EDP Sciences EDP Sciences EDP Sciences EDP Sciences

Global Well-Posedness for Some Fluid Models

de Yuming QIN (auteur), Jianlin ZHANG (auteur)
février 2023
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Présentation

This book presents recent results on global well-posedness including asymptotic behavior of global solutions to some fluid models, such as combustion model of radiative gas, radiation hydrodynamics model, Navier-Stokes equations with capillary and p-th power Newtonian fluid model. These models have the similar structures, which consist of Navier-Stokes equations coupled with other equation or with other effects. Results collected in this book are established by the authors and their collaborators in recent years.

Sommaire

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

Compléments

Caractéristiques

Langue(s) : Anglais

Public(s) : Etudiants, Recherche

Editeur : EDP Sciences & Science Press

Collection : Current Natural Sciences

Publication : 16 février 2023

EAN13 (papier) : 9782759829057

Référence eBook [PDF] : L29064

EAN13 eBook [PDF] : 9782759829064

Intérieur : Noir & blanc

Nombre de pages eBook [PDF] : 292

Taille(s) : 2,7 Mo (PDF)

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