TWO DIMENSIONAL INCOMPRESSIBLE VISCOUS FLOW AROUND A SMALL OBSTACLE
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English
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TWO DIMENSIONAL INCOMPRESSIBLE VISCOUS FLOW AROUND A SMALL OBSTACLE

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39 Pages
English

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TWO-DIMENSIONAL INCOMPRESSIBLE VISCOUS FLOW AROUND A SMALL OBSTACLE D. IFTIMIE, M. C. LOPES FILHO AND H. J. NUSSENZVEIG LOPES Abstract. In this work we study the asymptotic behavior of viscous incom- pressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity ?0 and the circulation ? of the initial flow around the obstacle. We prove that, if ? is sufficiently small, the limit flow satisfies the full-plane Navier-Stokes system, with initial vorticity ?0 + ??, where ? is the standard Dirac measure. The result should be contrasted with the corresponding invis- cid result obtained by the authors in [15], where the effect of the small obstacle appears in the coefficients of the PDE and not only in the initial data. The main ingredients of the proof are Lp?Lq estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato's fixed point method, energy estimates, renormalization and interpolation. Contents 1. Introduction 2 2. Estimates for the Stokes semigroup 4 3. The evanescent obstacle 7 4. Initial data asymptotics 11 5. The impulsively stopped rotating cylinder 13 6. Initial-layer and the nonlinear evolution 14 7. Global-in-time nonlinear evolution 17 8.

  • circulation ? uniquely

  • stokes semigroup

  • sufficiently small

  • given v0 ?

  • divergence-free vector

  • small obstacle

  • initial vorticity

  • boundary ?


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TWO-DIMENSIONALINCOMPRESSIBLEVISCOUSFLOW
AROUNDASMALLOBSTACLE

D.IFTIMIE,M.C.LOPESFILHOANDH.J.NUSSENZVEIGLOPES

Abstract.
Inthisworkwestudytheasymptoticbehaviorofviscousincom-
pressible2Dflowintheexteriorofasmallmaterialobstacle.Wefixthe
initialvorticity
ω
0
andthecirculation
γ
oftheinitialflowaroundtheobstacle.
Weprovethat,if
γ
issufficientlysmall,thelimitflowsatisfiesthefull-plane
Navier-Stokessystem,withinitialvorticity
ω
0
+
γδ
,where
δ
isthestandard
Diracmeasure.Theresultshouldbecontrastedwiththecorrespondinginvis-
cidresultobtainedbytheauthorsin[15],wheretheeffectofthesmallobstacle
appearsinthecoefficientsofthePDEandnotonlyintheinitialdata.The
mainingredientsoftheproofare
L
p

L
q
estimatesfortheStokesoperatorin
anexteriordomain,aprioriestimatesinspiredonKato’sfixedpointmethod,
energyestimates,renormalizationandinterpolation.

Contents
1.Introduction
2.EstimatesfortheStokessemigroup
3.Theevanescentobstacle
4.Initialdataasymptotics
5.Theimpulsivelystoppedrotatingcylinder
6.Initial-layerandthenonlinearevolution
7.Global-in-timenonlinearevolution
8.Velocityestimates
9.Compactnessinspace-time
10.Passingtothelimit
11.Uniquenessforthelimitproblem
12.Conclusions
References
1

24711314171912272235373

2IFTIMIE,LOPESFILHOANDNUSSENZVEIGLOPES
1.
Introduction
Thepurposeofthisworkistostudytheinfluenceofamaterialobstacleonthe
behavioroftwo-dimensionalincompressibleviscousflowswhenthesizeoftheob-
stacleissmallcomparedtothatofareferencespatialscale.Moreprecisely,wefix
bothaninitialvorticity
ω
0
,smoothandcompactlysupported,andthecirculation
γ
oftheinitialvelocityaroundtheboundaryoftheobstacle,whilehomothetically
contractingtheobstacletoapoint
P
outsidethesupportof
ω
0
.Theinitialvortic-
ity
ω
0
andthecirculation
γ
uniquelydetermineafamilyofdivergence-freeinitial
velocities
u
0
ε
withcurl
u
0
ε
=
ω
0
and
u
0
ε
(
x
)

