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AppliedMathematicsandMechanics(EnglishEdition),2007,28(5):581592

    ;@EditorialCommitteeofApp1.Math.Mech.,ISSN0253-4827

    ;Ontwo-dimensionallarge-scaleprimitiveequations

    ;inoceanicdynamics(I)

    ;HUANGDaiwen(黄代文),,GUOBo-ling(郭柏灵)

    ;(1.InstituteofAppliedPhysicsandComputationalMathematics,Beijing100088,P.R.China ;2.GraduateSchool,ChinaAcademyofEngineeringPhysics,Beijing100088,P|R,China) ;(ContributedbyGUOBo-ling)

    ;AbstractTheinitialboundaryvalueproblemforthetwo-dimensionalprimitiveequa- ;tionsoflargescaleoceanicmotioningeophysicsisconsidered.Itisassumedthatthe ;depthoftheoceanisapositiveconstant.Firstly.iftheinitialdataaresquareintegrable ;thenbyF{eo-Galerkinmethodtheexisteneeoftheglobalweaksolutionsfortheprob- ;lemisobtained.Secondly.iftheinitialdataandtheirverticalderivativesareallsquare ;integrable.thenbyFaedo-Galerkinmethodandanisotropicinequalities,theexisterceand ;uniquenessoftheglobalweaklystrongsolutionfortheaboveinitialboundaryproblem ;areobtalned.

    ;Keywordsprimitiveequationsoftheocean,globalweaklystrongsolution,existence ;uniqueness

    ;ChineseLibraryClassificationO175

    ;2000MathematicsSubjectClassification35B40,35M10,35Q30,86A10

    ;Digital0bjectIdentifier(DOI)10.1007/s10483-007-0503-x

    ;Introduction

    ;Inordertounderstandthemechanismoflongtermweatherpredictionandclimatechanges,

    ;onecanstudythemathematicalequationsandmodelsgoverningthemotionoftheatmosphere ;andtheocean.Inthelatesttwodecades,thereweresomemathematicians,suchasJ.L.Lions, ;R.TemamandS.,)v_ang,whobegantoconsidertheprimitiveequationsoftheatmosphere,the ;oceanandthecoupledatmosphere-ocean1一引.

    ;Lions.eta1.’2Jconcernedthemathematicalformulationsandattractorsoftheprimitiveequa-

    ;tions,theprimitiveequationswithverticalviscosityandtheBoussinesqequationsoftheocean. ;Theyobtainedtheexistenceofglobalweaksolutionfortheabovethreesystemsofequations ;oftheocean.Inaddition,undertheassumptionthatthereexistglobalstrongsolutions,they ;studiedtheattractorsofdynamicsfortheseequations.

    ;InRefs.f6-81,theauthorsstudiedthehydrostaticNavier.Stokesequationswhichcorrespond ;totheprimitiveequationswherethedensityisassumedtobeconstant.Guill6n-Gonz~lez.

    ;eta1.8Jobtainedglobalexistenceresultsofstrongsolutionbyassumingthedataaresmall ;enough.Moreover,undersomeadditionalregularityforweaksolutions,theyprovedaunique

    ;nessresult.Assumingthattheverticalderivativeoftheinitialdataissquareintegrable,Bresch, ;eta1.6Jconcernedtheexistenceanduniquenessofweaksolutionforthetwo-dimensionalhydro- ;staticNavierStokesequationswithafrictionconditionatthebottom.InRef.171,theauthors

;consideredthetwo-dimensionalhyrdrostaticNavierStokesequationswithDirichletboundary

    ;conditionsatthebottom,assumingabasinwithastrictlypositivedepth.Theyobtainedthe ;ReceivedMar.16,2006;RevisedMar.5,2007

    ;ProjectsupportedbytheNationalNaturalScienceFoundationofChina(No.90511009) ;CorrespondingauthOrHUANGDal-wen,Doctor,E-mail:hdw55@tom.com

    

    ;582HUANGDai-wenandGUOBo-ling

    ;existenceanduniquenessoftheglobalstrongsolutionandtheexponentialdecayintimeofthe ;energy..,

    ;-

    ;Petcu.eta1.9lconsideredsomeregularityresultsforthetwodimensionalprimitiveequations ;oftheoceanwithperiodicalboundaryconditions,wheretheequationsareobtainedfromthe ;three-dimensionalprimitiveequationsbyassumingthatallunknownfunctionsareindependent ;ofthelatitude.Theyprovedtheexistenceofweaksolution,theexistenceanduniqueness ;ofstrongsolutionandtheexistenceofmoreregularsolutionfortheprimitiveequationsin ;two-dimensionalspace.

