Hydrodynamic Limit for the Vlasov Navier Stokes Equations

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Hydrodynamic Limit for the Vlasov-Navier-Stokes Equations. Part II: Fine Particles Regime coro Thierry Goudon1, Pierre-Emmanuel Jabin2 and Alexis Vasseur3 1 Labo. Paul Painleve, UMR 8524 CNRS-Universite des Sciences et Technologies de Lille Cite Scientifique F-59655 Villeneuve d'Ascq cedex , France 2 Departement de Mathematiques et Applications, ENS 45, rue d'Ulm, F-75232 Paris 3 Labo. J.A. Dieudonne, UMR 6621 Universite Nice-Sophia Antipolis, Parc Valrose F-06108 Nice cedex 02 Abstract The paper is devoted to the analysis of a hydrodynamic limit for the Vlasov-Navier-Stokes equations.This system is intended to model the evolution of particles interacting with a fluid. The coupling arises from the force terms. The limit problem is the Navier-Stokes sys- tem with non constant density. The density which is involved in this system is the sum of the (constant) density of the fluid and of the macroscopic density of the particles. The proof relies on a relative entropy method. Key words. Fluid-particles interaction. Vlasov-Navier-Stokes equation. Hydrodynamic limits. Relative entropy method. AMS Subject classification. 35Q99 35B25 1

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HydrodynamicLimitfortheVlasov-Navier-StokesEquations.PartII:FineParticlesRegimecoroThierryGoudon1,Pierre-EmmanuelJabin2andAlexisVasseur31Labo.PaulPainleve´,UMR8524CNRS-Universite´desSciencesetTechnologiesdeLilleCite´ScientiﬁqueF-59655Villeneuved’Ascqcedex,Francethierry.goudon@univ-lille1.fr2De´partementdeMathe´matiquesetApplications,ENS45,rued’Ulm,F-75232Parisjabin@dma.ens.fr3Labo.J.A.Dieudonne´,UMR6621Universite´Nice-SophiaAntipolis,ParcValroseF-06108Nicecedex02vasseur@math.unice.frAbstractThepaperisdevotedtotheanalysisofahydrodynamiclimitfortheVlasov-Navier-Stokesequations.Thissystemisintendedtomodeltheevolutionofparticlesinteractingwithaﬂuid.Thecouplingarisesfromtheforceterms.ThelimitproblemistheNavier-Stokessys-temwithnonconstantdensity.Thedensitywhichisinvolvedinthissystemisthesumofthe(constant)densityoftheﬂuidandofthemacroscopicdensityoftheparticles.Theproofreliesonarelativeentropymethod.Keywords.Fluid-particlesinteraction.Vlasov-Navier-Stokesequation.Hydrodynamiclimits.Relativeentropymethod.AMSSubjectclassiﬁcation.35Q9935B251

1IntroductionInthispaper,weinvestigatemodelsofparticlesdispersedinanincompress-ibleviscousﬂuid.Theparticlesaredescribedthroughadensityfunctionf(t,x,v)≥0.TheevolutionoffisgovernedbythefollowingVlasov-typeequation∂tf+divx(vf)+divv(Ff)=rΔvf.(1.1)Theparticlesevolveinaﬂuidwhichisdescribedbyitsvelocityﬁeldu(t,x)∈RN.Thecloudofparticulesisassumedhighlydilutesothatwecansupposethatthedensityofthegasremainsconstant.Accordingly,uveriﬁesthefollowingincompressibleNavier-Stokesequation∂tu+Divx(u⊗u)+rxp−Δxu=F,div(u)=0.(1.2)xHereandbelow,foru=(u1,...uN)∈RN,weusethenotationu⊗utodenotethematrixwithcomponentsuiujwhereas,Abeingamatrixvaluedfunction,NNPDivx(A)=j=1∂xjAij∈R.Inviewoftheincompressibilitycondition,wehaveofcourse(atleastifuisregular)Divx(u⊗u)=(u∙rx)u.Here,equation(1.1)and(1.2)arewrittenindimensionlessform.Werefertothecompanionpaper[15]foradimensionanalysis.In(1.1),F(t,x,v)isassociatedtotheforcesactingontheparticlewhiletherighthandsidemodelsBrownianmotion.In(1.2)thefunctionF(t,x)isassociatedtothedensityofforcesexertedontheﬂuid.Equations(1.1)and(1.2)arecoupledthroughtheseforceterms.Theforcesactingontheparticlesaresupposedtoreducetothefrictionforceexertedbytheﬂuid,assumedtobeproportionaltotherelativevelocityF=F0(u−v),F0>0.TherighthandsidefortheﬂuidequationisthereforegivenbythesumZZF=−Ffdv=F0f(v−u)dv.(1.3)NNRRThispaperisdevotedtothestudyoftheasymptoticbehaviorofthiscoupledsystemwhenboththeforcetermsandthebrownianeﬀectsareverystrong,namely:r=F0=1/ε1.2

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