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MAXIMAL ATTRACTORS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS OF VISCOUS AND HEAT CONDUCTIVE FLUID

By Melissa Price,2014-01-11 15:22
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MAXIMAL ATTRACTORS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS OF VISCOUS AND HEAT CONDUCTIVE FLUID

    MAXIMAL ATTRACTORS FOR THE

    COMPRESSIBLE NAVIER-STOKES EQUATIONS OF VISCOUS AND HEAT

    CONDUCTIVE FLUID

290ACTAMATHEMATICASCIENTIAV0l_30Ser.B

    Cp(OtO+vOTO)一一

    

    fa_{j+12().

    +RPfu+

    1)v

    r2

    0

    whereP,n1,0arethedensity,velocity.andabsolutetemperature)respectively;andarethe

    constantviscositycoefficients.R,Cv,andarethegasconstant,specificheatcapacity,and

    thermalconductivity,respectively.TheysatisfythatR,,,>0,+2#/n0.In(1.2),we

    define=+2it.Weshallconsiderproblem(1.1)(1.3)intheregion{r?Gn.to)subject

    tothefollowinginitialandboundaryconditions or

    ,,(r,0)=po(r),fj(r,0)=UO(r),(r,0)=0(r),r?G

    tJ(f,t)=v(b,t)

    v(a,t)v(b,t)

    0,(.,t)=Or(6,t)=0,t0

    0,O(a,t)O(b,)To=const.>0,t0

    Theequations(1.1)(1.3)describethesphericallysymmetricmotionofaviscousandheat

    conductivepolytropicidealgasintheannulardomain{??la<Izi<6}inthecaseof

    n=2orofn=3(cf.f7,8,17,18,31,40,42,54]).Inwhatfollowswefirsttransferproblem (1.1)(1.5)or(1.1)(1.4)and(1.6)tothatintheLagrangiancoordinatesandobtainsome results.Then,wereturntotheEulercoordinatesanddrawthecorrespondingconclusions. ItisknownthattheEulercoordinates(r,t)areconnectedtotheLagrangiancoordinates (?,()bythefollowingrelation

    where(,t)=(r(?,t),t)and

    r(?,t).()+/(?,d0

    r0(?)一卵一(?),(r)np.()d,?Gt,0

    Itfollowsfromequation(1.1)andboundarycondition(1.5)or(1.6)that whichimplies

    kpl8.

    ?n

    t)ds

    andGnistransfermedtof2n:(0,L)with

    :

    p(,?0

    .np.(s)ds0

    t)ds

    ?

    .snp.(s)d

    ?0

    Moreover,differentiating(1.10)withrespectto?yields

    r(?,)=(?,t)n-1p(r(?,t),)]

    No.1Y.M.Qin&J.P.Song:ATTRACTORSFORNAVIER-STOKESEQUATIONS291

    Ingeneral,forafunction(r,t),ifwedenote(?,t)(r(?,t),t),then,weeasilyget

    (?,t)=(r,t)+vOT(r,(),

    )=0re())=

    Intheseque1,withoutdangerofconfusion,wedenote(,stillby(,u,)and(,)byx,).

Weuseu:1/ptodenotethespecificvolume.Thus,by(1.12),theequations(1.1)(1.3)in

    theEu1ercoordinatescanbewrittenintheLagrangiancoordinatesinthenewvariables(x,),

    z?Q,t0asfollows(seealso[17,18,40,42,54]):

    Cv0t=(r2n--20

    (

    (rn--1").

    9(rn--1u)R)

    )+((r1)R)(rn-lv)2(扎一1)(rn--2V2)

    subjecttothefollowinginitialandboundaryconditions or

    u(,0):u0(),u(z,0):u0(),o(x,0)=o(),?Qn

    v(o,t)=v(L,t)=0,Ox(0,t)=Ox(,t)=0,t0

    u(0,t)=((,t)=0,o(o,t)=O(L,t)=To,t0

    Inviewof(1.7)and(1.11),weeasilydeducethatr(x,t)isdeterminedby rt(x,t)(z,t),r}:.=()=an+nfoo~uo()d)

    (z,)(X,t)(z,t)

    (1.16)

    (1.17)

    (1.18)

    ThroughoutthisarticlewealwaysassumethatAandsatisfyseealso[18,40,42,54])

    nA+2>0

    (1.20)

