On stationary Navier-Stokes flows around a rotating obstacle in two-dimensions

0.37 MB·

by  Mitsuo Higaki

On stationary Navier-Stokes flows around a rotating
obstacle in two-dimensions
MitsuoHigaki∗ YasunoriMaekawa† YuuNakahara‡
Abstract Westudy the two-dimensional stationary Navier-Stokes equations describing the
7
1 flows around a rotating obstacle. The unique existence of solutions and their asymptotic
0
behavior at spatial infinity are established when the rotation speed of the obstacle and the
2
givenexteriorforcearesufficiently small.
n
a
J Keywords Navier-Stokes equations · two-dimensional exterior flows · scale-criticality ·

5 flowsaround arotatingobstacle

] MathematicsSubjectClassification (2000) 35B35·35Q30·76D05·76D17
P
A
.
h 1 Introduction
t
a
m
Inthispaper weconsider thetwo-dimensional Navier-Stokes equations forviscous incom-
[ pressible flowsaround arotatingobstacle intwo-dimensions:

1
∂ v−∆v+v·∇v+∇q = g, divv = 0, t > 0, y ∈ Ω(t),
v t
(1)
5 ( v = αy⊥, t > 0, y ∈ ∂Ω(t).
1
2
1 Herev = v(y,t) = (v1(y,t),v2(y,t))⊤ andq = q(y,t)arerespectively unknown velocity
0 field and pressure field, and g(t,y) = (g (t,y),g (t,y))⊤ is a given external force. The
1 2
.
1 timedependent domainΩ(t)isdefinedas
0
7 Ω(t) = y ∈ R2 |y = O(αt)x,x ∈Ω ,
1
: (cid:8)cosαt −sinαt (cid:9) (2)
v O(αt) = ,
sinαt cosαt
i
X (cid:18) (cid:19)
r whereΩisanexteriordomaininR2withasmoothcompactboundary,whiletherealnumber
a αrepresentstherotationspeedoftheobstacleΩc = R2\Ω. Weusethestandardnotationfor
derivatives: ∂ = ∂ ,∂ = ∂ ,∆ = 2 ∂2,divv = 2 ∂ v ,v·∇v = 2 v ∂ v.
t ∂t j ∂xj j=1 j j=1 j j j=1 j j
The vector x⊥ denotes the perpendicular: x⊥ = (−x ,x )⊤. The system (1) describes
P 2P1 P
the flow around the obstacle Ωc which rotates with a constant angular velocity α, and the
condition v(t,y) = αy⊥ ontheboundary ∂Ω(t)represents theno-slip boundary condition.
Toremovethedifficultyduetothetimedependenceofthefluiddomainitismoreconvenient
toanalyze thesystem(1)inthereference frame:
y = O(αt)x, u(x,t) = O(αt)⊤v(y,t), p(x,t) = q(y,t), f(x,t)= O(αt)⊤g(y,t).
∗DepartmentofMathematics,KyotoUniversity,Japan.E-mail:[email protected]
†DepartmentofMathematics,KyotoUniversity,Japan.E-mail:[email protected]
‡MathematicalInstitute,TohokuUniversity,Japan.E-mail:[email protected]
1
HereM⊤ denotesthetransposeofamatrixM. Then(1)isequivalentwiththeequationsin
thetime-independent domainΩ:
∂ u−∆u−α(x⊥ ·∇u−u⊥)+∇p = −u·∇u+f, divu = 0, t > 0, x ∈ Ω,
t
( u = αx⊥, t > 0, x ∈ ∂Ω.
In this paper we are interested in the stationary solutions to this system. Thus we assume
thatf isindependent oftandconsidertheellipticsystem
−∆u−α(x⊥ ·∇u−u⊥)+∇p = −u·∇u+f, divu = 0, x ∈ Ω,
(NS )
α
( u = αx⊥, x ∈ ∂Ω.