0atinfinity;here
ε
denotesthesize
oftheobstacle.Thesizeofthesupportoftheinitialvorticity
ω
0
canbeusedas
referencespatialscale.Let
u
ε
=
u
ε
(
x,t
)beasolutionoftheNavier-Stokesequa-
tionswithinitialdata
u
0
ε
andno-slipdataattheboundaryofthesmallobstacle.
Ourproblemistodeterminetheasymptoticbehaviorof
u
ε
as
ε

0.Wewillshow
that
u
ε
convergestoasolutionoftheNavier-Stokesequationsinthefullplanewith
initialvorticity
ω
0
+
γδ
(
x

P
),aslongas
γ
issufficientlysmall.Moreprecisely,
weprovethefollowingtheorem.

Theorem1.
Let
Ω
ε
=
ε
Ω
bea2Dsimply-connectedsmoothobstacle,
ω
0
asmooth
functioncompactlysupportedin
R
2
\{
0
}
,independentof
ε
and
γ
arealnumber
independentof
ε
.ConsidertheNavier-Stokesequationsintheexteriorof
Ω
ε
with
homogeneousDirichletboundaryconditionsandassumethattheinitialvelocityhas
vorticity
ω
0
andcirculationaroundtheobstacleequalto
γ
.Let
u
ε
denotethe
correspondingglobalsolution.Thereexistsaconstant
γ
0
>
0
suchthatif
|
γ
|≤
γ
0
,
then
u
ε
convergestothesolutionoftheNavier-Stokesequationsin
R
2
withinitial
vorticitygivenby
ω
0
+
γδ
0
.

Thereisasharpcontrastbetweenthebehaviorofidealandviscousflowsaround
asmallobstacle.In[15],theauthorsstudiedthevanishingobstacleproblemfor
incompressible,ideal,two-dimensionalflow.Theidealflowassumptionisphysically
incorrectinthepresenceofmaterialboundaries,andpartofthemotivationforthe
presentwork(andof[15])istoexploremorepreciselythisincorrectnessfroma
mathematicalstandpoint.Themainresultin[15]isthatthelimitvorticityinthe
idealcasesatisfiesamodifiedvorticityequationoftheform
ω
t
+
u
∙r
ω
=0,with
div
u
=0andcurl
u
=
ω
+
γδ
(
x

P
).Inotherwords,foridealflowthecorrection
duetothevanishedobstacleappearsastime-independentadditionalconvection
centeredat
P
,whereasintheviscouscase,thecorrectionappearsontheinitial
dataandgetsconvectedanddiffusedasitevolves.

2DINCOMPRESSIBLEVISCOUSFLOWAROUNDASMALLOBSTACLE3
ThesmallobstaclelimitisaninstanceofthegeneralproblemofPDEonsingu-
larlyperturbeddomains.Thereisalargeliteratureonsuchproblems,speciallyin
theellipticcase,see[23]forabroadoverview.Asymptoticbehavioroffluidflowon
singularlyperturbeddomainsisanaturalsubjectforanalyticalinvestigationwhich
isvirtuallyunexplored.Thepresentwork,togetherwith[15],mayberegardedas
afirstattempttoaddressthisclassofproblems.
Thereisanaturalconnectionbetweentheapproximationproblemaswehave
formulateditandtheissueofuniquenessforthelimitproblem.Infact,froma
technicalpointofview,ourworkiscloselyrelatedtotheclassicaluniquenessresult
duetoY.Giga,T.MiyakawaandH.Osada,onsolutionsoftheincompressible
2DNavier-Stokesequationswithmeasuresasinitialdata,see[14].Someofthe
morestrikingsimilaritiesare:thedifficultieswithlocallyinfinitekineticenergy,
theuseof
L
p
estimatesforthelinearizedproblemandtheuseofKato-typenorms
toestimatethenonlinearity.Thesmallnessconditiononthemassofthepoint
vorticesintheinitialdata,requiredintheuniquenessresult,iscloselyrelatedto
oursmallnessconditiononthecirculation.
Theremainderofthisworkisorganizedinelevensections.InSection2wesum-
marize
L
p
estimatesforthetime-dependentStokesproblemonexteriordomains.In
Section3weformulatepreciselytheproblemwewishtodiscussandwriteuniform
estimatesfortheinitialdata.InSection4westudytheasymptoticbehaviorof
theinitialdata.InSection5wediscussphysicalmotivationforourproblemand
weestablishthesmallobstacleasymptoticsforcircularlysymmetricflows,alinear
versionofourproblem.InSection6wederiveaprioriestimatesintheinitiallayer
forthenonlinearcorrectionterm.InSection7wededuceglobal-in-timeenergy
estimatesforthenonlinearcorrectionterm.InSection8weputtogethertheesti-
matesforthelinearpartwiththeestimatesforthenonlinearcorrection,obtaining
acompletesetofaprioriestimatesforvelocity.InSection9weprovecompactness
inspace-time,inSection10weperformthepassagetothelimit,inSection11we
discussuniquenessforthelimitproblemandinSection12weaddcommentsand
concludingremarks.
Weconcludethisintroductionwithafewremarksregardingnotation.Givena
vector
z
=(
z
1
,z
2
)

