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STRONG SOLUTIONS FOR THE INCOMPRESSIBLE FLUID MODELS OF KORTEWEG TYPE

By Clifford Garcia,2014-02-18 23:35
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STRONG SOLUTIONS FOR THE INCOMPRESSIBLE FLUID MODELS OF KORTEWEG TYPEOF,of,FOR,THE,FLUID,TYPE,the,for,type,The

    STRONG SOLUTIONS FOR THE

    INCOMPRESSIBLE FLUID MODELS

    OF KORTEWEG TYPE

    AvailableonlineatWWW.sciencedirect.com

    ScienceDirect

    ActaMathematicaScientia2010,30B(3):799809

    数学物理

    http://actams.wipm.ac.an

    STR0NGS0LUT10NSF0RTHEINCoMPRESSIBLE

    FLUIDM0DELS0FK0RTEWEGTYPE

    TanZhong(谭忠)WangYanjin(王焰金)

    SchoolofMathematicalSciences,XiamenUniversity,Xiamen361005,China E-mail:ztan85@163.corn;yj-wang19@sohu.corn

    AbstractInthisarticle,weareconcernedwiththestrongsolutionsfortheincompress

    iblefluidmodelsofKortewegtypeinaboundeddomainQCR.Weprovetheexistence anduniquenessoflocalstrongsolutionstotheinitialboundaryvalueproblem.Wepoint outthatinthisarticleweallowtheexistenceofinitialvacuumprovidedinitialdatasatisfy acompatibilitycondition.

    KeywordsKortewegmodel;strongsolutions;incompressiblefluids;NavierStokesequa-

    tions;vacuum

    2000MRSubjectClassification76D03;76D05;35Q53;76D45

    1Introduction

    Thepurposeofthisarticleistostudytheexistenceanduniquenessoflocalstrongsolutions forsomeincompressibleviscouscapillaryfluids,thatis,weareconcernedwiththefollowing incompressiblesystem

    ?u+V=kpVAp+pf

in(0,T)×Q(1)

    whereQc.isabounded,smoothenoughdomainandtheunknownfunctionsp(t,),u(t,)

    (?,)denotethedensity,velocity,andthepressureofthefluid,respectively,>0isviscosity coefficient,k0capillarycoefficient,andfisagivenexternalforce. Wecomplementthissystem(1)withtheinitialdata

    O

    andtheboundaryconditiononu

    P0,(pu)l0=P0u0in2,

    U=0,Oil(0,T)×Q

    ReceivedAugust15,2007;revisedApril11,2008.SupportedbyNSF(10531020)ofChinaandtheProgram

    of985InnovationEngineeringonInformationinXiamenUniversity(2004

    2007)andNCETXMU

    lCorrespondenceAuthor

    ,=

    00

    0

    

    ,?????,,???【

    800ACTAMATHEMATICASCIENTIAVb1.3OSer.B

    Whenk:0thesystemf11i8reducedtotheincompressibleNavierStokesequations,

    denotedbyfINS1.TheexistenceofweaksolutionstoINScarlbeconsideredtobesolvedwell, buttheuniquenessiSstillanopenproblem,seeLionsf1].Inthecaseofstrongsolutions,forthe initialdensitywithapositivelowerbound,theexistenceoftheuniquestrongsolutionglobalor localintimecanbefoundinLions[1],Ladyhzenskaya[2],Salvi[3],Simon[4

    forinstance.When

    51providedalocaluniquesolutionif aninitialvacuumexists,itismorecomplicated,Choe

    theinitialdataPo,tl0satisfyingsomeregularitiesandacompatibilitycondition -

#Auo+7r0pf0rsome(0,)?H(Q)×L2(Q)anddiv(uo)=0

    When>0.namely,KortewegmodelswasintroducedforthefirsttimebyKorteweg 61.Uptoourbestknowledge,thereisnoresultsofstrongsolutionstothefullincompressible

    system(1),butwecanfindasimplermodelstudiedinSy[7]

    {ut+u.Vu-=#.A,u+V7r:P?

    Theyprovedthatthereexistsauniquestrongs.luti.n,whichislocalintimeforgeneralin dataorglobalintimeforthedatasmallenough.

