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Index
Adherence condition, 4–5Anisotropic Sobolev spaces, 55Annihilator, 143Aperture domain, 20, 199, 368, 901,
928Navier–Stokes flow in, 928ff
existence and uniqueness for, 937asymptotic structure of, 965–966,
970see also Navier–Stokes flow in
aperture domainsStokes flow in, 393, 407
existence and uniqueness for, 407asymptotic behavior of, 414–415
see also Stokes flow in aperturedomains
Approximating solutionsin bounded domains, 599, 617in domains with noncompact
boundaries, 915, 945in exterior domains, 682, 767
Approximation of functionsin D1,q
0 ∩ Lr, 218, 225in H1
q ∩ Lr , 218, 222in Lq ∩D−1,q
0 , 456in Wm,q , 50, 51
Aronszajn-Gagliardo theorem, 742Asymptotic behavior, see Behavior,
Decay, Oseen flow, Stokes flow,Navier–Stokes flow
Behavior at large distances of functionsfrom D1,q , 86, 88–89
pointwise, 117
Bessel functions, 431, 887Body force, 2Bogovskiı formula, 163Boundary
bounded, 4, 37unbounded, 4Stokes flow, estimates near the, 271ff
Boundary inequalities, 61ffBoundary portion
of class Ck, 37of class Ck,λ, 37
Bounded domainNavier–Stokes flow in a, 583ffsee also Navier–Stokes flow in a
bounded domainOseen flow in a, 469Stokes flow in a, 231ffsee also Stokes flow in a bounded
domainBounded regions
flow in, 4Brouwer theorem, 597
Calderon-Zygmund theorem, 130Canonical basis in R
n, 26Carnot theorem, 127Cauchy inequality, 42Cauchy sequence, 30
weak, 32Cauchy stress tensor, 2Coincidence of H1
q and bH1q
in bounded domains, 196in domains with a noncompact
boundary, 198
1009
1010 Index
in exterior domains, 197Coincidence of D1,q
0 and bD1,q0 , 214ff
Compactness criterion, 73Compatibility condition for the exis-
tence of Navier-Stokes flow in abounded domain, 584
Compatibility condition for the exis-tence of q-generalized solutionsfor Navier-Stokes flow in exteriordomains, 743
Compatibility condition for the exis-tence of Stokes flow in exteriordomains, 338
Cone property, 170Constants, 27Continuity properties of trilinear form,
588, 592, 661, 908Convective term, 1Convergent sequence, 29
weakly, 32Convexity inequality, 42Convolution, 125Counterexample to solvability of
∇ · v = f , 173, 606Cross section, 10, 20, 192, 193, 370, 902
unbounded, 387Curl operator, 28Cut-off function
anisotropic, 93Sobolev, 102
D-solutions, 650two-dimensional, 803
asymptotic behavior of, 806ffpressure field associated to, 818ff,886derivatives of, 812, 828
total head pressure field associatedto, 831ffvelocity field, 824, 876derivatives of, 812, 827–828, 885
vorticity field, 826ff, 887D-solutions, three-dimensional,
see generalized solutionsDecay, pointwise
for functions from D1,q , 117for Navier–Stokes flow
in aperture domains,three-dimensional case, 960ff
two-dimensional case, 321in semi-infinite straight channels,
918ffin three-dimensional exterior
domains; irrotational casewith v∞ 6= 0, 688ff, 709ffwith v∞ = 0, 724ff
in three-dimensional exteriordomains; rotational casewith v0 · ω 6= 0, 777ffwith v0 · ω = 0, 795ff
in two-dimensional exteriordomains,with v∞ 6= 0, 876ffwith v∞ = 0, 857
see also Navier-Stokes flow in exteriordomains, and in domains with anunbounded boundary
for Oseen flow, 471, 472–473for Stokes flow in exterior domains,
313, 314for Stokes flow in a semi-infinite
straight channel, 379fffor Stokes flow in channels with
unbounded cross sections, 393fffor Stokes flow in aperture domains,
