16A hierarchy of hydrodynamic models for plasmas.

AHierarhyofHydrodynamiModelsforPlasmas.Quasi-NeutralLimitsintheDrift-DiusionEquationsAnsgarJungelFahbereihMathematik,TehnisheUniversitatBerlin,Straedes17.Juni136,D{10623Berlin,Germany,e-mail:jungelmath.tu-berlin.de,andLaboratoireDieudonne,UniversitedeNie,BP71,F{06108NieCedex2,FraneYue-JunPengLaboratoiredeMathematiquesAppliquees,CNRSUMR6620,UniversiteBlaisePasal,F{63177AubiereCedex,Frane,e-mail:pengufma.univ-bplermont.frAbstratThispaperisaontinuationofaseriesofpapersinwhih(quasi-)hydrodynamimodelsforplasmasarerigorouslyderivedbymeansofasymptotianalysis.Here,thequasi-neutrallimit(zero-Debye-lengthlimit)inthedrift-diusionequationsisperformedinthetwoases:weaklyionizedplasmasandnotweaklyionizedplasmas.Themodelonsistsoftheontinuityequationsfortheeletronsandions,theonstitutiverelationsforthepartileurrentdensities,andthePoissonequationfortheeletrostatipotentialinaboundeddomain.Intheaseofaweaklyionizedplasma,theontinuityequationfortheeletronsisreplaedbyarelationbetweentheeletrostatipotentialandtheeletrondensitysuhthatthePoissonequationbeomesnonlinear.TheequationsareomplementedbymixedDirihlet-Neumannboundaryonditionsandinitialonditions.Thequasi-neutrallimitsareshownwithoutassumingompatibilityonditionsontheboundarydensities.Theproofsrelyontheuseoftheso-alledentropyfuntionalwhihyieldsappropriateuniformestimates,andompensatedompatnessmethods.Keywords.Asymptotianalysis,singularperturbation,drift-diusionequations,plas-mas,entropymethod,boundarylayers.1991MathematisSubjetClassiation.35B25,35B40,35K57.Aknowledgments.TheauthorsaknowledgesupportfromtheDAAD-PROCOPEPro-gram.TherstauthoraknowledgessupportfromtheTMRProjet\Asymptotimeth-odsinkinetitheory,grantnumberERB-FMRX-CT97-0157,andfromtheGerhard-HessProgramoftheDFG,grantnumberJU359/3-1.TheresearhoftheseondauthorwaspartiallysupportedbytheTMRnetworkoftheProjet\Hyperbolisystemsofonser-vationlaws,grantnumberERB-FMRX-CT96-0033.TheseondauthorwouldliketothankE.Grenierforvaluabledisussions.11IntrodutionReently,westartedaprogramtoderiverigorouslyahierarhyofmarosopimodelsforplasmas.Inpartiular,weareinterestedinjustifyingmathematiallyvariousasymptotilimitsinthehydrodynamiandthedrift-diusionequationsusedinplasmaphysis,namely(i)thezero-relaxation-timelimit,(ii)thezero-eletron-masslimit,and(iii)thequasi-neutrallimit.Inthispaperweontinueourprogrambystudyingthequasi-neutrallimitinthedrift-diusionequations.Werefertothepapers[14,18,19,20℄forthelimits(i)and(ii).Consideranunmagnetizedplasmaonsistingofeletronswithdensityneandofasinglespeiesofpositivelyhargedionswithdensityni.Wesupposethatthepartiledensitiessatisfythedrift-diusionequationsoupledself-onsistentlytothePoissonequationfortheeletrostatipotential(system(DD-EI)):tnidiv(rpi(ni)+inir)=0;(1.1)tnediv(rpe(ne)ener)=0;(1.2)2=nine(1.3)inanopenandboundeddomainRd(d1).TheboundaryofthedomainissupposedtoonsistoftwodisjointsetsDandN.Thesystem(DD-EI)isom-plementedbymixedDirihlet-Neumannboundaryonditionsandinitialonditions:n=nD;;=Don(0;T)D;(1.4)rp(n)=r=0on(0;T)N;(1.5)n(0)=nI;in;(1.6)for=e;i.Here,pe(ne),pi(ni)denotethepressurefuntionsoftheeletronsandions,respetively.Thefuntionisthenormalunitvetorofwhihisassumedtoexistalmosteverywhere.Thephysialparametersarethe(onstant)mobilitiese0,i0andthe(saled)Debyelength0.Themodel(1.1){(1.3)anbederivedfromthehydrodynamiplasmaequationsinthezero-relaxation-timelimit[18,19℄.Ourrstmaingoalistoperformthelimit!0inthesystem(DD-EI)rigorously.Whenthekinetienergyoftheeletronsismuhsmallerthantheirthermalenergy,theeletronontinuityequation(1.1)reduesasymptotiallytotherelation(see[20℄fordetails)0=he(ne)e;wherehistheenthalpyfuntiondenedbysh0(s)=p0(s)fors0andh(1)=0;=e;i:Henetheeletrondensityisgivenbytherelationne=fe(),wherefe(s)=h1e(es)2andsolvesthenonlinearPoissonequation2=nife():Inthissituation(‘weaklyionizedplasma’),theiondensityandtheeletrostatipotentialsolvetheproblem(DD-I):tnidiv(rpi(ni)+inir)=0;(1.7)2=nife()(1.8)subjettotheboundaryandinitialonditions(1.4){(1.6)for=i.Ourseondmaingoalistostudyrigorouslythelimit!0in(DD-I).Beforestatingourmainresults,weperformthequasi-neutrallimit!0for-mallyinthesystems(DD-EI)and(DD-I)toderivethelimitproblems.In(DD-EI)wegetfor=0:ndef=ne=ni;tndiv(rpi(n)+inr)=0;tndiv(rpe(n)enr)=0:Addingandsubtratingtheseequationsleadsto2tndiv(r(pi(n)+pe(n))+(ie)nr)=0;div((i+e)nr)=(pi(n)pe(n)):Insertingtheseondequationintherst,weaneliminatethedriftterm,andthenalsystem(QN-EI)beomestnp(n)=0;(1.9)div((i+e)nr)=(pi(n)pe(n));(1.10)wherep(n)=ei+epi(n)+ii+epe(n):(1.11)Asanexample,letusassumethattheeletronsandionsaredesribedbythesamedensity-pressurerelation,butwithdierentdiusivities,whihaountsforthedierentmasses,i.e.p(s)=Dp0(s);=e;i:ThenEqs.(1.9){(1.10)beometnDap0(n)=0;div(nr)=Dbp0(n);whereDa=Dei+Diei+e;Db=DiDei+e:3Thismeansthatthenewdiusivityofthequasi-neutralplasmaisgivenbytheso-alledambipolardiusionoeÆientDa.Thisrelationiswell-knowninplasmaphysis[7,p.160℄.Thequasi-neutr