    Theaimofournoteistoprovetheexistenceanduniquenessoflocalstrongsolutionto thefullsystem(1).Forsimplicity,weassumef=0,andobservethefollowingmanipulation pVAp=v(pa,)VpAp

    So,includingthetermIvY(pAp)inthepressureterm,wecanrewritethesystemunder considerationasfollows,

    (pu)t+div(puti)一?u+7r

    pt+Vp?U=0

    div(u):0,

    kVpAp(4)

    (5)

    (6)

    supplementedwithinitialboundaryconditions(2)and(3).Werefertothissystemas(K

    M)

    Now,wecanstateourmainresults:

    Theorem1LetP?(3,6],assumetheinitialdatepo,UOsatisfy

    0P0?W2,p(f1),U0?月(2)nH(Q)

    andthecompatibilitycondition

    ?u.+7r.+p0?p.:P{I9fors.me71-9)?H(Q)×L2(2),divu.)=0(8)

    Then,thereexistsatime?(0,,suchthatthereisauniquestrongsolution(P,11,7r)of (KM)in(0,)satisfyingtheregularities

    P?L..(0,;W2,p(Q)),Pt?L(0,;,(Q))

    NO.3,I1an&Wang:STRONGSOLUTIONSFORINCoMPRESSIBLEFLUIDMODELS801

u?L..(0,;(Q)nH(Q)),U?L2(0,;Lp(Q)),ut?L2(0,;(Q))

    Vlr?o.(0,;L2(2))nL2(0,;Lp(2))

    Remark1WecancheckasinthenextsectionsthatTheorem1alsoholdsforthecase ofexternalforcef?0,whichsatisfiesf?L2(0,;H(Q)),,c?(0,;.(Q)).

    Remark2IfQisaboundedsmoothdomaininR.,

    underthesameassumptionsof

    Theorem1exceptwearrangeP?(2,?),wecanshowthesameresultsenjoyingtheadditional regularitiesVU,7r?L(0,;Lq(f1)),Vq?【1,..).

    2APrioriEstimates

    Inthissection,wewillderiveaprioriestimatesforsmoothsolutionsinsuitablehigher normswithP>0in(0,T)×Q.

    Inwhatfollows,wedenotethenormsofLP(f1),wk,p(a),H(2)byI?lLp,1.1Wk1.1Hk,

    respectively,andCisagenericpositiveconstantdependingonlyonT,,andthenormsofthe initialdata,butindependentofthelowerboundofp0.

    Westartwithanelementarycalculus.First,multiplyingthemomentumequation(4)by U,integratingover(0,t)×Q,alsousingthecontinuityequation(5)andtheincompressibility onu,weobtain

    u(?)l.+IVp(t)l~.+/IVu(s)f~.dsC,Vt?[0,J0

    Next,equations(5)and(6)immediatelyimplythat

    P(t)IL.:lPOlL.,foranylq?,Vt?[0,T

    Now,weintroduceafunction西(?),aswewillsee,whoseboundnesshelpsUSderivethea prioruniformboundsonsmoothsolutions,

    ()=lVu(t)l~.+lut()l.+Ip(t)1wz,+1

    Wemayremindreadersthatwewillusefrequentlythefacts:西(?)1and3<P6

    Byusualsobolev'Sinequality,wecandeducefromtheexpressionof(11)that Vp(t)lLco(t)for2<qCO

    U(t)IL.+lVu(t)IL:c(?)for1q6

    (12)

    (13)

    Wearegoingtoestimatetheuniformboundsinhighernormsforthesmoothsolutionsin

termsof.Wewillestimateeverytermoftoderiveanestimateinwhichiscontrolledby

    someintegralof~(see(25)),thenapplyLemma2toprovethat

    islocallybounded,ultimately

    derivetheuniformboundswerequired.