414–415Decomposition of Lq , 112ff, 141ffDerivative
generalized (or weak), 48normal, 67
Diameter of a set, 27Difference quotient, 59Differential inequality, 381Dirichlet integral
bounded, 7, 8, 13, 15unbounded, 20, 367–368
Dirichlet problem for the Poissonequation in a half-space
existence, 133uniqueness, 134
Dirichlet problem for the Poissonequation in exterior domains
existence and uniqueness, 349Distance, 26
regularized, 219Distorted channel, 902Divergence operator, 2, 28
generalized, 155
Index 1011
Domain, 27
of class Ck, 37
Ck-smooth, 37of class Ck,λ, 37
Ck,λ-smooth, 37
locally Lipschitz, 37
star-shaped, 38star-like, 38
with cylindrical ends, 370
Double-layer potentials for Stokes flowin a half-space, 247
Duality pairing
〈 , 〉, 60
[ , ], 112Dyadic product, 26
Ehrling inequality, 77
Embedding theorems, 57, 59Energy equation, 11, 602, 651, 660,
727, 757, 932
generalized, 662, 667Energy inequality, 602, 726, 761, 766
generalized, 639, 663, 669, 682
Erenhaft–Millikan experiment, 300
Estimatesfor generalized Oseen flow, 560ff
for Oseen flow, exterior domains,481ff
for Stokes flow, interior, 263fffor Stokes flow, near the boundary,
271ff
for Stokes flow, exterior domains,320ff, 337ff
for Stokes flow, in Holder spaces,287ff
for Stokes flow in a semi-infinitestraight channel, in Wm,q , 375ff
Exceptional solutions, 337
Existence see generalized Oseen flow,Navier-Stokes flow, Oseen flow,Stokes flow, generalized solutions
Extension, 57–58Leray–Hopf, 604
solenoidal 176, 181, 603, 616, 678
Extension Condition (EC), 603
counterexample to, 605–606Exterior domain, 37
Generalized Oseen flow in an, 485ff
Navier–Stokes flow in a three-dimensional,
irrotational case, 649ffrotational case, 747ff
Navier–Stokes flow in a two-dimensional, 799ff
see also Navier–Stokes flow in exteriordomains
Oseen flow in an, 417ffsee also Oseen flow in exterior
domainsStokes flow in an, 299ffsee also Stokes flow in exterior
domainsExterior regions, 4
flow in, 8
Fluid, 1plane flow of a, 14
Fluxthrough the boundary
of a bounded domain, 5, 604ffof an exterior domain, 678
three-dimensional, 13, 680, 686two-dimensional, 680–681, 687
through the aperture, 929through the cross section, 18, 900
Flux carrier, 900, 914, 938Force exerted by the liquid on the
boundary, 703, 741, 743, 854, 893Friedrichs inequality, 73Function spaces
of hydrodynamics, 155ff, 193ff, 214ffsee also SpacesFundamental solution
for the biharmonic equation, 239for the Laplace equation, 115for the Oseen equation, 429ff
three-dimensional, 434–435estimates, 436ff
two-dimensional, 439–440estimates, 440ff
n-dimensional, estimates, 443for the Stokes equation, 238ff
estimates at large distances, 240for the time-dependent Oseen
equation, 516asymptotic estimates, 517ff
truncated Oseen-Fujita, 470
1012 Index
truncated Stokes-Fujita, 310
Gagliardo theorem, 64Galerkin method, 424, 463, 497, 597,
682, 766, 914, 945Gauss divergence theorem, 61, 68
generalized, 159Generalized derivative, 48Generalized Oseen flow
generalized solutions for,existence of, 501uniqueness of, 505ff
q-generalized solutions for, 499asymptotic behavior of, 547, 555,
558–559existence, uniqueness and estimates
in R3 of, 547
existence, uniqueness and estimatesin exterior domains of, 555
pressure associated to, 500regularity of, 500–501representation of, 498
Generalized solutionssee Generalized Oseen flow, Navier–
