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1、NonbneorAnulys~s.Theory,Methods&Applications,Vol.26,No.4.pp.659-689,1996Copyright01995ElsevierScienceLtdPrmtedmGreatBritain.Allrightsreserved0362-546X/%$15.00+.oO0362-546X(94)00308-4VARIATIONALMETHODSFORSTATIONARYMHDFLOWUNDERNATURALINTERFACECONDITIONS-f
2、-A.J.MEIRandPAULG.SCHMIDTDepartmentofMathematics,AuburnUniversity,AL36849,U.S.A.(Received8February1994;receivedforpublication21October1994)Keywordsandphrases:Magnetohydrodynamics,Navier-Stokesequations,Maxwell’sequations,variationalmethods,boundaryinteg
3、ralmethods.0.INTRODUCTIONMagnetohydrodynamics(orMHD)isthetheoryofthemacroscopicinteractionofelectricallyconductingfluidswithamagneticfield.Itisofimportanceinconnectionwithmanyengineeringproblems,suchassustainedplasmaconfinementforcontrolledthermonuclear
4、fusion,liquid-metalcoolingofnuclearreactors,andelectromagneticcastingofmetals(see,e.g.[l-3]).Italsofindsapplicationsingeophysicsandastronomy,whereprominentexamplesaretheso-calleddynamoproblem,thatis,thequestionoftheoriginoftheEarth’smagneticfieldinitsli
5、quidmetalcore,andtheequilibriumandstabilityofmagneticstarsintheirowngravitationalfields(see,e.g.[4-61).Assumingthefluidtobeviscous,resistiveandincompressible(aswewilldothroughoutthispaper),MHDflowisgovernedbytheNavier-Stokesequations(forthefluidvelocity
6、)andMaxwell’sequations(forthemagneticfield);theequationsarenonlinearlycoupledviaOhm’slawandtheLorentzforce(cf.[7,Chapter21).Themagneticfieldwillingeneraltranscendthefluidregionand,ideally,extendtoallofspace,satisfyingdifferentequationsinthefluidandinthe
7、solidmaterialorvacuumoutside,andcertainmatchingconditionsattheinterface.Itismainlythisphenomenon,theelectromagneticprincipleofactionatQdistance,whichdistinguishesMHDfromordinaryhydrodynamics:typically,attentioncannotbeconfinedtotheconductingfluiditself;
8、itsinteractionwiththeoutsideworldmustbeanalyzedtoo.Thisaddsconsiderablytotheintricaciesoftheproblemandmayexplainwhysomuchtheoreticalworkhasbeendevotedtosituationswhere,duetospecialcircumstances,externalconditionsareeitherpre-impo