    N0w,letusrecallsomerelatedresultsintheliterature.Intheonedimensionalcase,forthe initia1boundaryvalueproblemsinboundeddomainsitisknownthat,forarbitrarylargedatu

    m,

    auniqueglobalfgeneralizedorsmooth)solutionexistsandconverges(exponentially)toastea

    dy

    stateastimegoestoinfinity(see1,2,15,16,19,20,22,30,32,3437,39,41Jandreferences citedtheref0rtheCOUnterpartinnonlinearthermoviscoelasticity(see,e.g.,

38,44,471)).For

    theonedimensionalCauchyproblem,see,e.g.,12,18

    21,52,55]andreferencescitedtherefor

    theresultsonglobalexistenceandasymptoticbehaviourastimegoestoinfinity. Intwoorthreedimensionalcase,theglobalexistenceandlarge-timebehaviourofsmooth solutionStotheequationsofaviscousandheatconductivepolytropicidealgasingerneral

    domainswereinvestigatedonlyforsufficientlysmallsmoothinitialdata(see,e.g.,3,24

    29,33,

    511).Forthesphericallysymmetricmotionofsuchagasinaboundedannulardomain(which 292ACTAMATHEMATICASCIENTIAVol30Ser.B

    isdescribedbyequations(1.1)(1.3))inanexteriordomain,theexistenceandumquenessof globalgeneralizedsolutionsforarbitrarylargeinitialdatawasprovedin(7,8,17,31

    forvarious

    boundarvconditions.Forthismodel,theasymptoticallyexponentialstabilityofglobalsolutions

    infi:1,2,4)wereestablishedinf18,40,42],andtheexistenceofmaxima1(uniVersa1) atrractorsinH.fi=1,2)for(1.5)or(1.17)wasobtainedinl541.

    Asfarastheassociatedinfinitedimensionaldynamicsisconcerned,wereferto[i0,49J

    andthereferencescitedtherefortheNavierStokesequationsofincompressiblefluid.For compressiblefluid,werefertotherecentworks[43,53,54]fortheresultsonexistenceof maximalfuniversa1)attractorsfortheviscousandheatconductivepolytropicidealgasand

    realgas.wehavenotedtheworksin4,5,13,14,45].In[13,l4,theauthorsestablished

    theexistenceofacompactattractorfortheone

    dimensionalisentropiccompressiblefluidina

    niteinterva1.In4,5]theisentropiccompressibleviscousflowinaboundeddomainin isinvestigated.However,sincetheywerebasedonthefundamentalresultsonglobalexistence

    ofweaksolutionsbyLionsf231,theuniquenessisnotknown,itisimpossibletoadoptthe usualsolutionsemigroupapproach.Asaresult,theauthoradoptedanapproachofbypassing

thepossiblenonuniquenessoftheweaksolutions,whichwasborrowedfrom[45wherethe

    globalattractorsforthethreedimensionalisentropicNavier——

    Stokesequationswasobtained.

    Thereforethepresentpaperisquitedifierentfrom

    4,5Iinthefollowingaspects:nonisentropic

    flowviaisentropicflow;sphericallysymmetricmotionvianon-sphericallysymmetricmotion;

    solutionsemigroupviasimpletimeshift.Recently,ZhengandQinestablishedtheexistenceof InaximalattractorsinH(=1,2)forapolytropicone

    dimensionalviscouspolytropicidealgas

    inf531andforasphericallysymmetricviscousidealgas(1.1)

    (1.5)inboundedannulardomains

    inn=2,3)in54.QinandRivera[43]provedtheexistenceofuniversalattractorsfor one

    dimensionalVISCOUSheatconductiverealgas.Moreover,wealsomentionedsomeother relatedresults1l46,481inthisdirection.

    Nowwewouldliketoexplainsomemathematicaldifficultiesinstudyingthedynamicsof thisproblem.

    First.fromphysicalreasons,thespecialvolumeuandtheabsolutetemperature0should beDositiveforalltime.Theseconstraintsgiverisetosomeseveremathematicaldifficulties. F0rinstance,wemustworkonincompletemetricspacesH()(i1,2,4),H(4)C(.)cH()

    whichareusualSobolevspaceswiththeseconstraintg.

    Secondly,thenonlinearsemigroups(t)definedbyproblem(1.12)(1.17),whereLisfixed,

    mapseachof(11,

    H('21,

    and(intoitself,asprovedin[40,42].Itisclearfromequations

    (1.14)and(1.15)thatwecannotcontinuouslyextendthesemigroups(t)totheclosureof H(1.