R
2
wedenoteitsorthogonalvectorby
z

=(

z
2
,z
1
).Weuse
thesubscript
c
infunctionspacestodenotecompactsupport,asin
C
c

,andwe
usestandardnotationforSobolevspaces,
W
k,p
,where1

p
≤∞
and
k

Z
,with
H
k
standingforthecase
p
=2.Weusethesubscript
loc
infunctionspaces
X
to
pdenotefunctionswhicharelocallyin
X
.Inparticular,
L
loc
([0
,

);
W
k,q
)denotes
functions
f
=
f
(
t,x
)

L
p
([0
,M
];
W
k,q
)forany
M>
0,whereas
L
lpoc
((0
,

);
W
k,q
)

4IFTIMIE,LOPESFILHOANDNUSSENZVEIGLOPES
denotesfunctions
f
=
f
(
t,x
)

L
p
([
δ,M
];
W
k,q
)forany
δ>
0andany
M>
0,but
notnecessarilyfor
δ
=0.Finally,
L
2
,

denotestheLorentzspaceoffunctions
f
whosedistributionfunctionsatisfies
λ
f
=
λ
f
(
s
)=
|{|
f
|
>s
}|
=
O
(
s

2
).
2.
EstimatesfortheStokessemigroup
InthissectionwewillputtogetherseveralresultsonestimatesfortheStokes
semigrouponexteriordomains.Letusbeginbyintroducingsomebasicnotation.
LetΩbeabounded,open,simplyconnectedsubsetof
R
2
withboundaryΓ,a
smoothJordancurve.WedenotebyΠtheunboundedconnectedcomponentof
R
2
\
Γ.Fix
ν>
0andlet
P
denotetheLerayprojectorontodivergence-freevector
fieldsonΠ.Let
A
≡−
P
ΔbetheStokesoperatoronΠanddenotetheStokes
semigroupby
S
ν
(
t
)=
e

νt
A
.Given
v
0

C
c

(Π),let
v
(
t,x
)=
S
ν
(
t
)
v
0
bethe
uniquesolutionofthesystem
∂
t
v

ν
Δ
v
=
−r
p,
in(0
,

)
×
Π


div
v
=0
,
in[0
,

)
×
Π
(2.1)
v
=0
,
on(0
,

)
×
Γ
lim
|
x
|→∞
v
(
t,x
)=0
,
forall
t

0

v
(0
,x
)=
v
0
(
x
)
,
on
{
t
=0

Π
.
Wedenoteby
X
p
(Π)theclosureofthespaceofdivergence-free,
C
c

(Π)vector
fieldswithrespecttothe
L
p
-norm.TheStokesoperatorin
X
p
generatesananalytic
semigroupofclass
C
0
on
X
p
(Π),forany1
<p<

,see[13],sothat,inparticular,
problem(2.1)iswell-posedin
X
p
(Π).
WewillrequiretwokindsofestimatesontheStokessemigroup,
L
p
estimates
andrenormalizedenergyestimates.Wefirststatethe
L
p
estimates.
Theorem2.
Let
1
<q<

