1D Radiative Fluid and Liquid Crystal Equations, livre ebook

This book presents recent results on nonlinear evolutionary fluid equations, in particular the global well-posedness and asymptotic behavior of solutions to 1D radiative fluid equations, as well as liquid crystal equations. Most of the material in this book was prepared by the author over the past few years.This book has two main features. Firstly, there are more known results on higher dimensional radiative fluid systems but only on the local existence and explosion of solutions; while the existing findings on the one-dimensional case present some shortcomings, this book introduces corrections and improvements of these shortcomings. Secondly, the current findings on the high-dimensional compressible liquid crystal fluid equations are few and include only globally existing solutions but not the asymptotic behavior of the solutions; the author developed not only the global existence and regularity of the solutions, but also the asymptotic behavior of the solutions for the one-dimensional case in the chapter 3 of this book. Therefore, this work provides the reader with complete elements related to the one-dimensional compressible liquid crystal fluid system.

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . VII

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 Poincaré 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 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 12

1.1.8 The Hölder Inequalities . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 13

1.1.9 The Minkowski Inequalities . . . . . . . . . . . . . . . . . . . .. . . . . . . 14

CHAPTER 2

Asymptotic Behavior of Solutions for the One-DimensionalInfrarelativistic

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 H1 . . . . . . . . . . . . . . . . . . . . . . . 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 H1 _H10_ H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.3 Proof of Theorem 3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 100

3.4 Proof of Theorem 3.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 103

3.5 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 109

CHAPTER 4

Large-time Behavior of Solutions to a One-Dimensional Liquid Crystal System. . . . . . . . 111

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 111

4.2 Uniform Estimates in Hi _Hi0_ Hi þ1 ði ¼1; 2Þ and H4 _H40_ H4 . . 113

4.3 Large-time Behavior in Hi _Hi0_ Hi þ1 ði ¼1; 2Þ and H4 _H40_ H4 . 122

4.4 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 134

Bibliography . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 135

Index . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 143