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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