Global Well-Posedness for Some Fluid Models

Jianlin ZHANG, Qin Yuming

Description

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.

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

Sujets

• Mathématiques fondamentales
• Mathématiques générales
• Mathématiques pures
• Mathématiques
• Mathematics
• Navier?Stokes equations
• Partial
• Sciences et techniques
• Differential Equations

Informations

• Publié par: EDP Sciences
• Date de parution: 16 février 2023
• Nombre de lectures: 0
• EAN13: 9782759829064
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Current Natural Sciences

Yuming QIN and Jianlin ZHANG

Global Well-Posedness

for Some Fluid Models

FLUID MODELS

FLUID MODELS

ISBN : 978-2-7598-2905-7

Current Natural Sciences

Global Well-Posedness for Some
Fluid Models

Yuming QIN and Jianlin ZHANG

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
andp-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.

Dr. Yuming QINprofessor, head of Mathematics Department is
and director of Institute of Nonlinear Sciences of Donghua
University. His research interests are global (local)
well-posedness of solutions and infinite dimensional dynamical systems
for nonlinear evolutionary equations including fluid equations
such as Navier-Stokes equations, MHD, and
thermos-viscoelastic equations.
Dr. Jianlin ZHANGassociate Professor in College of Science, is
Zhongyuan University of Technology since 2012.

www.edpsciences.org

Current Natural Sciences

Yuming QIN and Jianlin ZHANG

Global Well-Posedness
for Some Fluid Models

Printed in France

EDP Sciences–ISBN(print): 978-2-7598-2905-7–ISBN(ebook): 978-2-7598-2906-4
DOI: 10.1051/978-2-7598-2905-7

All rights relative to translation, adaptation and reproduction by any means whatsoever
are reserved, worldwide. In accordance with the terms of paragraphs 2 and 3 of Article 41
of the French Act dated March 11, 1957,“copies or reproductions reserved strictly for
private use and not intended for collective use”and, on the other hand, analyses and
short quotations for example or illustrative purposes, are allowed. Otherwise,“any
representation or reproduction–whether in full or in part–without the consent of the
author or of his successors or assigns, is unlawful”(Article 40, paragraph 1). Any
representation or reproduction, by any means whatsoever, will therefore be deemed an
infringement of copyright punishable under Articles 425 and following of the French
Penal Code.

The printed edition is not for sale in Chinese mainland. Customers in Chinese mainland
please order the print book from Science Press. ISBN of the China edition: Science Press
978-7-03-072138-9

Science Press, EDP Sciences, 2023

In memory of Yuming’s father, Zhenrong QIN
and mother, Xilan XIA
To Jianlin’s parents, Huaijian ZHANG and Xiuyun CHEN
To Yuming’s wife and son, Yu YIN and Jia QIN
To Jianlin’s wife, son and daughter, Huijun SUN, Lenan ZHANG
and Yuxin ZHANG

Contents

Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 1
Preliminary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 SobolevSpaces and Their Basic Properties. . . . . . . . . . . . . . . . . . . . .
1.1.1 WeakDerivatives and Their Properties. . . . . . . . . . . . . . . . . .
1.1.2 SobolevSpaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 SomeProperties of Sobolev Spaces. . . . . . . . . . . . . . . . . . . . . .
1.2 SomeInequalities in Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 TheYoung Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 ThePoincaréInequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 TheBellman–Gronwall Inequality. . . . . . . . . . . . . . . . . . . . . .
1.2.4 TheHölder Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.5 TheMinkowski Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.6 TheJensen Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.7 TheShen–Zheng Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.8 TheInterpolation Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.9 TheGagliardo–Nirenberg Interpolation Inequality. . . . . . . . . .
1.3C0-Semigroup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 2
Global Existence and Exponential Stability of Spherically Symmetric
Solutions to a Compressible Combustion Radiative and Reactive Gas. . . . . .
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Modeland Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 CombustionModel of Radiative and Reactive Gas. . . . . . . . . .
2.2.2 SphericalSymmetry Model of Combustion Model. . . . . . . . . . .
2.2.3 MainTheorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2.3 GlobalExistence and Exponential Stability inH. . . . . . . . . . . . . . . .
1
2.3.1 GlobalExistence of Solutions inH. . . . . . . . . . . . . . . . . . . . .
1
2.3.2 ExponentialStability of Solutions inH. . . . . . . . . . . . . . . . . .

