DOC

On_0

By Jorge Hernandez,2014-07-14 11:57
9 views 0
On_0On,on,ON,on_0,On_0,ON_0

    On

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?,