.Consider
v
0

X
q
(Π)
and
F

L
q
(Π;
M
2
×
2
(
R
))
.
Thenwehavethefollowingestimates.
(S1)
Let
q

p<

.Thereexists
K
1
=
K
1

,p,q
)
>
0
suchthat
11−k
S
ν
(
t
)
v
0
k
L
p

K
1
(
νt
)
pq
k
v
0
k
L
q
,
forall
t>
0
.
(S2)
Let
q

p

2
.Thereexists
K
2
=
K
2

,p,q
)
>
0
suchthat
111kr
S
ν
(
t
)
v
0
k
L
p

K
2
(
νt
)

2
+
p

q
k
v
0
k
L
q
,
forall
t>
0
.
(S3)
Assume
q

2
andlet
q

p<

.Thenthereexists
K
3
=
K
3

,p,q
)
>
0
suchthat
111k
S
ν
(
t
)
P
div
F
k
L
p

K
3
(
νt
)

2
+
p

q
k
F
k
L
q
,

2DINCOMPRESSIBLEVISCOUSFLOWAROUNDASMALLOBSTACLE5
forall
t>
0
,withthedivergencetakenalongrowsofthematrix
F
.
Thistheoremsummarizesseveralresultsalreadycontainedintheliterature,
whichwehavecollatedaboveforconvenience.
Proof.
Estimates(S1)and(S2)wereprovedin[5,20](seealso[6]forthecase
p
=

).Estimate(S3)followsfrom(S2)byduality.Indeed,theadjointof
S
ν
(
t
)
0on
X
p
isagain
S
ν
(
t
),definedon
X
p
,with1
/p
+1
/p
0
=1andthereforetheadjointof
r
S
ν
(
t
)is
S
ν
(
t
)
P
div.Thedependenceontheviscosityfollowsdirectlybyrescaling
time,since
S
ν
(
t
)=
S
1
(
νt
).

NextweaddressarenormalizedenergyestimatefortheStokessemigroup.Our
concernsincludeinfiniteenergysolutionstotheNavier-Stokesequationswhose
behavioratinfinityis
O
(1
/
|
x
|
).Inthefollowingresultwewillprovethatsolutions
totheStokessystemretainthebehavioratinfinityoftheirinitialdata.
Proposition3.
Let
v
0
beasmoothdivergence-freevectorfieldon
Π
vanishing
attheboundary
Γ
.Weassumealsothat
v
0

X
p
(Π)
forsome
p>
2
andthat
r
v
0

L
2
(Π)
.Then
1220S
ν
(
t
)
v
0

v
0

C
([0
,

);
L
(Π))

L
loc
([0
,

);
H
(Π))
.
Moreoverthefollowinginequalityholds
tZ222(2.2)
k
S
ν
(
t
)
v
0

v
0
k
L
2
+
ν
kr
[
S
ν
(
τ
)
v
0

v
0
]
k
L
2


νt
kr
v
0
k
L
2
.
0Proof.
Let
W
=
S
ν
(
t
)
v
0

v
0
.Then
W
satisfies:
∂
t
W

ν
Δ
W
=
−r
p
+
ν
Δ
v
0
,
in(0
,

)
×
Π


div
W
=0
,
in[0
,

)
×
Π
(2.3)
W
=0
,
on(0
,

)
×
Γ
lim
|
x
|→∞
W
(
t,x
)=0
,
forall
t

0
W
(0
,x
)=0
,
on
{
t
=0

Π
.
Itiswell-knownthat(2.3)admitsauniquesolution
W
f
in
C
0
([0
,

);
L
2
(Π))

L
l
2
oc
([0
,

);
H
1
(Π)),see,forinstance,TheoremIII.1.1in[26].Thefactthat
W

W
f
=0followsfromthewell-posednessof(2.1)in
X
p
.Thestandardenergyestimate
gives(2.2).

