A gradient flow approach to an evolution problem.pdf

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1、Agradientflowapproachtoanevolutionproblemarisinginsuperconductivity∗LuigiAmbrosio†SylviaSerfaty‡ScuolaNormaleSuperioreCourantInstitutePisaNewYorkJanuary31,2007AbstractWestudyanevolutionequationproposedbyChapman-Rubinstein-Schatzmanasamean-fieldmodelfortheevolutionofthevortex-densityinasupercon

2、ductor.Wetreatthecaseofaboundeddomainwherevorticescanexitorenterthedomain.Weshowthattheequationcanbederivedrigorouslyasthegradient-flowofsomespecificenergyfortheRiemannianstructureinducedbytheWassersteindistanceonprobabilitymeasures.Thisleadsustosomeexistenceanduniquenessresultsandenergy-dissi

3、pationidentities.Wealsoexhibitsome“entropies”whichdecreasethroughtheflowandallowtogetregularityresults(solutionsstartinginLp(p>1)remaininLp).1Introduction1.1PresentationoftheproblemWeareinterestedinstudyingthefollowing“mean-fieldmodel”(alsocalledhydrodynamiclimit)forsuperconductivitywhichwasde

4、rivedformallybyChapman,RubinsteinandSchatzmanin[CRS](seealsoE[E]):d(1)µ(t)−div(∇hµ(t)

5、µ(t)

6、)=0inΩ,dtwhereforalltimesµandhµarecoupledthroughtherelation−∆hµ+hµ=µinΩ(2)hµ=1on∂Ω.∗ThisworkwaspartiallysupportedbyaM.I.U.R.grant†l.ambrosio@sns.it‡serfaty@nyu.edu,supportedbyNSFCAREERaward#DMS0239121

7、andaSloanFoundationFellow-ship1Type-IIsuperconductors,submittedtoanexternalfield(herenormalizedtobeofin-tensity1),haveaveryparticularresponse:they“repel”theappliedfield,whichonlypen-etratesthrough“vortices”.Intheequationsabove,µrepresentsthesuitablynormalizeddensityofvortices:aprioriitshouldbe

8、asignedmeasure,butherewerestrictourselvestopositivemeasures.Thefunctionhµrepresentstheintensityoftheinducedmagneticfieldinthesample.Thisfunctioncanbeseenasapotentialgeneratedbythevorticesthroughthe“Londonequation”(2).Finally,thedomainΩisasmoothboundeddomainofR2,correspondingtoasectionofthesup

9、erconductingmaterial.Severalproblems,whichwewilladdresslater,appearintheformulation(1).First,theequationdoesnotalwayshaveameaningsince,whenµisonlyameasure,thefunction∇hµisnotcontinuousingeneralandthereforetheproductµ∇hµisnotwelldefined.Second,thecon