    ;InspiredbyRef.[91weareinterestingtoconsideringthetwo-dimensionallarge-scaleprimitive ;equationsoftheocean.SincetheEarthcurvatureisnotconsideredinthelarge-scaleoceanic ;motion.wecalluseCartesiancoordinatesinsteadofsphericalcoordinates.Thetwo-dimensional ;primitiveequationswillbegiveninSection1.Here.theboundaryconditionsforthetwo- ;dimensionalprimitiveequationsgiveninSection1,areunliketothoseinRef.9.Especially,

    ;weconsiderthetractionbywindandtheheatfluxatthesurfaceoftheocean.Acompatibility ;conditionaboutuzonthebottommustbefoundinordertoprovetheexistenceanduniqueness ;oftheglobalweaklystrongsolutionfthedefinitionisgiveninSection2)fortheequations ;withsuchboundaryconditions.ComparedwithRefs.f6_81,ourpaperdoesnotassumethat ;thedensityisconstant.Sotemperatureandsalinityshouldbeintroduced.Moreover.the ;boundaryconditionsinourpapercanbeinhomogeneons.However.wehavetoassumethat ;thedepthoftheoceanisstrictlypositive.InspiredbyRef.7,usingFaedo-Galerkinmethod

    ;andanisotropicinequalities,wecanobtaintheexistenceanduniquenessofglobalweakstrong ;solutionsfortheinitialboundaryvalueproblemforthetwo-dimensionalprimitiveequationsof ;large-scaleoceanicmotionwithaconstantdepth,whichwillbedenotedasthesystem(I1.In ;thecompanionarticleRef.[10,weshallconsidersequentiallythesystem(I)withanon-constant ;depthandobtaintheexistenceanduniquenessofglobalstrongsolutions.Meanwhile.weshall ;alsostudytheasymptoticbehaviorofsolutionsforthesystemfI).

    ;Thepaperisorganizedasfollows:InSection1.weshallposethetwo-dimensionalprimitive ;equationsoflarge-scaleoceanicmotionandformulateourmainresult.InSection2,weshall ;givethefunctionalsettingforthesystem(I1andsomepreliminaries.Section3isdevotedto ;provingourmainresult.

    ;1Two-dimensionalprimitiveequationsoflarge-scaleoceanicmotion

    ;Inthissection,weshallgivethephysicalbackgroundofthetwo-dimensionalprimitive ;equationsoflarge-scaleoceanicmotionandourmainresult.

    ;UndertheBoussinesqapproximation(thedensitydifferencesareneglectedexceptinthe ;buoyancytermandintheequationofstate)andthehydrostaticapproximationOz:,the

    ;oceanicmotionisdescribedbythefollowingcompletelynon-dimensionalequationsfordetails,

;seeRefs.9,11,121):

    

    ;p

    ;Ou

    ;+u

    ;Ov

    ;+u

    ;1Op

    (ox

    ;1Op

    (Oy

    ;+

    +o,

    ;++u+=%?s,++u+%?.’,

    ;=l?3u

    ;72/k3~)

    ;(1)

    ;(3)

    ;(4)

    ;(5)

    ;++

    ;U

    一一

    }

    舰一加一

    ;++

    ;c若一如踟一如

    ;U

    一一珧却一c苦一cg一珧

    

    ;Ontwo-dimensionallarge-scaleprimitiveequationsinoceanicdynamics(I)583 ;+

    ++u=?ss+++u?3,

    ;P=Pref(1(Tr.f)+(S.f)),

    ;(6)

    ;(7)

    ;wheretheunknownfunctionsare(,V,u),P,P,S,T.(,V,u)isthethree-dimensionalvelocity,

    ;Pthedensity,Pthepressure,Sthesalinity,Tthetemperature,gthegravity,EtheRossby

    ;number,and>0(1i4)istheviscositycoefficient.Pref,ef,efarethereference ;valuesofthedensity,thetemperatureandthesalinity,respectively;8T,8Saretheexpansion

    ;coefficients(constants),A3isthethree-dimensionalLaplacianoperator.Theequation(7)isan

    ;empiricalapproximations(seef.[12).Sinceweconsiderthelarge-scaleoceanicmotion,the ;EarthcurvatureisnotconsideredandwecanadoptaCartesiancoordinatesystemwiththree

    ;axesOx,Oy,Oz,whereOxistheeastwestdirection,Oythesouthnorthdirection,Ozthe

    ;verticaldirection. ;IfweassumethatallunknownfunctionsareindependentofYandthefunctionVisnonzero.

