1.13) kht ðtÞk  kht  ht k þ jht j  khtx ðtÞk þ Cd kvx ðtÞk: That is, 1 1 kht ðtÞk2  khtx ðtÞk2 þ Cd kvx ðtÞk2 : 4 2

ð7:3:57Þ

Similarly, we infer from (7.1.13) d 1 1 kht ðtÞk2 þ kht ðtÞk2 þ khtx ðtÞk2  Cd ðkht ðtÞk2 þ kvx ðtÞk2 þ kvtx ðtÞk2 Þ dt 4 2 implying o 1 dn ðt  t3 Þet=4 kht ðtÞk2 þ ðt  t3 Þkhtx ðtÞk2 et=4 dt 2 n o tt 3 2 t=4 kht ðtÞk2 et=4 : ð7:3:58Þ  Cd ðt  t3 Þe kvx ðtÞk þ kht ðtÞk2 þ kvtx ðtÞk2 þ 4

Global Attractor for a Nonlinear Thermoviscoelastic System

173

Thus integrating of (7.3.58) over ½t3 ; t, we have that for any t [ t3, Z 1 t s  t3 kht ðtÞk2 þ khtx ðsÞk2 eðtsÞ=4 ds 2 t3 t  t3 Z t Z t Z t s  t3  Cd kvx ðsÞk2 ds þ Cd eðtsÞ=4 kht ðsÞk2 ds þ Cd kvtx ðsÞk2 eðtsÞ=4 ds t  t 3 t3 t3 t3  Cd þ CB ec1 at=12 ; which gives (7.3.54). The proof is now complete.

h

Lemma 7.3.4. There exists some time t0 ¼ t0 ðBÞ [ S 0 such that as t  t0 ðBÞ, the orbit starting from any given bounded set B in Hd , t  t0 SðtÞB is relatively compact in Hd : Proof. Fixing a 2 ð0; 2=7Þ and choosing t0 ¼ t0 ðBÞ ¼ maxft3 ðBÞ; ðc1 aÞ1 logðC4d CB1 Þg þ 1, it follows from lemma 7.3.3 that as t [ t0 , kvt ðtÞk  2C4d ; kht ðtÞk  2C4d :

ð7:3:59Þ

Using (7.1.12), (7.1.13) and (7.3.59), we can derive from lemmas 7.3.2 and 7.2.13 that as t [ t3 , kvxx ðtÞk  kvt ðtÞk þ kr1x ðtÞk  kvt ðtÞk þ Cd ðkux ðtÞk þ khx ðtÞkÞ  Cd ; khxx ðtÞk  kht ðtÞk þ Cd ðkvx ðtÞk þ kvx ðtÞkkvx ðtÞkL1 Þ:

ð7:3:60Þ ð7:3:61Þ

Thus (7.3.60) and (7.3.61) along with (7.3.50) give that as t [ t0 , kðuðtÞ; vðtÞ; hðtÞÞkH1d  Cd ; which means the semiflow fSðtÞg is compact due to the compact embedding of H1d into Hd . This is thus to complete the proof. h Proof of Theorem 7.1.1. Exploiting lemmas 7.2.1, 7.2.13 and 7.3.4, we easily prove theorem 7.1.1. h

7.4

Bibliographic Comments

Since the system (7.1.1)–(7.1.3) is the same formally as the system (5.1.1)–(5.1.3) only with different constitutive relations, the essential mathematical difficulties are similar. Thus for mathematical difficulties for the system (7.1.1)–(7.1.3), we can refer to four points (1)–(4) in section 4.6 of chapter 4.

Chapter 8 Global Attractors for Nonlinear Reaction–Diffusion Equations and the 2D Navier–Stokes Equations In this chapter, we shall prove the existence of global attractor for nonlinear reaction–diffusion equations and the 2D Navier–Stokes equations, which are regarded as applications of the norm-to-weak continuous semigroup in a Banach space established in [163]. The content of this chapter is adapted from Zhong, Yang and Sun [163]. The methods used in this chapter are x-limit compact semigroup method (see theorems 1.2.29, 1.2.30, 1.2.37), condition-(C) method (see theorem 1.2.38), and the norm-to-weak continuous semigroup method (see theorem 1.2.51).

8.1

Global Attractor for Strong Solutions of Reaction–Diffusion Equations

This chapter is devoted to the asymptotic behavior for the reaction–diffusion equations with a polynomial growth nonlinearity of arbitrary order 8 @u > >
Du þ f ðuÞ ¼ g; ðx; tÞ 2 X  R þ ; > < @t ð8:1:1Þ u ¼ 0; ðx; tÞ 2 @X  R þ ; > > > : uðx; 0Þ ¼ u ; x 2 X; 0

where X is a bounded smooth domain in Rn , f is a C 1 function satisfying that f 0 ðsÞ 
l; C1 jsjp
C0  f ðsÞs  C2 jsjp þ C0 ; p  2 for any s 2 R, and either g 2 H
1 ðXÞ, leading to an attractor in Lp ðXÞ and H01 ðXÞ. DOI: 10.1051/978-2-7598-2702-2.c008 © Science Press, EDP Sciences, 2022

Attractors for Nonlinear Autonomous Dynamical Systems

176

8.1.1

Existence of Solutions and Uniqueness

In this section, we shall prove the existence and uniqueness of solutions by the normal Faedo–Galerkin methods. Here we only state the results in [125, 146]. Lemma 8.1.1. Assume that X is a bounded smooth domain in Rn , g 2 H
1 ðXÞ. Then for any initial data u0 2 L2 ðXÞ and any T [ 0, there exists a unique solution u for system such that u 2 L2 ð0; T ; H01 ðXÞÞ \ Lp ð0; T ; Lp ðXÞÞ; 8T [ 0; u 2 C ðR þ ; L2 ðXÞÞ; and the mapping u0 ! uðtÞ in L2 ðXÞ. If g 2 L2 ðXÞ and u0 2 H01 ðXÞ, then u 2 C ð0; T ; H01 ðXÞÞ \ L2 ð0; T ; H 2 ðXÞÞ; 8T [ 0: According to this lemma, we can define the operator semigroup fSðtÞg in L2 ðXÞ for both g 2 H
1 ðXÞ and g 2 L2 ðXÞ as SðtÞu0 : L2 ðXÞ  R þ ! L2 ðXÞ; which is continuous in L2 ðXÞ.

8.1.2

Global Attractor for the Semigroup in Lp ðXÞ

It follows from [162] that the semigroup associated with the solutions of system has a global attractor in L2 ðXÞ. Combining this result and the asymptotic properties of the semigroup, we shall prove the existence of global attractors in Lp ðXÞ. Lemma 8.1.2. Assume fSðtÞg is a semigroup on Lp ðXÞ and fSðtÞg have a bounded absorbing set in Lp ðXÞ. Then for any e [ 0 and any bounded subset B  Lp ðXÞ, there exist positive constants T ¼ TB and M ¼ M ðeÞ such that mðXðjSðtÞu0 j  M ÞÞ  e; 8 u0 2 B; t  T ; where mðeÞ denotes the Lebesgue measure of e  X and Xðjuj  M Þ , fx 2 Xjj uðxÞj  M g. The following lemma verifies that the semigroup fSðtÞg is x-limit compact in Lp ðXÞ ðp [ 0Þ. Lemma 8.1.3. If there exists a positive constant M ¼ M ðeÞ which depends on e, such that (i) (ii)

B has a finite ð3M Þðq
pÞ=q ð2e Þp=q -net in Lq ðXÞ for some q [ 0; R ð Xðjuj  M Þ jujp dxÞ1=p \2
ð2p þ 2Þ=p e for any u 2 B. Then for any e [ 0, the bounded subset B of Lp ðXÞ ðp [ 0Þ has a finite e-net in Lp ðXÞ:

Global Attractors for Nonlinear Reaction–Diffusion Equations

177

Noting the definition of x-limit compactness of the semigroup, from lemma 8.1.3, the following result follows easily. Remark 8.1.1. Assume that fSðtÞg is a semigroup on Lp ðXÞ and Lq ðXÞ, respectively, where p  q [ 0 and X  Rn is bounded, and fSðtÞg satisfy the following two assumptions: (i) fSðtÞg is x-limit compact in Lq ðXÞ; (ii) for any e [ 0 and any bounded subset B  Lq ðXÞ, there exist positive constants M ¼ M ðe; BÞ and T ¼ T ðe; BÞ such that Z jSðtÞu0 jp dx\e; 8 u0 2 B; t  T : XðjSðtÞu0 j  M Þ

Then fSðtÞg is x-limit compact in Lp ðXÞ. Combining the above result, we have the following theorem which can be used easily to prove the existence of the global attractor for some semigroup in Lp ðXÞ using theorems 1.2.29, 1.2.30, and 1.2.37. Theorem 8.1.4. Assume fSðtÞg is a semigroup on Lp ðXÞ and Lq ðXÞ, respectively, where p  q [ 0 and X  Rn is bounded, and fSðtÞg satisfy the following four assumptions: (i) fSðtÞg is x -limit compact in Lq ðXÞ; (ii) fSðtÞg has a bounded absorbing set B0 in Lp ðXÞ; (iii) for any e [ 0 and any bounded subset B  Lq ðXÞ, there exist positive constants M ¼ M ðe; BÞ and T ¼ T ðe; BÞ such that Z jSðtÞu0 jp dx\e; 8 u0 2 B; t  T ; XðjSðtÞu0 j  M Þ

(iv) fSðtÞg is norm-to-weak continuous on ðSðB0 Þ; j  jp Þ. Then fSðtÞg possesses a global attractor in Lp ðXÞ.

