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ON A RELAXATION APPROXIMATION OF THE INCOMPRESSIBLE NAVIER STOKES EQUATIONS

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Niveau: Supérieur, Doctorat, Bac+8
ON A RELAXATION APPROXIMATION OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS YANN BRENIER?, ROBERTO NATALINI†, AND MARJOLAINE PUEL‡ Abstract. We consider an hyperbolic singular perturbation of the incom- pressible Navier Stokes equations in two space dimensions. The approximating system under consideration, arises as a diffusive rescaled version of a standard relaxation approximation for the incompressible Euler equations. The aim of this work is to give a rigorous justification of its asymptotic limit toward the Navier Stokes equations using the modulated energy method. 1. Introduction Let us consider the incompressible Euler equations, namely (1.1) ? ? ? ? ? ? ? ? ? ? ? ∂tu +? · (u? u) = ??, ? · u = 0, u(0, x) = u0(x), for (t, x) ? [0, T ] ? T2, where T2 is the unit periodic square R2/Z2. This sys- tem describes a perfect incompressible fluid, the unknowns u and ? corresponding respectively to the velocity, which is valued in R2, and to the pressure of the fluid. To approximate these equations, most in the spirit of [14], we introduce its relaxed version, which is obtained by a singular perturbation of the nonlinear term (u? u), through a supplementary matrix valued variable V : T2 ? R4.

  • multiply equation

  • equations

  • u?

  • has been

  • energy

  • u? ?

  • let also

  • pressible navier

  • jin-xin relaxation


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ONARELAXATIONAPPROXIMATIONOFTHEINCOMPRESSIBLE
NAVIER-STOKESEQUATIONS
YANNBRENIER

,ROBERTONATALINI

,ANDMARJOLAINEPUEL

Abstract.
Weconsideranhyperbolicsingularperturbationoftheincom-
pressibleNavierStokesequationsintwospacedimensions.Theapproximating
systemunderconsideration,arisesasadiusiverescaledversionofastandard
relaxationapproximationfortheincompressibleEulerequations.Theaimof
thisworkistogivearigorousjusticationofitsasymptoticlimittowardthe
NavierStokesequationsusingthemodulatedenergymethod.

1.
Introduction
LetusconsidertheincompressibleEulerequations,namely

t
u
+
r
(
u

u
)=
r
,
(1.1)
r
u
=0
,
u
(0
,x
)=
u
0
(
x
)
,
for(
t,x
)

[0
,T
]

T
2
,where
T
2
istheunitperiodicsquare
R
2
/
Z
2
.Thissys-
temdescribesaperfectincompressibleuid,theunknowns
u
and

corresponding
respectivelytothevelocity,whichisvaluedin
R
2
,andtothepressureoftheuid.
Toapproximatetheseequations,mostinthespiritof[14],weintroduceits
relaxedversion,whichisobtainedbyasingularperturbationofthenonlinearterm
(
u

u
),throughasupplementarymatrixvaluedvariable
V
:
T
2

R
4
.Thisleads
tothefollowingsystem

t
u
+
r
(
V
)=
r
,
1(1.2)




t
V
+
a
r
u
=

(
V

u

u
)
,
r
u
=0
,
u
(0
,x
)=
u
0
(
x
)
,V
(0
,x
)=
V
0
(
x
)
.
Letusnoticethat,as

goestozero,weformallyrecoversystem(1.1).
1991
MathematicsSubjectClassication.
Primary:35Q30;Secondary:76D05.
Keywordsandphrases.
IncompressibleNavier-Stokesequations,relaxationapproximations,
hyperbolicsingularperturbations,modulatedenergymethod.
WorkpartiallysupportedbyEuropeanTMRprojectsNPPDE#ERBFMRXCT980201and
CNRShortTermVisitingprogram.

LaboratoireJ.A.Dieudonne,U.M.R.C.N.R.S.N6621,UniversitedeNiceSophia-Antipolis,
ParcValrose,F–06108Nice,France

IstitutoperleApplicazionidelCalcolo“MauroPicone”,ConsiglioNazionaledelleRicerche,
VialedelPoliclinico,137,I-00161Roma,Italy

UniversitePierreetMarieCurie,Laboratoired’analysenumerique,Boitecourrier187,F–75252
Pariscedex05,France.
1

2
Y.BRENIER,R.NATALINI,ANDM.PUEL
Letusconsidernowadiusivescaling,namely,for
ε>
0,weset
tx1u
ε
(
t,x
):=

u

,,
εεε

1

xt

(1.3)
V
ε
(
x,t
):=
V

,,
εεεεεε


ε
(
x,t
):=1


x,t.
Thereforesystem(1.2)becomes,settingfromnow

=1,

t
u
ε
+
r
(
V
ε
)=
r

ε
,
1a√(1.4)
εε


ε∂
t
V
ε
+
√r
u
ε
=

(
V
ε

u
ε

u
ε
)
,
r
u
ε
=0
,
u
ε
(0
,x
)=
u
0
ε
(
x
)
,V
ε
(0
,x
)=
V
0
ε
(
x
)
.
Inthispaperweshallprovethat,undersomesuitableassumptions,thesolutions
to(1.4)converge,when
ε
goesto0,tothe(smooth)solutionsoftheincompressible
NavierStokesequations

t
U
+
r
(
U

U
)

a

U
=
r
,
(1.5)
r
U
=0
,
U
(0
,x
)=
U
0
(
x
)
.
Thisresultcouldbepromptlyrecovered,atleastataformallevel,ifweassumethat,
insome(weak)topologies,notonly
u
ε

