3维非齐次不可压Navier-Stokes方程组在旋度边界条件下的消失粘性极限问题.pdf 16页

'˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn3维非齐次不可压Navier-Stokes方程组在旋度边界条件下的消失粘性极限问题陈鹏飞，肖跃龙湘潭大学数学与计算科学学院学院，湘潭市　邮编411105摘要：本文主要研究在一般光滑有界区域R3的情形时，考虑了3维非齐次不可压Navier-Stoke方程组在一类旋度边界条件下的消失粘性极限问题。在文章中建立了这类初边值问题强解的局部存在性，然后证明了消失粘性极限的过程，得到了强解的一个收敛率的结果。关键词：非齐次不可压Navier-Stokes方程；旋度边界条件；消失粘性极限中图分类号：35Q30;76D05OntheVanishingViscosityLimitforthe3DNonhomogeneousIncompressibleNavier-StokesEquationwithavorticityBoundaryConditionCHENPeng-Fei,XIAOYue-LongDepartmentofMathematicsandComputationalScienceUniversityofXiangtan,Xiangtan411105Abstract:Thispaperconcernsthethree-dimensionalnonhomogeneousincompressibleNavier-StokesequationwithaclassofvorticityboundaryconditiononsmoothboundeddomaininR3.Itestablishesthelocalwell-posednessofthestrongsolutionforinitialboundaryvalueproblemforsuchsystems.Furthermore,thevanishingviscositylimitofthenonhomogeneousincompressibleNavier-Stokessystemisprovedandarateofconvergenceestimatesisshownforthestrongsolution.Keywords:NonhomogeneousincompressibleNavier-Stokesequation,Avorticityboundaryconditions,Vanishingviscositylimit.Foundations:ThisworkwassupportedbyResearchFundfortheDoctoralProgramofHigherEducationofChi-na(20134301110008)andtheHunanProvincialInnovationFoundationForPostgraduate(CX2015B205)AuthorIntroduction:Correspondenceauthor：XIAOYue-Long（1961-），male，professor，majorresearchdirection：Partialdifferentialequation.CHENPeng-Fei（1987-），male，doctor，majorresearchdirection：Partialdifferentialequation.Email：cpfxtu@163.com-1- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn0IntroductionLetΩ⊂R3beaboundedsmoothdomain,theinitialboundaryvalueproblemofthenonhomogeneousincompressibleNavier-Stokesequationisgivenbyρ∂tu−ν∆u+ρu·∇u+∇p=0,inΩ,(1)∂tρ+u·∇ρ=0,inΩ,(2)∇·u=0,inΩ,(3)u(0,x)=u0,ρ(0,x)=ρ0,inΩ,(4)equippedwiththefollowingvorticityboundaryconditionsu·n=0,ω·n=0,n×(∆u)=0on∂Ω.(5)Heretheconstantν>0,n,ρ,u,prepresenttheviscositycoefficient,theoutwardunitnormalvector,themassdensity,thevelocityfieldandthepressureofthefluids,respectively.Theinitialdensityρ0(x)isassumedtosatisfytheconditionm≤ρ0(x)≤MwithmandMboundedpositiveconstants.ThevanishingviscositylimitforthenonhomogeneousincompressibleNavier-Stokesequa-tionwiththecauchyproblemandtheperiodicboundaryconditionshasbeeninvestigatedbyItoh[1],ItohandTani[2]andDanchin[3],respectlvely.Inthepresenceofaphysicalbound-ary,thevanishingviscositylimitproblemsbecomemorechallengingandsignificanceduetotheemergenceoftheboundarylayer.Formally,whentheviscoustermisvanishing,system(1)-(4)degeneratesintothenonhomogeneousincompressibleEulerequation000000ρ∂tu+ρu·∇u+∇p=0,inΩ,(6)000∂tρ+u·∇ρ=0,inΩ,(7)0∇·u=0,inΩ,(8)00u(0,x)=u0,ρ(0,x)=ρ0,inΩ,(9)withtheslipboundaryconditions0u·n=0,on∂Ω.