This up-to-date reference text examines the mathematical aspects of nonlinear wave propagation;emphasizing nonlinear hyperbolic problems;and introduces the most effective tools for the study of perturbation methods and for exploring global existence, singularity formation, and large-time behavior of solutions.
Author(s): Satyanad Kichenassamy
Series: Pure and Applied Mathematics M. Dekker
Publisher: CRC Press
Year: 1995
Language: English
Pages: 288
Contents......Page 14
Preface......Page 11
1 Linear Wave Propagation......Page 19
Generalities......Page 20
Basic properties......Page 21
First applications......Page 23
The standard Sobolev inequality......Page 25
The Lorentz and conformal groups......Page 28
1.3 The initial-value problem......Page 30
Elementary criteria for well-posedness......Page 31
Non-persistence results......Page 32
Hyperbolicity......Page 34
Representation formulae......Page 35
The method of stationary phase......Page 38
Use of representation formulae......Page 39
Use of global Sobolev inequalities......Page 43
Complex interpolation......Page 44
Local smoothing......Page 45
Inhomogeneous problems......Page 47
1.5 Propagation of singularities......Page 48
Propagation of discontinuities......Page 49
Propagation of C^\infty singularities......Page 51
1.6 What is a wave?......Page 53
1.7 Further results and problems......Page 55
Notes......Page 58
References......Page 60
2 Local and Global Existence......Page 64
An example......Page 66
Method of majorants......Page 67
Fuchsian equations......Page 68
Basic existence theorem......Page 69
Symmetric hyperbolic systems......Page 75
Examples......Page 78
Leray systems......Page 79
The Hille-Yosida theorem......Page 81
Application......Page 83
Nonlinear scattering......Page 84
Basic issues......Page 86
Compactification of Minkowski space......Page 87
Transformation formulae......Page 89
2.5 Other iteration techniques......Page 92
Nonlinear Klein-Gordon equation......Page 93
Perturbations of the wave equation......Page 94
2.6 Further results and problems......Page 95
Notes......Page 97
References......Page 99
3 Singularity Formation......Page 103
Scalar equations......Page 105
Quasilinear systems......Page 108
Blow-up via differential inequalities......Page 109
Other methods......Page 110
3.2 Propagation of weak singularities......Page 115
Action of nonlinear functions......Page 116
Application to nonlinear equations......Page 117
3.3 Shock waves......Page 119
Definitions......Page 120
Jump discontinuities......Page 121
Single conservation law......Page 123
3.4 Models for singularity formation......Page 124
The breakdown time......Page 125
The blow-up surface: semi-linear case......Page 128
The blow-up surface: quasi-linear case......Page 133
Integral invariants for first order ODEs......Page 135
The single first order equation......Page 137
3.6 Further results and problems......Page 140
Notes......Page 142
References......Page 144
4 Solitons and Inverse Scattering......Page 150
4.1 Universal equations......Page 152
Nearly monochromatic wavetrains......Page 153
Long waves......Page 155
Three-wave interaction......Page 156
Instabilities......Page 157
4.2 Isospectrality......Page 158
4.3 The Korteweg-deVries equation......Page 160
Eigenfunctions of L......Page 161
Scattering data......Page 164
Fourier transform of eigenfunctions......Page 165
Reconstruction procedure......Page 166
Application to the KdV equation......Page 168
Multi-solitons......Page 169
4.4 The AKNS systems......Page 170
Case of the AKNS systems......Page 175
Generalized AKNS systems......Page 176
Notions on the Cauchy integral and the B problem......Page 179
4.6 Criteria for integrability......Page 180
The WTC test......Page 181
Analytic characterizations......Page 188
Hirota's method......Page 192
Defining "integrability"......Page 193
4.7 Further results and problems......Page 194
Notes......Page 198
References......Page 199
5 Perturbation Methods......Page 203
A possible pathology......Page 204
"Type (A)" methods......Page 205
5.2 Implicit function theorems......Page 206
Nash-Moser theorem without smoothing......Page 207
Nash-Moser theorem with smoothing......Page 209
An application......Page 212
Nash-Moser theorem and paradifferential operators......Page 215
5.3 Nonlinear geometrical optics......Page 216
5.4 Whitham's theory......Page 218
Case of the 2 x 2 AKNS system......Page 220
Case of KdV......Page 222
Abstract results......Page 223
Examples......Page 224
5.7 Further results and problems......Page 225
Notes......Page 229
References......Page 231
6 General Relativity......Page 233
Metric and connection......Page 234
6.2 Einstein's equations......Page 244
Preliminary remarks......Page 250
Analytic Cauchy problem......Page 251
Non-analytic Cauchy problem......Page 252
Constraint equations......Page 254
Characteristic initial-value problem......Page 256
Linearization stability......Page 257
Other perturbation problems......Page 259
6.5 Singularities and cosmic censorship......Page 261
What is a singularity?......Page 262
Singularities and horizons......Page 263
Singularity theorems......Page 265
Spacetime symmetries......Page 266
Important exact solutions......Page 267
6.7 Further results and problems......Page 273
Notes......Page 279
References......Page 281
Index of Notation......Page 286
Index......Page 287
