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STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

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STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTIONOF,TO,of

    STRONG COMPACTNESS OF

    APPROXIMATE SOLUTIONS TO

    DEGENERATE

    ELLIPTIC-HYPERBOLIC EQUATIONS

    WITH DISCONTINUOUS FLUX

    FUNCTION

1574ACTAMATHEMATICASCIENTIAVo1.29Ser.B

    Keywordsdegeneratehyperbolicellipticequation;degenerateconvection

    diffusion

    equation;conservationlaw;discontinuousflux;approximatesolutions;corn

    pactness

    2000MRSubjectClassification35J70;35K65;35L65;35B25 1Introduction

    LetQbeanopensubsetofR".InthedomainQweconsiderthequasilinearelliptic

    equation

    div(,,")D2?B()+(,)=0(1)

    whereD.?B()=02,

    6()weusetheconventionalruleofsummationoverrepeatedin- dexes),B(u)={():1isasymmetricmatrix.Weshallassumethatthismatrixisonly continuous:bij(u)?c(R),,J=1,,.Inthiscasetheellipticityof(1)isunderstoodin thefollowingsense

    B(u1)B(u2)0,l,U2?R,l>2,(2)

    thatis,forall??Rwehave(B(u1)B("2))?-?0(hereu?"denotesthescalarproduct

ofvectorsu,?R).

    Wesupposethat(,,f)=(1(,u),,(,f))isaCaratheodoryvector(i.e.,itis continuouswithrespecttoandmeasurablewithrespecttoX)suchthatthefunctions aM(X)m

    .<

    axI~p(x,u)l?.(Q)

    f0rallM>0(hereandbelow1.1standsfortheEuclideannormofafinite

    dimensionalvector)

    WealsoassumethatforallP?P,wherePCRisasetoffullmeasure.thedistribution div(,P)=7p?MI..(Q)

    whereM1.c(Q)denotesthespaceoflocallyfiniteBorelmeasuresonQequippedwith

    dardlocallyconvextopologygeneratedbyseminormsp()=Var(),with

    Co(Q).

    Thefunction(,)isassumedtobeaCaratheodoryfunctionon2×R,and

    ()maxI~(,)f?c()forallM>0

    (4)

    thestan.

    =

    西()?

    (5)

    Let=+bethedecompositionoftheintothesumofregularandsingular measures,SOthat=p(x)dx,03p?}0c(),andisasingularmeasure(supportedona setofzeroLebesguemeasure).WedenotebyJJthevariationofthemeasure,whichisa

    non-negativelocallyfiniteBorelmeasureonQ.

    Asusua1wedenote

    I1,u>0,

    sign(u)={1,u<0,0,:0.

    Now,weintroduceanotionofentropysolutionof(1).

    No.6H.H0ldeneta1:DEGENERATEEQUATIONSWITHDISC0NTINU0USCOEFFICIENTS1575

    Definition1Ameasurablefunctionu(x)onQiscalledanentropysolutionofequation (1)ift(,()),6t(()),(,(z))?.(Q),,J=1,',,andforalmostallP?Pthe

    Kru~kovtypeentropyinequality(see[10])

    div(sign(u()p)((,,"())(,p)))D?(sign(u(x)p)(B(())B)))

    +sign(u(x)p)[up(z)+(,())]ff0

    holdsinthesenseofdistributionsonQ(inthespaceD());thatis,forallnon-negative functionsf(x)?(Q)

    sign(()p)[((,())(,p))v/()+(B((z))B(p))?..,

    (z)+))+)dIGI()>0.

    WeusethenotationD.,forthematrix{,:1and

    n

    P?Q=TrPQ=?Pijqij

    i,J:1

    denotesscalarproductofsymmetricmatricesP={p}J:1,Q={qij}ind:1.Inparticular, (B(())B(p))?D.,=(bij(u)(p)),

    Inthecasewhenthesecondordertermisabsent()0)ourdefinitionextendsthenotion oftheentropysolutionforfirst?

    orderbalancelawsintroducedforthecaseofonespacevariable inf6,817seealso[7foronedimensionaldegenerateconvection-diffusionequations. Wealsonoricethatwedonotrequireu(x)tobeaweaksolutionof(1).If(z)?LO*(Q)

    and=0forP?P,thenanyentropysolutionu()satisfies(1)in(Q),i.e.,u()isa weaksolutionof(1).Indeed,thisfollowsfrom(6)withP>IlooandP<Ilull~.Butin

    general,entropysolutionsarenotweaksolutions,eveninthecasewhenthesingularmeasures

    ,yareabsent.Forinstance,asiseasilyverified,()=signxJJ/.isanentropysolutionof

    thefirstorderequation(.)=0onthelineQ=R,butitdoesnotsatisfythisequationin (R).

