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On the Cauchy problem for Boltzmann equations global existence and weak stability

On the Cauchy problem for Boltzmann equations global existence and weak stability

ID：40391269

大小：2.38 MB

页数：46页

时间：2019-08-01

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1、AnnalsofMathematics,130(1989),321-366OntheCauchyproblemforBoltzmannequations:GlobalexistenceandweakstabilityByR.J.DIPERNA*ANDP.L.LIONSAbstractWestudythelarge-dataCauchyproblemforBoltzmannequationswithgeneralcollisionkernels.Weprovethatsequencesofsolutionswhichsatisfyonlythephysicallyna

2、turalaprioriboundsconvergeweaklyinL'toasolution.FromthisstabilityresultwededuceglobalexistenceofasolutiontotheCauchyproblem.Ourmethodreliesuponrecentcompactnessresultsforvelocityaver-ages,anewformulationoftheBoltzmannequationwhichinvolvesnonlinearnormalizationandananalysisofsubsolution

3、sandsupersolutions.Itallowsustoovercomethelackofstrongaprioriestimatesanddefineameaningfulcollisionoperatorforgeneralconfigurations.ContentsIntroductionI.Preliminaries:basicphysicalestimatesII.FormulationsofBoltzmannequationsandsequentialstabilityresultsIII.AprioriboundsandweakL'compac

4、tnessofthenormalizedinteractionIV.Velocityaveragesandapplications:averagedcontinuityofthecollisionoperatorV.Passingtothelimit:supersolution,regularityalongcharacteris-ticsVI.Passingtothelimit:subsolutionVII.GeneralcollisionkernelsVIII.GlobalexistenceAppendixA:Aremarkonlineartransportth

5、eoryAppendixB:OnformulationsofBoltzmannequations*TheAnnalseditorsweresaddenedtolearnofthedeathofProfessorDiPernaonJanuary8,1989.322R.J.DIPERNAANDP.L.LIONSIntroductionWestudytheCauchyproblemfortheBoltzmannequationaf(B)af+tvf=Q(f,f)in(0,no)XRNXRNwhereN?1,xERN,ERNandQ(f,f)isthecollisionop

6、eratorswhosestructureisdescribedbelow.Weshallprovethatsequencesofclassicalsolutionsof(B)withuniformaprioriboundsobtainedfromthestandardphysicalidentitiesassociatedwith(B)convergeweaklyinL'toarenormalizedsolutionof(B).Wededucefromthisstabilityresulttheexistenceofaglobalrenormalizedsolut

7、ionof(B)givenaninitialcondition(1)fIt=o=fo(x,)inRNXRNwhichsatisfies(2)fo0a.e.inRNXRN,ff1RN120(l+IX12++Iogfl)dxd

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