8.1.3

Global Attractor of System in Lp ðXÞ and H01 ðXÞ

In this subsection, we shall discuss the case where the external forcing term d 2 H
1 ðXÞ, and further prove the existence of global attractors in Lp ðXÞ and H01 ðXÞ, respectively. For convenience, we denote g by P Di f i þ hðxÞð¼ ni¼1 Di f i þ hðxÞÞ where f i ; h 2 L2 ðXÞ. Using the nonstandard estimate about the energy functional, the fact that the semigroup SðtÞ has a global attractor in L2 ðXÞ has been proved in [162]. Lemma 8.1.5. Assume that X is a bounded smooth domain in R2 , f i ; h 2 L2 ðXÞ ði ¼ 1; . . .; nÞ. Then the semigroup generated by the weak solution of system has a global attractor AH in L2 ðXÞ. The next lemma concerns the existence of absorbing sets in Lp ðXÞ and H01 ðXÞ.

Attractors for Nonlinear Autonomous Dynamical Systems

178

Theorem 8.1.6. The semigroup fSðtÞg has a bounded absorbing set in Lp ðXÞ and H01 ðXÞ respectively, that is, for any bounded subset B in L2 ðXÞ, there exists a positive constant T , depending only on the L2 -norm of B, such that juðtÞjpp  M ; 8u0 2 B; t  T and jruðtÞj22  M ; 8u0 2 B; t  T ; where M is a positive constant independent of B, uðtÞ ¼ SðtÞu0 . Proof. Multiplying (8.1.1) by u and integrating by parts, we arrive at Z Z 1d 2 2 i juj þ jjujj þ f ðuÞu ¼ \Di f ; u [ þ \h; u [ ¼
~f  ru þ \h; u [ ; 2 dt X X ð8:1:2Þ which implies that

Z d 2 2 juj þ jjujj þ jujp  C ðj~f j2 ; k1 ; jhj2 ; jXjÞ; ð8:1:3Þ dt X Rs P where ~f ¼ ðf 1 ; . . .; f n Þ, j~f j2 ¼ ni¼1 jf i j2 . Meanwhile, let FðsÞ ¼ 0 f ðsÞds, from the condition we can deduce that ~ 1 jsjp
k  FðsÞ  k þ C ~ 2 jsjp : C Therefore, ~1 C

Z X

Z jujp
kjXj 

X

~2 FðuÞ  kjXj þ C

ð8:1:4Þ Z X

jujp :

ð8:1:5Þ

Also noticing that jru þ ~f j2  2jruj2 þ 2j~f j2 ; by (8.1.4)–(8.1.6), we infer from (8.1.3) that Z d 2 juj þ C ðjru þ ~f j2 þ FðuÞÞ  C ðj~f j2 ; k1 ; jhj2 ; jXjÞ: dt X

ð8:1:6Þ

ð8:1:7Þ

Integrating (8.2.6) from over ½t; t þ 1, we have Z tþ1 Z 2 ~ ðjru þ f j þ FðuÞÞdx  C ðj~f j2 ; k1 ; jhj2 ; jXjÞ þ juðtÞj2 ; X

t

thus we can find T0 ¼ T ðju0 jÞ [ 0 and q0 [ 0 satisfying that juðtÞj2  q0 for any t  T0 . Thus we can derive that for any t  T0, Z tþ1 Z ðjru þ ~f j2 þ FðuÞÞ  C ðj~f j2 ; k1 ; jhj2 ; jXjÞ þ q20 : ð8:1:8Þ t

X

Global Attractors for Nonlinear Reaction–Diffusion Equations

Now, multiplying (8.1.1) by ut to give Z 1d d 2 2 jjujj þ FðuÞ ¼ \Di f i ; ut [ þ \h; ut [ jut j þ 2 dt dt X d ¼
\~f ; ru [ þ \h; ut [ : dt

179

ð8:1:9Þ

Using the Hölder and the Cauchy’s inequalities, we can derive from (8.1.6) and (8.1.9) that Z d ðjru þ ~f j2 þ FðuÞÞdx  jhj2 : ð8:1:10Þ dt X Combining with (8.1.8) and (8.1.9), and using the uniform Gronwall lemma (see theorem 1.1.8), we obtain Z FðuÞ  jhj2 þ C ðj~f j2 ; k1 ; jhj2 ; jXjÞ þ q20 : ð8:1:11Þ jru þ ~f j2 þ X

Thanks to jruj  2jru þ ~f j2 þ 2j~f j2 and (8.1.5), (8.1.11) implies that for t  T0 þ 1, ð8:1:12Þ jruðtÞj2  3jhj2 þ 2C ðj~f j2 ; k1 ; jhj2 ; jXjÞ þ 2q20 2

and

Z X

juðtÞjp  C ðjhj2 þ 2C ðj~f j2 ; k1 ; jhj2 ; jXjÞ þ 2q20 Þ:

ð8:1:13Þ

Now, taking M ¼ C ðjhj2 þ 2C ðj~f j2 ; k1 ; jhj2 ; jXjÞ þ 2q20 Þ and T ¼ T0 þ 1, we thus complete the proof. h Next, in order to get asymptotic estimates, we need to give a priori estimate for the unbounded part of the modular juj for the solution u of system in Lp -norm, which is stated in the next theorem. Theorem 8.1.7. For any e [ 0 and any bounded subset B  L2 ðXÞ, there exist two positive constants T ¼ T ðe; BÞ and M ¼ M ðeÞ such that Z juðtÞjp \C e; 8 u0 2 B; t  T ; XðjuðtÞj  M Þ

where the constant C is independent of e and B. Proof. For any fixed e [ 0, there exists d [ 0 such that if e  X and mðeÞ  d, then Z jf i ðxÞj2 dx\e; 1  i  n ð8:1:14Þ e

and

Z jhðxÞj2 dx\e: e

ð8:1:15Þ

Attractors for Nonlinear Autonomous Dynamical Systems

180

Moreover, from the lemmas above, it follows that there exist constants T ¼ T ðe; BÞ and M ¼ M ðeÞ such that for any u0 2 B and t  T , mðXðjuðtÞj  M1 ÞÞ  minfe; dg and

ð8:1:16Þ

Z XðjuðtÞj  M1 Þ

juðtÞj2 dx  8e:

ð8:1:17Þ

Moreover, we derive from the condition on f that s [ ðC0 =C1 Þ1=p . Let M ¼ maxfM1 ; ðC0 =C1 Þ1=p g and t  T . Multiplying (8.1.1) by ðu
M Þ þ and integrating over X to give Z Z 1d 2 2 jðu
M Þ þ j þ jjðu
M Þ þ jj dx þ f ðuÞðu
M Þ þ dx 2 dt XðjuðtÞj  M Þ X ¼\Di f i ; ðu
M Þ þ [ þ \h; ðu
M Þ þ [ Z ¼
~f  rðu
M Þ þ þ \h; ðu
M Þ þ [ ; X

where ðu
M Þ þ denotes the positive part of u
M , i.e., ðu
M Þ þ ¼ u
M ; for u  M ; ðu
M Þ þ ¼ 0; for u  M : Let X1 ¼ XðuðtÞ  M Þ, then

Z Z 1d jðu
M Þ þ j2 þ jjujj2 þ f ðuÞðu
M Þ 2 dt X1 X1 Z Z ~f  ru þ ¼
hðu
M Þ: X1

X1

By applying the Cauchy’s inequality and the Hölder inequality, it follows Z  Z d jðu
M Þ þ j2 þ C jjujj2 þ f ðuÞðu
M Þ dt X1 X1 Z  Z 2 2 C j~f j þ juj ; ð8:1:18Þ X1

X1

which, in combining with (8.1.14)–(8.1.17) and integrating over ½t; t þ 1, yields  Z t þ 1 Z Z 2 jjujj þ f ðuÞðu
M Þ  C e: ð8:1:19Þ t

X1

X1

Hence Z t

tþ1

Z Xðu  2M Þ

jru þ ~f j dx þ

!