Stokes flow, Oseen flow, Stokesflow
Gradient operator, 2, 28generalized, 175
Green’s identityfor the Laplace operator, 115for the Oseen system, 467for the Stokes system, 290for the time-dependent Oseen system,
531–532Green’s tensor for the Stokes problem
in bounded domains, 288ffestimates for, 289
in exterior domains, 349ffestimates for, 350–351
in half-space, 261ffestimates for, 263
Hagen–Poiseuille flow, 18, 366Hamel solution, 805Hardy inequality, 66Helmholtz–Weyl decomposition of Lq,
141ffHeywood’s problem, 20, 928
see also aperture domain
Holder continuous, 36Holder inequality, 41
generalized, 42Homogeneous Sobolev spaces, 80
Incompressibility condition, 2Inequality
Cauchy, 42convexity, 42differential 381Ehrling, 77Friedrichs, 73Hardy, 66Holder, 41
generalized, 42integro-differential 381–382Ladyzhenskaya, 55Minkowski, 42
generalized, 42Nirenberg, 51
generalized, 54Poincare, 69, 71, 72, 75
generalized, 175Poincare-Sobolev, 75Schwarz, 42Sobolev, 54trace 62–64, 68, 122Troisi, 55for vector functions with normal
component vanishing at theboundary, 71, 77
weighted, 85–86, 98, 135Wirtinger, 76Young, for convolutions, 125Young, for numbers, 42
Integral transform, 125ffIntegro-differential inequality, 381–382
Jeffery–Hamel solution, 971
Kernel, 125weakly singular, 126singular, 129
Kinematic viscosity coefficient, 4Kinetic energy,
finite, 703infinite, 698, 702ff, 750, 774
Ladyzhenskaya inequality, 55
Index 1013
Ladyzhenskaya’s variant of Leray’smethod, 643
Laminar motion, 584
Landau solution, 12, 728–729, 793, 797
Laplace operator, 2Laplace equation, fundamental solution
for, 115
Lebesgue spaces, 40ffLeray’s contradiction argument, 645ff
Leray–Hopf extension, 604
Leray’s problem, 18, 370ff, 903ffNavier–Stokes flow for,
generalized solutions for, 903
asymptotic behavior of, 927
existence of, 914regularity of, 904
uniqueness of, 911, 918
pressure associated to, 904Stokes flow for,
generalized solutions to, 371
asymptotic decay of, 379ffexistence and uniqueness of, 378
pressure associated to, 371
regularity of, 372Leray–Schauder theorem, 584, 644
Limit of vanishing Reynolds number,
for Oseen flow, 487fffor Navier–Stokes flow
in bounded domains, 640ff
in three-dimensional exteriordomains, 731ff
in two-dimensional exteriordomains, 887ff
Liouville theorem for generalizedsolutions
in R3, 12, 705, 729
in Rn, n > 3, 731
in R2, 808
Liquid, 1Lizorkin theorem, 446
Locally Lipschitz domain, 37
Minkowski inequality, 42
generalized, 54Multi-index, 28–29
Mollifier, 43–44
Mollifying kernel, 43Mozzi–Chasles transformation, 13, 496
Navier–Stokes equations, 2steady-state, 4
Navier–Stokes flow in aperture domains,see also Heywood’s problem, 928ff
generalized solutions for, 929asymptotic structure of, 965–966derivative of, 966pressure associated to, 970
existence of, 945global summability properties of,
951pressure field associated to, 929–930regularity of, 931uniqueness of, 951
Navier–Stokes flow in bounded domainsgeneralized solutions for, 587
existence of,with homogeneous boundary
data, 596ffwith nonhomogeneous boundary
data, 602ffnon-uniqueness of, 596pressure field associated to, 590regularity of, 621ff, 636–639uniqueness of, 592
with homogeneous boundarydata, 602
with nonhomogeneous boundarydata, 619
Navier–Stokes flow in distortedchannels, see Leray’s problem
Navier–Stokes flow in three-dimensionalexterior domains; irrotational case