    Step1EstimateforIVu(t)IL2

    Werewritethemomentumequation(4)inanonconservativeformas put+pu?Vu一?u+kVpAp

    802ACTAMATHEMATICASCIENTIAVoI.3OSer.B

    Multiplyingthisequationbyatandintegrating(byparts)overQ,wehave u)I.+lVu?l.(IIIIuIIulpII?pIIut1)

    lplLo.lulL.IYlllL.IlutlLe+klVpl~I?pILslu&e

    Then,integratingover(0,t)yields

    Vu(t)l~.

    c(()+1Vut(?)l.)

    c+(s)4d8+Cf0lut(s)l.ds

    Step2Estimateforlu(t)b/and1u(t)1w=,

    RecallingthattheclassicalregularityresultsonthestationaryStokesequations

    

    ?u+VTr:putp(u-Vu)kVpAp,div(u):0

    withtheDirichletboundaryconditiononuimply(seeTeman[8]) u()lClputp(u?u))kVpApI~.^)

    c(IprLo.f,/u}+fplo./I-I}uI2dx+Ivvl~..l?p}.)n

    andwecancompute

    /lul.lui.dxIu.}ulL.IuILeClVul~lulH.ClVul~.+Iu1刍—

    Then,theprevioustwoinequalitiesyield

    u()1((?)+(t).+(t))c().

    Inequality(17)andtheusualsobolev'Sinequalityimply u(t)lL+IVu(t)lLec(t).,6<q..

    UsingtheStokesequations(16)again,andthefact3<P6,weobtain

u(t)1.,CIputp(u?u)kVpApIL

    c(IplL}ut}+JlL..luJL..}VUlL+}pl..1?l,J)

    c(Ivut(?)lL+(t).+西(t))(?)+ClVut(t)l~z(19)

    Step3Estimatefor1v~U,(t)IL.andlVu~(t)lL.

    Becauseutinequation(14)ismultipliedbyPandPmayvanish,itseemsdifficulttoderive

    aprioriestimateforUinanapproPriatenorHi.Wemanaget0estimateVuinL2-Normand

    usesobolev'sinequalitytoovercomethisdifficulty. Differentiating(14)withrespecttot,multiplyingbyat,wehave p(l)+(Aut-ut+'ut

    :(p.u)(ut+u.Vu)utp(-t.u).ut一?p(p?u)?ut~A(vo?u)Vp-ut

    No.3Tan&Wang:STRONGSOLUTIONSFORINCOMPRESSIBLEFLUIDMODELS803

    Hence.integratingoverSZyields

    Iut12dx+utI.d

    /2plullutIlVutI+plullutIlVul.+plullutIIVul+luIIVulJVutl+plu~lIVuldx

    +IApV(vp.u)~utdxI+l一?(Vp.u)Vp"utdxI

    =

    ?厶,

    where

    121

    ..luIIuILo.IuI()+lu.,

    IplL~IuILo.IutIeIVuIL.IVuIL.西(t)+Iu()I.,

    IplL~IuIL..IuIL.IVuIL.IutIL.西()+IVut()I.,

    IplL..IulIVuLLLVutIL.c(t)+Iut(t)I.,

    ll1..IutIL.IutIIVuIL.c()+Iut(?).,

    /(IVpllull/XplluI+IVpllVull/XpllutI)I,n

    I.pl.lulL..IAplsIutI.+klVpl..IVulL.IAplLslutILe ().+lu.,

    (Vp-u)?(Vp.ut)

/pl.lulluI+IVpllV.pllVullutI+IpllVpllullVuf-I-IvplIVullVutIdxt,Q

    =

    ?,

    j=l

    IVpl.IuI..JuI().+IVu.,

    2IVplLo.IVplLsfvuILIuI().+lVu.,

    3IplL.IPlLo.IuIL..lutIL.?().+lut(t)I.,

    /73IVpl~lVulLzIVuI().+lut()..

    Summingupalltheaboveestimates,wederive pIutI(t)d+utI.()d西(.

    Then,integratingthisinequalityover(7-,t)c(0,T)yields

    plutI?d+/olu2dxdt<_II()dx+C9[~t)14(20)

    804ACTAMATHEMATICASCIENTIAV01.30Ser.B Incontrast,as

    plutl(pu?u+?u)不一kVpAp)-Ut

    usingYoung'Sinequality,weobtain

    u?d=-pu~Vu+#Au-VTr0-?u

    C/plulJuJ+pJ#AuV7r0)p?pJdxJ2

    Letting7-0+in(20),weconcludethat

    where

    u?d/0Iu2dxdt<_C+COo+C~oot)l4ds Co=C(9.,u.,)=l-#Auo+V7r0+kVpoAPo12dx=.<.. thankstothecompatibilitycondition(8). Step4Estimateforlp(t)1w~,p

    Toestimatethelasttermof(t),wedifferentiatethecontinuityequation(5)withrespect

    tOXi:

    (Px1)t+V.?U+Vp?11=0.

