An Elementary Proof of the Existence and Uniquenes

arXiv:math/9903042v2[math.AP]7Apr1999AnElementaryProofoftheExistenceandUniquenessTheoremfortheNavier-StokesEquationsJ.C.Mattingly∗andYa.G.Sinai†November19,1998,RevisedMarch10th,19991IntroductionThepurposeofthispaperistoshowthatsomeresultsconcerningsolutionsoftheNavier-Stokessystemscanbeprovenbypurelyelementarymethods.Intwo-dimensionswithperiodicboundaryconditions,theNavier-Stokessystemhastheform∂u1∂t+u1∂u1∂x1+u2∂u1∂x2=νΔu1−∂p∂x1+f1(x1,x2,t),(1)∂u2∂t+u1∂u2∂x1+u2∂u2∂x2=νΔu2−∂p∂x2+f2(x1,x2,t),∂u1∂x1+∂u2∂x2=0.Hereνistheviscosity,pisthepressure,andf1,f2arethecomponentsofanexternalforcingwhichmaybetime-dependent.Asoursettingisperiodic,thefunctionsu1,u2,∇p,f1,andf2areallperiodicinx.Forsimplicity,wetaketheperiodtobeone.Thefirstexistenceanduniquenesstheoremsforweaksolutionsof(1)wereprovenbyLeray([Ler34])inwholeplaneR2.LatertheseresultswereextendedbyE.Hopf(see[Hop51]).In1962,Ladyzenskayaprovedexistenceanduniquenessresultsforstrongsolutionsforgeneraltwo-dimensionaldomains[Lad69].V.Yudovich,C.Foias,R.Teman,P.Constantin,andothersdevelopedstrongmethodswhichprovideddeepinsightsintothedynamicsdescribedby(1)(see[Yud89,Tem79,Tem95,CF88]).Thepurposeofthispaperistopresentelementaryproofsofthreetheorems.Thesetheoremsimplytheexistenceanduniquenessofsmoothsolutionsof(1)andshedsomeadditionallightonthedissipativecharacterofthedynamics.Wewillalsodiscusswhatourtechniquescangiveinthethree-dimensionalsetting.Intwo-dimensions,itisusefultoconsiderthevorticityω(x1,x2,t)=∂u1(x1,x2,t)∂x2−∂u2(x1,x2,t)∂x1.Theequationgoverningωhastheform(see[CM93,DG95])∂ω∂t+u1∂ω∂x1+u2∂ω∂x2=νΔω+g(x1,x2,t)(2)whereg(x1,x2,t)=∂f1(x1,x2,t)∂x2−∂f2(x1,x2,t)∂x1.Wewillneedg(x1,x2)topossesamodicumofspatialsmooth-ness;thiswillbemadepreciseshortly.Inourtwo-dimensionalsetting,thesystems(1)and(2)areequivalent.ExpandingωinFourierserieswhereω(x1,x2,t)=Pk∈Z2ωk(t)e2πi(x,k)withx=(x1,x2),wecanwriteacoupledODE-systemforthemodesωk(t)(see[DG95]).dωkdt+2πiXl1+l2=kωl1ωl2(k,l⊥2)(l2,l2)=−4π2ν|k|2ωk+gk(t)(3)∗DepartmentofMathematics,StanfordUniversity,StanfordCA94305.†DepartmentofMathematics,PrincetonUniversity,PrincetonNJ08544.1wherek∈Z2,|k|=pk21+k22,l⊥=(l(1),l(2))⊥=(−l(2),l(1)),andgk(t)arethespatialFouriermodesofthefunctiong(x,t).Sinceωisreal,weknowω−k=¯ωk.Furthermore,wealwaysassumethatω0=0.Thesystem(3)istheGalerkinsystemcorrespondingto(2).AfinitedimensionalapproximationofthisGalerkinsystemcanbeassociatedtoanyfinitesubsetZofZ2bysettingωk(t)=0forallkoutsideofZ.Inthefollowing,wewillimplicitlyassumethatZiscentrally-symmetric,thatisifk∈Zthen−k∈Z.Infact,wewillstudyaslightlymoregeneralversionof(2)wheretheLaplacianisreplacedbytheoperator|∇|αwithα1.Thisleadstoaversionof(3)whichweindexbythechoiceofαandbythefiniteindexsetZ,Z⊂Z2,indicatingwhichmodesareincludedintheGalerkinapproximation.Inshort,weconsiderthefinitedimensionalODEsystemdωkdt+2πiXl1+l2=kl1,l2∈Zωl1ωl2(k,l⊥2)(l2,l2)=−4π2ν|k|αωk+gk.(3αZ)Wenowstatetheassumptionsonthecoefficientsgk(t)tobeusedatvarioustimesduringourdiscussion.Assumption1.Theforcingf(x,t)=(f1(x,t),f2(x,t))issuchthatg∗=supt∈[0,∞)|g(·,t)|L2∞.Assumption2.Forsomer,thereexistsaconstantG(r)0suchthatsupt∈[0,∞)|gk(t)|≤G(r)|k|r−α+ǫforsomeǫ0andallk∈Z2\0.Theconstantαisthesameasin(3αZ).Assumption3.Forsomerandγ0,thereexistsaconstantG(r,γ)0suchthatsupt∈[0,∞)|gk(t)|≤G(r,γ)|k|r−α+ǫe−γ|k|1+δforsomeδ0,ǫ0,andallk∈Z2\0.Again,theconstantαisthesameasin(3αZ).Observethatassumption3impliesassumption2.Criticaltoourdiscussionisthatfor(3αZ)wehavetheso-calledenstrophyestimate.Namely,ifE(0)=Rω2(x1,x2,0)dx1dx2=Pk|ωk(0)|2∞thenonecanfindE∗dependingonlyonE(0),ν,supt∈[0,∞)|g(·,t)|L2,andαsuchthatE(t)=Rω2(x1,x2,t)dx1dx2≤E∗forallsolutionsto(3αZ).ItisimportanttonotethatE∗isindependentofthesetZwhichdefinestheGalerkinapproximation.Thisenstrophyestimateholdsiftheforcingsatisfiesassumption1(seee.g.[CF88,DG95,Tem79]).Nowwearereadytoformulateourtheorems.Theorem1.Assumetheforcingsatisfiesassumption1and2forsomer1andG(r)0.IfforsomeD1∞|ωk(0)|≤D1|k|rthenonecanfindaD′1∞,dependingonlyonD1,ν,g∗,andG,suchthatanysolutionto(3αZ)withtheseinitialconditionssatisfies|ωk(t)|≤D′1|k|rforallt0.Inparticular,D′1isindependentofthesetZdefiningtheGalerkinapproximation.Anexistenceanduniquenesstheoremfor(3)followsfromtheorem1bynowstandardconsiderations(see[CF88,DG95,Tem79]).Webrieflyrecallthegenerallineoftheargument.BytheSobolevembeddingtheorem,theGalerkinapproximationsaretrappedinacompactsubsetofL2ofthe2-torus.Thisguaranteestheexistenceofalimitpointwhichcanbeshowntosatisfy(3).UsingthetheregularityinheritedfromtheGalerkinapproximations,onethenshowsthatthereisauniquesolutionto(3).Gallavotti[Gal96]containsasimilarproofofasimilarstatement.2Theorem2.Assumethatassumption3holdsforsomer1,γ0,andG(r,γ)0.Iftheinitialconditionssatisfy|ωk(0)|≤D2|k|re−γ2|k|forsomeD2∞andγ20,thenonecanfindaD′2∞andaγ′20,dependingonlyonD2,γ,r,ν,g∗,G,suchthatanysolutionto(3αZ)startingfromtheseinitialconditionssatisfies|ωk(t)|≤D′2|k|re−γ′2|k|forallt0.Inparticular,theconstantsD′2andγ′areindependentofthesetZdefiningtheGalerkinapproximation.Theorem2showsthatequation(2)preservesth