Tostateourresultletusintroducethefunctionspacesusedinthispaper. Asusual,theclass
C∞(Ω)isdefined as theset ofsmooth divergence free vector fields withcompact support
0,σ
in Ω, and the homogeneous space W˙ 1,2(Ω) is the closure of C∞(Ω) with respect to the
0,σ 0,σ
norm k∇fk . For a fixed number s ≥ 0 we also introduce the weighted L∞ space
L2(Ω)
L∞(Ω)anditssubspace L∞(Ω)asfollows.
s s,0
L∞(Ω) = f ∈ L∞(Ω)|(1+|x|)sf ∈ L∞(Ω) ,
s
(3)
L∞(Ω) = f ∈ L∞(Ω)| lim ess.sup |x|s|f(x)| = 0 .
s,0 (cid:8) s |x|≥R (cid:9)
R→∞
(cid:8) (cid:9)
Thesespacesareequippedwiththenaturalnormkfk = ess.sup (1+|x|)s|f(x)|.
L∞(Ω) x∈Ω
s
WedenotebyL2 (Ω)thesetoffunctions whichbelongtoL2(Ω∩K)foranycompactset
loc
K ⊂ R2, and Wk,2(Ω), k = 1,2,···, is defined in the similar manner. Finally, for r > 0
loc
thetruncated domainΩ isdefinedasΩ = {x ∈ Ω||x|< r}.
r r
Themainresultofthispaperisstatedasfollows.
Theorem 1.1 Thereexists ǫ = ǫ(Ω) > 0suchthat thefollowing statement holds. Assume
that f ∈ L2(Ω)2 is of the form f = divF with some F ∈ L∞(Ω)2×2 and F − F ∈
2 21 12
L1(Ω). Ifα 6= 0and
|α|12 log|α| + |α|−12 log|α| kfkL2(Ω)+kFkL∞(Ω)+kF21 −F12kL1(Ω) < ǫ, (4)
2
(cid:12) (cid:12) (cid:12) (cid:12)(cid:0) (cid:1)
then th(cid:12)ere exis(cid:12)ts a solutio(cid:12)n (u,∇(cid:12)p) ∈ W2,2(Ω)∩L∞(Ω) 2 ×L2 (Ω)2 to (NS ), which
loc 1 loc α
is unique in a suitable class of functions (see Theorem 4.1 for the precise description). If
(cid:0) (cid:1)
F ∈ L∞(Ω)2×2 inaddition, thenthesolution ubehavesas
2,0
x⊥
u(x) = β + o(|x|−1), |x|→ ∞, (5)
4π|x|2
where
β = y⊥· T(u,p)ν dσ + lim e−δ|y|2y⊥·f dy. (6)
y
Z∂Ω δ→0ZΩ
(cid:0) (cid:1)
Here T(u,p) = ∇u+(∇u)⊤ −pI, I = (δ ) , denotes the Cauchy stress tensor,
ij 1≤i,j≤2
andν istheoutwardunitnormalvectorto∂Ω.
2
Remark1.2 (i) The smallness condition on f and F in (4) can be slightly improved with
respecttothedependence onα;seeTheorem4.1fordetails.
(ii)BothconditionsF ∈L∞(Ω)2×2 andF −F ∈ L1(Ω)arecriticalinviewofscaling.
2 21 12
NotethattheL1 summability isneeded onlyfortheantisymmetric partofF. Thesecondi-
tionsarenotenoughtoensurethatubehaveslikethecircularflowβ x⊥ as|x| → ∞,and
4π|x|2
the additional decay condition F ∈ L∞(Ω)2×2 as in Theorem 1.1 is required to achieve
2,0
thisproperty.
(iii) The second term of the right-hand side of (6) is well-defined if F ∈ L∞(Ω) and
2,0
F −F ∈ L1(Ω). If F possesses an additional decay such as L∞ (Ω) with γ ∈ (0,1)
21 12 2+γ
thentheordero(|x|−1)in(5)isreplaced byO(|x|−1−γ)atleastwhen|α|andgivendataf
arefurther smalldepending onγ. Theprecise statement onthisresult isstatedinTheorem
4.1.
Asfarastheauthorsknow,Theorem1.1isthefirstgeneralexistenceresultoftheflows
aroundarotatingobstacle inthetwo-dimensional case. Beforestating theideaoftheproof
of Theorem 1.1, let us recall some known results on the mathematical analysis of flows
aroundarotating obstacle.
