Global Propagation of Regular Nonlinear Hyperbolic Waves

by Tatsien, Li; Libin, Wang

• ISBN13: 9780817642440
• ISBN10: 0817642447
• Format: Hardcover
• Copyright: 2009-06-25
• Publisher: Birkhauser

Summary

This monograph describes global propagation of regular nonlinear hyperbolic waves described by first-order quasilinear hyperbolic systems in one dimension. The exposition is clear, concise, and unfolds systematically beginning with introductory material and leading to the original research of the authors.A systematic theory is established'”by means of the concept of weak linear degeneracy and the method of (generalized) normalized coordinates'”for the global existence and blow-up mechanism of regular nonlinear hyperbolic waves with small amplitude for the Cauchy problem and many others, including the Cauchy problem on a semi-bounded initial data, the one-side mixed initial-boundary value problem, the generalized Riemann problem, and the generalized nonlinear initial-boundary Riemann problem.Topics are motivated with a number of physical examples from the areas of elastic materials, one-dimensional gas dynamics, and waves. This book will stimulate further research and help readers further understand important aspects and recent progress of regular nonlinear hyperbolic waves.

Table of Contents

Preface	p. v
Introduction	p. 1
Cauchy Problem	p. 1
Weak Linear Degeneracy	p. 5
Some Examples	p. 7
System of Nonlinear Elasticity	p. 7
System of Traffic Flow	p. 8
System of One-Dimensional Gas Dynamics	p. 9
System of Compressible Elastic Fluids with Memory	p. 11
System of the Motion of an Elastic String	p. 13
System of Finite Amplitude Plane Elastic Waves for Hyperelastic Materials	p. 15
Main Results for the Cauchy Problem	p. 16
Normalized Coordinates	p. 18
Weak Linear Degeneracy and Generalized Null Condition	p. 19
Nonstrictly Hyperbolic Case	p. 20
Cauchy Problem on a Semibounded Initial Axis	p. 21
One-Sided Mixed Initial-Boundary Value Problem	p. 22
Generalized Riemann Problem	p. 23
Generalized Nonlinear Initial-Boundary Riemann Problem	p. 24
Inverse Generalized Riemann Problem	p. 26
Inverse Piston Problem	p. 26
Preliminaries	p. 29
Definition of Quasilinear Hyperbolic System	p. 29
Invariance Under a Smooth Invertible Transformation of Unknown Variables	p. 31
Genuine Nonlinearity and Linear Degeneracy	p. 33
Normalized Coordinates	p. 34
Normalized Coordinates for Strictly Hyperbolic Systems	p. 34
Normalized Coordinates for Nonstrictly Hyperbolic Systems with Characteristics with Constant Multiplicity	p. 36
Generalized Normalized Coordinates for General Hyperbolic Systems	p. 37
Weak Linear Degeneracy	p. 38
Weak Linear Degeneracy for Strictly Hyperbolic Systems	p. 38
Weak Linear Degeneracy for Nonstrictly Hyperbolic Systems with Characteristics with Constant Multiplicity	p. 40
Weak Linear Degeneracy for General Hyperbolic Systems	p. 40
Decomposition of Waves	p. 41
Formulas on the Decomposition of Waves	p. 41
Formulas on the Decomposition of Waves (Continued)	p. 45
Two Lemmas on Ordinary Differential Equations	p. 48
The Cauchy Problem	p. 51
Necessary Condition to Guarantee the Global Existence and Uniqueness of the C1 Solution to the Cauchy Problem for the Strictly Hyperbolic System	p. 51
Some Uniform a Priori Estimates Independent of Normalized Coordinates and Weak Linear Degeneracy for the Strictly Hyperbolic System	p. 54
Some Uniform a Priori Estimates Depending on Normalized Coordinates and Weak Linear Degeneracy for the Strictly Hyperbolic System	p. 60
Sufficient Condition to Guarantee the Global Existence and Uniqueness of the C1 Solution to the Cauchy Problem for the Strictly Hyperbolic System	p. 63
Global C1 Solution to the Cauchy Problem for the Hyperbolic System with Characteristics with Constant Multiplicity	p. 67
Applications	p. 73
System of One-Dimensional Gas Dynamics	p. 73
System of Compressible Elastic Fluids with Memory	p. 74
System of the Motion of an Elastic String	p. 76
The Cauchy Problem (Continued)	p. 79
Some Uniform a Priori Estimates Independent of Weak Linear Degeneracy	p. 79
Formation of Singularities of the C1 Solution in the Noncritical Case ¿ <+&infinity;	p. 80
Some Uniform a Priori Estimates Depending on Weak Linear Degeneracy	p. 80
Sharp Estimate on the Life Span of the C1 Solution	p. 84
Blow-Up Mechanism of the C1 Solution in the Noncritical Case ¿ <+&infinity;	p. 92
Introduction and Main Results	p. 92
Proof of Main Results	p. 93
Applications	p. 100
System of Traffic Flow	p. 100
System of One-Dimensional Gas Dynamics	p. 102
System of Compressible Elastic Fluids with Memory	p. 104
Blow-Up Mechanism of the C1 Solution in the Critical Case ¿ =+&infinity;	p. 106
Introduction and Main Results	p. 106
Some Uniform a Priori Estimates Depending on Weak Linear Degeneracy	p. 107
Proof of Main Results	p. 110
Remarks	p. 113
Cauchy Problem on a Semibounded Initial Axis	p. 115
Introduction and Main Results	p. 115
Proof of Theorem 5.1.1	p. 117
Application	p. 124
One-Sided Mixed Initial-Boundary Value Problem	p. 127
Global Existence of the Classical Solution	p. 127
Introduction and Main Results	p. 127
Proof of Theorem 6.1.1	p. 130
Formation of Singularities of the C1 Solution	p. 144
Applications	p. 145
Planar Motion of an Elastic String with a Fixed End	p. 145
Planar Motion of an Elastic String with a Dissipative Boundary	p. 148
Generalized Riemann Problem	p. 149
Introduction and Main Results	p. 149
Preliminaries	p. 154
Decomposition of Waves	p. 154
Rankine-Hugoniot Condition	p. 155
Proof of Main Results	p. 158
Applications	p. 169
System of Traffic Flow	p. 169
System of One-Dimensional Gas Dynamics	p. 171
System of Plane Elastic Waves for Hyperelastic Material	p. 172
Generalized Nonlinear Initial-Boundary Riemann Problem	p. 175
Introduction and Main Results	p. 175
Preliminaries	p. 179
Proof of Theorem 8.1.1	p. 180
Proof of Theorem 8.1.2	p. 182
Inverse Generalized Riemann Problem	p. 191
Introduction and Main Results	p. 191
Generalized Cauchy Problem	p. 195
Proof of Theorem 9.1.1	p. 202
Inverse Piston Problem	p. 209
Inverse Piston Problem for the System of One-Dimensional Isentropic Flow	p. 209
Introduction and Main Results	p. 209
Proof of Theorem 10.1.1	p. 215
Related Problem in Eulerian Representation	p. 223
Generalized Cauchy Problem with Cauchy Data Given on a Semibounded Noncharacteristic Curve	p. 230
Inverse Piston Problem for the System of One-Dimensional Adiabatic Flow	p. 235
Introduction	p. 235
Inverse Piston Problem in Lagrangian Representation	p. 238
Inverse Piston Problem in Eulerian Representation	p. 240
References	p. 245
Index	p. 251
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