Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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708 18. Einstein’s Equations
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Index
A
absolute boundary condition, 559
area functional, 152
second variation, 168
arithmetic-geometric mean inequality, 272,
289
Ascoli’s theorem, 160
B
Bakelman’s estimate, 258
Banach space, 96, 588, 603
barrier, 292
lower, 214
upper, 214
Belousov–Zhabotinski system, 351
Bernoulli’s law, 432, 454, 539, 540
Bernstein’s theorem, 184
Besov space, 59
Bianchi identity, 625
bi-Lipschitz maps
volume preserving, 600
Birkhoff ’s theorem, 635
blow-up, 420
Bochner Laplacian, 561
Boltzmann equation, 457
boundary gradient estimate, 210, 236, 287
boundary layer, 589
boundary problem
regular, 191
Brusselator, 348, 352
C
Caccioppoli inequality, 244
calculus of variations, 222
Calderont–Zygmund theorem, 18
catenoid, 156
Cauchy–Kowalewsky theorem, 445, 521
circularly symmetric flows, 587
Codazzi equation, 167
completely nonlinear, 128, 139, 387,
423, 437
concave, 273
conformal, 112, 119
conformal Laplacian, 117
conservation law, 452, 537
scalar, 457
system, 472, 498, 509
constraint equations, 689
contraction mapping principle, 117, 314
convex, 108, 223
covering Lemma
Calderont–Zygmund, 19
Vitali, 239
Wiener, 87
D
Deformation tensor, 552, 562, 690
DeGiorgi–Nash–Moser theory, 196
difference scheme, 469
Dirichlet boundary condition, 591
Dirichlet problem, 107, 130, 194, 208, 332
div-curl lemma, 29, 83, 93, 511
divergence form, 143
divergence theorem, 620
Duhamel’s formula, 592, 594, 603
dust, 617
E
Eddington–Finkelstein coordinates, 636
eikonal equation, 505
Einstein’s equations, 435, 615, 634
initial-value problem, 677
Einstein tensor, 616, 631
elliptic, 47, 127
strongly, 229, 236
underdetermined, 132
uniformly, 215
very strongly, 229
712 Index
elliptic boundary problem, 185
energy, 325
conservation, 455
Enneper’s surface, 156
enstrophy, 552
entropy, 456, 485
condition, 501
inequality, 460
Kruzhkov’s condition, 460
entropy-flux pair, 498, 510
equation of state, 450, 456, 485
Euler characteristic, 119
Euler equations, 449, 476, 534, 539, 553,
586, 599
exterior derivative, 535
F
Finite propagation speed, 424, 440, 463,
687
Fisher equation, 346
Fitzhugh–Nagumo equations, 335, 341,
347, 349, 362
fixed-point theorem
Brouwer, 302
Leray–Schauder, 115, 294, 303, 697
Schauder, 221, 302
fluid flow
compressible, 427, 440, 448, 473, 482,
485, 503
incompressible, 532
irrotational, 454, 540
relativistic, 659
stationary, 539
steady, 454
viscous, 561
Fourier transform, 17
Friedrichs mollifier, 3, 377, 414, 542, 558
Frobenius’s theorem, 540
fuzzy function, 74
sharply defined, 75
G
Gagliardo–Nierenberg inequality, 8, 29
Galerkin method, 422
modified, 377
G˚arding inequality, 388, 392
Gauss–Bonnet formula, 119, 175, 625, 638
Gauss–Codazzi equations, 688
Gauss curvature, 112, 119, 170, 183, 282,
288, 294, 438, 631
Gauss map, 184
Geometrical optics, 504
nonlinear, 507
Glimm scheme, 495, 524
gravitational collapse, 670
gravitational constant, 669
Gronwall inequality, 358, 379, 416, 418,
421, 546
growth condition
controllable, 230
H
Hamiltonian, 649
Hardy space, 86, 256
harmonic, 154
harmonic coordinates, 146, 247, 678
harmonic map, 166, 225, 247, 255, 325
Harnack inequality, 199, 262, 401
Hausdorff measure, 237
helicity, 542
helicoid, 156
Helmholtz theorem, 453, 538
Hessian, 282
H¨odge decomposition, 94, 534, 540, 553,
559
H¨odge Laplacian, 545, 561
H¨older continuity, 99, 201, 269, 273, 402
H¨older inequality, 4
reverse, 244
H¨older space, 40, 135
holomorphic, 155
Hopf theorem, 567, 580
hyperbolic system, 703
genuinely nonlinear, 475, 520
linearly degenerate, 483
second-order, 432
strictly, 428, 472, 509, 518
symmetric, 414, 446, 542
symmetrizable, 426
I
Implicit function theorem, 120
instability
diffusion-driven, 347
Index 713
integral equation, 314, 363
interpolation, 5, 25, 589, 595
complex, 96
Marcinkiewicz, 18
inverse function theorem, 114
isometric imbedding, 133, 147, 153, 225,
325
isothermal coordinates, 112
K
Kelvin circulation theorem, 453, 537
Kepler problem, 650
Kerr metric, 648
Kerr–Newman metric, 659