U
,butalso
εV
ε

0and
u
ε

u
ε

U

U
.
Theaimofthispaperistoshowhowtoobtainthisresultinadierent(andsimpler)
waybyusingthemodulatedenergymethod[3],leadingtoadirecterrorestimate
inthestrong
L

([0
,T
]
,L
2
(
T
2
))norm,forallnitepositive
T
.
Letusrecallthatthediusivescaling(

x
,
t
)hasbeenlargelyinvestigatedin
εεtheframeworkofhydrodynamiclimitsoftheBoltzmannequations,seeforinstance
[8]andreferencestherein.Startingfromtheworksaboutthediusivelimitof
theCarlemanequationsbyKurtz[11]andMcKean[20],thisscalinghasalsobeen
systematicallyusedintheanalysisofhyperbolic-parabolicrelaxationlimitsforweak
solutionsofhyperbolicsystemsofbalancelawswithstronglydiusivesourceterms
bymeansofcompensatedcompactnesstechniquesbyMarcatiandcollaborators
[18,17,19,7].Forotherdiusivekineticmodelsandapproximations,wereferto
[15,13,12].Ageneralclassofkineticapproximationsfor(possiblydegenerate)
parabolicequationsinmulti-Dhasbeenconsideredin[4,1].Letusalsopointout
thatthesamescalingwasusedin[16]toanalyzethetime-asymptoticlimitofthe
Jin-Xinrelaxationmodel[14],towardsthefundamentalsolutionofthediusive
Burgersequation.
Finallyletusremarkthatourscalingcanbeconsideredasahyperbolicpertur-
bationoftheNavierStokesequations,whichissimilartotheCattaneo
hyperbolic
heatequation
[6],justeliminatingtheunknown
V
inequations(1.4)


t
u
ε
+
P
(
r
(
u
ε

u
ε
))

a

u
ε
+
ε∂
tt
u
ε
=0
,
)6.1(
r
u
ε
=0
,

ONARELAXATIONAPPROXIMATIONOFTHEINCOMPRESSIBLENAVIER-STOKESEQUATIONS
3
where
P
representstheprojectiononthedivergencefreevectors.Inthisregard,
wementionthatsomequitedierenthyperbolicperturbationsoftheNavierStokes
equationshasbeeninvestigatedin[21],byconsideringincompressibleviscoelastic
uidsofOldroydtype.Wealsopointoutthatasimilarapproximationhasbeen
alsorecentlyproposedin[2]fornumericalpurposes,asareducedkineticmodel.
Concerningthemethodofthemodulatedenergy,letusrecallthatithasbeen
usedbyBrenierin[3]toprovetheconvergenceinaquasi-neutrallimitofthecurrent
involvedintheVlasov-Poissonsystemtowardadissipativesolutionoftheincom-
pressibleEulerequations.Themethodconsistsinestimating,throughitstime
derivative,asuitablemodicationofthethestandardenergyfunctional,whichis
obtainedbyintroducingintheenergyamodulationbyawell-adaptedtestfunc-
tion,inpracticethe(smooth)solutiontothelimitequation.Thismethodhas
connectionswiththerelativeentropymethodusedbyYau[23],andthemodulated
hamiltonianmethodintroducedbyGrenier[9]tosolveboundarylayerproblems.
Herewecanusesomespecialenergyfunctionals,mostinthespiritofTzavaras
estimatesfortheJin–Xinrelaxationmodel[22].
Thepaperisorganizedasfollows.InSection2wegivesomeanalyticalback-
groundsandstateourmainresult.EstimatesandproofsaregiveninSection3.
2.
Analyticalbackgroundsandstatements
Firstweshallstatetheexistenceofsmoothlocalsolutionsforsystem(1.4).
Theorem2.1.
Supposetheinitialdata
(
u
0
ε
(
x
)
,V
0
ε
(
x
))
aresmoothfunctionsbe-
longingto
H
s
for
s

2
.Then,thereexistsapositivetime
T
ε
,whichdependsonly
ontheinitialdata,andasolution
(
u
ε
,V
ε
,
ε
)