    Differentiatethisidentitywithrespecttoxjagain (Pz,)t+Vpz,'U+Vpz'uz,+Vp,.U+Vp'Uzz,=0

    Multiplying(22)bylP,lP,,integratingoverQandusing(5),wederive Bytheincompressibility(6)onu,wehaveJ1=0.Andrecallingthat3<P?6,wehave 21Vuloo1.pClu1wz,lp

    1Vpl..IvulLIv.plCluLw.,IV.p

    SummingupforiandJleadsto

    l2pI<elul1p1

    Thus,integratingdirectlyover(0,t),weobtain p(t)IPl2.pLexp{/oIulw.,(s)ds)_Cexp{c/0Iul.,(s)ds) (22)

    (23)

    

    2(

    +

    u.

    厂如一.?

    d一出

    No.3Tan&Wang:STRONGSOLUTIONSFORINCOMPRESSIBLEFLUIDMOD

    ELS805

    Recalling(10)togetherwith(23),wefind ()1w~,p-<Cexp(/0fu.,(s)ds)+

    Step5Theuniformbounds

    Now,combining(15),(19),(21),and(24),ityields (?)c++t(

    s)14ds-~-Cexp(+G+t(s)4ds)

    Inviewofthisinequality,wecaneasilydeducethatthereexistsasmalltime?(0,T

    aconstantCwhosesizedependsonlyon,T,I91L2,andthenormsoftheinitialdata(Po,

    butnotonthelowerboundofPo,suchthat

    Indeed,ifwedefine()

    ()C,Vt?(0,

    (s)Mds,then,wehave

?()[+CC.+()+exp(+CC.+())]M

    Thus,(26)followsfromthisinequalityandthefollowingLemma(SimonLernma6)

    Lemma2Letg?W,(0,T)andk?L(0,T)satisfy

    F(9)+in[0.(0)9.

    (24)

    (25)

    (26)

    whereFisboundedonboundedsetsfromintoR.Then,fromeveryE>0.thereexists

    >0independentofg,suchthat

    g(t)go+E,Vt

    Consequently,(26)togetherwith(11),(17),(19),and(21)yields SUP

    0<t<{lu()lHz+Iut()IL.+Ip@)J.,)+/olu(s)I+u(s)12~,pd8<

    3TheLocalExistence

    (27)

    Becauseaprioriestimatesinhighernormshavebeenderived ,

    theexistenceofstrong

    solutionscanbeestablishedbyastandardargument:First ,

    letPo,u0satisfyingthehy-

    pothesesofTheorem1andassumeforthemomentthatPo?C()andP0>0,

    viatheSOcalled"semi

    discrete"Galerkinscheme,wecanconstructapproximatesolutions P?C([0,);C.())

    Kim[9]),

    Um?C([0,);X)totheinitialdata(,u0)seeSimon[4or

    herewetakeourbasicfunctionspaceas

    {?础(Q)nH.(Q)ldive=0inQ)

    anditsfinitedimensionalsubspacesXaredefinedas X=span{~,,)cnC(),m=1,2

    m

    u

    ACTAMATHEMATICASCIENTIAVo1.30Ser.B where西i8themtheigenfunctionoftheStokesoperatorA=PAinX.Pbeingtheusual

    projectionoperatoronthedivergencefreevectorfields.Forallv?X. (pmu+pmumt,2

    Vu一?u+kVpmApm)-vdx=0

    p+Vp?u=0,

    div(um)=0,

    m

    u(0)=?(uo,)L.,pm(0)=P0

    :1

    Asintheprevioussection,wedefine ():lu()l+l,/u()1+Ipm(?)1w.,+1

    (28)

    (29)

    (30)

    (31)

    Usingthesameargumentasabove(seeChoe[5]also),wecarlderivethefollowingestimates

    similarto(9),(10),(15),(17),(19),(21),and(24)respectively:

    where

    t

    ,//u(t)I.+IVp(?)l.+/Iu(s)l.dsC,Vt?【0,],(32)0

    pm(t)JLqPOJL.,forany1q?,Vt?[0,,

    u)fc+(s)s+fu(s)I.ds

    lu)iH.c[@).,

    u(t)1w.,c[(?).+ClVu?(t)lL.

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