So far the mathematical results on this topic have been obtained mainly for the three-
dimensionalproblem,aslistedbelow. Forthenonstationaryproblemtheexistenceofglobal
weak solutions is proved by Borchers [1], and the unique existence of time-local regular
solutions is shown by Hishida [17] and Geissert, Heck, and Hieber [15], while the global
strong solutions for small data are obtained by Galdi and Silvestre [14]. The spectrum of
the linear operator related to this problem is studied by Farwig and Neustupa [8]; see also
the linear analysis by Hishida [18]. The existence of stationary solutions to the associated
systemisprovedin[1],Silvestre[25],Galdi[11],andFarwigandHishida[5]. Inparticular,
in [11] the stationary flowswith the decay order O(|x|−1)are obtained, while the work of
[5] is based on the weak L3 framework, which is another natural scale-critical space for
the three-dimensional Navier-Stokes equations. Our Theorem 1.1 is considered as a two-
dimensional counterpart ofthethree-dimensional result of[11]. In3Dcase theasymptotic
profiles of these stationary flows at spatial infinity are studied by Farwig and Hishida [6,
7] and Farwig, Galdi, and Kyed [4], where it is proved that the asymptotic profiles are
described by the Landau solutions, stationary self-similar solutions to the Navier-Stokes
equations in R3 \{0}. It is worthwhile to mention that, also in the two-dimensional case,
the asymptotic profile is given by the stationary self-similar solution cx⊥, as is shown in
|x|2
Theorem 1.1. The stability of the above stationary solutions has been well studied in the
three-dimensional case; Theglobal L2 stability isprovedin[14], andthelocal L3 stability
isobtained byHishidaandShibata[20].
Allresults mentioned aboveareinthethree-dimensional case, whileonly afewresults
are known so far for the flow around a rotating obstacle in the two-dimensional case. Re-
centlyanimportantprogresshasbeenmadebyHishida[19],wheretheasymptoticbehavior
of the two-dimensional stationary Stokes flow around a rotating obstacle is investigated in
details. Theequations studiedin[19]arewrittenas
−∆u−α(x⊥ ·∇u−u⊥)+∇p = f, divu = 0, x ∈Ω,
(S )
α
( u = b, x ∈ ∂Ω.
Here b is a given smooth function on ∂Ω. It is proved in [19] that if the smooth external
3
forcef satisfiesthedecayconditions
|x||f|dx < ∞, f(x) = o |x|−3(log|x|)−1 , as|x| → ∞, (7)
ZΩ
(cid:0) (cid:1)
thenthesolution uto(S )decaying atspatialinfinityobeystheasymptotic expansion
α
c x⊥−c x
u(x) = 1 2 + (1+|α|−1)o(|x|−1), as |x| → ∞, (8)
4π|x|2
where
c = y⊥· T(u,p)+αb⊗y⊥ νdσ + y⊥·fdy,
1 y
Z∂Ω ZΩ (9)
(cid:0) (cid:1)
c = b·νdσ .
2 y
Z∂Ω
Theresultof[19]leadstoanimportantconclusion thattherotationoftheobstacle resolves
theStokesparadox(seeChangandFinn[3]fortherigorous description oftheStokespara-
dox) as in the Oseen resolution. We recall that when the obstacle is translating with a
constant velocity u ∈ R2 \{0} the Navier-Stokes flows have been constructed by Finn
∞
andSmith[9,10]forsmallbutnonzerou throughtheanalysisoftheOseenlinearization;
∞
seealsoGaldi[13]. Theresolution oftheStokesparadoxfor(S )isduetothefactthatthe
α
rotation removes thelogarithmic singularity of the associated fundamental solution, which
hasbeenwellknownfortheOseenproblem.