generalized solutions for, 654asymptotic structure of,
with v∞ 6= 0, 709ffderivative of, 717pressure associated to, 719vorticity of, 720
asymptotic structure of,with v∞ = 0, 726derivative of, 724, 726pressure associated to, 724, 726
existence of, 681global summability properties of,
701pressure field associated to, 655regularity of, 658–659uniqueness of, 668ff
1014 Index
with v∞ 6= 0, 709with v∞ = 0, 727
Navier–Stokes flow in three-dimensionalexterior domains; rotational case
generalized solutions for, 763–764asymptotic structure of,
with v0 · ω 6= 0, 777ffderivative of, 787pressure associated to, 790
asymptotic structure of,with v0 · ω = 0, 795ffderivative of, 796, 797pressure associated to, 796, 797
existence of, 765global summability properties of,
772pressure field associated to, 752, 756regularity of, 757uniqueness of, 760ff
with v0 · ω 6= 0, 777with v0 · ω = 0, 796
Navier–Stokes flow in two-dimensionalexterior domains, 799ff
generalized solutions for, 654asymptotic structure of,
with v∞ 6= 0, 876derivative of, 885pressure associated to, 886vorticity of, 828–829, 887
asymptotic structure of,with v∞ = 0, 857
existence of, 687, 838ffglobal summability properties of,
857ffnon-existence for large data of, 17,
856non-uniqueness of, 805–806pressure field associated to, 655regularity of, 658–659uniqueness of, 803ff
Non-existence for large data forNavier–Stokes flow, 17, 856
Non-uniqueness for Navier–Stokes flowin bounded domains, 596in exterior domains, 805–806
Neumann problem, generalized, 146ffNirenberg inequality, 51
generalized 54Norm, 29
‖ ‖q , 40‖ ‖m,q , 50| |m,q , 83‖ ‖(q,r),A,t , 526
Notation 6, 26ff
Olmstead–Gautesen drag paradox, 474Orthogonal complement, 143Oseen flow
generalized, see generalized Oseenflow
q-generalized solutions for, 420asymptotic behavior of, 471, 472asymptotic behavior of the vorticity
of, 475existence, uniqueness and estimates
in Rn of, 452, 459
existence, uniqueness and estimatesin exterior domains of, 481, 484
local representation of, 471pressure associated to, 421regularity of, 422representation of, 470
generalized solutions for,three-dimensional; existence of, 425two-dimensional; existence of, 465uniqueness of, 422ff
limit of vanishing Reynolds number,487ff
time-dependent, see time-dependentOseen flow
Oseen fundamental solution, 429ffthree-dimensional, 434–435
estimates of, 436fftwo-dimensional, 439–440
estimates of, 440ffn-dimensional; estimates of, 443paraboloidal wake region exhibited
by the,three-dimensional, 436two-dimensional, 440
Oseen volume potentials, 444time-dependent, 532
Oseen-Fujita truncated fundamentalsolution, 470
Paraboloidal wake regionthree-dimensional, 436, 714two-dimensional, 440, 881–882
Index 1015
Paradoxof Olmstead–Gautesen, 474of Stokes, 302, 309, 318, 319, 351ff,
887ffof Whitehead 417, 732within the Oseen approximation,
419, 474Partition of unity, 40Perturbation series around Stokes flow,
723–724Physically Reasonable (PR) solutions,
651,Plane flow in exterior domains, 14,
799ffPoincare constant, 70Poincare inequality, 69, 70, 71, 75
generalized, 175, 191Poincare–Sobolev inequality, 75Poiseuille solution, 18, 366, 900Poiseuille constant, 369Poisson integral, 133PR solutions, see Physically Reasonable
solutionsPressure field associated to a q-gene-
ralized solution, 235, 305, 371,390, 421, 500, 590, 655, 752, 904,929–930
