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SOME DISCUSSIONS ON WELLPOSEDNESS OF THE EULER EQUATION

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SOME DISCUSSIONS ON WELLPOSEDNESS OF THE EULER EQUATIONON,OF,of,SOME,THE,EULER,SoME,the,ofthe,The

    SOME DISCUSSIONS ON

    WELLPOSEDNESS OF THE EULER

    EQUATION

    Ann.ofD.Eqs.

    23:2(2007),127130

    SoMEDISCUSSIONSONWELLPoSEDNESSoF

    THEEULEREQUATIoN

    ChenMin

    (Dept.ofMath.,ShanohaiUniversity,Shanghai2004) Abstract

    InthispaperanewformulationforEulerequationispresented,thewellpo- sednessforthenewequationinSobolevspacesisdiscussed. KeywordsEulerequations,wellposedness

    2000MathematicsSubjectClassification35L45,35L60,35L65 1Introduction

    FirstlookattheincompressibleEulerequation

    wherethedivergenceofis0.

    03

    +.V=p,

    diV=0,t=0,=(z),?Rn.

    (1)

    (2)

    Ithasbeenpointedoutinpreviousworkthat(1),(2)iswellposedlocallyinthe

    SobolevspaceHfor亿=3,furthermore,inthetwodimensionalcase,thereiSa universalsolutionundersomeconditions,thatis,ifC1iSbounded,seef10.1l1.Due

    totheimportanceoftheincompressibleEulerequation,thetheoreticalresearchaS

    wellasapplicationsofthiskindofequationhasbeenoneofthemostattractivetask formathematicians.YetitiSfarfrombeingsolvedtooursatisfactory.Amongmany approaches,thevorticitymethodishelpfulincomputation;thereisarelativelynew wayofdescribingtheincompressibleEulerequationWhichiSrelatedtothevorticity equation.ItWasfirstfoumulatedin11,andlater,studiedin121,theoutlineofthe

    formulationiSasfollows.

    Denition1.1ThevectorfieldMiScalledvelicity,ifitsatisfies: whereisthevelocity,satisfies

    M=+,

    ?lp=V?M.

    Ifisthesolutionofincompressible

    (3)

    (4)

    Eulerequation(1),thenthevelicityequation

    'ManuscriptreceivedJanuary17,2005;RevmedMarch17,2006

    127

128ANN.OFDIFF.EQS

    

    OMi+ff.VMMjOuj

    ,

    t=0,=(

    Vr0J.23

    (5)

    (6)

    follows,pleasereferto919fordetails.

    ThenewformulationhasaHamitonianstructurewhichwasconvenientfornu. mericalcomputation.Thereforeitdrewinterestsfromanscientists.Inf127,Buttke firstcomputedthevelicityequationandpresentedthenumericalschemebasedon thisvelicityequationin1992.ThenumericalmethodtransformsEulerequationinto

    thesolutionofODE,somerelatedworkwerediscussedinmanypapers,suchasf21. Ialsoconductedsomeresearchonsolvingthevehcityequationin91,itWaspointed

    outalsointhispaperthatthevehcityequationisweaklyhyperboliC.Thestudyof theexistenceanduniquenessofvelicityequationsmayaddnewunderstanding,in thatpapertheauthorusedconvergenceprincipleswhichisrelativelynewtostudy wellposednessofveUcityequation,anddifferentfromthetraditionalmethods.The purposeofthispaperistoshowthatthevehcityformulationcanalsobeusedto provetheexistenceofthesolutionundersimilarconditions.Theproofdependson someinequalities.Tobeself-contained,someestablishedinequalitiesareused,as jnf8.Ourdiscussionjsmaybeconductivetoourunderstandingandsubsequent studyiSneeded.InthisPaper.werestrictedtothetwodimensionalcase.Themain theoremiSasfollows.

    Theorem1.1Theinitialvalueproblem(5).(6)hasauniquesolution?

    C(O,TI,风一1)undertheassumptionMo?风(R2),S>2.

    2Proofofthe

    Toprovetheexistenceofthesolution

    thecurlofM,itsatisfiestheequation

    0f(5),(6),first,itisconvenienttotake

    

    +,

    +.:0,+'u,

    t=0,W=(.

    Definition2.1TheoperatoractingonS(R'I)isdefinedasfollows (9-)=(,(z)=(),j0(Z)=0,1.)=(?),(),(z)=0(),=,IzI1.

    (7)

    (8)

    Supposethat(7),(8)arenonlinear,ournextstepistostudythemodifiedequation :

    .

vJ~w,Ot-JeffVJeW——='

    t=0,W=(,

    (9)

    (1O)

N0.2ChertM..DISCUSSIONSONWELLPOSEDNESSOFEULEREQ.129

    where

    z)=J厂后(,(d=(,0),():lln,

    itistheonlypossiblesolutionifthefluidisstillatinfinity.Nowifweassumethat thefluidisstillatinfinity.

    Inthissection,wewillprovethewellposednessof(5),(6),ourproofdependson theconvergenceofthesolutionsof(9),(10).Toestablishthisresult,anestimateof commutatoroperatorswithH3normisfirstgiven,forsimplicitytheproofisgiven forthecasestobeaninteger.

