Contents Foreword i CHAPTER 1 Preliminary 1 1.1 Sobolev Spaces and Their Basic Properties 1 1.1.1 Weak Derivatives and Their Properties 1 1.1.2 Sobolev Spaces 4 1.1.3 Some Properties of Sobolev Spaces 5 1.2 Some Inequalities in Analysis 12 1.2.1 The Young Inequality 12 1.2.2 The Poincare Inequality 13 1.2.3 The Bellman-Gronwall Inequality 13 1.2.4 The Holder Inequality 14 1.2.5 The Minkowski Inequality 15 1.2.6 The Jensen Inequality 15 1.2.7 The Shen-Zheng Inequality 16 1.2.8 The Interpolation Inequality 19 1.2.9 The Gagliardo-Nirenberg Interpolation Inequality 19 1.3 Co-Semigroup 20 CHAPTER 2 Global Existence and Exponential Stability of Spherically Symmetric Solutions to a Compressible Combustion Radiative and Reactive Gas 23 2.1 Introduction 23 2.2 Model and Main Results 24 2.2.1 Combustion Model of Radiative and Reactive Gas 24 2.2.2 Spherical Symmetry Model of Combustion Model 25 2.2.3 Main Theorems 28 2.3 Global Existence and Exponential Stability in H1 30 2.3.1 Global Existence of Solutions in Hl 31 2.3.2 Exponential Stability of Solutions in H1 55 2.4 Global Existence and Exponential Stability in H2 67 2.4.1 Global Existence of Solutions in H2 67 2.4.2 Exponential Stability of Solutions in H2 74 2.5 Global Existence and Exponential Stability in H4 75 2.5.1 Global Existence of Solutions in H4 75 2.5.2 Exponential Stability of Solutions in W4 87 2.6 Bibliographic Comments 90 CHAPTER 3 Global Existence, Uniqueness and Exponential Stability of Solutions for the One-Dimensional Navier-Stokes Equations with Capillarity 93 3.1 Introduction 93 3.2 Model and Main Results 94 3.2.1 Korteweg-Type Model 94 3.2.2 Main Theorems 97 3.3 Global Existence and Exponential Stability in 99 3.3.1 Global Existence in * 99 3.3.2 Asymptotic Behavior in * 114 3.3.3 Exponential Stability in * 120 3.4 Global Existence and Exponential Stability in * 122 3.4.1 Global Existence and Uniqueness in * 122 3.4.2 Exponential Stability in * 125 3.5 Global Existence and Exponential Stability in * 126 3.5.1 Global Existence in * 126 3.5.2 Asymptotic Behavior of Solutions in* 132 3.5.3 Exponential Stability in * 134 3.6 Bibliographic Comments 137 CHAPTER 4 Exponential Stability of Solutions for the Compressible p-th. Power Newtonian Fluid with Large Initial Data 39 4.1 Introduction 39 4.2 Model and Main Results 40 4.3 Global Existence and Exponential Stability in H1 43 4.3.1 Global Existence in H1 43 4.3.2 Exponential Stability in H1 62 4.4 Global Existence and Exponential Stability in H2 63 4.5 Global Existence and Exponential Stability in H4 65 4.6 Bibliographic Comments 66 CHAPTER 5 Global Existence and Asymptotic Behavior of Spherically Symmetric Solutions for the Multi-Dimensional Infrarelativistic Model 169 5.1 Introduction 169 5.2 Reformulation and Main Theorems 171 5.2.1 Reformulation of Model 171 5.2.2 Main Theorems 174 5.3 Global Existence and Asymptotic Behavior in H1 177 5.3.1 Global Existence in Ti} 177 5.3.2 Asymptotic Behavior in H1 196 5.4 Global Existence and Asymptotic Behavior in H2 201 5.4.1 Global Existence in Ti.2 201 5.4.2 Asymptotic Behavior in H2 205 5.5 Global Existence and Asymptotic Behavior in H4 206 5.5.1 Global Existence in H4 206 5.5.2 Asymptotic Behavior in 219 5.6 Bibliographic Comments 221 CHAPTER 6 Global Existence and Asymptotic Behavior of Cylindrically Symmetric Solutions for the 3D Infrarelativistic Model with Radiation 223 6.1 Introduction 223 6.2 Reformation and Main Results 225 6.2.1 Reformation of Model 225 6.2.2 Main Theorems 228 6.3 Global Existence and Asymptotic Behavior in H1 229 6.3.1 Global Existence in H1 230 6.3.2 Asymptotic Behavior in H1 256 6.4 Global Existence and Asymptotic Behavior in H2 257 6.4.1 Global Existence in H 257 6.4.2 Asymptotic Behavior in H2 260 6.5 Global Existence and Asymptotic Behavior in H4 261 6.5.1 Global Existence in H4 261 6.5.2 Asymptotic Behavior in 273 6.6 Bibliographic Comments 276 Bibliography 277
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