IX

1
1
1
4
5
12
12
13
13
14
15
15
16
19
19
20

23
23
24
24
25
28
30
31
55

VI

2.4

2.5

2.6

Contents

2
Global Existence and Exponential Stability inH. . . . . . . . . . . . . . . .
2
2.4.1 GlobalExistence of Solutions inH. . . . . . . . . . . . . . . . . . . . .
2
2.4.2 ExponentialStability of Solutions inH. . . . . . . . . . . . . . . . . .
4
Global Existence and Exponential Stability inH. . . . . . . . . . . . . . . .
4
2.5.1 GlobalExistence of Solutions inH. . . . . . . . . . . . . . . . . . . . .
4
2.5.2 ExponentialStability of Solutions inH. . . . . . . . . . . . . . . . . .
Bibliographic Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 3
Global Existence, Uniqueness and Exponential Stability of Solutions
for the One-Dimensional Navier–Stokes Equations with Capillarity. . . . . . . .
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Modeland Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Korteweg-TypeModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 MainTheorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
3.3 GlobalExistence and Exponential Stability inH. . . . . . . . . . . . . . .
þ
1
3.3.1 GlobalExistence inH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
þ
1
3.3.2 AsymptoticBehavior inH. . . . . . . . . . . . . . . . . . . . . . . . . . .
þ
1
3.3.3 ExponentialStability inH. . . . . . . . . . . . . . . . . . . . . . . . . . .
þ
2
3.4 GlobalExistence and Exponential Stability inH. . . . . . . . . . . . . . .
þ
2
3.4.1 GlobalExistence and Uniqueness inH. . . . . . . . . . . . . . . . . .
þ
2
3.4.2 ExponentialStability inH. . . . . . . . . . . . . . . . . . . . . . . . . . .
þ
4
3.5 GlobalExistence and Exponential Stability inH. . . . . . . . . . . . . . .
þ
4
3.5.1 GlobalExistence inH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
þ
4
3.5.2 AsymptoticBehavior of Solutions inH. . . . . . . . . . . . . . . . .
þ
4
3.5.3 ExponentialStability inH. . . . . . . . . . . . . . . . . . . . . . . . . . .
þ
3.6 BibliographicComments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 4
Exponential Stability of Solutions for the Compressiblep-th Power
Newtonian Fluid with Large Initial Data. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Modeland Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
4.3 GlobalExistence and Exponential Stability inH. . . . . . . . . . . . . . . .
1
4.3.1 GlobalExistence inH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
4.3.2 ExponentialStability inH. . . . . . . . . . . . . . . . . . . . . . . . . . .
2
4.4 GlobalExistence and Exponential Stability inH. . . . . . . . . . . . . . . .
4
4.5 GlobalExistence and Exponential Stability inH. . . . . . . . . . . . . . . .
4.6 BibliographicComments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67
67
74
75
75
87
90

93
93
94
94
97
99
99
114
120
122
122
125
126
126
132
134
137

139
139
140
143
143
162
163
165
166

Contents

CHAPTER 5
Global Existence and Asymptotic Behavior of Spherically Symmetric
Solutions for the Multi-Dimensional Infrarelativistic Model. . . . . . . . . . . . . .
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Reformulationand Main Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Reformulationof Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 MainTheorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
5.3 GlobalExistence and Asymptotic Behavior inH. . . . . . . . . . . . . . . .
1
5.3.1 GlobalExistence inH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
5.3.2 AsymptoticBehavior inH. . . . . . . . . . . . . . . . . . . . . . . . . . .
2
5.4 GlobalExistence and Asymptotic Behavior inH. . . . . . . . . . . . . . . .
2
5.4.1 GlobalExistence inH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
5.4.2 AsymptoticBehavior inH. . . . . . . . . . . . . . . . . . . . . . . . . . .
4
5.5 GlobalExistence and Asymptotic Behavior inH. . . . . . . . . . . . . . . .
4
5.5.1 GlobalExistence inH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
5.5.2 AsymptoticBehavior inH. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 BibliographicComments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 6
Global Existence and Asymptotic Behavior of Cylindrically Symmetric
Solutions for the 3D Infrarelativistic