Z

2

Xðu  2M Þ

f ðuÞudx

 C e:

ð8:1:20Þ

Global Attractors for Nonlinear Reaction–Diffusion Equations

181

We now multiply (8.1.1) by ½ðu
2M Þ þ t and denote by X2 ¼ Xðu  2M Þ, then Z  Z d 2 ~ jru þ f j dx þ FðuÞdx  C e: ð8:1:21Þ dt X2 X2 From (8.1.20) and (8.1.21), using the uniform Gronwall lemma, it easily follows Z  Z jru þ ~f j2 dx þ FðuÞdx  C e: X2

Hence,

X2

Z X2

Z jruj2 dx  2

and

X2

jru þ ~f j2 dx þ 2

Z X2

j~f j2 dx  C e

ð8:1:22Þ

Z X2

FðuÞdx  C e:

ð8:1:23Þ

Repeat the same steps above, just choose ðu þ M Þ
½ðu þ 2M Þ
t instead of ðu
M Þ þ ½ðu
2M Þ þ t , respectively, to be able to Z jjujj2 dx  C e ð8:1:24Þ Xðu 
2M Þ

and

Z Xðu 
2M Þ

FðuÞdx  C e:

Then from (8.1.22) to (8.1.24), we infer that Z jjuðtÞjj2 dx  C e; XðjuðtÞj  2M Þ

ð8:1:25Þ

ð8:1:26Þ

and Z XðjuðtÞj  2M Þ

FðuðtÞÞdx  C e:

Thus, using (8.1.5) and (8.1.16), the conclusion follows naturally.

ð8:1:27Þ h

Using the above conclusion, we can obtain the following theorem of the existence of global attractors in Lp ðXÞ and H01 ðXÞ. Theorem 8.1.8. Assume that X is a bounded smooth domain in Rn , g 2 H
1 ðXÞ. Then the semigroup generated by the solution of system with initial data u0 2 L2 ðXÞ has a global attractor Ap in Lp ðXÞ.

Attractors for Nonlinear Autonomous Dynamical Systems

182

Next, we shall prove the existence of a global attractor in H01 ðXÞ. Now we first give some a priori estimates about ut endowed with L2 -norm. Theorem 8.1.9. Assume f i ði ¼ 1; 2; . . .; nÞ, h 2 L2 ðXÞ, and fSðtÞg is the semigroup associated with system. Then for any bounded subset B in L2 ðXÞ, there exists a positive constant T ¼ TB such that jut ðsÞj2 \M ; 8 u0 2 B; s  T ; where jut ðsÞ ¼ dtd ðSðtÞu0 Þjt¼s and M is a positive constant which is independent of B. Proof. Differentiating (8.1.1) in time and denoting v ¼ ut , we know vt
Dv þ f 0 ðuÞv ¼ 0:

ð8:1:28Þ

Multiplying the above equality by v, we get 1d 2 jvj þ jjvjj2  ljvj2 : 2 dt

ð8:1:29Þ

Integrating (8.1.9) over ½t; t þ 1 and by (8.1.11) to yield Z t

t þ1

jvj2 dx  C ðj~f j2 ; k1 ; jhj2 ; jXjÞ þ q20

ð8:1:30Þ

as t large enough. Combining (8.1.29) with (8.1.30), and using the uniform Gronwall lemma, we conclude jvðt þ 1Þj2  ðC ðj~f j2 ; k1 ; jhj2 ; jXjÞ þ q20 Þel

ð8:1:31Þ h

as t large enough.

Next, we shall prove that fSðtÞg satisfies condition ðC Þ in Let Hm ¼ spanfx1 ; . . .; xm g, Pm be the canonical projector on Hm and I be the identity. Then for any u 2 L2 ðXÞ or u 2 H01 ðXÞ, u has a unique decompose: u ¼ u1 þ u2 , where u1 ¼ Pm u 2 Hm ; u2 ¼ ðI
Pm Þu 2 Hm? . H01 ðXÞ.

Theorem 8.1.10. Assume f i ði ¼ 1; 2; . . .; nÞ, h 2 L2 ðXÞ, and fSðtÞg is the semigroup associated with system. Then fSðtÞg fulfills condition ðC Þ in H01 ðXÞ, that is, for any e [ 0 and any bounded subset B in L2 ðXÞ, there exist T ¼ TB and m 2 N such that jjðI
Pm ÞuðtÞjj2  e; 8 u0 2 B; provided that t  T and m  m0 .

Global Attractors for Nonlinear Reaction–Diffusion Equations

183

Proof. Denoting by u2 ¼ ðI
Pm ÞuðtÞ for any u 2 L2 ðXÞ, and multiplying (8.1.1) by u2 , we know Z 1d ju2 j2 þ jju2 jj2 þ f ðuÞu2 ¼ \Di f i ; u2 [ þ \h; u2 [ 2 dt X and

Z 2

jju2 jj  jut2 jju2 j þ

X

jf ðuÞjju2 j þ ðC ðj~f2 j2 þ jh2 j2 Þ

2 ~ 2  jut2 jju2 j þ jjujjp
1 Lp jju2 jjLp þ ðC ðjf2 j þ jh2 j þ ju2 jÞ;

and the proof is complete.

ð8:1:32Þ h

Based on above conclusions and using theorem 1.2.38, we can immediately prove the existence of global attractor in H01 ðXÞ which is stated in the next theorem. Theorem 8.1.11. Assume that X is a bounded smooth domain in Rn , g 2 H
1 ðXÞ. Then the semigroup generated by the solution of system with initial data u0 2 L2 ðXÞ has a global attractor AV in H01 ðXÞ.

8.2

Global Attractors for the 2D Navier–Stokes Equations in H01 ðXÞ

This section shall prove the existence of global attractors for the 2D incompressible Navier–Stokes equations in H01 ðXÞ. Again, we also need to verify the uniform compactness for the existence of global attractors is weak condition, Condition-(C), which can be verified with energy estimates in H01 itself. Let X be an open bounded domain of R2 with smooth boundary @X, the equations are as follows @u þ ðu  rÞu
mDu þ rp ¼ f ; @t divu ¼ 0; u ¼ 0; on @X; ð8:2:1Þ uðx; 0Þ ¼ u0 ðxÞ; x 2 X; which, in weak form, is equivalent to the following operator equations of the system (8.2.1) du þ mAu þ Bðu; uÞ ¼ f ; dt ð8:2:2Þ uð0Þ ¼ u0 : We recall here some classical results on the existence of solutions for the 2D Navier–Stokes equations (see [146]) shown in the next theorem.

184

Attractors for Nonlinear Autonomous Dynamical Systems

Theorem 8.2.1. Assume that f ; u0 2 H . Then there exists a unique solution u of (8.2.2) such that, for any T [ 0, u 2 C ð0; T ; H Þ \ L2 ð0; T ; V Þ. Moreover, for t  0, u is analytic in t with values in DðAÞ, and the mapping u0 ! uðtÞ is continuous from H into DðAÞ; If u0 2 V , then for any T [ 0, u 2 C ð0; T ; H Þ \ L2 ð0; T ; DðAÞÞ. Note that the above theorem allows us to define a C 0 semigroup fSðtÞg by SðtÞ : u0 ! uðtÞ and the next existence theorem of global attractors in H was established in [45]. Theorem 8.2.2. The semigroup fSðtÞg associated with (8.2.2) has a global attractor A 2 V that is compact, connected and maximal in H , and attracts all bounded set of H. The semigroup fSðtÞg has an absorbing set BV ð0; q1 Þ  V , centered at 0, with radius q1 , satisfying that, for any bounded set B of H , there exists some t0 [ 0 such that SðtÞB  BV ð0; q1 Þ  V for all t  t0 . Thus we can prove our main result in this section as follows. Theorem 8.2.3. The global attractor A of he semigroup fSðtÞg associated with (8.2.2) is compact, connected and maximal in V , and attracts all bounded set of V . Consequently it attracts all bounded sets of H in V norm. Proof. Let m 2 N be a fixed integer, and P ¼ Pm is the the projector in H onto the space spanned by the first m eigenvectors of operator A, w1 ; . . .; wm . Also, we set Q ¼ Qm ¼ I
Pm , and for the sake of simplicity k ¼ km ; K ¼ km þ 1 : Let p ¼ Pu; q ¼ Qu, p represents a superposition of large eddies of size larger , and q represents small eddies of size smaller than k
1=2 . Projecting than k
1=2 m m (8.2.2) on PH and QH , we have, due to PA ¼ AP and QA ¼ AQ dp þ mAp þ PBðp þ qÞ ¼ Pf ; dt dq þ mAq þ QBðp þ qÞ ¼ Qf : dt