As a reference to the 2D exterior problem related with ours, the reader is referred to a
recentworkbyHillairetandWittwer[16],wherethestationary problem of(1)isdiscussed
whenΩ(t) = Ω= {y ∈ R2 ||y| > 1}andtheboundarycondition isgivenasv = αy⊥+b
withasmoothandtime-independent b. Wenotethatthestationaryflowαy⊥ exactlysolves
|y|2
thisproblemwhenb = 0. In[16]thestationarysolutionsareconstructedaroundthisexplicit
solution forsufficiently smallbwhenthenumberαislarge enough. Although theproblem
discussed in [16] is in fact different from ours due to the time-independent given data b in
theoriginalframe(1),thesolutionsobtainedin[16]shareacommonpropertywiththeones
inTheorem1.1inviewoftheirasymptotic behaviors atspatialinfinity.
ItiswellknownthattheexistenceofstationaryNavier-Stokesflowsintwo-dimensional
exterior domains (hence, the obstacle is rest) is an open problem ingeneral. Partial results
related to this problem have been obtained by Galdi [12], Russo [24], Yamazaki [26], and
Pileckas and Russo[23], wherethesolutions areconstructed under somesymmetry condi-
tionsonbothdomainsandgivendata. Inparticular,theNavier-Stokesflowsdecayinginthe
scale-critical order O(|x|−1) are obtained in [26] in this category. The uniqueness is also
available againundersomesymmetryconditions, seeNakatsuka[22].
Thestabilityofthestationarysolutionsobtainedin[26,16]orinTheorem1.1isahighly
challenging issue due to their spatial decay in the scale-critical order in two-dimensions,
and it is still an open question in general. The difficulty is brought from the fact that the
Hardy inequality k 1 fk ≤ Ck∇fk , f ∈ W˙ 1,2(Ω), does not hold when Ω is an
|x| L2(Ω) L2(Ω) 0
exterior domain in R2. As far as the authors know, the only result available so far is [21]
bythesecondauthorofthispaper,wherethelocalL2 stability isestablished forthespecial
solution αx⊥ ,|α| ≪ 1,whenΩistheexteriordomaintotheunitdisk.
|x|2
Finally,letusstatethekeyideafortheproofofTheorem1.1. Ourapproachismotivated
bythelinearanalysis developedin[19],where(9)isobtainedthroughthedetailedanalysis
4
ofthefundamentalsolutionassociatedtothesystem(S )inR2. Theexpansion(8)strongly
α
indicates that the similar asymptotics is valid also for the Navier-Stokes flow, since the
leading profile in (8) is a stationary self-similar solution to the Navier-Stokes equations
in R2 \ {0}. However, it is far from trivial to justify this idea directly from the results
of [19]. Indeed, the condition (7) is slightly restrictive in handling the nonlinear term u·
∇u, and more seriously, the singularity on |α| in (8) for 0 < |α| ≪ 1 can prevent us
closing thenonlinear estimates. Toovercomethisdifficultywerevisittheargument of[19]
and improve the estimates of the remainder term for the linear problem by analyzing the
fundamental solution to (S ) in R2; see Theorem 3.1, Lemma 3.3, and Theorem 3.8. The
α
nonlinear problem (NS ) is solved by applying the Banach fixed point theorem, however,
α
the argument becomes complicated since we have to control two kinds of norms; the one
bounds thelocalquantity, whiletheotheronecontrols thespatialdecay. Thismachinery is
needed since the flowinafar fieldregion (|x| ≫ 1)exhibits adifferent dependence on|α|
fromtheflowinafinitefluidregion. Inordertoclosethenonlinearestimatesitisimportant
to distinguish these two dependences on |α| and to estimate their interaction through the
nonlinearity carefully. We also note that the structure of the nonlinear term ∇·(u⊗u) is
essentialtosolve(NS )inthescale-critical framework. Indeed,thesymmetryofthetensor
α
u⊗uleadstoacrucialcancellationforthecoefficient“ y⊥·(u·∇u)dy”,whichremoves
Ω
apossible singularity caused bythescale-critical decayoftheflow.
R
Thispaperisorganizedasfollows. InSection2thebasicresultsontheoscillatoryinte-
grals are collected, which are used to establish the pointwise estimates of the fundamental
solution to(S )withamildersingularity on|α|,|α| ≪ 1. InSection3thelinearized prob-
α
lem(S )withb = 0isstudiedindetails. Section3.1isdevotedtotheanalysisinR2,while
α
the exterior problem is discussed in Section 3.2. Finally the nonlinear problem (NS ) is
α
solvedinSection4.
