Contents
Preface i
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 Q-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
41 Our Problem 67
4.2 Nonlinear Semigroup on 69
4.3 Existence of an Absorbing Set in 73
4.4 Existence of an Absorbing Set in 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 Co-Semigroup on 95
5.3 Existence of an Absorbing Set in 97
5.4 Existence of an Absorbing Set in 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 119
6.3 Existence of an Absorbing Set in 120
6.4 Existence of an Absorbing Set in 129
6.5 Existence of an Absorbing Set in 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 in Hs 152
7.3 Compactness of the Orbit in Hs 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 176
8.1.3 Global Attractor of System in and 177
8.2 Global Attractors for the 2D Navier-Stokes Equations in 183
CHAPTER 9 Global Attractors for an Incompressible Fluid Equation and a Wave Equation 187
9.1 An Incompressible Fluid Equation 187
9.2 A Wave 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
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