OneconsequenceofthenontrivialtopologyofΠistheexistenceofharmonic
vectorfields,i.e.divergence-freeandcurl-freevectorfieldswhicharetangenttoΓ
andvanishatinfinity.Wedenoteby
H
Π
theuniqueharmonicvectorfieldonthe
exteriordomainΠwhichsatisfiesthecondition
IH
Π

ds
=1
,
Γ

6IFTIMIE,LOPESFILHOANDNUSSENZVEIGLOPES
wherethecontourintegralistakeninthecounterclockwisesense.Itisanelementary
applicationofHodgetheorythatthevectorspaceoftheseharmonicvectorfields
onΠisonedimensional,andwecanuse
H
Π
asabasis.InthecasewhereΠisthe
exterioroftheunitdiskcenteredattheorigin,wewilldenote
H
Π
simplyby
H
,
andwehave:
⊥x(2.4)
H
=
2
.
|x|π2Wewillrequiredetailedinformationonthebehaviorof
H
Π
bothatinfinityand
neartheboundaryΓ,whichweobtainbymeansofaconformalmapping.We
denote
U≡{|
x
|
>
1
}
andswitchtocomplexvariablesnotationintheresultbelow.
Lemma4.
Thereexistsasmoothbiholomorphism
T

→U
,extendingsmoothly
uptotheboundary,mapping
Γ
to
{|
z
|
=1
}
.Furthermore,thereexistsanonzero
realnumber
β
andaboundedholomorphicfunction
h


C
suchthat:
(2.5)
T
(
z
)=
βz
+
h
(
z
)
.
Additionally,
10(2.6)
h
(
z
)=
O
2
,
as
|
z
|→∞
.
|z|ThisLemmaisanexcerptfrom[15].Itsproofisanexerciseincomplexanalysis.
Itwasobservedin[15](seeidentity(2.10)in[15])that
1
DT
t
(
x
)(
T
(
x
))

(2.7)
H
Π
=
H
Π
(
x
)=
2
.
2
π
|
T
(
x
)
|
FromLemma4,weseethat
|
H
Π
|
is
O
(1
/
|
x
|
)forlarge
|
x
|
.Thisimpliesthat
H
Π
belongstotheLorentzspace
L
2
,

(Π).
WeclosethissectionwithanestimatefortheStokessemigroupactingoninfinite
energyinitialdata.
Proposition5.
Let
2
<p<

andlet
v
0

L
2
,

(Π)

X
p
(Π)
.Thereexistsa
constant
K
5
>
0
suchthat
11k
S
ν
(
t
)
v
0
k
L
p

K
5
(
νt
)
p

2
k
v
0
k
L
2
,

.
Inparticular,thisestimateholdstruefor
v
0
=
H
Π
(
x
)
.
Proof.
ThisestimateiscontainedinProposition2.2,item(4),of[17].Toseethat
itholdsfor
H
Π
,wefirstshowthat
H
Π

X
p
(Π)forany
p>
2.Thisiseasyto
proveinthecaseΠ=
U
because,foranyfunction
ϕ

C
c

((0
,

)),
ϕ
(
|
x
|
)
H
(
x
)is

2DINCOMPRESSIBLEVISCOUSFLOWAROUNDASMALLOBSTACLE7
smooth,compactlysupportedanddivergence-free,and,bytaking
ϕ
ε
asequence
ofcutoffsfortheinterval(1+
ε,
1

),itiseasytoseethat
ϕ
ε
(
|
x
|
)
H

H
in
L
p
,for
p>
2.ForgeneralΠ,weusetheconformalmapping
T
,approximating
H
Π
by
ϕ
ε
(
|
T
(
x
)
|
)
H
Π
(
x
),where
ϕ
ε
isthesamefamilyofcutoffsusedinthecase
oftheexteriorofthedisk.Thisstrategyworksbecause
ϕ
ε
(
|
T
(
x
)
|
)
H
Π
(
x
)isalso
divergence-free.

3.
Theevanescentobstacle
Thepurposeofthissectionistosetdownaprecisestatementofthesmall
obstacleproblem.Manyofthekeyissuesregardingthesmallobstaclelimitand
incompressibleflowhavebeendiscussedindetailin[15],sothatwewillfocuson
issuesspecificallyrelatedtoviscousflowandbrieflyoutlinetherest.
Asin[15],fix
ω
0

C
c

(
R
2
)andassumethattheorigindoesnotbelongtothe
supportof
ω
0
.LetΩbeabounded,open,connectedandsimply-connectedsubset
oftheplanewhoseboundaryΓisa
C

