Nonlinear hyperbolic partial differential equations

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