2 Preliminaries
Inthissectionwecollecttheresults oftheoscillatory integrals usedinSection3.1.
Lemma2.1 Letα∈ R\{0}andletm,r > 0. Thenwehave
∞eiαte−rt2 dt + ∞eiαt ∞e−rs2 ds dt ≤ Cmin 1 , 1 ,
tm sm+1 |α|r2m 1 2m2
(cid:12)Z0 (cid:12) (cid:12)Z0 Zt (cid:12) |α|m+1rm+1
(cid:12) (cid:12) (cid:12) (cid:12) (cid:8) ((cid:9)10)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
whereC = C(m)isindependent ofrandα. Moreover,form > 1wehave
∞e−rt2 dtdt = γ(m−1) , ∞ ∞e−rs2 ds dt = γ(m−1) , (11)
tm r2(m−1) sm+1 r2(m−1)
Z0 Z0 Zt
whereγ(·)denotes theEulergammafunction.
Proof: The proof of (11) is a straightforward computation, and we omit the details. To
show(10)letustakeapositiveconstantl = l(r,α)whichwillbedeterminedlaterandsplit
theintegralas
∞eiαte−rt2 dt = leiαte−rt2 dt + ∞eiαte−rt2 dt .
tm tm tm
Z0 Z0 Zl
5
Thefirsttermisestimatedwithoutusingtheeffectofoscillation:
leiαte−rt2 dt = 1 le−rt2 r2 mdt ≤ Cl .
tm r2m t r2m
(cid:12)Z0 (cid:12) (cid:12)Z0 (cid:18) (cid:19) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Forthesecond te(cid:12)rmweusetheeff(cid:12)ectofosc(cid:12)illation toobtain (cid:12)
∞eiαte−rt2 dt = 1 ∞ d eiαt e−rt2 dt
tm iα dt tm
Zl Zl (cid:20) (cid:21)
= 1 eiαte−rt2 t=∞− 1 ∞eiαt r2e−rt2 − me−rt2 dt,
iα tm iα tm+2 tm+1
(cid:20) (cid:21)t=l Zl (cid:18) (cid:19)
whichyields
∞eiαte−rt2 dt ≤ 1 e−rl2 + 1 ∞ r2 +m r2 m+1e−rt2 dt . (12)
tm |α| lm r2(m+1) t t
(cid:12)Zl (cid:12) (cid:18) Zl (cid:18) (cid:19) (cid:19)
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
By(cid:12)takingthelimitof(cid:12)l = 0weobservethattheleft-handsideof(12)isthenboundedfrom
above by C in virtue of (11). On the other hand, the right-hand side of (12) is also
|α|r2m
bounded fromaboveby C . Takingl = rm2m+1|α|−m1+1,wehavearrivedat
|α|lm
∞eiαte−rt2 dt ≤ C .
tm 1 2m2
(cid:12)Z0 (cid:12) |α|m+1rm+1
(cid:12) (cid:12)
(cid:12) (cid:12)
Theestimateoftheintegral(cid:12) (cid:12)
∞eiαt ∞e−rs2 ds dt
sm+1
Z0 Zt
isobtainedexactlyinthesamemanner,andhencethedetailsareomittedhere. Theproofis
complete. ✷
Lemma2.2 Letm > 1. Thenwehave
∞ e−|O(αt4)tx−y|2 −e−|x4|t2 dt + ∞ ∞ e−|O(αt4)sx−y|2 −e−|x4s|2 ds dt
tm sm+1
Z0 Z0 Zt (13)
(cid:12) |y| (cid:12) (cid:12) (cid:12)
≤ C(cid:12) , |x| > 2|(cid:12)y|, (cid:12) (cid:12)
|x|2m−1
and
∞eiαte−|x4|t2 dt ≤ Cmin 1 , 1 , |x|> 0. (14)
tm |α||x|2m |x|2(m−1)
(cid:12)Z0 (cid:12)
(cid:12) (cid:12) (cid:8) (cid:9)
Moreover, fo(cid:12)rm > 1wehave (cid:12)
(cid:12) (cid:12)
∞eiαt ∞e−|x4|s2 ds dt ≤ Cmin 1 , 1 , |x| > 0. (15)
sm+1 |α||x|2m |x|2(m−1)
(cid:12)Z0 Zt (cid:12)
(cid:12) (cid:12) (cid:8) (cid:9)
(cid:12) (cid:12)
Here(cid:12)C = C(m)isindependent of(cid:12)x,y,andα.
