Foreword................................................... IX
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 Poincaré Inequality.............................. 13
1.2.3 The Bellman–Gronwall Inequality...................... 13
1.2.4 The Hölder 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 C0-Semigroup........................................... 20
CHAPTER 2
Global Existence and Exponential Stability of Spherically Symetric 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 H1..................... 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 H4.................. 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 Stabilityin H1 þ ............... 99
3.3.1 Global Existence in H1 þ .............................. 99
3.3.2 Asymptotic Behavior in H1 þ ........................... 114
3.3.3 Exponential Stability in H1 þ ........................... 120
3.4 Global Existence and Exponential Stabilityin H2 þ ............... 122
3.4.1 Global Existence and Uniqueness in H2 þ .................. 122
3.4.2 Exponential Stability in H2 þ ........................... 125
3.5 Global Existence and Exponential Stabilityin H4 þ ............... 126
3.5.1 Global Existence in H4 þ .............................. 126
3.5.2 Asymptotic Behavior of Solutions in H4 þ ................. 132
3.5.3 Exponential Stability in H4 þ ........................... 134
3.6 Bibliographic Comments................................... 137
CHAPTER 4
Exponential Stability of Solutions for theCompressible p-th Power Newtonian Fluid with Large Initial Data........................... 139
4.1 Introduction............................................ 139
4.2 Model and Main Results................................... 140
4.3 Global Existence and Exponential Stability in H1 ................ 143
4.3.1 Global Existence in H1 ...............................143
4.3.2 Exponential Stability in H1........................... 162
4.4 Global Existence and Exponential Stability in H2 ................ 163
4.5 Global Existence and Exponential Stability in H4 ................ 165
4.6 Bibliographic Comments................................... 166
CHAPTER 5
Global Existence and Asymptotic Behavior of Spherically Symetric 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 H1............................... 177
5.3.2 Asymptotic Behavior in H1........................... 196
5.4 Global Existence and Asymptotic Behavior inH2 ................ 201
5.4.1 Global Existence in H2............................... 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 H4........................... 219
5.6 Bibliographic Comments................................... 221
CHAPTER 6
Global Existence and Asymptotic Behavior ofCylindrically Symetric Solutions for the 3D Infrarelativistic Modelwith 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 H2............................... 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 H4........................... 273
6.6 Bibliographic Comments................................... 276
Bibliography................................................. 277