    Weassumethatequation(1)isnondegenerateinthefollowingsense:

    Deftnition2Equationf1)issaidtobenon.degenerateifforalmostallx?Qforall

    ??R,??0thefunctionsH??(,),_?B()???arenotconstantsimultaneouslyon non-degenerateintervals.

    Inthispaper,weestablishthestrongprecompactnesspropertyforsequencesofentropy solutions.Thisresultgeneralizespreviousresultsof[1215,17]tothecaseofquasilinear

    ellipticequations.

    Theorem3Supposethatu,k?N,isasequenceofentropysolutionsofthenon

    degenerateequation(1)suchthat

    l(,"())l+l(z,u())l+IB(,"k())I+m(())

    isboundedinc(Q),wherem()isanonnegativesuperlinearfunction?.Thenthereexists

    asubsequenceofk,whichconvergesinc(Q)toanentropysolutionu(x)of(1). 1)Anonnegativesuperlinearfunctionmsatisfiesm(u)-_'..38u__+..

    1576ACTAMATHEMATICASCIENTIAVl01.29Ser.B

    WeusehereandeverywherebelowthenotationIBIfortheEuclideannormofasymmetric matrixB,thatisIBI=B?B.

    Moregenerally,weestablishthestrongprecompactnessofapproximatesequencesuk(x)for non-degenerateequation(1).Theonlyassumptionweneedisthatthesequenceofdistributions

    div(,s.,b(())).D.?B(s.,6((z)))

    isprecompactintheSobolevspacec(Q)forsomed>1,foreach0,b?R,<b

    relation(78)below).Throughoutthispaperweuses.,b(U)todenotethecutofffunction

    s.I6()=max(a,min(u,6))

    (see

    ObservethatthenondegeneracyconditionisessentialforthestatementofTheorem3. Inthecaseoftheequationdiv~o(u)D-B(u)=0thisconditionisnecessaryforstrong

    precompactnessproperty.Forinstance,if??()andB()???areconstantonthesegment

    [a,bwith??R,f?0thenthesequenceuk(x)=[a+b+(ba)sin(k{?x)]/2ofentropy

    solutionsdoesnotcontainstronglyconvergentsubsequences.

    Wealsostressthatforsequencesofweaksolutions(withoutadditionalentropyconstraints) thestatementofTheorem3doesnothold.Forexample,thesequenceu=signsinzconsists ofweaksolutionsfortheBurgersequationut+()z=0(aswellasforthecorresponding stationaryequation(u2)z=0)andconvergesonlyweakly,whilethenon

    degeneracycondition

isevidentlysatisfied.

    Theorem3willbeprovedinthelastsection.Theproofisbasedongenerallocalization

    propertiesforultra-parabolicH--measurescorrespondingtoboundedsequencesofmeasure

    --

    valuedfunctions.Italsofollowsfromthesepropertiesthestrongconvergenceofvariousap

    proximatesolutionsforequation(1).

    Wedescribebelowoneusefulapproximationprocedure.Weassumeforsimplicitythat

    (,)0,bid(u)?C(R),i,J=1,,n.Asshownin[17],thereexistsasequencemx,u)?

    C~fft×R)suchthat(,f")(,)in(Q,C(R,R))whileforeachP?P,

    divzm(,p)=()+(),where()=j(z)inL1(Q)1Iz~(x)r:lweakly

    in.(Q).

    BytheellipticityassumptionA(u)=B(u)0,wecanchooseasequenceofsmooth symmetricmatricesA()={0())lsuchthatAcmI,>0(hereIistheidentity matrix),andforeachM>0

    Em/2

    l

    m

    .

    ax

    M

    lAin(u)(r")l0

    Thenwehavethelimitrelation

    (A(u)(u))(A(u))?/I0inc(a)

    MoreoverpassingtoasubsequenceofAmifnecessary,wemayachievethatforeachM>0

    andeverycompactKCQ

    I(m()())(m())/2l=0(/m(K,M+1)-1/.),m

    +..\/

    b

    <

?

    <<>

    UU

    m

    ,??I??,,?????\

    No.6H.Holdeneta1:DEGENERATEEQUATIONSWITHDISCONTINUOUSCOEFFICIENTS1577

    where

    ImKM,+Jdivxg~mp)ldpd

    Ingeneral,thesequence(,M)maytendtoinfinityasm__+...Weconsidertheapproxi

    mateequation

    divx,)A(u)Vu]=0,

    andsupposethatu=?2rn()isaboundedweaksolutionof(8)(forinstance,wecantake u=Ura)beingaweaksolutiontotheDirichletproblemwithaboundeddataat0a).This means(see[11,Chapter4])thatit?..(Q)nl.(Q),wherel.c(Q)istheSobolevspace

    consistingoffunctionswhosegeneralizedderivativesareinLLc(Q),andthefollowingstandard

    integralidentityissatisfied:Forallf=f(x)?(Q)wehave

    /[x,())A(u)V."()].Vf(x)dx=0

    t,Q

    WealsoassumethatthesequenceUmisboundedino.(Q).Undertheaboveassumptionswe establishthestrongconvergenceoftheapproximations.