Problem ∇ · v = fin bounded domains, 161ffin domains with noncompact
boundary, 191ffin exterior domains, 188ffin a half-space, 261
Projection operator Pq, 142
q-generalized solutionsfor generalized Oseen flow, 499for Navier–Stokes flow 587, 644for Oseen flow 420for Stokes flow in bounded domains,
234for Stokes flow, interior estimates,
266, 270for Stokes flow, estimates near the
boundary, 278for Stokes flow in R
n; existence anduniqueness of, 244
for Stokes flow in a half-space;existence and uniqueness of, 257
for Stokes flow in exterior domains,341
asymptotic behavior of, 313, 314regularity of, 265, 266, 276, 306, 372
q-weak solutions, see q-generalizedsolutions
Regularity of generalized solutionssee generalized Oseen flow,Navier–Stokes flow, Oseen flow,Stokes flow
Regularized distance, 219Regularizer, 44Representation formulas
for Navier–Stokes flow in exteriordomains
three-dimensional case 692–693two-dimensional case 867–868
for Navier–Stokes flow in aperturedomains 965–966
for Oseen flow, 472–473local, 471
for Stokes flow in aperture domains,411, 414
for Stokes flow in bounded domains,292, 294
for Stokes flow in exterior domains,315
local, 312for time-dependent Oseen flow, 532
Reynolds number, 231, 420, 496effective, 497
see also limit of vanishing Reynoldsnumber
Riesz potential, 126
Scalar potential, 141Schauder estimates, 287Schmidt orthogonalization procedure,
425Schwarz inequality, 42Segment property, 51Self-propelled body 352, 364, 653, 703Semi-infinite straight channel
Stokes flow in; estimates in Wm,q ,375
Stokes flow in; asymptotic decay, 379Sequence
Cauchy, 30
1016 Index
weak, 32
convergent, 29weakly convergent 32
Singular kernel, 129
Sobolev “cut-off” function, 102Sobolev theorem, 128
Sobolev space, 50Space
Banach, 30Ck(Ω), 35
C∞(Ω), 35Ck
0 (Ω), 35
C∞0 (Ω), 35
Ck,λ(Ω), 36Dm,q(Ω), 80
Dm,q(Ω), 83Dm,q
0 (Ω), 84
D−m,q0 (Ω) 109
D(Ω), 142
D1,q0 (Ω), 214bD1,q
0 (Ω), 214Gq(Ω), 142Hq(Ω), 142
H1q (Ω), 193bH1
q (Ω), 193Lq(Ω) 40
Lqloc(Ω),Lq
loc(Ω), 43Lq(At), 526
Lr,q(At), 526Wm,q(Ω), 49Wm,q
0 (Ω), 50
W−m,q′ (Ω), W−m,q′0 (Ω), 60
Wm−1/q,q (∂Ω), 64, 67Wm,q
loc (Ω),Wm,qloc (Ω), 81
Sobolev, 50anisotropic, 55
homogeneous, 80negative, 60
trace, 64, 67–68Star-shaped or star-like 38
Steady-state Navier-Stokesequations, 4
Steady fall of a body, 653Stein theorem
on extension maps, 58on singular transforms in weighted
spaces, 131on regularized distance, 219
Stokes flow in an aperture domain, seealso Heywood’s problem, 407
generalized solutions for, 388asymptotic behavior of, 414–415existence and uniqueness of, 392,
407pressure associated to, 390representation of, 411, 414
existence and uniqueness in D1,q,407
Stokes flow in bounded domains, 231ffgeneralized solutions for, 234
existence and uniqueness of, 237regularity of, 267, 277
q-generalized solutions for, 234estimates of
in Holder spaces,in Wm,q , 227 234interior, 263ffnear the boundary, 271ff
existence and uniqueness ofin Holder spaces, 287ffin Wm,q , 279ff
pressure field associated to, 186uniqueness of, 228
maximum modulus theorem for, 298Green’s tensor, 288
estimates for 289Stokes flow in channels with unbounded
cross sections 387ffgeneralized solutions for, 388
asymptotic behavior of, 393ffexistence and uniqueness of, 390pressure associated to, 390
Stokes flow in exterior domains, 299ffexistence and uniqueness in Dm,q ,
334–335generalized solutions, 304