(10)Theinitialboundaryvalueproblemoftheequation(6)-(10)hasasmoothsolutionatleastlocalintime,ithasbeenaddressedbyseveralauthors,see,e.g.[2,4,5].ConceringthenonhomogeneousincompressibleNavier-Stokesequation,oneofthemostcommonphysicalboundaryconditionsistheclassicalno-slipboundaryconditionsu=0,on∂Ω,(11)-2- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnwhichmeansthatfluidparticlesareadherenttotheboundaryduetothepositiveviscosity,itwasproposedbyStokesin[6].ThisDirichlettypeproblemhasbeenaddressedin[7,8]andreferencestherein.However,theasymptoticconvergenceofthesolutionisoneofthemajoropenproblemexceptsomespecialcases,themainchallengingisadiscrepancybetweentheno-slipboundaryconditionsforthenonhomogeneousincompressibleNavier-StokesequationandthetangentialboundaryconditionsforthenonhomogeneousincompressibleEulerequation.AnotherclassoffamiliarboundaryconditionsistheNavier-slipboundaryconditions,whichcanbeshownasfollowingu·n=0,2(S(u)n)τ=−γuτ,on∂Ω,(12)itwasfirstintroducedin[9],where2S(u)n=(∇u+(∇u)⊤)istheviscousstresstensor,γisagivensmoothfunctionontheboundary.Wecanalsowritetheequivalentlyformasthefollowingvorticity-slipconditionu·n=0,n×ω=βu,on∂Ω.(13)TheresultofweakconvergencehavebeenconsideredbyFerreiraandPlanas[10].Asβ=0,thespecialvorticity-slipconditionshaveinitiallybeenappliedtothree-dimensionalincompressibleNavier-Stokesequationin[11].Basedontheaboveworks,theauthorandcoauthorfoundanadditionalconditionforthedensitytoobtainthestrongconvergencerateforthenonhomo-geneousNavier-Stokesequationontheflatdomainin[12].However,toourbestknowledge,itisstillunknownifthesimilarstrongconvergenceresultscanbeestablishedinageneralboundedomain.TherearemanyreferencesoninviscidlimitforNavier-StokesequationwithNavier-slipboundaryconditions,thereaderscanreferredin[13,14,15,16,17,18,19,20,21].Ourmaingoalinthispaperistoshowthevanishingviscositylimitproblemwiththevor-ticityboundarycondition(5).Thistypeofboundarycondition,whichwasinitiallyestablishedin[22]forthehomogeneousincompressibleNavier-Stokesequation,wheretheauthorestab-lishedthemathematicalresultonrateofconvergenceforstrongsolution.Ourapproachhereismotivatedbytheideals[22]tostudytheproblemforthenonhomogeneousincompressibleNavier-Stokesequationandisbasedonthefollowingobservations:First,weneedtoaddthesomeadditionalboundaryconditionsforthedensity,whichisdescribedby∇ρ=0,on∂Ω.(14)Theboundaryconditions(14)canbalancewellthemomentumequation(19)withboundaryconditions(21),wecanobtainthestrongsolutionslocalintime.Second,weneedtoconstructanewsystem(18)-(24),whichcanberegardedasarelaxedvorticitysystemofnonhomogeneousincompressibleNavier-Stokesequation.Thefactshowsthatthepressurevanishesinthenewsystem,yetthenewsystemisindeedthevorticitysystemoftheequations(1)-(5).