    Theorem4Supposethatequation(1)isnondegenerate?Thenthesequencem(m--

    +oo

    u(x)in(Q),whereu=u()isanentropyandadistributionalsolutionof(1). WeremarkthatTheorem4allowstoestablishtheexistenceofentropysolutionsofbound

    aryvalueproblemsforequation(1)(aswellasinitialorinitialboundaryvalueproblemsfor evolutionaryequationsofthekind(1)).

    Forexample,in[17]weuseapproximationsandthestrongprecompactnesspropertyin ordertoprovetheexistenceofentropysolutionstotheCauchyproblemforanevolutionary

    hyperbolicequationwithdiscontinuousmultidimensionalflux.Thisextendsresultsof9],where

    thetwodimensionalcaseistreatedbythecompensatedcompactnessmethod. Wealsoremarkthatanotherapproachtoprovethestrongprecompactnesspropertyfor equation(1)basedonthekineticformulationandaveraginglemmaswasdevelopedin[21.

    But

    thisapproachcanbeappliedonlywhentheflux=(u)doesnotdependonx?R,and

    whenthefluxvectoraswellasthediffusionmatrixaresufficientlyregular. InSections2,3wedescribethemainconcepts,inparticulartheconceptofmeasurevalued

    functions,andintroduceanotionoftheHmeasure.Mostofthestatementsinthesectionsare takenfromf16].F0rcompletenesswealsoreproducetheproofsofthesestatements.InI161we consideredthestrongprecompactnesspropertyforthegeneralultra-parabolicequation div~(x,u)D?B(x,u)+(,u)=0,(10)

    whereitisassumedthatB(x,1isaCaratheodorymatrixvaluedfunction,whichsatisfiesthe

    ellipticityconditionsign(ul一札2)(B(z,U1)

    B(x,u2))0,anddegeneratesonafixedsubspace

    X(thatis,Xcker(B(x,u)B(x,0))).

    Wehavemorecomplicatedsituationin(1)sincethediffusionmatrixB=B(u),?R,

    degeneratesonasubspaceX=Z(u)dependingon?R.Still,sincethematrixB=

    B(p),P?R,iscontinuous,wewillbeabletoreduceourinvestigationonthebehaviorof thecorrespondingH?measureinaneighborhoodofafixedpointP0?Rfseethestatement

    1578ACTAMATHEMATICASCIENTIAVo1.29Ser.B

    ofTheorem25).Therefore,wewillbeabletousetechniquesfrom[16](ofcourse,inarather nontrivialmanner).

    ObservethatresultsanalogoustoTheorems3and4wereprovedin[16]forequation(10) underthestrongernondegeneracyassumption:

    Foralmostallz?Qandforall??X,??Xsuchthatf?0,专?0,thefunctions

    H??(,),HB(x,)-arenotconstantonnondegenerateintervals.

    HereXdenotestheorthogonalcomplementtothesubspaceX.

InSection4weprovethelocalizationpropertyfortheabovedefinedH-measurescorre--

    spondingtosequencesofmeasure

    valuedfunctions.Finally,inthelastSection5,theseresults areappliedtoproveourmaintheorems.

    2MainConcepts

    Recallfsee2,3,22])thatameasurevaluedfunctiononQisaweaklymeasurablemap HofthesetQintothespaceofprobabilityBorelmeasureswithcompactsupportin R.Theweakmeasurabilityof/2xmeansthatforeachcontinuousfunction|()thefunction

    xHr,()d()isLebesguemeasurableonQ.

    Remark5Ifisameasurevaluedfunction,then,aswasshownin13,thefunctions

    fg(A)dux(A)aremeasurablein2forallboundedBorelfunctions9().Moregenerally,if f(x,)isaCaratheodoryfunctionandg()isaboundedBorelfunctionthenthefunction rf(x,A)g(A)dux(A)ismeasurable.ThisfollowsfromthefactthatanyCaratheodoryfunction

    isstronglymeasurableasamapHf(x,?)?C(R)(see5,Ch.2])and,therefore,isa

    pointwiselimitofstepfunctions,mx,)=?gmi(x)hmt()withmeasurablefunctionsgmi(x) andcontinuous^i()sothatforz?Q.x,?)}f(x,?)inc(R).