    (.

and(.

    Noticethefollowingsignificantdifferencesbetweenthestudyofglobal

    existenceandthestudyofexistenceofa(maxima1)universalattractor:forthestudyofglobal existence.theinitialdatumisgivenwhileforthestudyofexistenceofa(maxima1)universal attractorincertainmetricspace,theinitialdataarevaryinginthatspace.Sincethe(maxima1) universalattractorisjustthelimitsetofanabsorbingsetinweaktopology(see[9]forthe

    moreprecisedefinition),therequirementoncompletenessofspacesisneeded.Toovercome thisseveremathematicaldifficulty,werestrictourselvestoasequenceofclosedsubspacesof (11,

    H(,and(4)fseethedefinitionbelow).Itturnsoutthatitisverycrucialtoprovethat theorbitstartingfromanyboundedsetofthisclosedsubspacewillreenterthissubspaceand

    No.1Y.M.Qin&J.P.Song:ATTRACTORSFORNAVIER-STOKESEQUATIONS2

    93

    staythereafterafinitetimewhichshouldbeuniformwithrespecttoallorbitsstartingfrom boundedset;otherwise,thereisnogroundtotalkabouttheexistenceofabsorbingsetand maximal(universa1)attractorinhissubspace.Theproofofthisfactbecomesanessentialpart ofourpaperanditwillbedonebydelicateaprioriestimates,usingthespiritsfrom[42,43, 53,54]_

    Thirdly,twoquantities,i.e.,thetotalmassandenergy(for(1.17))areconserved(see (2.1)(2.2)).Thesetwoconservationsindicatethattherecanbenoabsorbingsetforinitial datavaryinginthewholespace.Instead,weshouldratherconsiderthedynamicsinasequence ofclosedsubspacesdefinedbysomeparameters.Inthisregard,thesituationisquitesimilarto thoseencounteredforsingleCahn

    Hilliardequationintheisothermalcase(see[501),andforthe

    coupledCahnHilliardequations(see[46])andforone

    dimensionalandmultipledimensional

    polytropicviscousidealgas(see[43,53,541).Therefore,oneofkeyissuesinthepresentpaper ishowtochoosetheseclosedsubspaces.

    Fourthly,(1.12)(1.14)isahyperbolicparaboliccoupledsystem.Itturnsoutthatingen

    eraltheorbitisnotcompact.Toovercomethisdifficulty,wewilladoptanapproachmotivated

bytheideasin91and43,53,54].

    Finally,unliketheonedimensionalcase,equations(1.13)(1.15)lookmorecomplicated

    thantheonedimensionalcounterpartandtheyexplicitlyinvolver,which,inturn,should satisfy(1.19)

    (1.20).Inotherwords,weareessentiallyconsideringasystemoffourequations withfourdependentvariables,v,0andr.Ontheotherhand,itseemsthatthetreatmentfor (1.18)ismorecomplicatedthanthatfor(1.17)andmorehigherderivativeswillbeinvolved.It turnsoutthatmuchmoredelicateestimatesareneeded.

    Theaimofthisarticleistoestablishtheexistenceofmaximal(universa1)attractorsinH f0r(1.17)andinH()(i=1,2,4)for(1.18),byexploitingtheabstractframeworkin[9]and someideasin[42,43,53,54].

    Now,wefirstbegintostudytheproblem(1.13)(1.17)or(1.13)(1.16)and(1.18)where L>0isfixed.andintroducethreespaces

    (1)

    H(2)

    {(,fJ,0)?H[0,L]×H0,L]×H0,L:u(x)>0,O(x)

    vl:o=vl:L:0,01:o=01:L=To>0for(1.18)),

    {(,",0)?H[0,L】×H[0,L】×H[0,L:u(x)>0,O(x)

    1:o=ul:L=0,02l:o=I:,J=0for(1.17)

    or01:o=01:,J=Tofor(1.18)),

    >0,x?[0,L]

    >0,x?[0,,

    ()={(",u,0)?H[0,L]×H[0,L]×H0,L:()>0,O(x)>0,x?[0,,

    ul:o=uI:L=0,l:o=1:L=0for(1.17)

    or01:o=01:L=TOfor(1.18)),

    whichbecome

    norms.Inthe

    Let(i=

    three

    above

metricspaceswhenequippedwiththemetricsinducedfromtheusual

    H.H.andHaretheusualSobolevspaces.

