Benzoni-Gavage S., Serre D. Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications
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INDEX
acoustics, 166
adiabatic exponent, 386
adjoint
BVP, 216
IBVP, 114
operator, 15
argument
T
∗
T ,37
Holmgren, 22
weak=strong, 68, 273
balance law, 291
Bethe–Weyl fluid, 387
block structure, 141
condition, 237
boundary
characteristic, 96, 103
contractible, 221
non-characteristic, 94, 103
boundary condition
conservative, 93
dissipative, 86
maximal dissipative, 86
number of, 100
strictly dissipative, 94,
226
boundary point
elliptic, 186
hyperbolic, 205, 223
boundary value problem
(see BVP), 123
BVP, 125
adjoint, 216, 255
Cauchy problem, 3
backward,9,23
homogeneous, 3
causality, 123
central subspace, xxiii
characteristic
boundary, 96, 103
bundle, 49
cone, xiv, 26
hyperplane, 29
hypersurface, 73
speed, 340
characteristic field
genuinely nonlinear, 394
linearly degenerate, 394
characteristics
incoming, 100
method of, 16
chemical potential, 397
commutator, 454
compatibility conditions, 270, 319
compensated compactness, 43
compression wave, 198
cone
characteristic, 26
forward, 26
of dependence, 18
of influence, 18
conservation law, 17, 291
conservative
boundary condition, 93
contact discontinuity, 400
continuous semigroup, 87
Dalembertian, 36
∆
q
, 460
density, 291
differential constraints, 41
Dirac operator, 37
discontinuity
compressive, 402
contact, 400
dynamical, 400
expansive, 402
dispersion, 197
dispersion relation, 415
displacement, 43
dissipative
boundary condition, 86
strictly-() boundary condition, 94, 226
symmetrizer, 145, 177, 187
distributions, xxiv
domain
influence, 20
of dependence, 19
Duhamelformula,4,93
Dunford–Taylor formula, xxiii, 234
dynamical discontinuity, 400
elastodynamics, 43, 173, 198
electromagnetism, 44
energy estimate
para-linearized, 349
weighted, 227
506 Index
energy identity, 17
entropy
-flux pair, 305
condition, xv
convex, 305
mathematical, 305
physical, 292
solution, 305
equation
in divergence form, 291
transport, 27
wave, 39
equation of state
complete, 385
incomplete, 388
equations
Cauchy–Riemann, 6
Euler, 291, 391
Maxwell, 42, 44
estimate
a priori,51
energy, 51, 183
Moser, 294
weighted, 81, 95, 96
Euler equations, 291, 391
exterior
algebra, 132
product, 132
Fa´a di Bruno’s formula, 321
FBP, 331
fluid, 291
Bethe–Weyl, 387
isentropic, 391
isothermal, 391
Smith, 388
van der Waals, 199, 389
forward cone, 26, 205
free boundary problem (see FBP), 330
Friedrichs
symmetric, 13, 85
symmetrizable, 13, 17, 52, 292
Frobenius norm, xxii
G˚arding inequality, 454, 487
sharp, 455, 459
with parameter, 458
γ (adiabatic exponent), 386
γ-law, 389
Γ (Gr¨uneisen coefficient), 386
gas
dusty, 390
ideal, 388
polytropic, 388
gas dynamics, 223
genuinely nonlinear, 394
glancing point, 157, 207, 237, 420
good unknown, 336
Gr¨uneisen coefficient, 386
Grassmannian, 136
Green formula, 88
Gronwall lemma, 443
basic, 443
discrete, 444
multidimensional, 444
guided wave, 198
Hadamard instability, 6, 107
Hardy–Littlewood–Sobolev inequality, 40
heat capacity, 386
Hille–Yosida theorem, 87
Holmgren principle, 22
homogeneous
IBVP, 85
Horn’s conjecture, 25
Hugoniot
adiabat, 401
locus, 401
Hugoniot set, 331
Huyghens principle, 198
hyperbolic
boundary point, 205, 223
constantly, 13, 139
first-order system, 8
in a direction (ξ
0
,λ
0
), 28
matrix, xxiii
operator, 8
parabolic-() operator, 35
polynomial, 30
strictly, 14
symmetric, 13
