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1、JournalofMathematicalAnalysisandApplications263,626–636(2001)doi:10.1006/jmaa.2001.7640,availableonlineathttp://www.idealibrary.comonGlobalExistenceofRegularSolutionsfortheVlasov–Poisson–Fokker–PlanckSystemKosukeOnoDepartmentofMathematicalandNaturalSciences,Universit
2、yofTokushima,Tokushima770-8502,JapanE-mail:ono@ias.tokushima-u.ac.jpSubmittedbyMariaClaraNucciReceivedAugust23,2000WestudytheglobalexistenceanduniquenessofregularsolutionstotheCauchyproblemfortheVlasov–Poisson–Fokker–Plancksystem.Twoexistencetheoremsforregularsolutio
3、nsaregivenunderslightlydifferentinitialconditions.OneofthemcompletelyincludestheresultsofP.Degond(1986,Ann.Sci.EcoleNorm.Sup.19,519–542).ã2001AcademicPressKeyWords:kinetictheory;Vlasovplasmaphysics;Vlasov–Poisson–Fokker–Plancksystem;regularity.1.INTRODUCTIONPlasmamea
4、nscompletelyionizedgases.TheVlasov–Poisson–Fokker–Plancksystem,weoftensayVPFPforshort,appearsinVlasovplasmaphysicsandstemsfromtheLiouvilleequationcoupledwiththePoissonequationfordeterminingtheself-consistentelectrostaticorgravitationalforces(see[7,11]).Inthispaper,we
5、considertheglobalexistenceanduniquenessofreg-ularsolutionstotheCauchyproblemfortheVPFPsystem.Letfxvtdescribethemicroscopicdensityofparticleslocatedatpositionx∈Nwithvelocityv∈Nattimet>0.Then,theVPFPsystemcanbewrittenas∂tf+v·∇xf+E·∇vf−vf=0(1.1)forf=fxvtxv
6、∈N×Nt>0,γxExt=∗fxvtdv(1.2)SN−1xN6260022-247X/01$35.00Copyrightã2001byAcademicPressAllrightsofreproductioninanyformreserved.globalexistenceofsolutions627withinitialdatafxv0=φxvwhere∇x=∂x1∂xN∇v=∂v1∂vNvistheLaplacianinthevvariable,γ=±1
7、SN−1isN−1-dimensionalvolumeoftheN-dimensionalunitsphere,andthesymbol∗istheconvolutioninthexvariable.Extistheforcefield(theelectricfield)actingontheparticle.Letρxtdescribethemacroscopicdensityofparticleslocatedatposi-tionx∈Nattimet>0;thatis,ρxt=fxvtdv
8、Equation(1.2)canbewrittenalternativelyasthePoissonequationE=−∇Uwith−U=γρ.Then,weseeUxt=2−N−1cx2−N∗ρxtxx0withc0=γ/SN−1.Thesi