asymptotic behavior of, 313, 314pressure field associated to, 305regularity of, 306
q-generalized solutions for, 304existence and uniqueness of, 341
Green’s tensor, 349estimates for, 350–351
representation of linear functionals,345
Stokes flow in Rn, 238ff
existence and uniqueness
Index 1017
in Dm,q, 243of q-generalized solutions, 244
Stokes flow in Rn+, 247ff
existence and uniquenessin Dm,q, 256of q-generalized solutions, 257
Green’s tensor, 261–262estimates for, 263
Stokes flow in a semi-infinite straightchannel
asymptotic behavior of, 379ffestimates in Wm,q , 374–375
Stokes flow in an unbounded distortedchannel, see Leray’s problem
Stokes flow, transition tofrom Navier-Stokes flow
in bounded domains, 640ffin three-dimensional exterior
domains, 731ffin two-dimensional exterior
domains, 887fffrom Oseen flow, 487ff
Stokes fundamental solution, 239–240Stokes paradox, 302, 309, 318, 319,
351ff, 839, 854, 894for generalized solutions, 309
Stokes potentialvolume, 240double-layer in a half-space, 247
Stokes solutionpast a sphere, 300past a cylinder, 302
Stokes-Fujita truncated fundamentalsolution, 310
Stream function, 301Stretching tensor, 2Stress tensor, 2Support of a function, 28Symmetric flow, 826, 855
Tensorstretching, 2Cauchy stress, 2
Time-dependent Oseen flowCauchy problem for
existence of solutions to, 527, 538,540–541
uniqueness of solutions to, 527, 536,539, 540–541
representation of solutions to, 532Time-dependent Oseen fundamental
solution, 515–516estimates of
integral, 517ffpointwise, 517
Total head pressure, 819Trace inequalities, 62–64, 68, 122Trace of a function, 63–64
on a bounded boundary, 61ffon a bounded portion of the
boundary, 68defined in a half-space, 121ff
Trace operator, 64, 67Trace space of functions
from Wm,q , 64, 67from D1,q(Rn
+), 122, 125Transition to the Stokes flow
from Oseen flow, 487fffrom Navier–Stokes flow,
in bounded domains, 640ffin three-dimensional exterior
domains, 731ffin two-dimensional exterior
domains, 887ffTrilinear form, 588
continuity of, 588, 592, 661, 908Troisi inequality, 55Truncated fundamental solution
Oseen-Fujita, 470Stokes-Fujita, 310
Unbounded Dirichlet integral, 20, 367Unbounded regions with unbounded
boundaryflow in, 17Stokes flow in, 365ffsee also Stokes flow in a half
space, in semi-infinite channels,in channels with unbounded crosssection, in aperture domains
Navier–Stokes flow in, 899ffsee also Heywood’s problem, Leray’s
problemUniqueness
see generalized Oseen flow, Navier–Stokes flow, Oseen flow, Stokesflow
Unsteady flow, 4
1018 Index
Unsteady Oseen flow, 514ff
Variational formulation, 233, 304, 420,586, 653, 751, 903
Vector potential, 141Very weak solution, 297, 644–645Viscosity
infinite limit, see limit of vanishingReynolds number
kinematic, 4Vorticity, 474, 806
asymptotic behavior of, 475, 720,828–829, 887
see also Navier–Stokes flow, Oseenflow
Wake regiongeneralized Oseen flow, 554, 555Navier–Stokes flow
irrotational, three-dimensional 714
rotational, 749, 777two-dimensional, 881–882
Oseen flowthree-dimensional, 436two-dimensional, 440
Weak compactness, 32Weak convergence, 32Weak derivative, 48Weak solution, see generalized solutionWeakly complete, 32Weakly divergence free, 155Weakly singular kernels, 126Weierstrass kernel, 515Weighted inequalities, 85–86, 98, 135Whitehead paradox, 417, 432Wirtinger inequality, 76
Young inequalityfor convolutions, 125for numbers, 42