ON A RELAXATION APPROXIMATION OF THE INCOMPRESSIBLE NAVIER STOKES EQUATIONS

Mauro Picone

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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

Sujets

• Équation
• Has Been
• Energy

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 8
• Langue: English

Extrait

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.