6
Proof: ByusingtheTaylorformulawithrespecttoy aroundy = 0,wesee
e−|O(αt4)tx−y|2 = e−|x4|t2 + hO(αt)x,yie−|x4|t2 + hy,Qyie−|O(αt)4xt−θy|2 , (16)
2t 8t2
where Q = O(αt)x −θy ⊗ O(αt)x −θy −2tI with θ = θ(α,t,x,y) ∈ (0,1) and
hx,yi = x·y. From
(cid:0) (cid:1) (cid:0) (cid:1)
|x|
|O(αt)x−θy|≥ |x|−|y| > for |x| > 2|y|,
2
Lemma2.1leadsto
∞ e−|O(αt4)tx−y|2 −e−|x4|t2 dt
tm
Z0
≤ C(cid:12)(cid:12) |x||y| ∞e−|x4|t2 dt(cid:12)(cid:12) +(|x|2|y|2+|x||y|3+|y|4) ∞e−|1x6|t2 dt
tm+1 tm+2
(cid:18) Z0 Z0 (cid:19)
C|y|
≤ , |x| > 2|y|.
|x|2m−1
SimilarlywehavefromLemma2.1,
∞ ∞ e−|O(αt4)sx−y|2 −e−|x4s|2 ds dt ≤ C|y| , |x|> 2|y|.
sm+1 |x|2m−1
Z0 Zt
(cid:12) (cid:12)
The proof of (13) i(cid:12)s complete. Since m >(cid:12) 1, Est.(14) and (15) are consequences of (10)
and(11). Theproofiscomplete. ✷
3 Stokes system with a rotation effect
Thissectionisdevotedtotheanalysisofthelinearizedproblem(S ),introduced inSection
α
1,withb = 0.
3.1 Linear estimateinthe wholeplane
Inthissectionletusconsider thelinearproblem inwholeplaneforα∈ R\{0}:
−∆u−α(x⊥ ·∇u−u⊥)+∇p = f, divu = 0, x ∈ R2. (Sα,R2)
Thevelocity u ∈ Wl1o,c2(R2)2 issaid tobeaweak solution to(Sα,R2)if(i)divu = 0inthe
senseofdistributions, and(ii)usatisfies
∇u·∇φ−α(x⊥·∇u−u⊥)·φdx = f ·φdx, forall φ∈ C∞(R2). (17)
0,σ
R2 R2
Z Z
The fundamental solution to (Sα,R2) plays a central role throughout this paper, which is
definedas
∞
Γ (x,y) = O(αt)TK(O(αt)x−y,t)dt,
α
Z0
where
∞
K(x,t) = G(x,t)I+H(x,t), H(x,t) = ∇2G(x,s)ds,
Zt
7
andG(x,t)isthetwo-dimensional Gausskernel
G(x,t) = 1 e−|x4|t2 .
4πt
The next theorem is the main result of this section, which extends the result of [19] to our
functional setting. Forf ∈ L2(R2)2 andF ∈ L2(R2)2×2 weformallyset
c[f] = lim e−ǫ|x|2x⊥ ·fdx,
ǫ→0ZR2 (18)
c˜[F] = lim e−ǫ|x|2(F −F )dx.
21 12
ǫ→0 R2
Z
Note that if f ∈ L2(R2)2 is of the form f = divF with some F ∈ L1(R2)2×2 then
c[f]= c˜[F]. Moreover, ifF issymmetricthenc˜[F] = 0.
Theorem 3.1 Letα ∈ R\{0}. Weformallyset
L[f](x) = lim e−ǫ|y|2Γ (x,y)f(y)dy. (19)
α
ǫ→0 R2
Z
Thenthefollowing statementshold.
