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
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