Killing field
conformal, 690
timelike, 639
Kolmogorov–Petrovskii–Piskunov
equation, 346
Kruskal coordinates, 637
Krylov–Safanov estimates, 258
L
Laplace operator, 112, 154, 232, 591
lapse function, 699
layer potential, 605
Legendre–Hadamard condition, 229
Leray projection, 533
Lie derivative, 451, 534
linearization, 127
Liouville’s theorem, 184
liquid crystal, 226
Littlewood–Paley theory, 22, 28, 51
local solvability, 129
longitudinal wave, 480
Lorentz force law, 618
Lorentz metric, 616
lower solution, 118, 283
M
Maximal function
grand, 86
Hardy-Littlewood, 87
maximum principle, 100, 110, 177, 182,
210, 214, 236, 283, 319, 384,
596, 598
Maxwell–Einstein equations, 656, 683
Maxwell’s equations, 617
mean curvature, 153, 219
prescribed, 257
vector, 153
membrane, 216
method of continuity, 113, 209, 234, 281
minimal submanifold, 153, 332
minimal surface, 152, 441
associated, 156
conjugate, 156
stable, 174
Weierstrass–Enneper representation,
156
minimal surface equation, 177, 195, 213,
235
minimization, 222, 234
constrained, 225
Monge–Ampere equation, 128, 282
hyperbolic, 438
Morrey’s lemma, 212, 241, 300
Morrey space, 202, 297, 299
Moser estimates, 13, 62, 66, 338, 378, 415,
430, 543
Moser iteration, 197, 397
Murat’s lemma, 511
N
Nash imbedding theorem, 151
Nash–Moser estimates, 387, 396
Navier–Stokes equations, 561, 575, 586,
591, 599
Neumann problem, 334, 555
nonlinear Klein–Gordon equation, 441
no-slip boundary condition, 575, 586
O
Oleinik condition (E), 465
Oppenheimer–Volkov equation, 668
orbit
heteroclinic, 345, 493
homoclinic, 345
P
Palatini identities, 620
parabolic, 314, 376, 387
Petrowski, 390
714 Index
paradifferential operator, 62, 67, 140, 387
parametrix, 36, 135
paraproduct, 73
partition of unity
Littlewood–Paley, 22, 41, 52, 60, 300
pattern formation, 349
Plateau problem, 158, 257
Poincar´e disk, 124
Poiseuille flows, 591
Poisson integral, 160
Prandtl’s principle, 599
predator-prey equations, 352
pressure, 450, 534
pressure driven, 591
proper time, 616
pseudodifferential operator, 17, 50, 102,
428, 545
elliptic, 607
Q
Quasi-linear, 128, 137, 208, 376, 389, 396,
414
R
Rankine–Hugoniot condition, 464, 476,
493
rarefaction wave, 467, 474
reaction-diffusion equations, 335, 407
regularity, 135, 185
partial, 236
Schauder, 137
Reissner–Nordstr¨om metric, 658
relativity, 616
Rellich’s theorem, 16, 322
Ricci tensor, 145, 173, 247, 555, 616
Riemann invariant, 503, 511, 514
Riemann mapping theorem, 157
Riemann problem, 466, 474
Riemann surface, 124
Riemann tensor, 554
S
Scalar curvature, 616
Schwartz kernel, 18
Schwarzschild metric, 636, 649, 656
second fundamental form, 167, 286, 332,
555
sectional curvature, 326
semigroup
heat, 589
holomorphic, 31
semilinear, 107, 244, 314, 424
shock interaction, 496
shock speed, 476
shock wave, 466, 479
spherically symmetric, 498
simple wave, 498
sine-Gordon equation, 442
Sobolev estimates, 595
Sobolev imbedding theorem, 4, 26, 37, 43,
318, 329, 379, 416
Sobolev space, 2, 97
soliton equations, 507
spacelike surface, 434, 678, 680, 689
spacetime, 615
spherically symmetric, 626
static, 635, 644
stationary, 639
splitting method, 353, 362
Stefan problem, 362
Stirling’s formula, 16
Stokes operator, 576
stream function, 454
streamline, 454, 539
stress-energy tensor, 616, 625
stress tensor, 562
strong energy condition, 624
symbol, 17, 50
symbol smoothing, 56, 65, 67, 387
symmetrizer, 426, 508
Synge’s theorem, 175
T
Theorema Egregium, 169
timelike, 616
transport equation, 505
traveling wave, 343, 495
Triebel space, 59
Trotter product formula
nonlinear, 341–342, 353
Trudinger’s estimate, 29
Trudinger’s inequality, 15, 121
Index 715
U
Uniformization theorem, 124, 157
upper solution, 118, 290
V
Vacuum, 491
vibrating membrane, 524
vibrating string, 482, 514, 515, 517
viscosity method, 457, 493
viscous profile, 493
vortex line, 539
vortex tube, 453, 538
strength of, 453
vorticity, 451, 535, 660
vorticity equation, 451, 535, 539, 662
Lichnerowicz, 663
W
Wave map, 444
Weingarten formula, 152, 169
Weitzenbock formula, 561, 562, 690
Y
Yamabe problem, 116
Young measure, 75, 509
Young’s inequality, 54, 59
Z
Zygmund space, 42, 47, 97, 99, 135,
300, 547