简介:Themainpurposeofthispaperistwo-fold:(i)togeneralizeanexistenceresultforacompressiblegas-liquidmodelwithafrictiontermrecentlypublishedbyFriisandEvje[SIAMJ.Appl.Math.,71(2011),pp.2014–2047];(ii)toderiveauniquenessresultforthesamemodel.Amainingredientintheexistencepartistheobservationthatwecanconsiderweakerassumptionsontheinitialliquidandgasmass,andstillobtainanexistenceresult.Comparedtotheabovementionedwork,werelyonamorerefinedapplicationoftheestimatesprovidedbythebasicenergyestimate.Concerningtheuniquenessresult,weborrowideasfromFangandZhang[NonlinearAnal.TMA,58(2004),pp.719–731]andderiveastabilityresultunderappropriateconstraintsonparametersthatdeterminerateofdecaytowardzeroattheboundaryforgasandliquidmasses,andgrowthrateofmassesassociatedwiththefrictiontermandviscouscoefficient.
简介:LetB(resp.K,Bl,Kl)denotethesetofallnonemptybounded(resp.compact,boundedconvex,compactconvex)closedsubsetsoftheBanachspaceX,endowedwiththeHausdorffmetric,andletGbeanonemptyrelativelyweaklycompactclosedsubsetofX.LetB°standforthesetofallF∈Bsuchthattheproblem(F,G)iswell-posed.Weprovedthat,ifXisstrictlyconvexandKadec,thesetKl∩B°isadenseG_δ-subsetofKl\G.Furthermore,ifXisauniformlyconvexBanachspace,wewillprovemore,namelythatthesetB\B°(resp.K\B°,Bl\B°,Kl\B°)isa-porousinB(resp.K,Bl,Kl).Moreover,weprovethatformost(inthesenseoftheBairecategory)closedboundedsubsetsGofX,thesetK\B°isdenseanduncountableinK.
简介:Inthispaperwell-conditioningofboundaryvalueproblemsforsystemsofsecondorderdifferenceequa-tionsisstudied.First,asufficientconditionfortheexistenceofauniqueboundedsolution(forlargeenoughnumberofsteps)ofanassociatedhomogeneoussystemisgiven.Finally,asufficientconditionforwell-condi-tioning,intrinsicallyrelatedtotheproblemdataisproposed.
简介:Inthispaper,westudytheCauchyproblemforthe3DgeneralizedNavier-Stokes-Boussinesqequationswithfractionaldiffusion:{ut+(u·▽)u+v∧2αu=-▽p+θe(3),e3=(0,0,1)T,θt+(u·▽)θ=0,Dicu=0.Withthehelpofthesmoothingeffectofthefractionaldiffusionoperatorandalogarithmicestimate,weprovetheglobalwell-posednessforthissystemwithα≥5/4.Moreover,theuniquenessandcontinuityofthesolutionwithweakerinitialdataisbasedonFourierlocalizationtechnique.Ourresultsextendonesonthe3DNavier-Stokesequationswithfractionaldiffusion.
简介:Inthispaper,weconsiderahydrodynamicflowofnematicliquidcrystalsystem.Weprovethelocalwell-posednessforthesysteminthecriticalLebesguespace,andstudythespace-timeregularityofthelocalsolution.
简介:SlowmotionforscalarAllen-Cahntypeequationisawell-knownphenomenon,precisemotionlawforthedynamicsoffrontshavingbeenestablishedfirstusingthesocalledgeometricapproachinspiredfromcentralmanifoldtheory(seetheresultsofCarrandPegoin1989).Inthispaper,theauthorspresentanalternateapproachtorecoverthemotionlaw,andextendittothecaseofmultiplewells.Thismethodisbasedonthelocalizedenergyidentity,andistherefore,atleastconceptually,simplertoimplement.Italsoallowstohandlecollisionsandroughinitialdata.