Jordancurve.Theevanescentobstacle
isthefamilyofdomains
ε
Ω,with0
<ε<ε
0
.Theparameter
ε
0
ischosensmall
enoughsothatthesupportof
ω
0
doesnotintercept
ε
Ωforany0
<ε<ε
0
.
Fix0
<ε<ε
0
.LetΠ
ε

R
2
\
ε
ΩandΓ
ε
=

Π
ε
.Weusetheconformalmapping
T

1
→U
,giveninLemma4,todefineafamilyofsmoothbiholomorphisms
(3.1)
T
ε
=
T
ε
(
x
)

Tx.
εThroughoutwewrite
H
ε
for
H
Π
ε
and
G
ε
=
G
ε
(
x,y
)willbetheGreen’sfunction
oftheLaplacianinΠ
ε
.Let
K
ε
(
x,y
)=
r
x

G
ε
(
x,y
)bethekerneloftheBiot-
SavartlawonΠ
ε
anddenotetheassociatedintegraloperatorby
f
7→
K
ε
[
f
]=

K
ε
(
x,y
)
f
(
y
)
dy
.Both
K
ε
and
H
ε
arerelatedto
K
U
and
H
U
respectively,
εthroughtheconformalmapping
T
ε
,inawaywhichwasmadeexplicitin[15].The
relevantfactisthewaythatboththeBiot-Savartkernelandthebasicharmonic
vectorfieldscalewith
ε
,seeidentities(3.5)and(3.6)in[15].
Fix
α

R
andlet
(3.2)
u
0
ε

K
ε
[
ω
0
]+
αH
ε
.
Weconsidertheproblem
∂
t
u
ε
+
u
ε
∙r
u
ε

ν
Δ
u
ε
=
−r
p
ε
,
in(0
,

)
×
Π
ε


div
u
ε
=0
,
in[0
,

)
×
Π
ε
(3.3)
u
ε
=0
,
on(0
,

)
×
Γ
ε
lim
|
x
|→∞
u
ε
(
t,x
)=0
,
forall
t

0

u
ε
(0
,x
)=
u
0
ε
(
x
)
,
on
{
t
=0

Π
ε
.

8IFTIMIE,LOPESFILHOANDNUSSENZVEIGLOPES
Webeginbyobservingthat
u
0
ε

L
2
,


ε
)

L
p

ε
)forany2
<p
≤∞
.Indeed,
u
0
ε
issmooth,andthereforelocallybounded,sothatweonlyrequireknowledgeon
thebehaviorof
u
0
ε
atinfinity.ByLemma4andidentity(2.7)
|
H
ε
|
has
O
(1
/
|
x
|
)
behavioras
|
x
|→∞
,andthereforeitbelongsto
L
2
,


ε
)

L
p

ε
)forany2
<
p
≤∞
.Infact,the
L
2
,

boundon
|
H
ε
|
isindependentof
ε
,ascanbereadily
seenbyrescalingtoafixeddomainandusingthefactthat
H
Π
belongsto
L
2
,

.
In[15]itwasshownthat
|
K
ε
[
ω
0
]
|
hasbehavior
O
(1
/
|
x
|
2
)atinfinity(seeestimate
(2.8)in[15])andthereforeitbelongsto
L
p

ε
),forany
p

2,and,inparticular,
to
L
2
,


ε
).
Global-in-timewell-posednessforproblem(3.3)wasestablishedbyKozonoand
Yamazakiin[17].TheexistencepartofKozonoandYamazaki’sresultrequires
thattheinitialvelocitysatisfyasmallnessconditionoftheform
limsup
R
|{
x

Π
ε
||
u
0
ε
(
x
)
|
>R
}|
1
/
2

1
.
∞→RSince
u
0
ε
isbounded,thelimsupaboveisalwayszero,forany
ε>
0.Uniqueness
holdsfordivergence-freeinitialdatain
L
2
,

+
X
p
withoutanyadditionalconditions.
Theevanescentobstacleproblemconsistsofunderstandingtheasymptoticbe-
haviorofKozonoandYamazaki’ssolution
u
ε
(
x,t
)forsmall
ε
.Morepreciselywe
willshowthat,underappropriateassumptions,
u
ε
hasalimit,andwewillidentify
anequationsatisfiedbythislimit.
Fix
ϕ
:
R

[0
,
1]asmooth,monotonefunctionsuchthat
ϕ
(
s
)