    ;wecanobtainthefollowingtwo-dimensionallarge-scaleprimitiveequations:

    ;P=Pref(1(Tr.f)+(SSr.f)) ;Thespacedomainoftheaboveequationsis

    ;Q=(0,1)×(-h(x),0), ;(8)

    ;(9)

    ;(10)

    ;(11)

    onsaregivenby

    aa”OTOS

    Z’Z,u【】,

    ;OZ

    s,O

    ;Z

    uons’

    ;d

    ;(,V,u)=0,T=Th,S=Shonb, ;(.,OT=.on2,

    ;(15)

    ;(16)

    ;(17)

    ;where,arethewindstress,qsistheheatfluxonthesurfaceoftheocean,,,q8?

    ,arethegiventemperatureandsautheseaon

    ;:

    ;th

    eb

    ;oceanwhicharesmoothenoughfunctionsofthevariableand1I:10-t,Iz:0.

    ;=

    ;Iz:0.=.

    ;U

    ?

    ;=

    

    

    ;+=

    ;U

    一一

    一十

    钆一如一

    ;u

    ;++

    ;c{耋一如一如

;UU

    ;++

    ;0

    ;=

    

    ;=+

    ;==

    一一

    ;u

    卯一如一如

    +

    ;c言一如一况却一c言一c{;一一

    ;584HUANGDai-wenandGUOBo-ling

    ;FromEqs.(10)and(14),wecanobtain

    ;0p

    ;=

    ;Ops

    

    ;.

    ;(z),=一,JI”1,一”?I.88jz,8’8(18)

    ;wherePsisthepressureonthesurfaceoftheocean,l=Pref/~F,2=Pref/3S.Inthispaper,

    ;wedenotek?dxdzanddsbyl?,?,respectively.Ontheotherhand,wecangetfrom

    ;Eqs.(11)and(15) ;,,

    ;f00

    ;(,Z,t)=,.j

    ;z

    ;ox

    ;Theequations(8)-(14)canbewrittenasthefollowingequations

    ;Ou)Ou

    ;t,+

    ;Ox

    ;.

    ;.

    ;(z)=7.?,J一一一.,yl?, ;oOU).OV+

    ;=7z?,

    ++(.Ou).

    

    ;OT

    ;Ot+/,)

    ++(.Ou).OSOt+./,) =“/3?T.

;=14?s.

    ;(19)

    ;(20)

    ;(21)

    ;(22)

    ;(23)

    ;Forsimplicity,wewillonlyconsidertheabovetwo-dimensionallarge-scaleoceanicequations

    ;(20)-(23)withthehomogeneousboundaryvalueconditions ;andtheinitialconditions

    ;0V

    ;-

    ;O,OT=0,0S=0.ns

    ;T=0.S=0onb.

    ;_0{:0onf{u,uon,

    ;(U,V,T,S)t:0=(0,u0,,5b).

    ;Wedenotetheaboveinitialboundaryvalueproblems ;Inthepaper,ourmainresultisgotasfollows: ;(20)(27)asthesystem(I)

    ;(24)

    ;(25)

    ;(26)

    ;(27)

    ;Theorem1.1h1,Uo=(UO,VO,To,So)?H,thenthereexistsaglobalweaksolution ;U=(U,V,T,S),0rthesystem(I)suchthat

    ;(U,u,T,S)?L..(0,co;H)nL(0,co;),

    ;wherethedefinitionsofH,VwillbegiveninSection2.Moreover,UO,OzVO,OzTo,OzSo?

    ;.(Q),thenthesolutionUisuniqueand

    ;Ozu,OzV,OzT,OzS?L..(0,co;L(Q))nL(0,co;H(Q)), ;thatis,Uisaglobalweaklystrongsolution/orthesystem(I). ;Remark1.1Forthegeneralcases(15)-(17),bythehomogenizationoftheboundary

    ;valueconditionsinRef.[21,wecansimilarlysolvetheinitialboundaryvalueproblems(15)

    ;(17),(20)(23),and(27)atthepriceofsometechnicalcomplications.

