Preface 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 Q-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 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 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 3.3 Upper Estimate on the Fractal Dimension of Global Attractors 3.4 Bibliographic Comments CHAPTER 4 Maximal Attractor for the Equations of One-Dimensional Compressible Polytropic Viscous Ideal Gas 4.1 Our Problem 4.2 Nonlinear Semigroup on 4.3 Existence of an Absorbing Set in H(2) 4.4 Existence of an Absorbing Set in H(2)β 4.5 Proof of Theorem 4.2.1 4.6 Bibliographic Comments CHAPTER 5 Universal Attractors for a Nonlinear System of Compressible One-Dimensional Heat-Conducting Viscous Real Gas 5.1 Main Results 5.2 Nonlinear Co-Semigroup on Hi+(i=1,2) 5.3 Existence of an Absorbing Set in H1δ 5.4 Existence of an Absorbing Set in H2δ 5.5 Proof of Theorem 5.1.1 5.6 Bibliographic Comments CHAPTER 6 Global Attractors for the Compressible Navier-Stokes Equations in Bounded Annular Domains 6.1 Main Result 6.2 Nonlinear Semigroup on 6.3 Existence of an Absorbing Set in H(1)δ 6.4 Existence of an Absorbing Set in H(2)δ 6.5 Existence of an Absorbing Set in H(4)δ 6.6 Bibliographic Comments CHAPTER 7 Global Attractor for a Nonlinear Thermoviscoelastic System in Shape Memory Alloys 7.1 Main Result
7.2 An Absorbing Set Bδ in Hδ 7.3 Compactness of the Orbit in Hδ 7.4 Bibliographic Comments 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(Ω) 8.1.3 Global Attractor of System in Lp(Ω) and H10(Ω) 8.2 Global Attractors for the 2D Navier-Stokes Equations in H10(Ω) 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 References Index
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