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一维辐射流体方程组和液晶方程组解的整体存在性、渐近性和正则性（英文）
作者：秦玉明 著
出版社：科学出版社
出版时间：2022-10-01
ISBN：9787030721372
定价：¥78.00
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内容简介
《一维辐射流体方程组和液晶方程组解的整体存在性、渐近性和正则性（英文）》旨在介绍一些非线性演化流体方程的*新结果，包括辐射流体方程、液晶方程解的整体适定性和吸引子的存在性。《一维辐射流体方程组和液晶方程组解的整体存在性、渐近性和正则性（英文）》的大部分材料都基于作者及其合作者近年来进行的研究。其中一些以前只在原始论文中发表，有些材料直到现在才发表。《一维辐射流体方程组和液晶方程组解的整体存在性、渐近性和正则性（英文）》有四章。在第一章中，我们回顾了Sobolev空间的一些基本性质，分析中的一些微分积分不等式，其中一些将在后面的章节中使用。第二章研究了可压缩次相对论模型，证明了该模型解的整体存在性和渐近性。本章的创新之处在于：（1）使用适当的比容表达式和精确的先验估计，我们建立了比容的正下界和正上界。（2）利用嵌入定理和精细插值不等式，我们克服了高阶偏导数带来的一些数学困难，证明了高正则空间中解的整体存在性和渐近性。值得注意的是，我们在第二章中遇到的困难是如何处理辐射项，这使得本《一维辐射流体方程组和液晶方程组解的整体存在性、渐近性和正则性（英文）》的分析不同于以往的分析。第3-4章致力于研究一维可压缩液晶流体方程。在第三章中，我们在拉格朗日坐标下证明了全局解的存在性。在第四章中，我们首先建立了一维可压缩液晶流体方程解的大时间行为。本章的创新之处在于，我们使用了一个合适的比容表达式，通过嵌入定理和一系列微妙的公式，建立了比容的一致界。然后利用沈征不等式证明了系统解的渐近性态。
作者简介
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目录
Contents
Foreword i
CHAPTER 1 Preliminary 1
1.1 Some Basic Inequalities 1
1.1.1 The Sobolev Inequalities 1
1.1.2 The Interpolation Inequalities 5
1.1.3 The Poincare Inequality 6
1.1.4 The Classical Bellman-Gronwall Inequality 7
1.1.5 The Generalized Bellman-Gronwall Inequalities 8
1.1.6 The Uniform Bellman-Gronwall Inequality 9
1.1.7 The Young Inequalities
1.1.8 The Holder Inequalities 13
1.1.9 The Minkowski Inequalities 14
CHAPTER 2 Asymptotic Behavior of Solutions for the One-Dimensional Infrarelativistic Model of a Compressible Viscous Gas with Radiation 17
2.1 Main Results 17
2.2 Global Existence and Uniform-in-Time Estimates in H1 22
2.3 Asymptotic Behavior of Solutions in Hi 48
2.4 Global Existence and Uniform-in-Time Estimates in H2 53
2.5 Asymptotic Behavior of Solutions in H2 60
2.6 Global Existence and Uniform-in-Time Estimates in H4 62
2.7 Asymptotic Behavior of Solutions in H4 81
2.8 Bibliographic Comments 85
CHAPTER 3 Global Existence and Regularity of a One-Dimensional Liquid Crystal System 89
3.1 Main Results 89
3.2 Global Existence in 91
3.3 Proof of Theorem 3.1.2 100
3.4 Proof of Hheorem 3.1.3 103
3.5 Bibliographic Comments 109
CHAPTER 4 Large-time Behavior of Solutions do p One-Dimensional Liquid Crystal System 111
4.1 Introduction 111
4.2 Uniform Estimates in 113
4.3 Laxge-time Behavior in 122
4.4 Bibliographic Comments 134
Bibliography 135
Index 143
Foreword i
CHAPTER 1 Preliminary 1
1.1 Some Basic Inequalities 1
1.1.1 The Sobolev Inequalities 1
1.1.2 The Interpolation Inequalities 5
1.1.3 The Poincare Inequality 6
1.1.4 The Classical Bellman-Gronwall Inequality 7
1.1.5 The Generalized Bellman-Gronwall Inequalities 8
1.1.6 The Uniform Bellman-Gronwall Inequality 9
1.1.7 The Young Inequalities
1.1.8 The Holder Inequalities 13
1.1.9 The Minkowski Inequalities 14
CHAPTER 2 Asymptotic Behavior of Solutions for the One-Dimensional Infrarelativistic Model of a Compressible Viscous Gas with Radiation 17
2.1 Main Results 17
2.2 Global Existence and Uniform-in-Time Estimates in H1 22
2.3 Asymptotic Behavior of Solutions in Hi 48
2.4 Global Existence and Uniform-in-Time Estimates in H2 53
2.5 Asymptotic Behavior of Solutions in H2 60
2.6 Global Existence and Uniform-in-Time Estimates in H4 62
2.7 Asymptotic Behavior of Solutions in H4 81
2.8 Bibliographic Comments 85
CHAPTER 3 Global Existence and Regularity of a One-Dimensional Liquid Crystal System 89
3.1 Main Results 89
3.2 Global Existence in 91
3.3 Proof of Theorem 3.1.2 100
3.4 Proof of Hheorem 3.1.3 103
3.5 Bibliographic Comments 109
CHAPTER 4 Large-time Behavior of Solutions do p One-Dimensional Liquid Crystal System 111
4.1 Introduction 111
4.2 Uniform Estimates in 113
4.3 Laxge-time Behavior in 122
4.4 Bibliographic Comments 134
Bibliography 135
Index 143
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