hyperbolicity, 7
constrained, 43
weak, 4
IBVP, 85
adjoint, 114, 137
characteristic, 158
homogeneous, 85, 182, 255
non-homogeneous, 85
normal, 102, 221
robust class of, 201
strongly well-posed, 109, 111
symmetric dissipative, 85
variable coefficients, 220
variational, 93
ideal gas, 388
incoming
characteristic, 100
flow, 411
mode, 104
influence domain, 20
initial boundary value problem
(see IBVP), 85
internal energy, 291
invariant subspace, xxiii
Index 507
isentropic fluid, 391
isothermal fluid, 391
Kreiss
matrix theorem, 10
symmetrizer, 120
Kreiss–Lopatinski˘ı condition, 105
uniform, 111, 235
lacunæ, xv
Lax
entropy criterion, 402
shock, 409
shock inequalities, 407
Lax shock, 341
Legendre transform, 398
Legendre–Hadamard inequality, 44
Lidskii theorem, 26
lifting of initial data, 286, 323
linearly degenerate, 394
Littlewood–Paley decomposition,
459
Liu criterion, 406
localization method, 21
Lopatinski˘ı
condition, 105, 234
determinant, 106, 130, 208, 236
Mach number, 409
mass transfer flux, 400
Maxwell’s equations, 42
nonlinear, 45
mixed problem, 315
mode
incoming, 104
outgoing, 104
monotone operator, 87
maximal, 87
neutral mode, 235
Newton’s polygon, 24
non-characteristic, 90
boundary, 103
non-glancing points, 186
non-homogeneous
IBVP, 85
normal
derivative, 167, 253
IBVP, 102
matrix, 86
mode, 102, 234, 337
Okhubo’s case, 175
Op(a), 452
operator
para-differential, 485
pseudo-differential, 452
stabilizable, 160
outgoing
flow, 411
outgoing mode, 104
overcompressive shock, 344
P-wave, 198
Paley–Wiener Theorem, 447
para-differential operator, 485
para-linearization, 348, 473
para-product, 466
Pauli matrices, 37
persistence, 329
Petrowsky’s
example, 12
point
interior, 146
polyconvex function, 44
polytropic gas, 388
Poynting identity, 45
pressure, 291
pressure law, 389
Mie–Gr¨uneisen-type, 391
propagation
speed of, 218
quasiconvex function, 44
quasilinear system, 292
R(u, v), 466
Rankine–Hugoniot condition, 312
Rayleigh wave, 198
real type subspace, 203
reflexivity, 158
relaxation model, 36
remainder, 466
Riesz–Thorin theorem, 38
robust class, 201
S
q
, 460
S-wave, 198
semi-group, 4
contraction, 92
shear wave, 198
shock
expansive, 437
Lax, 341
layer, 403
overcompressive, 344
state behind the, 409
undercompressive, 343
shock wave
planar, 313
Smith fluid, 388
solution
continuation of, 302
entropy, 305
weak, 304
sound speed, 292, 387
508 Index
space-like hypersurface, 73
spectrum, xxi
speed
of sound, 292, 387
stable subspace, xxiii
Strichartz estimates, xiv, 36
subsonic
flow, 411
state, 409
subspace
central, xxiii
invariant , xxiii
of real type, 203
stable, xxiii
unstable, xxiii
supersonic
flow, 411
state, 409
surface wave, 196
symbol, 450
homogeneous, 450
Sobolev, 451
symbolic symmetrizer, 59
symmetrizer
dissipative, 145, 177
functional, 54
functional boundary, 225
Kreiss, 120, 230
refined Kreiss, 247
symbolic, 56, 300
weakly dissipative, 187
system
in canonical form, 307
T
∗
T argument, 40
T
u
, 466
T
χ
a
, 485
tangential derivative, 166, 251
temperature, 292
critical, 389
test functions, xxiv
trace, 88
normal, 88
transport equation, 27
undercompressive shock, 343
uniformly diagonalizable, 11
unstable subspace, xxiii
vacuum, 399
van der Waals fluid, 389
van der Waerden inequality, 32
velocity, 291
vortex sheet, 400
wave
compression, 198
guided, 198
P-, 198
Rayleigh, 198
S-, 198
shear, 198
wave equation, 39
well-posedness
strong, 183
Weyl inequalities, 25
WR class, 209