TWO DIMENSIONAL INCOMPRESSIBLE VISCOUS FLOW AROUND A SMALL OBSTACLE

Description

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 ?

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 13
• Langue: English

Extrait

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