    ,8)beanygivenconstantswith51<0,58>0,52>0,0<55<52,54

    294ACTAMATHEMATICASCIENTIAVo13OSer.B

    n1aX

    where

    2(262/CvL).y

    .)

    3I>0,56>0,7>0,andlet ,

    )?日(i):/oL(l.g()+Rlog())d,

    

    L

    (/2)r(1_17),/0(/2)s

    for(1.18)Ludx<($4,254 ,e?

    6

    66<To<67for(1.18) d5

    2CvL7:f0r)CO

    v

    

    L.it.1rJ

    i=1.2.4

    E(_",o)=(",o)(,To)(,)(札一面)Cfo(u,)(口一To) =

    (,0)=e=CvOO(Cvlog0+Rlogu) ee(",0):e(,S)=CvOinternalenergy S:S(,0):Cvlog0+Rloguentropy

    (1.22)

(1.23)

    f1.24)

    (1.25)

    (1.26)

    (1.27)

    Clearly,.isasequenceofclosedsubspacesof(.)(i=1,2,4).Wewillseelateronthat

    thefirstfourconstraintsareinvariant.However,thelasttwoconstraintsarenotinvariant, whichareustintroducedtoovercomethedifficultycausedbythefactthattheoriginalspaces H(0fi:l,2,4)areincomplete.Weshouldpointoutherethatitisverycrucialtoprovethat theorbitstartingfromanyboundedsetof(i=1,2,4)willreenter(i=1,2,4)after

    afinitetime.

    Thenotationinthisarticlewillbeasfollows.

    (p1<p<+..,",m?N,=r='denotetheusual(Sobolev)spaces

    on(0,L).Inaddition,I1.1lBdenotesthenorminthespaceB;wealsoputll_ll=lj.1lL2. WedenotebyC(,B),k?N0,thespaceofk-timescontinuouslydifferentiablefunctionsfrom IcintoaBanachspaceB,andlikewisebyLP(I,B),1P..thecorresponding

    Lebesguespaces.Subscriptstandxdenotethe(partia1)derivativeswithrespecttotandx, respectively.Weuse(i=1,2,3,4)tostandfortheuniversalconstantdependingonlyon the()I1()rmofinitia1data_

    ?

    m

    

    in1u0(z)anQ

    ?

    m

    [

    in

    ]

    ()?Ca

    xoL0L

den.eheuniVealconant

    ?『1?I

    dependingonlyon(J=1,,8),butindependentofinitialdata.Ca,B(i:1,2,4)denotes theuniversalconstantdependingonboth6j(j=1,2,,8),normofinitialdatawith

    fi("0,.,00)IIHB100()jand部】"0()'

    o1rrnaljnres11tsread.a.sflouows.

    Theorem1.1Thenonlinearsemigroups(t)definedbythesolutionto(1.13)?(1.17)or

    (1.13)(1.16)and(1.18)mapsH)(i=,2,4)intoitself.M,

    oreover,forany(i=1,,8)

    with1<0,2>0,8>0,4maxl互互i,3I>0,0<5<82,0<6<7 definedin(1.22)(1.23),itpossessesinamaximal(universa1)attractori,(i=1,2,4for (1.18)andi=4f0r(1.17)).

    Remark1.1Fortheboundaryconditions(1.5)or(1.17),theexistenceofmaximal attractorsinH(i=1,2)wasestablishedinf541.

    NO.1Y.M.Qin&J.P.Song:ATTRACTORSFORNAVIER-STOKESEQUATIONS295

    Remark1.2Seef9]andalsoSection3inthepresentarticleformoreprecisedefinition of(maxima1)universalattractor.

    Remark1.3ThesetAi=U,(i=1,2,4)isaglobalnoncompact

    1,2,3,4,65,66,67,as

    attract0rinthemetricspace{.inthefollowingsensethatitattractsanyboundedsetsof H:.)withconstraintsit1,0_,2with1,2beinganygivenpositiveconstants. Now.wegobacktotheEulercoordinatesandconsiderproblem(1.1)(1.5)or(1.1)(1.4)

    and(1.6)withG=(n,b)beingfixed.Let

    H{(p,?j,)?日[n,酬×何[n,6]×【n16]:/snpds0

    Ja

    z?[a,6],ul:.u!:b=0,oln

    L,p()>0,O(x)>0

    :6=Tofor(1.6))

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