Galdi G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems
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1016 Index
weak, 32
convergent, 29
weakly convergent 32
Singular kernel, 129
Sobolev “cut-off” function, 102
Sobolev theorem, 128
Sobolev space, 50
Space
Banach, 30
C
k
(Ω), 35
C
∞
(Ω), 35
C
k
0
(Ω), 35
C
∞
0
(Ω), 35
C
k,λ
(Ω), 36
D
m,q
(Ω), 80
˙
D
m,q
(Ω), 83
D
m,q
0
(Ω), 84
D
−m,q
0
(Ω) 109
D(Ω), 142
D
1,q
0
(Ω), 214
b
D
1,q
0
(Ω), 214
G
q
(Ω), 142
H
q
(Ω), 142
H
1
q
(Ω), 193
b
H
1
q
(Ω), 193
L
q
(Ω) 40
L
q
loc
(Ω), L
q
loc
(Ω), 43
L
q
(A
t
), 526
L
r,q
(A
t
), 526
W
m,q
(Ω), 49
W
m,q
0
(Ω), 50
W
−m,q
0
(Ω), W
−m,q
0
0
(Ω), 60
W
m−1/q,q
(∂Ω), 64, 67
W
m,q
loc
(Ω), W
m,q
loc
(Ω), 81
Sobolev, 50
anisotropic, 55
homogeneous, 80
negative, 60
trace, 64, 67–68
Star-shaped or star-like 38
Steady-state Navier-Stokes
equations, 4
Steady fall of a body, 653
Stein theorem
on extension maps, 58
on singular transforms i n weighted
spaces, 131
on regularized distance, 219
Stokes flow in an aperture domain, see
also Heywood’s problem, 407
generalized solutions for, 388
asymptotic behavior of, 414–415
existence and uniqueness of, 392,
407
pressure associated to, 390
representation of, 411, 414
existence and uniqueness in D
1,q
,
407
Stokes flow in bounded domains, 231ff
generalized solutions for, 234
existence and uniqueness of, 237
regularity of, 267, 277
q-generalized solutions for, 234
estimates of
in H¨older spaces,
in W
m,q
, 227 234
interior, 263ff
near the boundary, 271ff
existence and uniqueness of
in H¨older spaces, 287ff
in W
m,q
, 279ff
pressure field associated to, 186
uniqueness of, 228
maximum modulus theorem for, 298
Green’s tensor, 288
estimates for 289
Stokes flow in channels with unbounded
cross sections 387ff
generalized solutions for, 388
asymptotic behavior of, 393ff
existence and uniqueness of, 390
pressure associated to, 390
Stokes flow in exterior domains, 299ff
existence and uniqueness in D
m,q
,
334–335
generalized solutions, 304
asymptotic behavior of, 313, 314
pressure field associated to, 305
regularity of, 306
q-generalized solutions for, 304
existence and uniqueness of, 341
Green’s tensor, 349
estimates for, 350–351
representation of linear functionals,
345
Stokes flow in R
n
, 238ff
existence and uniqueness
Index 1017
in D
m,q
, 243
of q-generalized solutions, 244
Stokes flow in R
n
+
, 247ff
existence and uniqueness
in D
m,q
, 256
of q-generalized solutions, 257
Green’s tensor, 261–262
estimates for, 263
Stokes flow in a semi-infinite straight
channel
asymptotic behavior of, 379ff
estimates in W
m,q
, 374–375
Stokes flow in an unbounded distorted
channel, see Leray’s problem
Stokes flow, transition to
from Navier-Stokes flow
in bounded domains, 640ff
in three-dimensional exterior
domains, 731ff
in two-dimensional exterior
domains, 887ff
from Oseen flow, 487ff
Stokes fundamental solution, 239–240
Stokes paradox, 302, 309, 318, 319,
351ff, 839, 854, 894
for generalized solutions, 309
Stokes potential
volume, 240
double-layer i n a half-space, 247
Stokes solution
past a sphere, 300
past a cylinder, 302
Stokes-Fujita truncated fundamental
solution, 310
Stream function, 301
Stretching tensor, 2
Stress tensor, 2
Support of a function, 28
Symmetric flow, 826, 855
Tensor
stretching, 2
Cauchy stress, 2
Time-dependent Oseen flow
Cauchy problem for
existence of solutions to, 527, 538,
540–541
uniqueness of solutions to, 527, 536,
539, 540–541
representation of solutions to, 532
Time-dependent Oseen fundamental
solution, 515–516
estimates of
integral, 517ff
pointwise, 517
Total head pressure, 819
Trace inequalities, 62–64, 68, 122
Trace of a function, 63–64
on a bounded boundary, 61ff
on a bounded portion of the
boundary, 68
defined in a half- space, 121ff
Trace operator, 64, 67
Trace space of f unctions
from W
m,q
, 64, 67
from D
1,q
(R
n
+
), 122, 125
Transition to the Stokes flow
from Oseen flow, 487ff
from Navier–Stokes flow,
in bounded domains, 640ff
in three-dimensional exterior
domains, 731ff
in two-dimensional exterior
domains, 887ff
Tril inear form, 588
continuity of, 588, 592, 661, 908
Troisi inequality, 55
Truncated fundamental solution
Oseen-Fujita, 470
Stokes-Fujita, 310
Unbounded Dirichlet integral, 20, 367
Unbounded regions with unbounded
boundary
flow in, 17
Stokes flow in, 365ff
see also Stokes flow in a half
space, in semi-infinite channels,
in channels with unbounded cross
section, in aperture domains
Navier–Stokes flow in, 899ff
see also Heywood’s problem, Leray’s
problem
Uniqueness
see generalized Oseen flow, Navier–
Stokes flow, Oseen flow, Stokes
flow
Unsteady flow, 4
1018 Index
Unsteady O seen flow, 514ff
Variational formulation, 233, 304, 420,
586, 653, 751, 903
Vector potential, 141
Very weak solution, 297, 644–645
Viscosity
infinite limit, see limit of vanishing
Reynolds number
kinematic, 4
Vorticity, 474, 806
asymptotic behavior of, 475, 720,
828–829, 887
see also Navier–Stokes flow, Oseen
flow
Wake region
generalized Oseen flow, 554, 555
Navier–Stokes flow
irrotational, three-dimensional 714
rotational, 749, 777
two-dimensional, 881–882
Oseen flow
three-dimensional, 436
two-dimensional, 440
Weak compactness, 32
Weak convergence, 32
Weak derivative, 48
Weak solution, see generalized solution
Weakly complete, 32
Weakly divergence free, 155
Weakly singular kernels, 126
Weierstrass kernel, 515
Weighted inequalities, 85–86, 98, 135
Whitehead paradox, 417, 432
Wirtinger inequality, 76
Young inequality
for convolutions, 125
for numbers, 42