0if
s

2and
ϕ
(
s
)

1if
s

3.Foreach
ε>
0and
λ>
0weintroducetheadaptedcut-off
functions:
(3.4)
ϕ
ε,λ
(
x
)

ϕε
|
T
ε
(
x
)
|
,
λNotethatthecutofffunction
ϕ
ε,λ
vanishesinaballofradius
O
(
λ
)anditisidenti-
callyequalto1outsidealargerballofradius
O
(
λ
),forlarge
λ
.Furthermore,the
radiioftheannuluswhere
ϕ
ε,λ
isnotconstantcanbemadeindependentof
ε
.This
followseasilyfromthefactthat
T
isasymptoticallyaffineatinfinity,see(2.5).
Wewillnowintroduceapairofparametersthatareusefultodescribetheas-
ymptoticbehaviorof
u
0
ε
when
ε

0.Consider
IZ(3.5)
m

ω
0
dx
and
γ

u
0
ε

ds.
2ΓRεByStokes’Theoremwehavethat
γ
=
α

m
,andtherefore,thecirculation
γ
does
notthedependon
ε
,seetheproofofLemma3.1in[15].
Foreach
λ>
0,weintroduceaconvenientdecompositionoftheinitialvelocity
sau
0
ε
=
b
0
ε
+
i
0
ε
+
o
0
ε
,

2DINCOMPRESSIBLEVISCOUSFLOWAROUNDASMALLOBSTACLE9

htiwb
0
ε

K
ε
[
ω
0
]+
m
(1

ϕ
ε,λ
)
H
ε
,
i
0
ε

γ
(1

ϕ
ε,λ
)
H
ε
,
dnao
0
ε

αϕ
ε,λ
H
ε
.
Weneedtounderstandthebehaviorofeachofthecomponentsofthisdecom-
position,inthelimit
ε

0.Thisisthecontentofournextresult.Theproofuses
alargepartoftheworkdonein[15].
Lemma6.
Thereexists
λ
0
>
0
,independentof
ε
,forwhich
k
b
0
ε
k
L
p
1
,
k
i
0
ε
k
L
p
2
and
k
o
0
ε
k
L
p
3
areuniformlyboundedin
ε
,forany
1
<p
1
≤∞
,
1

p
2
<
2
and
2
<p
3
≤∞
.Thevectorfields
b
0
ε
,
i
0
ε
and
o
0
ε
aredivergence-free,thefirsttwoare
tangentto
Γ
ε
andthelastonevanisheson
Γ
ε
.Moreover,
kr
o
0
ε
k
L
2

ε
)
isbounded
independentof
ε
and

(3.6)

o
0
ε

αϕβ
|
x
|
H


0
as
ε

0
,
λ
0
L
2

ε
)
where
H
=
x

/
(2
π
|
x
|
2
)
and
β
isasinLemma4.Wealsohavethat
1k
i
0
ε
k
L
2

ε
)

C
|
log
ε
|
2
.
Proof.
Choose
λ
0
suchthattheradiioftheannuluswhere
ϕ
ε,λ
0
isnotconstantare
uniformin
ε<ε
0
.
The
L

boundon
b
0
ε
comesfromTheorem4.1in[15].The
L
p
1
boundon
b
0
ε
,
1
<p
1
<

followsfromthelocal
L

boundabove,togetherwithtwofacts:(1)
m
(1

ϕ
ε,λ
0
)
H
ε
hassupportinacompactsetindependentof
ε
and(2)
|
K
ε
[
ω
0
]
|
=
O
(1
/
|
x
|
2
)atinfinity,uniformlyin
ε
.Asmentionedpreviously,fact(2)isestimate
(2.8)in[15].
The
L
p
3
boundon
o
0
ε
followsfromthefactthat
ϕ
ε,λ
0
isconstantoutsidean
annulusindependentof
ε
,fromformula(2.7),fromthescaling
H
ε
(
x
)=
ε
1
H
Π
(
x/ε
)
andfromthebehaviorof
T
farfromtheobstaclegivenbyLemma4.
For
i
0
ε
,boththelogarithmicestimateandthe
L
p
2
estimatefollowfromadapting
theargumentusedforestimate(3.7)of[15]inastraightforwardmanner.
Toestimate
r
o
0
ε
weobservethat
|r
o
0
ε
|
=
O
(1
/
|
x
|
2
)nearinfinity,uniformlyin
ε
.Thisestimateeasilyreducestoanestimateon
DH
ε
,whichinturnreducesto
calculatingderivativesoftheconformalmapping
T
using(2.5).
Finally,(3.6)reducestoshowingthat
H

H
ε
goestozeroin
L
2
nearinfinity,
whichcanbedonebyacomputationsimilartotheonecarriedoutintheproofof
Lemma4.2in[15].

