Nonlinear hyperbolic partial di.erential equations describe many physical phenomena. Particularly, important examples occur in gas dynamics, shallow water theory, plasma physics, combustion theory, nonlinear elasticity, acous-tics, classical or relativistic .uid dynamics and petroleum reservoir engineering etc. For linear hyperbolic equations with suitably smooth coe.cients, it is well-known that Cauchy problem always admits a unique global classical solution on the whole domain, provided that the initial data are smooth enough. For nonlinear hyperbolic equations, however, the situation is quite di.erent. Gen-erally speaking, in this case, the classical solutions to Cauchy problem exist only locally in time and singularities may occur in a .nite time, even if the initial data are su.ciently smooth and small.This book is concerned with the classical solution to nonlinear hyperbolic partial di.erential equations. The greatest part of the book is the fruit aca-demic research on the part of the author. Some of what contained in the book has been published for the .rst time, and what was previously published in the form of separate papers has also been revised and upgraded.There are 7 chapters in this book. Chapter 1 is a preliminary chapter in which we give some basic concepts of nonlinear hyperbolic system: genuinely nonlinear, linearly degenerate, weak linear degenerate, matching condition etc.In chapter 2, we shall investigate the .rst order nonlinear hyperbolic equa-tion in two independent variables, and give some results on the classical solu-tions.Chapter 3 is devoted to the study of the mechanism and the character of singularity caused by eigenvectors are investigated for nonlinear hyperbolic system, and some new concepts on nonlinear hyperbolic system are proposed.Chapter 4 will concern the Cauchy problem and mixed initial boundary value problem for hyperbolic geometric .ow. Some geometric properties of hyperbolic geometric .ow on general open and closed Riemannian surfaces are also discussed.In chapter 5, we shall investigate the life-span of classical solutions to the hyperbolic geometric .ow in two space variables with slow decay initial data.Chapter 6 will be concerned the dissipative e.ect of the relaxation. The convergence of approximate solution to nonlinear hyperbolic conservation laws with relaxation is proved.In chapter 7, we shall consider some applications of nonlinear hyperbolic system.The whole approach to the problems under discussion is primarily based on the theory on the local solution. For more comprehensive information, the reader may refer to the book by Li Tatsien and Yu Wenci: Boundary Value Problems for Quasilinear Hyperbolic Systems (Duke University Mathematics Series V, 1985).Because the local classical solution theory has been established well, the key point of this method is how to establish some uniform a priori estimates on the solution.This work was partially supported by plan for scienti.c innovation Talent of North China University of Water Resources and Electric Power.AuthorAugust, 2016Zhengzhou, China
劉法貴，教授，理學(xué)博士，華北水利水電大學(xué)教務(wù)處處長(zhǎng)。碩士生導(dǎo)師。河南省數(shù)學(xué)學(xué)會(huì)理事，鄭州市數(shù)學(xué)學(xué)會(huì)理事。河南省學(xué)術(shù)技術(shù)帶頭人，河南省優(yōu)秀中青年骨干教師，省級(jí)重點(diǎn)學(xué)科帶頭人，河南省555省級(jí)人選。從事擬線性雙曲偏微分方程的研究，在國(guó)內(nèi)外重要學(xué)術(shù)期刊上發(fā)表論文50余篇。
Preface
...................................................................................................I
Chapter 1 Introduction.....................................................................
1
1.1 Intention
and Signi.cances ....................................................... 1
1.2 Basic
Concepts ........................................................................
7
1.3 Some
Examples.......................................................................14
1.4 Preliminaries
..........................................................................18
Chapter 2 Cauchy
Problem for Nonlinear Hyperbolic Systems in Diagonal Form
...........................................................25
2.1 The
Single Nonlinear Hyperbolic Equation ...............................25
2.2 The
Classical Solutions to Single Nonlinear Hyperbolic Equation ................................................................................32
2.3 Nonlinear
Hyperbolic Equations in Diagonal Form....................40
Chapter 3 Singularities
Caused by the Eigenvectors ....................50
3.1 Introduction
...........................................................................50
3.2 Completely
Reducible Systems.................................................55
3.3 2-Step
Completely Reducible Systems ......................................59
3.4 m(m>
2)-Step Completely Reducible Systems with Constant Eigenvalues
..............................................................67
3.5 Non-completely
Reducible Systems ..........................................74
3.6 Examples
...............................................................................76
Chapter 4 Hyperbolic
Geometric Flow on Riemannian
Surfaces...........................................................................85
4.1 Introduction
...........................................................................85
4.2 Cauchy
Problem for Hyperbolic Geometric Flow.......................87
4.3 Mixed
Initial Boundary Value Problem for Hyperbolic Geometric Flow
......................................................................99
4.4 Dissipative
Hyperbolic Geometric Flow .................................. 107
4.5 Explicit
Solutions..................................................................119
4.6 Radial
Solutions to Hyperbolic Geometric Flow ...................... 124
Chapter 5 Life-Span
of Classical Solutions to Hyperbolic Geometric Flow in Two Space Variables with
Slow Decay Initial Data .............................................. 127
5.1 Intention
and Signi.cances .................................................... 127
5.2 Some
Useful Lemmas ............................................................ 130
5.3 Lower
Bound of Life-Span ..................................................... 143
Chapter 6 Nonlinear
Hyperbolic Systems with Relaxation ...... 153
6.1 Introduction
......................................................................... 153
6.2 Global
Classical Solutions...................................................... 155
6.3 Applications
.........................................................................162
6.4 Convergence
of Approximate Solutions...................................165
Chapter 7 Applications..................................................................
175
7.1 One Dimensional Hydromagnetic
Dynamics............................175
7.2 Fluid Flow on a Pipe
............................................................ 187
7.3 Heat Conduction with Finite of
Propagation .......................... 189
7.4 A Nonlinear Systems in
Viscoelasticity...................................191
Bibliography
......................................................................................
202
Index
..................................................................................................
209
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