holder continuity of solutions of__ 2D Navier-Stokes equations with singular forcing

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1、H¨olderContinuityofSolutionsof2DNavier-StokesEquationswithSingularForcingPeterConstantin∗GregorySeregin†January21,2009DedicatedtoNinaNikolaevnaUraltsevaAbstractWediscusstheregularityofsolutionsof2DincompressibleNavier-Stokesequationsforcedbysingularforces.Theproblemismotivatedbythestudyofcomp

2、lexfluidsmodeledbytheNavier-StokesequationscoupledtoanonlinearFokker-Planckequationdescribingmicroscopiccorporaembeddedinthefluid.ThisleadsnaturallytoboundedaddedstressandhencetoW−1,∞forcingoftheNavier-Stokesequations.1991Mathematicalsubjectclassification(Amer.Math.Soc.):35K,76D.KeyWords:Navier-

3、Stokesequations,H¨oldercontinuity,singularforcing.1IntroductionWediscusstheregularityofsolutionsof2DincompressibleNavier-Stokesequationsforcedbysingularforces.TheproblemismotivatedbythestudyofcomplexfluidsmodeledbytheNavier-Stokesequationscoupledtoanon-linearFokker-Planckequationdescribingmicr

4、oscopiccorporaembeddedin∗TheUniversityofChicago†OxfordUniversity1thefluid.ThisleadsnaturallytoboundedaddedstressandhencetoW−1,∞forcingoftheNavier-Stokesequations.Amoredetaileddescriptionoftheprobleminquestion,togetherwithanapplicationoftheresultsinthepresentpapercanbefoundinourforthcomingpaper

5、[2].Inthispaperwefocusonthe2DNavier-Stokesissues.Theglobalex-istenceofenergysolutionsandtheiruniquenessarewellknownasclassicalresultsofJ.LerayfortheCauchyproblemandO.Ladyzhenskayaforinitialboundaryvalueproblemsinboundeddomains.Theseresultsremaintobetrueforsingularforcesaswell.Theregularityofe

6、nergysolutionswithrelativelysmoothforcesisalsoknown.Regularitycanbeestablished,forinstance,byscalarmultiplicationoftheNavier-StokesequationbytheStokesoperatorofthevelocityfield,integrationbyparts,andapplicationofLadyzhenskay’sinequality√2∞2kukL4(R2)≤2kukL2(R2)k∇ukL2(R2),∀u∈C0(R).Thisprocedurey

7、ieldssummabilityofthesecondspatialderivatives.Furtherregularitycanbeobtainedperturbatively,withthehelpofthelineartheory.Theregularityofenergysolutionswithsingularforcingislimited.ThebestonecanexpectisH¨oldercontinuityofthevelocityfield.Weprove