-3- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnOurfirstmainresultisconcernedwiththelocalwell-posednessoftheinitialboundaryvalueproblemfortheequations(1)-(5).Theorem1.LetΩbetheboundedsmoothdomain,denoteHbythespace{u∈L2(Ω);∇·u=0,inΩ,u·n=0on∂Ω},u∈H1(Ω)∩H,ρ∈H2(Ω),ω∈H1(Ω)∩H.Thenthereexists000Tν=Tν(ω)>0,suchthattheinitialboundaryvalueproblem(1)-(2)hasauniquesolution0(ρ,u,p)satisfying23ν2u∈L(0,T;H(Ω))∩C([0,T);H(Ω)),ν2′2ρ∈C([0,T);H(Ω)),u∈L(0,T;V),foranyT∈(0,Tν),and−∆p=ρ∂iuj∂jui,∂np=(∆u−ρu·∇u)·n,∫p=0,Ωfort∈[0,Tν).Remark1.Inordertoobtaintheresultsabove,weneedtoconstructanewinitialboundaryvalueproblem(18)-(24).Sincethereisonemoreconditionin(5)thanthatnormallyNavier-slipboundaryconditions,thusitisnon-trivialtoshowtheconsistencyoftheboundaryconditionstogetthewell-posedness.Astheviscositycoefficientνtendstobezero,weshowthefollowingconvergenceofrate.Theorem2.Letρ∈H4(Ω),u∈H∩H4(Ω)satisfy∇ρ·n=0,∇×u∈H,ρ0(t),u0(t)be0000thesolutiontotheEulerequationsfornonhomogeneousfluidson[0,T]withinitialdataρ0,u0,ρ(t),u(t)bethesolutioninTheorem1.Then,wehavethefollowing∫t0202021−s∥ρ−ρ∥2+∥u−u∥2+ν∥u−u∥3dt≤cν(15)0ontheinterval[0,T]withT=T(σ,s)>0independentofν∈(0,σ)fors>0andν∈(0,σ).Remark2.Underthevorticityboundaryconditions,wecangetaresultmathematicallyofstrongconvergenceestimatetothesolutions.Therateofconvergence(15)isbetterthanthosefortheNavier-slipboundaryconditionscasesin[10].Comparedwiththecaseofco-normaluniformestimateasin[23,24],ourproblemheredoesnotsotediousandcomplicated,itcanbeprovedonlybystandardenergyestimates.Therestofthepaperisorganizedasfollows:Section2,werecallsomenotations,defini-tions,andpreliminaryfacts.Section3,wegivethelocalwell-posednesstotheinitialboundaryvalueproblemforthenonhomogeneousNavier-Stokesequations(1)-(5).Section4,weestablishtherateofconvergencetothesolutions.-4- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn1PreliminariesLetusstartingbyrecallingthestandardnotationofsomefunctionspacesandoperatorswhicharefamiliarinthemathematicaltheoryoffluidsmodelledbyNavier-Stokessystem,see[11,22].Forconvenience,notetheinnerproductby(·,·)andthenormofthestandardHilbertspaceL2(Ω),Hs(Ω)by∥·∥,∥·∥,respectively.Wealsodenote[A,B]=AB-BA,thecommutatorsbetweentwooperatorsAandB.Set2H={u∈L(Ω);∇·u=0,inΩ,u·n=0onΩ},1V=H(Ω)∩H2W={u∈H(Ω);n×(∇×u)=0onΩ}.Letψ,ϕbethevectorfunction,thefollowingformulaholdbydirectcalculations:∇×(ψ×ϕ)=ϕ·∇ψ−ψ·∇ϕ+ψ∇·ϕ−ϕ∇·ψ,(16)⊥∇×(ψ·∇ϕ)=ψ·∇(∇×ϕ)+∇ψ·∇ϕ,(17)where∇ψ⊥isexpressedincomponentby⊥j+1j+2(∇ψ·∇ϕ)j=(−1)∂j+1ψ·∇ϕj+1+(−1)∂j+