    Lemma1lj[),IL2cII"l+~1II"l1,j'=m,m>-2,whereDfisthedrential

    operatoractingonD(n),suchthatDr/=f,misapositiveinteger?

    ProofBytheLeibnitzformulaff]flL2=C?[DiffDi'_;flL2,thelemma

    followsbytheSobolevimbeddinglemma.

    Corollarylj[)@]flL2C(1~lcIflk1

    Lemma2Ifl.<I/Is'-1I,I.

    +I~[klflL),Ifl=2.

    ProofI=_f(西Ij1dCbyholderinequality,

    ^=((81)Ij1,=Ij1(1'(8(1一引,Iraf21d<I^III.

    ByLemmas1and2,thewellposednessof(9),(10)iSobtained.

    ThereisaBanachspaceYC8,foranyw0?Y,thereexists.

    aT,suchthat

    (9),(10)hasasolutionin(0,,.).InfactY={叫?J?2,Jwit=

    II+IIL2).Firstlyitisclearthattherighthandsideof(9)isamappingfromY intoitself.LetW?Y,bythepropertyoftheFridrichsmodifier,I?wIk

    J?J,?//k+l,sOtheconclusionfollows.Secondly,asimilarargument showsthatitiSalsoLipschtzcontinuouswithrespectto.Theconstant,however, dependson,bythefixedpointtheorem,thereisaW?c(0,T,Hk),whereTis

    determinedbyII,.If1w01kC,forany,thereexistsatc,suchthatthe

    solutionexistsinttc,furthermore,sincetheequationremainstobeunchanged undertransformationt=ttthesolutionmaybeextendedtoanytas

    .

    1ongaS

    IIyC.

    =

    d

    

    (?vJ,w,J,W)=0,

    d

    

    (D14~,D14~)=

    (Dfi.JeW,DpleW)=(..JeW,DpleW),

    dlWl~c

130ANN.OFDIFF.EQs.,/,0j.23

    ThereforethereexistsaTwhichisindependentof,suchthatsolution(9),(10)

    exists.Sincein[0,,JJC,JJ七十lC.

    BythepropertyofweaksequentialconvergenceinHs,thereexistsaWsuch thatWweaklyin日七,k>-2.

    ItisclearthatW?..([0,,Hk)nLip([O,,Hk1).weaklyinHk+1.

    Wisthesolutionof(7),(8),thenthesolutionMof(5),(6)isobtained.The prooffortherealsissimilarandtheexistencehasbeenproved.Theuniqueness canbesimilarlyobtained.Inordertoestabhshtheresultforthecasek=2,by approximation,let?日3,suchthatW0inH2,thenthereexistsaTo,

fortTo,andeach,thereisasolution?风,correspondingto?凰,

    forfixedt,satisfying(7),(8).Thenbyequation(7),(t,:const,alongthe streamhne=(t,.Ontheotherhand,bythedefinitionof(t,,andthe

    propertyoftheZygmandintegraloperatorJJC,whereCisindependentof

    at[0,.ItiseasytoseethatII2C,whereCisindependentof.Therefore

    thereexistsaW?日2,?巩,suchthat,M,weakly,whichisthe

    solutionof(7),(8),Missolvedfrom(5),(6).Theproofiscomplete. References

    [1]OseledetsVIOnanewwayofwritingtheNavierStokesequation,TheHamiltonian

    formalism,Russ.Math.Surveys,44(1989),210211.

    E.Weinan,LiuJ.G.,Finitedifferenceschemesforincompressibleflowsinthevelocity

    impulsedensityformulation,J.Comput.Phys.,130:1(1997),6%76. MiaoC.X.,HarmonicAnalysisandPartialDifferentialEquations,Beijing:Scientific PublishingHouse,1999.

    MiaoC.X.,LectureNotesinHarmonicAnalysis,Beijing:ScientificPublishingHouse, toappear.

    f51MachioroC.,PulvirentiM.,MathematicalTheoryofIncompressibleNonviscousFluids, Berlin:SpringerVerlag.1994.

    f61ChenM.,Proceedingsofthe10thConf.onMordernMathematicsandMechanics, MMMIX,ShanghaiUniversityPress.2004:589.592.

    f71NishitaniT.,BookSeries:InternationalSocietyforAnalysis,ApplicationsandCom- putation,V.o1.i0(GilbertR.P.etceds.),GluwerAcademicPublishing,2003,16. I8TaylorM.E.,PseudodifferentialOperators,Princeton:PrincetonUniversityPress,

    1981.

    9ChenM.,mericaMathematicaSinica,21:2(1999),171180.

    [10]TaylorM.E.,PartialDifferentialEquations,Berlin:Springer,1996. [1lMajdaA.,CompressibleFluidFlowandSystemsofConcervationLawsinSeveralSpace Variables,NewYork:SpringerVerlag,1984.

    [12T.F.Buttke,VelicityMethods:LagrangianNumericalMethodswhichPreservethe HamiltonoanStructureofIncompressibleFluidElow,inJ.T.Beale,G.H.Cottetand

S.Huberson,VortexFlowsandRelatedNumericalMethods,KluwerVerlag,Norwell,

1993.

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