Contents Preface iCHAPTER 1 Preliminary 11.1 Some Useful Inequalities 11.2 Basic Theory of Infinite-Dimensional Dynamical Systems for Autonomous Nonlinear Evolutionary Equations 101.2.1 Uniformly Compact Semigroups 101.2.2 Weakly Compact Semigroups 161.2.3 Q-Limit Compact Semigroups 171.2.4 Asymptotically Compact Semigroups 221.2.5 Asymptotically Smooth Semigroups 271.2.6 Norm-to-Weak Continuous Semigroups 281.2.7 Closed Operator Semigroups 301.3 Basic Theory of Finite-Dimensional Attractors 321.3.1 The Fractal Dimension of Global Attractors 321.3.2 The Estimate on Fractal Dimension of Global Attractors 33CHAPTER 2 Global Attractors for the Navier-Stokes-Voight Equations with Delay 372.1 Global Wellposedness of Solutions 372.2 Existence of Global Attractors 432.2.1 Dissipation: Existence of Absorbing Sets 432.2.2 Asymptotical Compactness and Existence of Attractor 442.3 Bibliographic Comments 46CHAPTER 3 Global Attractor and Its Upper Estimate on Fractal Dimension for the 2D Navier-Stokes-Voight Equations 473.1 Global Existence of Solutions 473.2 Existence of Global Attractors 553.2.1 Existence of Absorbing Sets 553.2.2 Some Compactness and the Existence of Global Attractors 563.3 Upper Estimate on the Fractal Dimension of Global Attractors 583.4 Bibliographic Comments 64CHAPTER 4 Maximal Attractor for the Equations of One-Dimensional Compressible Polytropic Viscous Ideal Gas 6741 Our Problem 674.2 Nonlinear Semigroup on 694.3 Existence of an Absorbing Set in 734.4 Existence of an Absorbing Set in 834.5 Proof of Theorem 4.2.1 864.6 Bibliographic Comments 88CHAPTER 5 Universal Attractors for a Nonlinear System of Compressible One-Dimensional Heat-Conducting Viscous Real Gas 915.1 Main Results 915.2 Nonlinear Co-Semigroup on 955.3 Existence of an Absorbing Set in 975.4 Existence of an Absorbing Set in 1065.5 Proof of Theorem 5.1.1 1085.6 Bibliographic Comments 111CHAPTER 6 Global Attractors for the Compressible Navier-Stokes Equations in Bounded Annular Domains 1156.1 Main Result 1156.2 Nonlinear Semigroup on 1196.3 Existence of an Absorbing Set in 1206.4 Existence of an Absorbing Set in 1296.5 Existence of an Absorbing Set in 1356.6 Bibliographic Comments 146CHAPTER 7 Global Attractor for a Nonlinear Thermoviscoelastic System in Shape Memory Alloys 1497.1 Main Result 1497.2 An Absorbing Set in Hs 1527.3 Compactness of the Orbit in Hs 1657.4 Bibliographic Comments 173CHAPTER 8 Global Attractors for Nonlinear Reaction-Diffusion Equations and the 2D Navier-Stokes Equations 1758.1 Global Attractor for Strong Solutions of Reaction-Diffusion Equations 175 8.1.1 Existence of Solutions and Uniqueness 1768.1.2 Global Attractor for the Semigroup in 176 8.1.3 Global Attractor of System in and 1778.2 Global Attractors for the 2D Navier-Stokes Equations in 183CHAPTER 9 Global Attractors for an Incompressible Fluid Equation and a Wave Equation 1879.1 An Incompressible Fluid Equation 1879.2 A Wave Equation with Nonlinear Damping 1939.2.1 Wellposedness of Solutions 1949.2.2 Dissipativity 1969.2.3 Asymptotic Compactness and Existence of Global Attractor 200References 203Index 211
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