10IFTIMIE,LOPESFILHOANDNUSSENZVEIGLOPES
Intheremainderofthisarticle,wewillfix
λ
0
,independentof
ε
,asinLemma6,
therebyfixingthebounded,innerandouterpartsoftheinitialvelocity,
b
0
ε
,
i
0
ε
and
o
0
ε
,respectively.
LetusdenotetheStokessemigrouponΠ
ε
by
S
νε
(
t
),sothat
S
νε
(
t
)[
v
0
ε
]isthe
solutiontotheStokessystem(2.1)onΠ
ε
withinitialdata
v
0
ε
.Weintroducethe
notation
τ
ε
=
τ
ε
(
x
)=
εx
,thecontractionby
ε
.Weobservethefollowingfunda-
mentalrelationbetweentheStokessystemonΠ
ε
andonΠ:
(3.7)(
S
ν
1
(
t
)[
v
0
ε

τ
ε
])(
x
)=(
S
νε
(
ε
2
t
)[
v
0
ε
])(
εx
)
,
if
x

Π
.
Ourstrategytostudythesmallobstaclelimitbeginsbyconsideringthesolu-
tion
u
ε
of(3.3)asaperturbationof
v
ε

S
νε
(
t
)
u
0
ε
.Thefirstthingwerequireis
informationon
v
ε
,whichwededuceintheresultbelow.
Lemma7.
Let
b
ε

S
νε
(
t
)
b
0
ε
,
i
ε

S
νε
(
t
)
i
0
ε
,
o
ε

S
νε
(
t
)
o
0
ε
andlet
2
<p<

.Then
thereexistsaconstant
K
=
K
(
p,ω
0
)
>
0
suchthatforany
ε>
0
wehave:
11(i)
k
b
ε
k
L
p

ε
)

K
(
νt
)
p

2
,
11(ii)
k
i
ε
k
L
p

ε
)

K
|
γ
|
(
νt
)
p

2
,
11(iii)
k
o
ε
k
L
p

ε
)

K
|
α
|
(
νt
)
p

2
.
Proof.
By(3.7)wehavethat
b
ε
(
ε
2
t,εx
)=(
S
ν
1
(
t
)[
b
0
ε

τ
ε
])(
x
),for
x

Π.Now,by
Theorem2,item(S1),itfollowsthatthereexists
K
1
>
0suchthat
11k
S
ν
1
(
t
)[
b
ε
0

τ
ε
]
k
L
p
(Π)

K
1
(
νt
)
p

2
k
b
0
ε

τ
ε
k
L
2
(Π)
.
Item(i)abovefollowsfromthisestimate,togetherwith(3.7)andthefactthat
11k
b
0
ε

τ
ε
k
L
2
(Π)
=
k
b
0
ε
k
L
2

ε
)

C,
εεwherewehaveusedLemma6inthelastinequality.Items(ii)and(iii)followinan
analogousmannerusingProposition5togetherwiththefactthat
|γ|1k
i
0
ε

τ
ε
k
L
2
,

(Π)
=
k
i
0
ε
k
L
2
,


ε
)

C
εεdna|α|1k
o
0
ε

τ
ε
k
L
2
,

(Π)
=
k
o
0
ε
k
L
2
,


ε
)

C.
εεWehaveusedthescaling
H
ε
(
x
)=(1

)
H
1
(
x/ε
)above,seeidentity(3.6)in[15].

Remark8.
Usingtherescaling(3.7)wemaydeducethattheestimates(S1),(S2)
and(S3)inTheorem2arevalidinΠ
ε
withconstants
K
1
,
K
2
and
K
3
independent
.εfo