    AmeasurevaluedfunctionissaidtobeboundedifthereexistsM>0suchthat suppC『一M,M]foralmostall?Q.Wedenotebyl1IIo.thesmallestvalueofMwith thisproperty.

    Finally,measurevaluedfunctionsoftheform/ix()=(一?()),where()isthe

    Diracmeasureconcentratedataresaidtoberegular;weidentifythemwiththecorresponding

    functions().Thus,thesetM(Q)ofboundedmeasurevaluedfunctionson52containsthe

    spaceo.fQ1.Notethatforaregularmeasurevaluedfunction()=(A())thevalue

    IIll?=ll1I?.ExtendingtheconceptofboundednessinL(2)tomeasure

    valuedfunctions,

    weshallsaythatasubsetAofM(Q)isboundedifsupl1_Io.<?.

    ??A

    Belowwedefinetheweakandthestrongconvergenceofsequencesofmeasure-valued functions.

Definition6Let?MV(a),k?N,andlet?M(Q).Then

    1)thesequencekconvergesweaklyto/2xifforeachf(A)?(R),

    /)d()/)()weakstinQ)

    2)thesequencekconvergestostronglyifforeachf(A)?(R)

    /.m)d()"--+00/m)dvx()in(Q)

    No.6H.HoldenetahDEGENERATEEQUATIONSWITHDISCONTINUOUSCOEFFICIENTS1579

    Thenextresultwasprovedin[22forregularfunctionsk.Theproofcanbeeasily extendedtothegeneralcase,aswasdonein[1a].

    TheoremTLetk,k?N,beaboundedsequenceofmeasurevaluedfunctions.Then thereexistasubsequence=k,k=,andameasurevaluedfunction?M(Q)such

    that__+//'xweaklya8r_...

    TheoremTshowsthatboundedsetsofmeasurevaluedfunctionsareweaklyprecompact. Ifuk(x)?(Q)isaboundedsequence,treatedasasequenceofregularmeasurevalued

    functions,anduk(x)convergesweaklytoameasurevaluedfunction/2xthenP'xisregular, L()=(u()),ifandonlyifuk(x)u(x)in.(Q)(see[22]).Obviously,ifuk(x) convergesto/'Ixstronglythenuk(x)__?u(x)=J'Ad(A)inL1.(Q)andthen()=(

    ()).

    WeshallstudythestrongprecompactnesspropertyusingTartar'stechniqueofHmeasures.

    T

    F(,")(?)/e27r'u()d,?R

    betheFouriertransformextendedasunitaryoperatoronthespaceu(x)?L(R),andlet

    S=Sn=(??Rn:l?l:1)betheunitsphereinRn.Denotecomplexconjugationof乱?C

    by西.

    TheconceptofH--measureassociatedtoasequenceofvector..valuedfunctionsboundedin

    L(Q)wasintroducedbyTartar[23]andGergrd[4]onthebasisofthefollowingresult.Fora

    fixedl?N,let()=((),,())?L.(Q,R)beasequenceweaklyconvergentto thezerovectorask__??.

    {

    osition7([2a,Thm.1.1)ThereisafamilyofcomplexBorelmeasures= onQ×Sandasubsequencex)=Uk(),=r,suchthat (,西l()西2(z)))./RF(1)(?)F(2)(?)()

    forau1(),O2(x)?Co(Q)and)?c(s).

    Thefamily={}:1iscalledthe日一measureassociatedto().

    Here,weshal1needmoregeneralvariantoftheHmeasuresdevelopedin161andbased ontheconceptoftheparabolicH-measuresrecentlyintroducedin[1J_

    SupposethatXcRisalinearsubspace,Xisitsorthogonalcomplement,P1,P2are orthogonalprojectionsonX,X,respectively.For??R,wewrite

    ?=?,?=P2?

    80that?X,?X,?=+.Let

    s(?R:ll+lI

    ThenSxisacompactsmoothmanifoldofcodimension1.InthecasewhenX:{0)or

    =

    RitcoincideswiththeunitsphereS=<?R:=l}.Letusdefinetheprojection 71"X:R"\{0)Sxby

    7rx()(

    +)1/

    p

    olI

    1580AcTAMATHEMATICASC!IENTIA

    ObservethatinthecasewhenX:{0)orX=R,

    W_edenote

    p(?)=(+).

    Thefollowingusefulpropertyoftheprojectionholds: Lemma8([16,Lemma1])Let,7?R,malX(p(?),p())1.Then

    lx()一不x()I61~-w1.

    ProofWedefinefor??R,>0.=+.Observethatforall>07r(?):

    7rx().Withoutloseofgeneralitywemaysupposethatp()p(),and,inparticular,p(?)1.

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