ð8:2:3Þ ð8:2:4Þ

Multiplying (8.2.4) by q 2 H, we have 1d 2 jqj þ mkqk2 ¼ ðQf ; qÞ
ðQBðp þ qÞ; qÞ; 2 dt Using the properties of trilinear form operator, we also derive that !1=2 jApj2 2 jQf jjqj þ ckpk jqj 1 þ ln þ cjqjkqkkpk k1 kpk2 !1=2 jApj2 2  jQf jjqj þ ckpk jqj 1 þ ln þ cK
1=2 kqk2 kuk: k1 kpk2

ð8:2:5Þ

ð8:2:6Þ

Global Attractors for Nonlinear Reaction–Diffusion Equations

185

We denote now a bound of juj (resp. kuk; jAuj), on I ¼ ðt0 ; 1Þ under consideration, by M0 (resp. M1 ; M2 ) M0 ¼ sup juðsÞj; M1 ¼ sup kuðsÞk; M2 ¼ sup jAuðsÞj; s2I

s2I

ð8:2:7Þ

s2I

we observe that jApj2  km kpk2  kkpk2 ; and let

 L¼

 km þ 1 : 1 þ ln k1

ð8:2:8Þ

We have d 2 jqj þ ð2m
C K
1=2 M1 Þkqk2  jQf jjqj þ CM12 L1=2 jqj: dt Hence, assuming that C K
1=2 M1  m, we have   2CM1 2 km þ 1 ¼ K1  ; m

ð8:2:9Þ

ð8:2:10Þ

which yields d 2 3m jqj þ kqk2  K
1=2 ðjQf j þ 4M12 L1=2 Þkqk dt 2 m 1 ðjQf j2 þ CM14 LÞ;  kqk2 þ 2 mK and d 2 1 jqj þ mkqk2  ðjQf j2 þ CM14 LÞ; dt mK

ð8:2:11Þ

d 2 1 jqj þ mKkqk2  ðjQf j2 þ CM14 LÞ: dt mK

ð8:2:12Þ

i.e.,

We infer easily from (8.2.13) that for t  t1 ; t1 ; t 2 I ; jqðtÞj2  jqðt1 Þj2 expð
mKðt
t1 ÞÞ þ

1 ðjQf j2 þ CM14 LÞ: m2 K2

ð8:2:13Þ

Before interpreting this inequality, we derive a similar inequality for the V norm. Multiplying (8.2.4) by Aq in H we find 1d kqk2 þ mjAqj2 ¼ ðQf ; AqÞ
ðBðp þ qÞ; AqÞ: 2 dt

Attractors for Nonlinear Autonomous Dynamical Systems

186

We use Schwarz inequality together with properties of trilinear form operator to majorize the right-hand side of this equation by jQf jjAqj þ C kpkL1=2 jAqjðkpk þ kqkÞ þ C jqj1=2 jAqj3=2 ðkpk þ kqkÞ 

m 1 CM14 L CM02 M14 jAqj2 þ jQf j2 þ þ : 2 m m m3

Thus,   d 1 M 4L M 2M 4 kqk2 þ mjAqj2  C jQf j2 þ 1 þ 0 3 1 ; dt m m m

ð8:2:14Þ

  d 1 M14 L M02 M14 2 2 2 kqk þ mKjAqj  C jQf j þ þ ; dt m m m3

ð8:2:15Þ

and we conclude that

  C 1 M14 L M02 M14 2 jQf j þ þ kqðtÞk  kqðt1 Þk expð
mKðt
t1 ÞÞ þ : mK m m m3 2

2

ð8:2:16Þ

In (8.2.13) and (8.2.16), we can bound jqðt1 Þj2 and kqðt1 Þk2 by M02 and M02 . Then after a time depending only on M0 (or M1 ), m and K ¼ km þ 1 , the term involving t becomes negligible and thus we obtain 8 2 2 2 4 > > < jqðtÞj  2 2 ðjQf j þ CM1 LÞ; mK   ð8:2:17Þ 2C 1 M 4L M 2M 4 > > : kqðtÞk2  jQf j2 þ 1 þ 0 3 1 mK m m m for t large, and condition-(C) is a direct consequence. Based on the existence of global absorbing set and using theorem 1.2.38, the semigroup fSðtÞg associated with (8.2.1) has the global attractor A. h

Chapter 9 Global Attractors for an Incompressible Fluid Equation and a Wave Equation This chapter shall establish the existence of global attractors for both an incompressible fluid equation and a wave equation using the energy equation method (see [93, 151]), see theorem 1.2.45 and the closed operator method ([103]), see theorem 1.2.53, respectively. The first part, section 9.1, is chosen from [93] and the second part, section 9.2, is adapted from [103].

9.1

An Incompressible Fluid Equation

In this section chosen from [93], we shall reveal the energy equation method (see theorem 1.2.45) to establish the existence of global attractor for the following second-order incompressible fluid equations a bounded domain X
R2 : 8 < @ ðu  aDuÞ  mDu þ crulðu  aDuÞ  u ¼ f þ rp; in X; @t ð9:1:1Þ : div u ¼ 0 in X: Here u ¼ uðx; tÞ is the velocity and p ¼ pðx; tÞ is the modified pressure given by 1 a p ¼ ~ p  juj2 þ au  Du þ trððruÞ þ ðruÞT Þ2 ; 2 4 and f is the external body force, the density of the fluid is q ¼ 1 and the parameters m and a are given positive constants. Assume that the fluid subjects to the boundary @X, uj@X ¼ 0;

ð9:1:2Þ

and initial data uðx; 0Þ ¼ u0 ðxÞ;

x 2 X:

ð9:1:3Þ

DOI: 10.1051/978-2-7598-2702-2.c009 © Science Press, EDP Sciences, 2022

Attractors for Nonlinear Autonomous Dynamical Systems

188

Let X be a simply connected, bounded, open set with smooth C 3 and connected boundary. Consider the following functional spaces: V ¼ fu 2 ½C01 ðXÞ2 ; div u ¼ 0g; Set

Z ðf ; gÞ ¼

X

1

f  g dx; jf j ¼ ðf ; f Þ2 ;

V ¼V

X

grad f  grad g dx;

:

8f ; g 2 ½L2 ðXÞ2 ;

Z ððf ; gÞÞ ¼

H01 ðXÞ

1

kf k ¼ ððf ; f ÞÞ2 ;

8f ; g 2 V :

Here V is a Hilbert space with the scalar product ðu; vÞV ¼ ðu; vÞ þ aððu; vÞÞ

ð9:1:4Þ

and consider another the Hilbert space W ¼ fu 2 V ; curlðu  aDuÞ 2 L2 ðXÞg; endowed with the scalar product ðu; vÞW ¼ ðu; vÞV þ ðcurlðu  aDuÞ; curlðv  aDvÞÞ;

8u; v 2 W :

ð9:1:5Þ

Thus we can prove that W ¼ fu 2 ½H 3 ðXÞ \ H01 ðXÞ2 ; div u ¼ 0g and that there exists a constant C ðaÞ [ 0 such that jujH 3  C ðaÞjcurlðu  aDuÞÞ;

8u 2 W :

ð9:1:6Þ

Now in a usual way, we can identify V with its dual space V 0 , i.e., V ¼ V 0 , and thus W
V  V 0
W 0 , with continuous injections and each space is dense in the following one. The weak formulation of the problem (9.1.1)–(9.1.3) as follows. For any given u0 and f , find u such that ( ðu 0 ; vÞV þ ððu; vÞÞ þ bðu; u; vÞ  abðu; Du; vÞ þ abðv; Du; uÞ ¼ ðf ; vÞ; 8v 2 V ; ð9:1:7Þ uð0Þ ¼ u0 : Therefore, the above result indicates that for any given u0 2 W and f 2 ðH 1 ðXÞÞ2 , there exists a unique solution u to problem (9.1.7) such that u 2 L1 ðR þ ; W Þ; u 0 2 L1 ðR þ ; V Þ:

ð9:1:8Þ

Using the standard Faedo–Galerkin method, choosing a special basis in V , namely, the spectral basis fxj gj 1 satisfying ðxj ; vÞW ¼ kj ðxj ; vÞV ; 8v 2 W ; 8j 1;

ð9:1:9Þ

where 0\k1  k2     and kj ! 1 as j ! 1, we easily prove the proof of the above existence of solution to problem (9.1.1). Note that fxj gj 1 is also an orthogonal

Global Attractors for an Incompressible Fluid Equation and a Wave Equation 189

basis in W . Moreover, if X is of class C 3 , then xj 2 ðH 4 ðXÞÞ2 . The approximate solutions um fulfill 0 fum g 2 L1 ðR þ ; W Þ; fum g 2 L1 ðR þ ; V Þ;

ð9:1:10Þ

and we obtain the following energy equations in V and W , respectively, 1d jum j2V þ mkum k2 ¼ ðf ; um Þ; 2 dt 1d m 1 jum j2W þ jum j2W ¼ K ðum Þ; 2 dt a 2

ð9:1:11Þ ð9:1:12Þ

where m  1 m K ðum Þ ¼ jum j2 þ ðf ; um Þ þ curlum þ curlf ; curlðum  aDum Þ : 2 a a

ð9:1:13Þ

Hence it follows from (9.1.10) that there exists a subsequence of fum g, still denoted by fum g, satisfying that um * u star-weakly in L1 ðR þ ; W Þ; um * u star-weakly in L1 ðR þ ; V Þ; um ! u strongly in L2 ð0; T ; V \ H 2 Þ; 8T [ 0: Thus we conclude that the above convergence allows to pass to the limit when m goes to infinity and find that u is the solution of problem (9.1.7), the uniqueness proof is standard. From u 2 L1 ðR þ ; W Þ and u 0 2 L1 ðR þ ; V Þ, it follows that u is a.e. equal to a continuous function from ½0; T  into V for all T [ 0. Moreover, u 2 L1 ðR þ ; W Þ \ C ð½0; T ; V Þ; 8T [ 0, yields u 2 Cw ð½0; T ; W Þ;

ð9:1:14Þ

juðtÞjW  jujL1 ð0;T ;W Þ ; 8t 2 ½0; T :

ð9:1:15Þ

Note that (9.1.14) and (9.1.15) also hold for um . The above results are known. Now we need to show that u 2 C ð½0; T ; W Þ. To this end, we need to prove the lemma. Lemma 9.1.1. um ðtÞ*uðtÞ weakly in W , for all t 0. Proof. Noting that um ! u strongly in L2 ð0; T ; V \ H 2 Þ; 8T [ 0, we conclude the existence of subsequence fum0 g satisfying that um0 ðtÞ ! uðtÞ strongly in V ; for a:e: t 2 ½0; T ; 8T [ 0: For 0  t  t þ a  T , we easily get

Z

um0 ðt þ aÞ  um0 ðtÞ ¼ t

t þa

0 um 0 ðsÞds; in V ;

ð9:1:16Þ

Attractors for Nonlinear Autonomous Dynamical Systems

190

whence

Z jum0 ðt þ aÞ  um0 ðtÞjV ¼

t þa

t

0 0 jum 0 ðsÞjV ds  ca; 8m 1:

ð9:1:17Þ

Using (9.1.16) and (9.1.17), and noting the fact that um0 and u are in C ð½0; T ; V Þ, we derive that um0 ðtÞ ! uðtÞ strongly in V ; for all t 2 ½0; T :

ð9:1:18Þ

From the previous convergence and using the spectral basis of V defined by (9.1.9), we get ðum0 ðtÞ; xj ÞW ! ðuðtÞ; xj ÞW ; 8j 1; t 2 ½0; T ;

ð9:1:19Þ

and by the density of fxj g in W , we derive um0 ðtÞ*uðtÞ; weakly in W ; 8t 2 ½0; T :

ð9:1:20Þ

Then, by a contradiction argument, we conclude that the whole sequence fum ðtÞgm converges to uðtÞ weakly in W for every t 2 ½0; T . We now prove an energy inequality in W for the solution u. Integrating (9.1.12) over ½0; 1, we obtain Z t 2m 2m jum ðtÞj2W ¼ ju0m j2W e a t þ Kðum ðsÞÞe a ðtsÞ ; 8t 2 ½0; T : ð9:1:21Þ 0

Since u0m ! u0 strongly in W , um *u in L1 ðR þ ; W Þ, and K is weakly continuous on W , using the Lebesgue-dominated convergence theorem, we can pass to the limit in (9.1.21) to get Z t 2m 2 2 2m t a lim jum ðtÞjW ¼ ju0 jW e þ K ðuðsÞÞe a ðtsÞ ; 8t 2 ½0; T : ð9:1:22Þ m!1

0

From lemma 9.1.1, it follows juðtÞj2W  lim inf jum ðtÞj2W ; 8t 2 ½0; T : m!1

Thus lim juðtÞj2W  ju0 j2W e a t þ 2m

m!1

Z

t

KðuðsÞÞe a ðtsÞ ds; 8t 2 ½0; T : 2m

ð9:1:23Þ

0

Reversing the time in equation (9.1.1), we obtain the next problem 8@ ðv  aDuÞ  mDv  curlðv  aDvÞ  v ¼ f þ rq; in X; > > < @t divv ¼ 0 in X; > > : vj@X ¼ 0; vðx; 0Þ ¼ v0 ðxÞ; x 2 X:

ð9:1:24Þ

Global Attractors for an Incompressible Fluid Equation and a Wave Equation 191

Similarly, we obtain the finite time estimates in V and W for v, and the following energy inequality in W : Z t1 2m 2m jvðtÞj2W  jv0 j2W e a t þ K ðvðsÞÞe a ðtsÞ ds; 8t 2 ½0; T : ð9:1:25Þ 0

If vð0Þ ¼ uðt1 Þ 2 W for some t1 2 ½0; T , we know from (9.1.14) that uðtÞ 2 W for all t 2 ½0; T , then by the uniqueness of the solutions, we conclude that vðtÞ ¼ uðt1  tÞ for t 2 ½0; t1 . Thus, it follows from (9.1.25) for t ¼ t1 that Z t1 2m 2m juðtÞj2W  jut1 j2W e a t1 þ K ðuðt1  sÞÞe a ðt1 sÞ ds; 8t1 2 ½0; T ; ð9:1:26Þ 0

which gives Z

2m

juðtÞj2W  jut1 j2W e a t1 þ

t1

2m

K ðuðsÞÞe a s ds; 8t1 2 ½0; T ;

0

or, equivalently, juðt1 Þj2W juð0Þj2W e a t1 þ 2m

Z

t1

K ðuðsÞÞe a ðt1 sÞ ds; 8t1 2 ½0; T : 2m

ð9:1:27Þ

0

Therefore, from (9.1.23) and (9.1.27), we conclude the following theorem. Theorem 9.1.1. For any u0 2 W and f 2 ½H 1 ðXÞ2 , the solution u of the problem (9.1.7) verifies the energy equality Z t 2m 2m juðtÞj2W ¼ juð0Þj2W e a t þ K ðuðsÞÞe a ðtsÞ ds; 8t [ 0; ð9:1:28Þ 0

and, u 2 C ðR þ ; W Þ:

ð9:1:29Þ

Obvious, from (9.1.28), it follows that juðtÞjW ! juðt0 ÞjW as t ! t0 , which, with u 2 Cw ðR þ ; W Þ, yields (9.1.29). It follows from theorem 9.1.1 that we can define the semigroup fSðtÞg in W by SðtÞu0 ¼ uðtÞ; 8t 0:

ð9:1:30Þ

We now introduce some properties of the semigroup fSðtÞg in the following theorem. Theorem 9.1.2. The operator fSðtÞg are continuous and weakly continuous on W for all t 0. Proof. Consider a sequence u0n 2 W such that u0n u0 weakly in W . Let un ðtÞ ¼ SðtÞu0n ; uðtÞ ¼ SðtÞu0 ; 8t 0:

Attractors for Nonlinear Autonomous Dynamical Systems

192

From a prior estimates in V and W , it follows fun g 2 L1 ðR þ ; W Þ; fun0 g 2 L1 ðR þ ; V Þ;