(i)Letγ ≤ 1. Suppose thatf ∈ L2(R2)2 satisfies suppf ⊂ B (0)forsomeR > 0. Then
R
u= L[f]isaweaksolutionto(Sα,R2)andiswrittenas
x⊥
u(x) = c[f] +R[f](x), x 6= 0, (20)
4π|x|2
whereR[f]satisfies forγ ∈ [0,1],
sup |x|1+γ|R[f](x)| ≤ C1 |α|−1+2γkfkL1({|y|≤R}) +k|y|1+γfkL1({|y|≤R}) , (21)
|x|≥2R (cid:18) (cid:19)
whileforγ < 0,
sup |x|1+γ|R[f](x)| ≤ C1 |α|−1+2γ +|α|−12Rγ kfkL1({|y|≤R})
|x|≥2R (cid:18) (22)
(cid:0) (cid:1)
+k|y|1+γfk .
L1({|y|≤R})
(cid:19)
HereC isanumericalconstant, andisindependent ofγ,α,R,andf.
1
(ii) Let γ ∈ (−1,1]. Suppose that f ∈ L2(R2)2 is of the form f = divF with some
F ∈ L∞ (R2)2×2, and in addition that c˜[F] in (18) converges when γ ∈ (−1,0]. Then
2+γ
u= L[f]isaweaksolutionto(Sα,R2)andiswrittenas
x⊥
u(x) = c˜[F] +R[f](x), x 6=0, (23)
4π|x|2
whereR[f]satisfies forR > 0,
sup |x|1+γ|R[f](x)| ≤ C k|y|2+γFk + sup |x|−1+γkyFk
2 L∞({|y|≥R}) L1({2|y|≤|x|})
y
|x|≥2R (cid:18) |x|≥2R
1
+ sup min ,|x|γ kFk
|α||x|2−γ L1y({2|y|≤|x|})
|x|≥2R
(cid:8) (cid:9)
+ sup |x|γ lim e−ǫ|y|2(F −F )dy .
21 12
|x|≥2R ǫ→0Z2|y|≥|x| (cid:19)
(cid:12) (cid:12) (24)
(cid:12) (cid:12)
8
Foreachδ ∈ [0,1)theconstant C isindependent ofγ ∈ [−δ,1],α,R,andF.
2
Remark3.2 Undertheassumptionsof(i)or(ii)inTheorem3.1itisnotdifficulttoseethat
L[f]belongs toW2,2(R2). Set
loc
x−y
p(x) = f(y)dy. (25)
R2 2π|x−y|2
Z
Then, ∇p belongs to L2(R2)2 under the assumptions of (i) or (ii) in Theorem 3.1 by the
Caldero´n-Zygmundinequality, andasisshownin[19,Proposition3.2],thepair(L[f],∇p)
satisfies(Sα,R2)inthesenseofdistributions. Invirtueoftheuniquenessresultstatedin[19,
Lemma3.5], foranysolution (u,p) ∈ W˙ 1,2(R2)2×S′(R2)satisfying (Sα,R2)inthesense
of distributions, the velocity u is expressed as u = u + L[f] for some constant vector
∞
u ∈ R2.
∞
Wenotethatin(ii)ofTheorem3.1thecoefficientc˜[F]isalwayswell-definedwhenγ >
0. The asymptotic expansion (20) for the case (i) is firstly established by [19, Proposition
3.2]. Indeed, forthecase(i)itisshownin[19,Proposition 3.2]thatR[f]decaysatinfinity
as O(|x|−2), while the singularity |α|−1 appears in the coefficient of the estimates there.
The new ingredients in Theorem 3.1 are (21) and (24), where both the consistency in the
weighted L∞ spacesandthemildersingularity onαforsmall|α|areessential tosolvethe
nonlinearprobleminSection4. Ontheotherhand,asin[19],thekeysteptoproveTheorem
3.1istheexpansionandthepointwiseestimateofthefundamentalsolutionΓ (x,y),which
α
are stated in Lemma 3.3 below. The fundamental solution Γ (x,y) is studied in details in
α
[19,Proposition3.1]andwewillrevisittheargumentdevelopedby[19]intheproofofthis
lemma.