    ;++

    钍一,c蔷一如一c;

    ;U

    ;++

    ;a耄,珧一况

    ;00

    ;0II

    ;:,==,,==,

    

    

    ;Ontwo-dimensionallarge-scaleprimitiveequationsinoceanicdynamics(I)585

    ;2Functionalsettingandpreliminaries

;Inthissection,weshallgivesomefunctionspaces,thedefinitionsofglobalweaksolu-

    ;tion,globalweaklystrongsolutionandglobalstrongsolution,somepreliminariesforproofof

    ;Theorem1.1.

    ;Atfirst,inordertogivethefunctionalsettingofthesystem(I),weintroducethefollowing

    ;functionspaces:

    ;(Q):={;U?co.(Q),suppuisacompactsubsetinQ\buf), ;(Q):={;乱?co.(Q),suppuisacompactsubsetinQ\6), ;,0

    ;1:={;?c(Q),/dz=0),

    ;Jh(x)

    ;:=1×(Q)×(Q)×(Q),

    ;f0

    ;H1:={;?L2(Q),/dz=0),

    ;Jh(x

    ;f0

    ;::{;?H(Q),/dz=0,ulbu,:0),

    ;Jh(x)

    ;::{;U?H(Q),ulbu,:0),

    ;::{;U?H(Q),ulb:0),

    ;V:=××v3×v3withthenorm

    ;IIUIl~=llull~,(Q)+Ilvll~,(Q)+IITII~(Q)+IISlI~(Q), ;H::H1×L(Q)×L(Q)×L(Q)withthenorm

    ;IlVll~=Ilulll:(Q)+Ilvlll:(Q)+IIII:(Q)+IISlI~:(Q). ;Definition2.1WesaythatU=(U,V,T,S)isaweaksolutionIorsystem(I)in(0,7.)

    ;U?Lo.(0,7.;H)nL(0,7.;V)satisfiesthevariationalformulation:forV:(l,2,3,4)?

    ;C(0,7.;)with(7.):0,

    

    ;~f(Otto1+um?1/o

    ;+7(?+?)=.?(.),

    

    ~o2+wa+u~2+72

    ;:

    ;(0)j

    

    ;Q,

    ;0ts+(f13-t-Ws)+%(s+TOz~3)=s(.),

    

    ;+~4--}-(a)azs+1.s8z4+8zs8z=11So4- ;wherew(x,,t):andU=u,s)satisfiesthefollowingenergyinequalityforalmost

    

    ;HUANGDai-wenandGUOBo.-ling

    ;everyt?(0,7_),

    ;(.Q+7./0llVull~L2(n))2+’72J0Q))2+/0llVQ))2

    ;+sII~L2(n)e<1(.OT,悒,

    ;whereCisapositiveconstant.WesaythatUisaglobalweaksolutionsystem(I)i/Uis aweaksolutionsystem(I)in(0,7_),1’orV7_<+...WesaythatUisaglobalweaklystrong

    ;solutionsystem(I)Uisaglobalweaksolutionsystem(I)suchthat ;U=(U,,T,S)?L(0,7_;L(Q))nL(0,7_;H(Q))

    ;Moreover,wesaythatUisaglobalstrongsolutionforsystem(I)theglobalweaksolutionU ;satisfiestheollowingregularity:

    ;U?L(0,7_;V)nL(0,7_;H(Q)n),

    ;,0rV7_<wesaythatafunctionU

    ;andllu(x,?)IlL(h(),0)?Lp(0,1).Moreover,the

    ;llL:L!(Q)=llllu(x,?)IlL(h(),0)ILL(0,1).

    ;Lemma2.1(Twoanisotropicinequalities[8J)ForU?H(Q),

    ;(i)L(Q)211uzIIL:(a)L.(Q),gull:0,

    ;(ii)L(Q)211uz(Q)llullL.(Q),//ulb=0;

    ;moveover,forsomeconstantC:c(a)>0,

    ;(iii)lL(Q)cIIvull(~.(Q)).llullL.(Q),

    ;(iv)lllIL(Q1cIIvull(~.(Q)).llullL.(Q).