ð9:1:31Þ

and from theorem 9.1.1, u; un 2 C ð½0; T ; W Þ; 8T [ 0:

ð9:1:32Þ

From (9.1.31) and (9.1.32), we can extract a subsequence fun0 g such that un0 * u~ weakly star in L1 ðR þ ; W Þ; un0 0 * u~ 0 weakly star in L1 ðR þ ; V Þ; un0 ! u~ strongly in L2 ð0; T ; H 2 \ V Þ; 8T [ 0; un0 ! u~ strongly in L2 ð0; T ; H 2 \ V Þ; 8t 2 ½0; T ;

ð9:1:33Þ

for some u~ 2 L1 ðR þ ; W Þ \ Cw ð0; T ; W Þ with u~ 0 2 L1 ðR þ ; V Þ. Then by the uniqueness of the solutions, we get u~ ¼ u. By a contraction argument, we derive that the whole sequence fun g converges to u in the sense of (9.1.33). Particularly, SðtÞu0n *SðtÞu0 ; weakly in W ; 8t 0:

ð9:1:34Þ

Now consider u0n ! u0 strongly in W . The energy equation (9.1.28) for u0 reads Z t 2m 2 2 2m t a jSðtÞu0n jW ¼ ju0n jW e þ K ðSðsÞu0n Þe a ðtsÞ ds; 8t 0: ð9:1:35Þ 0

and lim jSðtÞu0n j2W n!1

¼

2m ju0n j2W e a t

Z þ

t

K ðSðsÞu0 Þe a ðtsÞ ds; 8t [ 0: 2m

ð9:1:36Þ

0

From the energy equation for u, the right-hand side term in (9.1.36) is jSðtÞu0n j2W . Thus lim jSðtÞu0n j2W ¼ jSðtÞu0 j2W ;

n!1

ð9:1:37Þ

which, with (9.1.34), yields SðtÞu0n ! SðtÞu0 strongly in W as n ! 1; 8t 0:

ð9:1:38Þ h

Theorem 9.1.3. Let X
R2 be a simply connected, bounded, open set with C 3 smooth and connected boundary, and let m [ 0; a [ 0, and f 2 ½H 1 ðXÞ2 be given. Then the semigroup fSðtÞgt 0 (which is actually a group) in W associated to the problem (9.1.1)–(9.1.3) possesses a global attractor in W . Proof. Using the a priori estimates in V and W , we can get the existence of bounded absorbing sets in V and W , respectively. Combining theorems 9.1.1, 9.1.2 and 1.2.45,

Global Attractors for an Incompressible Fluid Equation and a Wave Equation 193 we can conclude the existence of the global attractor. This completes the proof of this theorem. h

9.2

A Wave Equation with Nonlinear Damping

In order to exhibit applications of the method of the closed operator semigroup (see theorem 1.2.53), in this section we shall study a wave equation with nonlinear damping. To this end, let X
R2 be a bounded domain with smooth boundary @X and consider the wave equation with nonlinear damping 8 < @tt u þ rðuÞ@t u  Du þ uðuÞ ¼ f ; t [ 0; uð0Þ ¼ u0 ; @t uð0Þ ¼ u1 ; ð9:2:1Þ : uj ¼ 0: @X Here, f 2 L2 ðXÞ is independent of time, the function u
C 2 ðRÞ with uð0Þ ¼ 0, satisfies ju00 ðuÞj  cð1 þ jujp Þ; c 0; p 0;

ð9:2:2Þ

u0 ðuÞ  l; l 0;

ð9:2:3Þ

lim inf juj!1

uðuÞ [  k1 ; u

ð9:2:4Þ

where k1 [ 0 is the first eigenvalue of D on L2 ðXÞ with Dirichlet boundary conditions, and r 2 C 1 ðRÞ satisfies rðuÞ r0 [ 0;

ð9:2:5Þ

jr0 ðuÞj  c½rðuÞ1d ; d 2 ð0; 1:

ð9:2:6Þ

Note that (9.2.6) implies that ju0 ðuÞj  cð1 þ jujq Þ; q ¼

1d : d

ð9:2:7Þ

Now denote by Hs ¼ DðAs=2 Þ and introduce the family of product Hilbert spaces H s ¼ Hs þ 1  Hs endowed with the standard inner products and norms. The symbols c and Q stand for a generic positive constant and a generic positive increasing function, respectively. Now, for any given function zðtÞ, we may write for short nz ðtÞ ¼ ðzðtÞ; @t zðtÞÞ.

Attractors for Nonlinear Autonomous Dynamical Systems

194

9.2.1

Wellposedness of Solutions

In this subsection, we shall first prove the next theorem of wellposedness of solutions. Theorem 9.2.1. Equation (9.2.1) generates a semigroup SðtÞ on the phase space H0 . Proof. The proof is carried out by means of a Galerkin approximation scheme. Existence is obtained exploiting the uniform energy estimate established in the next proposition, and then passing to the limit in a standard way. h Lemma 9.2.2. For every t 0, there holds knu ðtÞkH0  Qðknu ð0ÞkH0 Þ þ Qðkf kÞ: Proof. Introduce the energy functional E0 ¼ knu k2H0 þ 2\UðuÞ; 1 [  2\f ; u [ ; where

Z

u

UðuÞ ¼

/ðyÞdy: 0

From (9.2.4), it follows kruk2 þ 2\UðuÞ; 1 [ 2bkruk2  c;

ð9:2:8Þ

for some b [ 0. Thus, (9.2.1) implies knu ðtÞkH0  Qðknu kH0 Þ þ Qðkf kÞ: Multiplying (9.2.1) by @t u, we find d E0 þ 2\rðuÞ@t u; @t u [ ¼ 0; dt

ð9:2:9Þ

and the conclusion follows integrating on ð0; tÞ: Integrating equality (9.2.9) on ð0; 1Þ, and using theorem 9.2.1 and (9.2.1), we also obtain the existence of suitable dissipation integrals, namely, Lemma 9.2.3. There holds Z 1 Z 1 r0 k@t uðtÞk2 dt  \rðuðtÞÞ@t uðtÞ; @t uðtÞ [ dt  Qðknu ð0ÞkH0 Þ þ Qðkf kÞ: 0

0

Note that E0 is a Lyapunov function for SðtÞ, and uniqueness is a consequence of the following continuous dependence result. h

Global Attractors for an Incompressible Fluid Equation and a Wave Equation 195 Lemma 9.2.4. For every T [ 0 and every R 0, any two solutions u 1 and u 2 to problem (9.2.1) satisfy the estimate knu 1 ðT Þ  nu2 ðT Þk2H1  QðRÞeQðRÞT knu 1 ð0Þ  nu2 ð0Þk2H1 ; for all initial data knu i ð0ÞkH0  R: Rt Proof. Define wðtÞ ¼ 0 uðsÞds. Integrating (9.2.1) on ð0; tÞ yields Z t @tt wðtÞ þ RðuðtÞÞ  DwðtÞ ¼  /ðuðsÞÞds þ Rðuð0ÞÞ þ @t uð0Þ þ tf ; 0

where

Z RðuÞ ¼

u

rðyÞdy: 0

Let now u 1 ; u 2 be two solutions to problem (9.2.1) with initial data knu i ð0ÞkH0  R, and denote their difference by u ¼ u 1  u 2. From the uniform estimate of theorem 9.2.1, nui ðtÞkH0  QðRÞ; 8t 0: Then, the corresponding difference w ¼ w 1  w 2 solves the equation @tt w þ Rðu 1 Þ  Rðu 2 Þ  Dw ¼ F þ G;

ð9:2:10Þ

where Z FðtÞ ¼ 

t

½/ðu 1 ðsÞÞ  /ðu 2 ðsÞÞds

0

and G ¼ Rðu 1 ð0ÞÞ  Rðu 2 ð0ÞÞ þ @t uð0Þ: The monotonicity of R implies that \Rðu 1 Þ  Rðu 2 Þ; u [ 0: Hence, multiplying (9.2.10) by @t w ¼ u, we have 1d d d d nw k2H0  \F; w [ þ \G; w [  \@t F; w [ : 2 dt dt dt dt Integrating on ð0; T Þ, we arrive at Z knw ðT Þk2H0

2

 kuð0Þk þ 2\FðT Þ; wðTÞ [ þ 2\G; wðTÞ [  2 0



1 kn ðT Þk2H0 þ 4kFðT Þk2H1 þ kuð0Þk2 þ 4kGk2H1 2 w Z T þ2 k@t FðtÞkH1 knw ðTÞkH0 dt: 0

