Contents ix
5.5.3. Small time propagation of singularities 188
5.5.4. Global propagation of singularities 192
§5.6. An application to stabilization 195
Appendix 5.I. Hamilton–Jacobi theory for the eikonal equation 206
5.I.1. Introduction 206
5.I.2. Determining the germ of φ at the initial manifold 207
5.I.3. Propagation laws for φ, 209
5.I.4. The symplectic approach 212
Chapter 6. The Nonlinear Cauchy Problem 215
§6.1. Introduction 215
§6.2. Schauder’s lemma and Sobolev embedding 216
§6.3. Basic existence theorem 222
§6.4. Moser’s inequality and the nature of the breakdown 224
§6.5. Perturbation theory and smooth dependence 227
§6.6. The Cauchy problem for quasilinear symmetric hyperbolic
systems 230
6.6.1. Existence of solutions 231
6.6.2. Examples of breakdown 237
6.6.3. Dependence on initial data 239
§6.7. Global small solutions for maximally dispersive nonlinear
systems 242
§6.8. The subcritical nonlinear Klein–Gordon equation in the
energy space 246
6.8.1. Introductory remarks 246
6.8.2. The ordinary differential equation and non-
lipshitzean F 248
6.8.3. Subcritical nonlinearities 250
Chapter 7. One Phase Nonlinear Geometric Optics 259
§7.1. Amplitudes and harmonics 259
§7.2. Elementary examples of generation of harmonics 262
§7.3. Formulating the ansatz 263
§7.4. Equations for the profiles 265
§7.5. Solving the profile equations 270
Chapter 8. Stability for One Phase Nonlinear Geometric Optics 277
§8.1. The
Hs(Rd)
norms 278
§8.2.
Hs
estimates for linear symmetric hyperbolic systems 281
§8.3. Justification of the asymptotic expansion 282
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