Singular and regular solutions of a non-linear par

arXiv:math/0302129v1[math.AP]11Feb2003Singularandregularsolutionsofanon-linearparabolicsystemPetrPlech´aˇc†,Vladim´ırˇSver´ak‡†MathematicsInstitute,UniversityofWarwick,Coventry,CV47AL,UK‡SchoolofMathematicalSciences,UniversityofMinnesota,Minneapolis,MN55455,USAE-mail:[e-mail]-mail:[e-mail]nearequationmodellingcertainfeaturesoftheNavier-Stokesequations.Weprovethattheevolutionofradiallysymmetriccompactlysupportedinitialdatadoesnotleadtosingularitiesindimensionsn≤4.Fordimensionsn4wepresentstrongnumericalevidencesupportingexistenceofblow-upsolutions.Moreover,usingthesametechniqueswenumericallyconfirmaconjectureofLepinregardingexistenceofself-similarsingularsolutionstoasemi-linearheatequation.Submittedto:NonlinearityAMSclassificationschemenumbers:35K55,35B05,76A02Singularsolutions21.IntroductionInthispaperwestudysolutionsofthefollowingmodelequationforthetime-dependentvectorfieldu(x,t)=(u1(x,t),...,un(x,t))onRn×(0,T)∂u∂t+au∇u+12(1−a)∇|u|2+12(divu)u=Δu+κ∇divu,(1.1)wherea∈(0,1)andκ≥0aregivenparameters.Theequation(1.1)isofinterestforvariousreasons.Forexample,ithasthesamescalingpropertiesandthesameenergyestimateastheNavier-Stokesequation(NSE):Ifu(x,t)isasolutionof(1.1)thenalsoλu(λx,λ2t)isasolutionforλ0and,forsufficientlyregularsolutionswithasuitabledecayatinfinity,wehaveZRn12|u(x,t)|2dx=ZRn|u(x,t′)|2dx(1.2)+Zt′tZRn|∇u(x,s)|2+κ(divu(x,s))2dxds.Heuristically,solutionsof(1.1)shouldconvergetothesolutionsoftheNSEasκ→∞.SimilarpenalizationschemeshavebeenusedinnumericalalgorithmsforsolutionofNSE,see,e.g.,[5].Indimensionn=2equation(1.1)is“critical”(i.e.thecontrolledquantitiesareinvariantunderthescalingsymmetriesoftheequation)andhenceitisnaturaltoexpectthatthefullregularityofsolutionswithfiniteenergycanbeprovedbystandardmethods.Inthispaperweshallconcentrateonthesuper-criticalcasen≥3.ItisnaturaltoexpectthatthetheoryofLeray’sweaksolutionsappliesinthiscase.Moreover,itislikelythatforn=3thepartialregularityresultsinthespiritofScheffer[10]andCaffarelli-Kohn-Nirenberg[3]canbeprovedhereaswell.Wenotethatfora=1/2thenon-linearpartin(1.1)canbewritteninthedivergenceformandconsequentlyonecandirectlyapplytheknownregularitytheoryfortheNSEinthatcase.However,mostquestionsregardingfullregularityofsolutionsto(1.1)inthecasen≥3appeartobeopen.Ouraimhereistoinvestigatetheproblemoffinite-timeblow-upforaspecialclassofsolutionsto(1.1).Westudysolutionsgivenbyu(x,t)=−v(r,t)x,(1.3)wherer=|x|,andv(r,t)isascalarfunction.Suchvectorfields,usuallycalledradialvectorfields,arenotdivergencefreeunlessv≡0andhencetherelevanceofsuchsolutionsforthetheoryoftheNSEmaybelimited.Nevertheless,thebehaviourofthesesolutionsprovidesaninterestinginsightintovariousscenariosofsingularityformation.Usingtheradialvectorfieldansatzandsubstitutingintheequation(1.1)weobtainvt=(1+κ)vrr+n+1rvr+3rvvr+(n+2)v2,(1.4)wheresubscriptsdenotecorrespondingpartialderivatives.Replacingv(r,t)by(1+κ)v(r,(1+κ)t)weseethat,whenstudyingtheradialsolutions,onecanassumeκ=0withoutlossofgenerality.Singularsolutions3Ourfirstresultisthatindimensionn≤4thesolutionsto(1.4)donotexhibitblow-upifthereexistsC0suchthattheinitialconditionv(r,0)=v0(r)satisfies−C≤v0(r)≤C(1+r)−(n+2)/3whenn4,and(1.5)−C≤v0(r)≤1r243logr+Cwhenn=4.(1.6)Ontheotherhand,whenv(r,0)=v0(r)=c0(wherecisaconstant),thenv(r,t)=v(t)solvesdvdt=(n+2)v2,v(0)=c,andthesolutionblowsupattimet=1/(c(n+2)).Thereforesomecontrolofv0(r)atinfinityisnecessarytopreventformationofsingularities.Theproof,thatconditions(1.5)-(1.6)aresufficientforpreventingblow-up,isbasedonananalysisofsteady-statesolutionstotheequation(1.4).Thesteadystatescanbeanalyzedmoreorlesscompletelysincetheequationv′′+n+1rv′+3rvv′+(n+2)v2=0,(1.7)canbetransformedtoanautonomoustwo-dimensionaldynamicalsystem.Webrieflyoutlinethebehaviourofthesteady-statesolutions:Equation(1.7)hasasolutionV:[0,∞)→(0,∞)withV(0)=1,V′(0)=0andthefollowingasymptoticsatinfinityV(r)∼r−(n+2)/3,when1n4V(r)∼r−243logr+C,whenn=4V(r)∼r−2,whenn4.Usingthescalingsymmetryweobtainaone-parameterfamilyofsolutionsvλ(r)=λ2V(λr).Therearealsootherinterestingsteady-statesolutions.Itturnsoutthatevenforradialsolutions,weaksolutionsof(1.1)canexhibitthefollowingnon-trivialbehaviour:(i)formationofsingularitieswithadifferentrateofblow-upthansuggestedbyscaling,(ii)violationoflocalenergyinequality,(iii)significantnon-uniqueness.WepresentmorespecificdiscussionofthesephenomenainSection2.Thesecondgroupoftheresults,weshalldiscuss,concernstheblow-upbehaviourofsolutionstotheequation(1.4)indimensionsn4.Inthiscaseourresultsarebasedoncombinationofanalyticalargumentsandnumericalcalculations.Wewillpresentstrongevidencethatforn4andsuitablecompactlysupportedinitialdatathereexistsolutionsof(1.4)thatformasingularityinfinitetime.Anaturalclassofsingularitiesfortheequation(1.4)areself-similarsingularitiesofthetypev(r,t)=12κ(T−t)wrp2κ(T−t)!,whereκ0,T0areparametersandwisafunctiondefinedon[0,∞).Theequationforwisw′′+n+1rw′−κrw′+3rww′+(n+2)w2−κw=0,(1.8)Singularsolutions4togetherwiththenaturalboundaryconditionsw(0)=α0,w′(0)=0