T

\@t FðtÞ; wðtÞdt [

Attractors for Nonlinear Autonomous Dynamical Systems

196

Using now the growth restrictions (9.2.2) and (9.2.7) on / and r, we get at once the controls Z T Z T kuðtÞk2 dt  QðRÞT knw ðtÞk2H0 dt; 4kFðT Þk2H1  QðRÞT 0

kuð0Þk

2

þ 4kGk2H1

0

 QðRÞknu ð0Þk2H1 ;

k@t FðtÞkH1  QðRÞkuðtÞk  QðRÞknw ðtÞkH0 : Therefore, Z knw ðT Þk2H0  QðRÞknu ð0Þk2H1 þ QðRÞð1 þ T Þ

T 0

knw ðtÞk2H0 dt;

and from the Gronwall lemma, theorem 1.1.9, we conclude that kuðT Þk2  knw ðT Þk2H0  QðRÞeQðRÞT knu ð0Þk2H1 : Finally, from (9.2.10), we derive k@t ukH1 ¼ k@tt wkH1  kRðu 1 Þ  Rðu 2 ÞkH1 þ krwk þ kFkH1 þ kGkH1 ; which, with the next immediate control kRðu 1 Þ  Rðu 2 ÞkH1  QðRÞkuk; implies k@t uðT Þk2H1  QðRÞeQðRÞT knu ð0Þk2H1 : h

The proof is then completed.

9.2.2

Dissipativity

We now investigate the asymptotic properties of problem (9.2.1). We have previously shown the existence of a bounded absorbing set B0
H0 . Lemma 9.2.5. For every R 0, there exists a time t0 ¼ t0 ðRÞ [ 0 such that knu ðtÞkH0  Qðkf kÞ; 8t t0 ; whenever knu ð0ÞkH0  R. Proof. For e 2 ð0; 1Þ to be fixed later, introduce the energy functional Ee ¼ knu k2H0 þ 2\UðuÞ; 1 [  2\f ; u [ þ 2e\cðuÞ; 1 [ þ 2e\@t u; u [ ; with UðuÞ as in theorem 9.2.1 and

Z

cðuÞ ¼

u

yrðyÞdy: 0

Global Attractors for an Incompressible Fluid Equation and a Wave Equation 197 Notice that, from (9.2.5), it follows \cðuÞ; 1 [ 0. Thus, on account of (9.2.2), (9.2.7) and (9.2.8) we see that bknu k2H0  Qðkf kÞ  Ee  Qðknu kH0 Þ þ Qðkf kÞ;

ð9:2:11Þ

for some b [ 0, and when e is small enough. Multiply (9.2.1) by @t u þ eu to get d Ee þ 2ekruk2 þ 2\r@t u; @t u [  2ek@t uk2 þ 2e\/ðuÞ; u [ ¼ 2e\f ; u [ : dt Using (9.2.4), we conclude 2ekruk2 þ 2e\/ðuÞ; u [ 2bekruk2  c; whereas (9.2.5) yields 2\r@t u; @t u [  2ek@t uk2 bek@t uk2 ; provided that e is small enough. Thus, the right-hand side of the differential equality can be estimated as 2e\f ; u [ bekruk2 þ ckf k2 ; which gives d Ee þ beknu k2H0  Qðkf kÞ: dt

ð9:2:12Þ

Fixing now the parameter e to satisfy all the above relationships hold, the claim follows from theorem 1.1.9 readily. Moreover, we find theorem 9.2.1 and lemma 9.2.4 can be written as the following unitary fashion. Lemma 9.2.6. For every t 0, there holds knu ðtÞkH0  Qðknu ð0ÞkH0 Þet þ Qðkf kÞ: The next step is to demonstrate higher order dissipativity. Lemma 9.2.7. For every t 0, there holds knu ðtÞkH1  Qðknu ð0ÞkH1 Þee1 t þ Qðkf kÞ; for some e1 [ 0 and some positive increasing function Q. Proof. Noting that the absorbing set B0 has been obtained, it suffices to prove that for every R [ 0, there exists m ¼ mðRÞ such that knu ðtÞkH1  Qðknu ð0ÞkH1 Þemt þ QðR þ kf kÞ;

ð9:2:13Þ

whenever knu ð0ÞkH0  R. Fix then R [ 0 and choose knu ð0ÞkH0  R. From lemma 9.2.6, we derive that knu ðtÞkH0  QR ;

ð9:2:14Þ

Attractors for Nonlinear Autonomous Dynamical Systems

198

where we wrote for short QR ¼ QðR þ kf kÞ: Setting g ¼ @t u, differentiating (9.2.1) in time implies @tt g þ rðuÞ@t g þ r0 ðuÞg2  Dg þ /0 ðuÞg ¼ 0: Then, for e 2 ð0; 1Þ to be fixed later, define the functional K ¼ kng k2H0 þ 2e\g; @t g [ ; satisfying 1 kn k2  K  2kng k2H0 ; 2 g H0

ð9:2:15Þ

provided that e is small enough. Multiplying the above equation by @t g þ eg, we arrive at d K þ 2ekrgk2 þ 2\rðuÞ@t g; @t g [ þ 2\r0 ðuÞg2 ; @t g [  2ek@t gk2 dt  2e\rðuÞg; @t g [  2e\r0 ðuÞg2 ; g [  2e\r0 ðuÞg; g [  2\r0 ðuÞg; @t g [ : Using (9.2.2), (9.2.7) and (9.2.14), we can control the terms in the right-hand side as e 2e\rðuÞg; @t g [ krgk2 þ eQR k@t gk2 ; 3 e 2e\r0 ðuÞg2 ; g [  2e\r0 ðuÞg; g [ krgk2 þ QR ; 3 and 2e\r0 ðuÞg; @t g [  QR kgkL4 k@t gk  QR krgk1=2 k@t gk e QR  krgk2 þ ek@t gk2 þ 2 : 3 e Therefore, we conclude d QR K þ ekrgk2 þ 2\rðuÞ@t g; @t g [ þ 2\r0 ðuÞg2 ; @t g [  eð3 þ QR Þk@t gk2  2 : dt e We now estimate the terms on the left-hand side. Obviously, 2\r0 ðuÞg2 ; @t g [  \rðuÞ@t g; @t g [  \½r0 ðuÞ2 ½rðuÞ1 g2 ; g2 [ : We now fix e ¼ eðRÞ small enough such that (9.2.15) holds r0  eð3 þ QR Þ e:

Global Attractors for an Incompressible Fluid Equation and a Wave Equation 199

Hence, using (9.2.5) and (9.2.15), we obtain d e K þ  \½r0 ðuÞ2 ½rðuÞ1 g2 ; g2 [ þ QR : dt 2

ð9:2:16Þ

Finally, we need to control the remaining term on the left-hand side. By (9.2.5) and (9.2.6), without the loss of generality, we assume d  1=2. Note that, from (9.2.6), ð½r0 ðuÞ2 ½rðuÞ1 Þ1=ð12dÞ  crðuÞ: Thus, applying the Hölder inequality with exponents 1=p1 þ 1=p2 ¼ 2d, we obtain



1 12d ; p1 ; p2



and

\½r0 ðuÞ2 ½rðuÞ1 g2 ; g2 [ ¼ \½r0 ðuÞ2 ½rðuÞ1 jgj24d ; g2 jgj4d [ ¼ c\rðuÞg; g [ 12d kgk2L2p1 kgk4d L4dp2 : Exploiting the interpolation inequality, we find the controls kgk2L2p1  ckgk2=p1 krgk22=p1  QR krgk22=p1 and 2=p2 kgk4d krgk4d2=p2  QR krgk42=p2 : L4dp2  ckgk

Making use of (9.2.7) and (9.2.14), we get \rðuÞg; g [ QR krgk: Applying (9.2.15) to obtain \½r0 ðuÞ2 ½rðuÞ1 g2 ; g2 [  QR \rðuÞg; g [ 12d krgk2  QR \rðuÞg; g [ 12d þ QR \rðuÞg; g [ 12d K e e  QR þ krgk2 þ K þ QR \rðuÞg; g [ K 8 8 e  QR þ K þ QR \rðuÞg; g [ K: 4 Thus, (9.2.16) becomes d e K þ K  QR þ QR \rðuÞg; g [ K: dt 4 Due to lemma 9.2.2, we apply theorem 1.1.10, which, with (9.2.15), gives kng ðtÞkH0  Qðkng ð0ÞkH0 Þemt þ QR ; for some m ¼ mðRÞ [ 0. Thus comparing with the original equation (9.2.1), we obtain the required inequality (9.2.13). h

