Index 361

definition variable coeﬃcient, 47

images, method of, 30–33

inequality of stationary phase, 106

influence curve, 79–83

integration by parts, justification, 61

inviscid compressible fluid dynamics, xv

Keller’s blowup theorem, 249

Klein–Gordon equation, 14, 19, 73, 78,

146, 246–257

Kreiss matrix theorem, xii, 88

lagrangian manifold, 213

Lax parametrix, 177–195

Liouville

Liouville number, 312

Liouville’s theorem, 238, 311

Littlewood–Paley decomposition, 109,

115, 222, 254

maximally dispersive, 92, 104, 242–246

Maxwell’s equations, xiv

characteristic variety, 67

circular and elliptical polarization, 73

eikonal equation, 154

introduction, 44–46

plane waves, 72

propagation cone, 68

rotation of polarization, 288–289

self phase modulation, 287–288

microlocal

analysis, xi, xviii

elliptic regularity theorem, xviii,

136–140

propagation of singularities theorem,

177–195

applied to stabilization, 195–205

Moser’s inequality, 224, 325

Hs,

284

nondegenerate phase, 182

nondispersive, 99

nonstationary phase, xii

and Fourier integral operators,

180–188

and group velocity, 16–20

and resonance, 293, 324

and the stationary phase inequality,

121–122

observability, xviii

operator

pseudodifferential, xii, 136, 186

transposed, 134

oscillations

creation of, 310

homogeneous, 302, 336–338

oscillatory integrals, 180–188

partial inverse

for a single phase, 155

multiphase

on quasiperiodic profiles, 307

on trigonometric series, 306

of a matrix, 117

perturbation theory, 239

for semisimple eigenvalues, 117–119,

164

generation of harmonics, 262–263

quasilinear, 239

semilinear, 227–230

small oscillations, 259–262

phase velocities, 74, 145

piecewise smooth

definition d = 1, 10

function, wavefront set of, 140

solutions for refraction, 38

solutions in d = 1, 11

plane wave, 17, 142–143

polarization

in nonlinear geometric optics, 268

linear, circular, and elliptical, 73

of plane waves, 72

rotation of axis, xviii, 288

polyhomogeneous, 182

prinicipal symbol, 64

profile equations

quasilinear, 302–314

semilinear, 265–275, 314–315

projection (a.k.a. averaging) operator

E, 267–275

propagation cone, 64–71, 75

propagation of singularities, xviii, 1

d = 1 and characteristics, 10–11

d = 1 and progressing waves, 12–16

using Fourier integral operators,

177–195

pulse, see wave

purely dispersive, 99–100

quasiclassical limit of quantum

mechanics, xvii, 149–151, 160–161,

278