C
([0
,T
];(
H
s
)
3
)
tosystem(1.4).
Moreover,if
T
ε
<

,then
(2.1)lim
ε
||
(
u
ε
,V
ε
)
||
H
2
→∞
.
Tt→Theprooffollowseasilybyarguingasfortheclassicalwaveequation,byusing
energyestimatesandtheGagliardo–Nirenberginequalities,seeforinstance[10],
anditisomitted.
Inthefollowingweshallusethenorm
|
u
|
H
2
(
T
2
)
=
||
u
||
L
2
(
T
2
)
+
||
curl
u
||
L
2
(
T
2
)
+
||r
(curl
u
)
||
L
2
(
T
2
)
.
Letusrecallthat,since
r
u
=0,thisnormisequivalenttothe
H
2
n

orm.Moreover
weshalldenoteby
C
0
agivenpositiveconstantsuchthat
C
0
<a
.Finally
K
s
istheconstantwhichappearsintheSobolevinequalityintwospacedimensions,
underthenorm
||
H
2
(
T
2
)
.
Thestudyoftheasymptoticbehaviorofthesequence
u
ε
,as
ε
goestozero,leads
tothestatementofourmainresult.
Theorem2.2.
Let
T

0
and
U
0
beasmoothdivergencefreevectoreldon
T
2
.
Letalso
(
u
0
ε
,V
0
ε
)
beasequenceofsmoothinitialdataon
T
2
forproblem(1.4).
Assumemoreoverthatthereexistsaconstant
C
independentof
ε
suchthat
(2.2)
||
u
0
ε
||
H
1
(
T
2
)

C
C(2.3)
||
V
0
ε
||
H
2
(
T
2
)

εC0(2.4)
|
u
0
ε
|
H
2
(
T
2
)
<

εKs√Z(2.5)
|
u
0
ε
(
x
)

U
0
(
x
)
|
2
dx

Cε.
2T

4
Y.BRENIER,R.NATALINI,ANDM.PUEL
Then,
u
ε
isaglobalsolutionoftherelaxedsystem(1.4)andconverges,as
ε

0
,
in
L

([0
,T
]
,L
2
(
T
2
))
towardsthe(uniquesmooth)solution
U
oftheincompressible
NavierStokesequations(1.5)with
U
0
asinitialdata.Inaddition
√Zsup
|
u
ε

U
|
2
dx

C
T
ε,
t

[0
,T
]
T
2
where
C
T
dependsonlyon
T
,
U
,
C
and
C
0
.
3.
Proofofthetheorem
3.1.
Preliminaries.
First,weshallprovesomeenergyestimatesunderanapriori
assumptiononthe
L

normof
u
ε
.Thereforeweshallverifythatthisassumption
holdsactuallytrue.
3.1.1.
Theenergyestimate.
Letusgiveourbasicenergyestimate.
Proposition3.1.
Assumethatthereexists
T>
0
suchthat
||
u
ε
||
L


εa
forall
pt

T
.Then,setting
w
ε
:=
curl
u
ε
,wehavethefollowingestimates
1dZ(3.1)(
|
u
ε
+
ε∂
t
u
ε
|
2
+
ε
2
|

t
u
ε
|
2
+
εa
|r
u
ε
|
2
)
dx

0
,
2tddna1dZ(3.2)(
|
w
ε
+
ε∂
t
w
ε
|
2
+
ε
2
|

t
w
ε
|
2
+
εa
|r
w
ε
|
2
)
dx

0
,
2tdforall
t

T
.
Proof.
Letusmultiplyequation(1.6)by(
u
ε
+2
ε∂
t
u
ε
)toobtain,afterintegration
bypartsinspaceandwriting

t
u∂
tt
u
=

t
(
u∂
t
u
)

(

t
u
)
2
,
d
(1
|
u
ε
+
ε∂
t
u
ε
|
2
+
ε
2
|

t
u
ε
|
2
+
εa
|r
u
ε
|
2
)
dx
Z2td(3.3)
ZZ
+
ε
|

t
u
ε
+
r
(
u
ε

u
ε
)
|
2
dx
+
a
|r
u
ε
|
2

ε
|r
(
u
ε

u
ε
)
|
2
dx
=0
.
Then,since
||
u
ε
||
L


εa
,weobtain(3.1).
pForthesecondestimate,weconsidertheequationsatisedby
w
ε
.Sinceintwo
spacedimensionswehave
w
=

2
u
1


1
u
2
,then
(3.4)

t
w
ε
+
u
ε
r
w
ε

a

w
ε
+
ε∂
tt
w
ε
=0
.
Ifwemultiplythisequationby(
w
ε
+2
ε∂
t
w
ε
),weobtain
d
(1
|
w
ε
+
ε∂
t
w
ε
|
2
+
ε
2
|

t
w
ε
|
2
+
εa
|r
w
ε
|
2
)
dx
Z2td(3.5)
ZZ
+
ε
|

t
w
ε
+
u
ε
r
w
ε
|
2
+(
a
|r
w
ε
|
2

ε
|
u
ε
r
w
ε
|
2
)=0
.
Theconclusionfollowsaspreviously.

3.1.2.
L

bounds.
Letusproveauniform
L

boundfor
u
ε
,whichimpliesthe
assumptionmadeininthepreviousstatement.
Proposition3.2.
UndertheassumptionsofTheorem2.2,if
C0ε|
u
0
|
H
2
(
T
2
)
<

,
εKs

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