INDEX 753

maximum principles, 391–400

regularity, 382–390

weak solution of, 375–381

Segal, I., 698

semiconcavity, 129, 130

semigroup, 204

linear, 11, 435–447, 450, 451

nonlinear, 11, 562, 567–575, 578

semilinear PDE, 2

separable space, 723

separation of variables, 167–172, 245

shallow water equations, 614, 656, 657

shape from shading, 607

shock set, 627, 656

shock wave, 8, 139, 142, 617, 627, 633,

644, 649, 656, 657

nonphysical, 140, 633

short time existence, 668–672

signature of matrix, 222

similarity solutions, 45, 176–187

simple

function, 736

waves, 623

sine-Gordon equation, 696

singular

integral, 94

perturbation, 212

Sobolev inequalities, 277–288, 668

Sobolev space, 10, 257

approximation, 266–270

compactness, 288–291

completeness of, 265

convergence in, 261

deﬁnition of, 260

diﬀerentiability a.e., 297

extensions, 270–273

fractional, 300

norm, 261

traces, 273–277

soliton, 180

solution

classical, 7

entropy, 149, 165, 649, 652, 658

integral, 136, 614

viscosity, 583, 608

weak, 8, 546, 584

Sommerfeld radiation condition, 371

sound speed, 623, 640

spectrum, 325, 725

compact, symmetric operator, 731

compact operator, 730

complex, 363

point, 725

spherical means, 67, 70

star-shaped

level sets, 556

set, 521, 553, 554

state

of system, 593

space, 612

stationary phase, 178, 220–229

Stokes’

problem, 499

rule, 88

strain, stress, 622

strict hyperbolicity, 617, 656

subdiﬀerential, 562

subharmonic function, 85

subscripts, superscripts, 462, 705

subsolution, 347, 369, 546, 576

heat equation, 88

Laplace’s equation, 85

summable function, 734

supersolution, 347, 545, 576

system, 2

analytic PDE, 240

conservation laws, 611–651

constant coeﬃcient, hyperbolic,

201–203, 431–435

Euler’s equations, 208

Euler–Lagrange, 461–465, 477–482

irrotational flow, 209

ODE, 181

reaction-diﬀusion, 172, 537

semilinear hyperbolic, 617

Taylor’s formula, 13, 31, 32, 238

telegraph equation, 4, 89, 196, 450

terminal-value problem

adjoint equations, 448, 659

Hamilton–Jacobi equation, 599

heat equation, 63

transport equation, 150

test function, 258

traces, 273–277, 309

transport equation, 3, 9, 18–19, 67

traveling wave, 176, 178, 180, 181, 246,

247

conservation law, 617, 644–647

Turing instability, 172–175

uniform integrability, 674

uniqueness