    ;Lemma2.2(UniformGronwallLemma)Let,,bethreepositivelocallyintegrable ;functionson[to,+..)suchthatislocallyintegrableon[to,+..)andwhichsatisfy ;fort,

    ;+

    ;(s)ds.?,+(s)ds.,+(

    s)ds.s,l0rt?t.,

    ;wherer,01,02,a3arepositiveconstants.Then

    ;(t+r)(+.2)exp(.1),Vtt.

    ;ForthedetailofproofforLemma2.5,wereferthereadertoRef.[14](P.91) ;3ProofofTheorem1.1

    ;TheproofforexistenceofglobalsolutionUforthesystemfI1isclassicallyobtainedby ;Faedo-Galerkinmethodt13j.Sincethismethodisnowstandard.weonlygivethepriorestimates.

    

    ;Ontwo-dimensionallarge.sceprimitiveequationsinoceanicdynamics(I)587 ;Energyestimatesonu,,T,S.BychoosingUasatestfunctioninEq.(20),andfrom ;10

    ;.

    ;(.(

    ;#iT-#2S))uxdxdzI.(Q)+c.(Q)+c.(Q),

    ;wecanget

    ;d)ll

    ;.(Q)

    ;1IVIEII.

    ;(Q)+clITQ)+cl.(Q).(28)

    ;Inthisarticle,Cdenotesthepositiveconstantandcanbedeterminedinconcreteconditions ;Eisasmallenoughpositiveconstant.ChoosingasatestfunctioninEq.(21),weobtain ;d)ll

    ;1

;+IVI<0.

    ;ChoosingT,SasatestfunctioninEqs.(22)and(23),respectively,wehave

    ;dT?

    ;.?

    ;dIQ+

    ;ByEqs.(30)and(31),weobtain ;VTI0,

    ;VSl0

    ;(29)

    ;(30)

    ;(31)

    ;T?L..(0,oc;L(Q))nL(0,?;),S?L..(0,?;L(Q))nL(0,?;).(32)

    ;FromEqs.(28),(29)and(32),wecanhave ;U?L..(0,co;H1)nL(0,oc;),?L..(0,oc;L(Q))nL(0,?;).(33)

    ;CompatibilityconditionInordertoensuretheexistenceanduniquenessofglobal

    ;weaklystrongsolutionandtheexistenceofglobalstrongforthesystem(I),weshouldfind

    ;anappropriateboundaryconditionforU2onthebottom.

    ;IntegratingEq.(20)from1to0,weget

    ;2

    ;COu

    

    ;if.+#cox1L,/~.(?COS))=--’71Uz

    ;Bytakingthetraceonz:lofEq?(20),weobtain ;10p8

    

    ;if.(OTCOS)=~lUzzl::1.

    ;Bydeleting,wegetthefollowingcompatibilitycondition:

    ;2:一一?

    ;COS

    ;-

    ;~-a/lUzz-z=--I”91-”~lUz?:..c34

    ;Energyestimatesonu2,,,Sz.TakingthederivativeofEqs.(20)(23)withrespect

    ;toZ,wegetthefollowingequations:

    ;COUz

    ;+UUxz+(

    ;.

    ;.COU

    ;:+(.一筹)=7.?:,(35)

    ;COVz+UzVx+U%z--UxVz+(.COU ;:+:::,

    ;a

    ;Ot

    ;COS:

;Ot

    ;+:+(.:

    ;+tt2Sx+t2UzS+c. ;:?,

    ;(36)

    ;(37)

    ;(38)

    

    ;HUANGDai-wenandGUOBo-ling

    ;BychoosinguzasatestfunctioninEq.(35)andapplyingthecompatibilitycondition(34),we

    ;have

    ;u:cQ

    ;1

    ;vzuz+~/1lu ;:+

    ;+2u:一一?z

    

    ;Let

    ;;??cuOT

    =?uc?z. ;Atfirst,letUSestimatethetermI1.From

    ;onecanhave =

    2)dz=-2/~_ ;.

    ;zzd

    ;11

    ;uzlz:?lcz

    ;211uzlIL2lluzzlIL2,

    ;I1llu:(Q)+c(Q). ;Inthefollowing.weshallestimatetheterm/2.ByLemma2.1,wehave

    ;ll<lluzllL2(Q)zlIL2(Q),

    ;thatis,

    ;L!

    ;1

    ;Q)Q).

    ;Since

    ;and

    ;weget

    ;u=l,

    ;=:

    ;,.0

    ;/t:l.,

    ;1

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