Lemma3.3 Set
x⊥⊗y⊥
L(x,y) = . (26)
4π|x|2
Thenform = 0,1thekernelΓ (x,y)satisfies
α
|∇m Γ (x,y)−L(x,y) |
y α
1 1 1 1 |y|2−m
≤ C(cid:0) δ min ,(cid:1) +|x|1−mmin , + , (27)
(cid:18) 0m |α||x|2 |α|12|x| |α||x|3 |x| |x|2 (cid:19)
(cid:8) (cid:9) (cid:8) (cid:9)
for |x|> 2|y|.
Hereδ istheKronecker deltaandC isindependent ofx,y,andα.
0m
Remark3.4 The case m = 0 of (27) is obtained in [19, Proposition 3.1] but with |α|−1
dependence of the coefficients in the estimate. The case m = 1 is not stated explicitly in
[19], although it can be handled in the similar spirit as in the case m = 0. In this sense
Lemma 3.3 is not completely new, and is an improvement of [19, Proposition 3.1] with
respecttothesingularity on|α|for|α| ≪ 1.
Proof of Lemma3.3: In principle, our proof of Lemma3.3will proceed along the line of
[19,Proposition 3.1]. Infact,theonlykeydifference ofoutproofforthecasem = 0isthe
9
application ofLemmas2.1,2.2insuitableparts. Intheproofforthecasem = 1Ineq. (13)
willbeessentially usedinaddition.
Followingtheargumentof[19,Section3],wedecompose Γ (x,y)as
α
Γ (x,y)
α
= Γ0(x,y)+Γ11(x,y)+Γ12(x,y)
α α α
∞
:= O(αt)TG(O(αt)x−y,t)dt
Z0 (28)
∞ ∞ ds
+ O(αt)T(O(αt)x−y)⊗(O(αt)x−y) G(O(αt)x−y,s) dt
4s2
Z0 Zt
∞ ∞ ds
− O(αt)T G(O(αt)x−y,s) dt.
2s
Z0 Zt
Wealsodecompose L(x,y)asfollows.
x⊗y+x⊥⊗y⊥ −3(x⊗y)+x⊥⊗y⊥ x⊗y x⊗y+x⊥ ⊗y⊥
L(x,y) = + + −
4π|x|2 8π|x|2 4π|x|2 8π|x|2
=: L0(x,y)+L111(x,y)+L112(x,y)+L12(x,y).
(29)
Then,byLemma2.1thefollowingrepresentations hold:
∞ dt x·y x⊥·y
L0(x,y) = G(x,t) ,
4t −x⊥·y x·y
Z0 (cid:18) (cid:19)
∞ ∞ ds −3(x⊗y)+(x⊥ ⊗y⊥)
L111(x,y) = G(x,s) dt ,
4s2 2
Z0 Zt (cid:18) (cid:19) (30)
∞ ∞ ds
L112(x,y) = G(x,s) dt|x|2(x⊗y),
16s3
Z0 Zt
∞ ∞ ds x·y x⊥·y
L12(x,y) = − G(x,s) dt ,
8s2 −x⊥·y x·y
Z0 Zt (cid:18) (cid:19)
wherewehaveusedtheequality
x·y x⊥ ·y
x⊗y+x⊥⊗y⊥ = .
−x⊥·y x·y
(cid:18) (cid:19)
Toprove(27)weobservethat
|∇m Γ (x,y)−L(x,y) |
y α
≤ |∇(cid:0)my Γ0α(x,y)−L0(x(cid:1),y) |+|∇my Γ1α1(x,y)−L111(x,y)−L112(x,y) |
+|∇my(cid:0) Γ1α2(x,y)−L12(x,(cid:1)y) |. (cid:0) (cid:1)
(cid:0) (cid:1)
Letusestimate each term inthe right-hand side of theabove inequality. Thekeyidea isto
usetheTaylorformulaforG(O(αt)x−y,t′)aroundy = 0asfollows.
hO(αt)x),yi hy,Qyi
G(O(αt)x−y,t′) = G(x,t′)+ G(x,t′)+ G(O(αt)x−θy,t′) ,
2t′ 8t′2
(31)
10