Attractors for Nonlinear Autonomous Dynamical Systems, livre ebook

This book introduces complete and systematic theories of infinite-dimensional dynamical systems and their applications in partial differential equations, especially in the models of fluid mechanics. It is based on the first author’s lecture “Infinite dimensional dynamical systems on nonlinear autonomous systems” given tograduate students in Donghua University since 2004. This book presents recent results that have been carried out by the authors on autonomous nonlinear evolutionary equations arising from physics, fluid mechanics and material science such as the Navier–Stokes equations, Navier–Stokes–Voight systems, the nonlinearthermoviscoelastic system, etc.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX

CHAPTER 1

Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Some Useful Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Basic Theory of Infinite-Dimensional Dynamical Systems for Autonomous Nonlinear Evolutionary Equations . . . . . . . . . . . . . . . 10

1.2.1 Uniformly Compact Semigroups . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.2 Weakly Compact Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.3 X-Limit Compact Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.4 Asymptotically Compact Semigroups . . . . . . . . . . . . . . . . . . . . 22

1.2.5 Asymptotically Smooth Semigroups . . . . . . . . . . . . . . . . . . . . . 27

1.2.6 Norm-to-Weak Continuous Semigroups. . . . . . . . . . . . . . . . . . . 28

1.2.7 Closed Operator Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.3 Basic Theory of Finite-Dimensional Attractors . . . . . . . . . . . . . . . . . . 32

1.3.1 The Fractal Dimension of Global Attractors . . . . . . . . . . . . . . . 32

1.3.2 The Estimate on Fractal Dimension of Global Attractors . . . . . 33

CHAPTER 2

Global Attractors for the Navier–Stokes–Voight Equations with Delay . . . . . 37

2.1 Global Wellposedness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Existence of Global Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.1 Dissipation: Existence of Absorbing Sets . . . . . . . . . . . . . . . . . 43

2.2.2 Asymptotical Compactness and Existence of Attractor . . . . . . . 44

2.3 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

CHAPTER 3

Global Attractor and Its Upper Estimate on Fractal Dimension

for the 2D Navier–Stokes–Voight Equations . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1 Global Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Existence of Global Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Existence of Absorbing Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.2 Some Compactness and the Existence of Global Attractors . . . 56

3.3 Upper Estimate on the Fractal Dimension of Global Attractors . . . . . . 58

3.4 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

CHAPTER 4

Maximal Attractor for the Equations of One-Dimensional Compressible

Polytropic Viscous Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Our Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Nonlinear Semigroup on Hð2Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Existence of an Absorbing Set in Hð1Þ b . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Existence of an Absorbing Set in Hð2Þ b . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5 Proof of Theorem 4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.6 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

CHAPTER 5

Universal Attractors for a Nonlinear System of Compressible

One-Dimensional Heat-Conducting Viscous Real Gas . . . . . . . . . . . . . . . . . . 91

5.1 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Nonlinear C0-Semigroup on Hiþ ði ¼ 1; 2Þ. . . . . . . . . . . . . . . . . . . . . . . 95

5.3 Existence of an Absorbing Set in H1d . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4 Existence of an Absorbing Set in H2d . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.5 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.6 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

CHAPTER 6

Global Attractors for the Compressible Navier–Stokes Equations in Bounded Annular Domains . . . . . . . . . . . . . . . . . . . . . . . 115

6.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2 Nonlinear Semigroup on Hð2Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3 Existence of an Absorbing Set in Hð1Þ d . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.4 Existence of an Absorbing Set in Hð2Þ d . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.5 Existence of an Absorbing Set in Hð4Þ d . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.6 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

CHAPTER 7

Global Attractor for a Nonlinear Thermoviscoelastic System in Shape Memory Alloys . . . . . . . . . . . . . .. . . . . . . . . . . 149

7.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2 An Absorbing Set Bd in Hd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.3 Compactness of the Orbit in Hd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.4 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

CHAPTER 8

Global Attractors for Nonlinear Reaction–Diffusion Equations and the 2D Navier–Stokes Equations . . . . . . . .. . . . . . . . . . . . . . . . . 175

8.1 Global Attractor for Strong Solutions of Reaction–Diffusion Equations . . 175

8.1.1 Existence of Solutions and Uniqueness . . . . . . . . . . . . . . . . . . . 176

8.1.2 Global Attractor for the Semigroup in LpðXÞ . . . . . . . . . . . . . . 176

8.1.3 Global Attractor of System in LpðXÞ and H1

0 ðXÞ . . . . . . . . . . . . 177

8.2 Global Attractors for the 2D Navier–Stokes Equations in H1

0 ðXÞ . . . . . 183

CHAPTER 9

Global Attractors for an Incompressible Fluid Equation and a Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.1 An Incompressible Fluid Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.2 AWave Equation with Nonlinear Damping . . . . . . . . . . . . . . . . . . . . . 193

9.2.1 Wellposedness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.2.2 Dissipativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9.2.3 Asymptotic Compactness and Existence of Global Attractor . . 200

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211