Attractors for Nonlinear Autonomous Dynamical Systems

200

9.2.3

Asymptotic Compactness and Existence of Global Attractor

This subsection will use decomposition of the solution to obtain the asymptotic compactness. First, using (9.2.2) and lemma 9.2.5, for every z 2 H1, every t 0 and every solution uðtÞ with nu ð0Þ 2 B0 , choose h l large enough such that there holds that 1 krzk2 þ ðh  lÞkzk2  \r0 ðuðtÞÞz; z [ 0: 2

ð9:2:17Þ

Then, set wðrÞ ¼ /ðrÞ þ hr: We know that condition (9.2.2) still holds with w replaced by /, and using (9.2.3), w0 ðrÞ 0:

ð9:2:18Þ

We now consider initial data nu ð0Þ belonging to the bounded absorbing set B0 produced by lemma 9.2.4, and decompose the solution to problem (9.2.1) into the sum u ¼ w þ v, where w and v solve the equations 8 < @tt w þ rðwÞ@t w  Dw þ wðwÞ ¼ hu þ f ; nw ð0Þ ¼ ð0; 0Þ; ð9:2:19Þ : wj ¼ 0; @X and

8 < @tt v þ rðuÞ@t v  Dv þ ðrðuÞ  rðwÞÞ@t w þ wðuÞ  wðwÞ ¼ 0; nv ð0Þ ¼ nv ð0Þ; : vj ¼ 0: @X

ð9:2:20Þ

Arguing exactly as in theorem 1.1.9 and lemma 9.2.4, we obtain the uniform bound knw ðtÞkH0  c; 8t 0:

ð9:2:21Þ

In addition, multiplying (9.2.19) by @t w and integrating on ðs; tÞ, thanks to lemma 9.2.2, we readily see that for every t s 0 and every w [ 0, Z t Z t c 2 k@t wðsÞk ds  \rðwðsÞÞ@t wðsÞ; @t wðsÞ [ ds  wðt  sÞ þ : ð9:2:22Þ r0 w s s Lemma 9.2.8. For every t 0, we have that knw ðtÞkH1  c. Lemma 9.2.9. For every t 0 and some m [ 0, we know that knv ðtÞkH0  cemt .

Global Attractors for an Incompressible Fluid Equation and a Wave Equation 201 Proof. For e 2 ð0; 1Þ to be determined later, define K ¼ knv k2H0 þ 2\wðuÞ  wðwÞ; v [  \w0 ðuÞv; v [ þ 2e\@t v; v [ : Using (9.2.3) and (9.2.17), together with the uniform bounds on kruk and krwk, the functional K satisfies the inequalities 1 kn k2  K  cknv k2H0 ; 4 v H0

ð9:2:23Þ

provided that e is small enough. Multiplying (9.2.20) d K þ 2ekrvk2 þ 2\rðuÞ@t v; @t v [  2ek@t vk2 þ 2e\wðuÞ  wðwÞ; v [ dt ¼ 2\ðw0 ðuÞ  w0 ðwÞÞ@t ; v [  \w00 ðuÞ@t u; v 2 [  2\ðrðuÞ  rðwÞÞ@t w; @t v [  2e\ðrðuÞ  rðwÞÞ@t w; v [  2e\rðuÞ@t v; v [ : We now reconstruct K in the right-hand side. Indeed, it is easily to see that, for e small enough, 2ekrvk2 þ 2\rðuÞ@t v; @t v [  2ek@t vk2 þ 2e\wðuÞ  wðwÞ; v [ e eK þ krvk2 þ r0 k@t vk2 : 2 Therefore, we get the following differential inequality. d e K þ eK þ krvk2 þ r0 k@t vk2 dt 2 ¼ 2\ðw0 ðuÞ  w0 ðwÞÞ@t ; v [  \w00 ðuÞ@t u; v 2 [  2\ðrðuÞ  rðwÞÞ@t w; @t v [  2e\ðrðuÞ  rðwÞÞ@t w; v [  2e\rðuÞ@t v; v [ : Then, we continue to control the terms in the right-hand side. Using estimates of the first two terms, we conclude 2\ðw0 ðuÞ  w0 ðwÞÞ@t ; v [  \w00 ðuÞ@t u; v 2 [

e krvk2 þ ðk@t uk2 þ k@t wk2 ÞK: 4

Concerning the remaining terms, we get 2\ðrðuÞ  rðwÞÞ@t w; @t v [  c\ð1 þ jujq þ jwjq Þj@t wj1=2 j@t wj1=2 jvj; j@t vj [  ck@t wk1=2 krvkk@t vk  ck@t wk1=2 K e  K þ ck@t wk2 K: 4 Similarly, e 2e\ðrðuÞ  rðwÞÞ@t w; v [ K þ ck@t wk2 K: 4

Attractors for Nonlinear Autonomous Dynamical Systems

202

Finally, e 2e\rðuÞ@t v; v [ cek@t vkkrvk  krvk2 þ cek@t vk2 : 4 Collecting the above inequalities, we arrive at d e e K þ K þ ðr0  ceÞk@t vk2  krvk2 þ ðk@t uk2 þ k@t wk2 ÞK: dt 2 4 Now fix e [ 0 small enough satisfying the above conditions and r0  ce 0, then we can obtain d e K þ K þ  cðk@t uk2 þ k@t wk2 ÞK: dt 2 Applying the integral estimates in lemma 9.2.2 and (9.2.22), the conclusion follows by applying theorem 1.1.10 along with (9.2.23). As it is shown in [103], we know this problem generates a semigroup SðtÞ on the phase space X ¼ H01 ðXÞ  L2 ðXÞ. However, for any two initial data x1 ; x2 2 X with kxj k  ., we conclude the next only a continuous dependence estimate of the form kSðtÞx1  SðtÞx2 kW  kekt kx1  x2 k for some k ¼ kðqÞ with W ¼ L2 ðXÞ  H 1 ðXÞ. Hence, we can prove the weaker continuity SðtÞ 2 C ðX ; WÞ. On the other hand, lemma 9.2.4 clearly gives the existence of a bounded absorbing set B0 , while lemmas 9.2.7 and 9.2.8 show that SðtÞB0 is (exponentially) attracted by a bounded subset C
H1 , this implies C is a compact attracting set. By theorem 1.2.53, we conclude that there exists a unique compact global attractor A
C. This completes the following theorem. h Theorem 9.2.10. Equation (9.2.1) generates a semigroup SðtÞ on H01 ðXÞ  L2 ðXÞ which has a unique compact global attractor A. Moreover, A is a bounded subset of ½H 2 ðXÞ \ H01 ðXÞH01 ðXÞ, and coincides with unstable set of the stationary points of SðtÞ.

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Index

C0-Semigroups, 19 C0-Semigroups, 11 E–E Uniform Boundedness, 11 γ-Contraction, 24 Ω-Limit Compactness, 18 A Absorbing Sets, 11 Agmon’s Inequality, 5 Analytic Semigroups, 11 Asymptotic Smoothness Semigroups, 27 Asymptotically Compact Semigroups, 22 Attracting Sets, 12 Attractors, 12 C Closed Operator Semigroup, 30 Compact Semigroups, 12 Condition-(C), 19 Contraction Semigroups, 11 E Energy Equation, 27 F Fractal dimension, 32 G Gronwall’s Inequality, 4 H Hölder Inequality, 3 I Invariant Sets, 12

J Jensen’s Inequality with Integration, 4 L Ladyzhenskaya’s Inequality, 5 M Minkowski’s Inequality, 3 N Navier-Stokes-Voight Equations, 37 Norm-To-Weak Semigroups, 28 O Omega-Limit Set, 12 P Poincaré’s Inequality, 4 Point Dissipation, 11 S Semi-Flows, 14 Semigroups, 10 Set-Contraction, 18 Sobolev’s Inequality, 5 T The The The The The The The The The

Bernstein Inequality, 8 Cauchy-Schwarz Inequality, 3 Gagliardo-Nirenberg Inequality, 5, 8, 9 Interpolation Inequalities, 9 Kozono and Tamiuchi Inequality, 7 Kuratowski Non-Compact Measure, 17 Lemarié-Rieusset Inequality, 7 Special Bernstein Inequality, 8 Uniform Bellman-Gronwall Inequality, 4

212

U Uniform differentiability, 33 Uniformly Convex Space, 19 W Weakly Compact Semigroup Method, 17 Y Young’s Inequality, 3

Index