2ψ·∇ϕj+2withtheindexmodulatedby3.WedenotebyA=−∆theStokesoperatorwithD(A)=W⊂Vistheself-adjointexten-sionofthepositiveclosedwithitsinversebeingcompact,andthereisacountableeigenvalues{λj}suchthat0<λ1≤λ2···→∞,thecorrespondingeigenvector{e}⊂W∩C∞(Ω)makesanorthogonalcompletebasisofH.jWeshowthefollowingestimate:Lemma1.Lets≥0beaninteger.Letu∈Hsbeavector-valuedfunction,then∥u∥s≤C(∥∇×u∥s−1+∥∇·u∥s−1+|n·u|s−1),2∥u∥s≤C(∥∇×u∥s−1+∥∇·u∥s−1+|n×u|s−1+∥u∥s−1).2Assumingthatϕ(t),ψ(t),f(t)aresmoothnon-negativefunctionddefinedforallt≥0,weshowthefollowingdifferentialinequality.dϕ(t)Lemma2.Supposeϕ(0)=ϕ0and+ψ(t)≤g(ϕ(t))+f(t)fort≥0,wheregisanon-dtnegativeLipschitzcontinuousfunctiondefinedforϕ≥0.Thenϕ(t)≤F(t;ϕ0)fort∈[0,T(ϕ0))dF(t)whereF(·;ϕ0)isthesolutionoftheinitialvalueproblemdt=g(F(t))+f(t);F(0)=ϕ0and[0,T(ϕ0))isthelargestintervaltowhichitcanbecontinued.Also,ifgisnondecreasing,then∫tψ(τ)dτ≤Fe(t;ϕ0)0-5- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnwith∫tFe(t;ϕ0)=ϕ0+[g(F(τ;ϕ0))+f(τ)]dτ.02Thelocalwell-posednessresultsOurmainpurposeinthissectionistosolvetheinitialboundaryvalueproblem(1)-(5).Webeginbyintroducingthefollowingadditionalboundaryconditionfordensity:Lemma3.Lettheinitialdensitysatisfiesthecondition∇ρ0=0ontheboundary,thenthedensityhavethepersistencepropertythat∇ρ(t,·)=0ontheboundary.Proof.Applyingthegradientoperator∇tothetransportequation(2),itfollowsthatD(∇ρ)+∇u·∇ρ=0,dttheordinarydifferentialequationsislinearandtheinitialdatasatisfies∇ρ0=0,wecanprovethelemma.Ontheotherhand,inordertoobtainthestrongsolution,weneedtoconstructthefollowingsystem,whichiscalledarelaxedvorticityequationof(1)-(5):ρt+u·∇ρ=0,inΩ,(18)ρ(∂tω+u·∇ω−ω·∇u)+∇ρ×(∂tu+u·∇u)−ν∆ω+∇q=0,inΩ,(19)∇·ω=0,inΩ,(20)ω·n=0,n×(∇×ω)=0,on∂Ω,(21)withu=Tωgivenby∇×u=ω,inΩ,(22)∇·u=0,inΩ,(23)u·n=0,on∂Ω,(24)WherethelinearoperatorsatisfyT:H→Vwithu=Tω,whichistheuniquesolutionoftheequations(22)-(24),iscontinuous.Weclaimthattheinitialboundaryvalueproblem(18)-(24)possessesexactlyonestrongsolutioninamaximaltimeinterval.LetPktheorthogonalprojectofHontothespaceHkspannedbythekfirsteigenfunctionse1,···ekofA.Thenthesolutionsofthesystem(18)-(24)canbeobtainedbyusingaSemi-GalerkinapproximationsmethoddeterminedbythespacesHkandtheoperatorsPk.Foreachfixedk,weconsiderthe-6- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnfollowingfinitdimensionalproblem:FindTk∈(0,T]suchthat(m)(m)(m)(m)(m)(m)(m)(m)Pm(ρ∂tω+ρTω·∇ω−ρω·∇Tω)(m)(m)(m)(m)(m)+Pm(∇ρ×(∂tTω+Tω·∇Tω))−ν∆Pmω=0,ρ(m)+Tω(m)·∇ρ(m)=0,t(m)(m)ω(0,x)=Pmω0(x),ρ(0,x)=ρ0(x),em·n=0,n×(∇×em)=0.Wehaveaninitialboundaryvalueproblemforasystemofordinarydifferentialequationscou-pledtoatransportequation.Byusingthecharacteristicsmethod,itcanprovethesystempossessesexactlyonesolution(ρ(m),ω(m))definedinatimeinterval[0,T).Thekthapproxi-kmatedproblemcanalsobewrittenintheform:(m)(m)(m)(m)(m)(m)(m)(m)(ρ∂tω+ρTω·∇ω−ρω·∇Tω,v)(m)(m)(m)(m)(m)+(∇ρ×(∂tTω+Tω·∇Tω),v)−ν(∆ω,v)=0,ρ(m)+Tω(m)·∇ρ(m)=0,t(m)(m)ω(0,x)=Pmω0(x),ρ(0,x)=ρ0(x),em·n=0,n×(∇×em)=0.ThroughtheSemi-Galerkinapproximationmethod,therestoftheprocesstoestimatethesolutionsof(18)-(24)isratherstandard.Wedonotgivethedetailedproof,thereadercanbereferredtoChapter3in[25].Themaintheoreminthissectionisthefollowing:Theorem3.Letρ∈H2(Ω)andω∈V,thenthereexistsTν=Tν(ρ,ω)>0,suchthatthe0000problem(18)-(24)haveauniquesolution(ρ,ω,q)ontheinterval[0,Tν)satisfyingνρ∈C([0,T);W),2νν′2νω∈L(0,T;W)∩C([0,T);V),ω∈L(0,T;H),andtheenergyequation∫t∫t2222∥ρ(t)∥2+∥∇×ω(t)∥+ν∥∂tω∥dx+ν∥ω(s)∥2ds≤c(25)00holdon[0,t]foranyt∈(0,Tν),andqisgivenuniquelyby∆q=0,(26)∂nq=−ρ(u·∇ω−ω·∇u)·n,(27)∫q=0,(28)∂Ωfora.e.t∈(0,Tν).-7- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnLemma4.Letω∈V,∇ρ=0onboundary.Thenρ(Tω·∇ω−ω·∇(Tω))∈H.Proof.Sinceω∈V,itcandeducethatTω∈H2(Ω)∩V,thenω×Tω∈H1(Ω).TheboundaryconditionTω·n=0andω·n=0impliesn×(ω×Tω)=0,on∂Ω.Thiscompletethelemma.ItfollowsfromLemma4thefollowingcorollary.Corollary1.Thesolutionqintheorem3satisfiesq=0,fora.e.t∈(0,Tν).ProofofTheorem1:Fromtheanalysisabove,itfollowsthattheequation(19)isthecurloftheequation(1).Theorem1isproved.Remark3.Itshouldbenotedthattheconstructingsystem(18)-(23)isnecessary.Iftheboundaryconditionisreplacedbythenonslipboundaryω=0,then(∆ω)·nmaynotbezero,fromtheequations(26)-(28),hence∇qmaynotbezero.Thenthemomentumequationshouldbetheformρ∂tu−ν∆u+ρu·∇u+F(q)+∇p=0,inΩ,forsomevectorfunctionFofq.3ConvergenceOfTheSolutionsInthissectionweproveTheorem2.Letusshowthefollowinglemmabeforegivingtheconvergenceestimate.Lemma5.Letρ,ubeasmoothsolutiontothenonhomogeneousincompressibleEulerequationsontheinterval[0,T]withinitialρ∈H3(Ω),u∈H3(Ω)∩Hand∇ρ=0,∇×u∈H.Then0000itholdsthat(∇×u0)·n=0,on∂Ωforallt∈[0,T].Proof.Notethattheparticlepathformsadiffeomorphismontheboundary.ThevorticityequationsofthenonhomogeneousincompressibleNavier-Stokesequationisthefollowing:000ρt+u·∇ρ=0,inΩ,(29)0000000000ρ(∂tω+u·∇ω−ω·∇u)+∇ρ×(∂tu+u·∇u)=0,inΩ,(30)DuetotheLemma3,itfollowsthat∇ρ0×(∂u0+u0·∇u0)vanishesontheboundary.Thetequation(30)multiplyingtheunitoutwardnormvector,itobtainsD(ω0·n)0000=(ω·∇)u·n+ω·(u·∇)n.dt-8- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnFollowingfromtheLemma3.1inXiao[22],thereexistα,βsuchthatD(ω0·n)0=(α+β)(ω·n).dtSinceω0·n=0on∂Ω,onehas0ω(x,t)·n=0on∂Ω.Thiscompletetheproof.Remark4.Inordertoobtaintheasymptoticconvergenceofthesolutions,weneedsomeadditionalconditionsfornonhomogeneousEulerequationtoovercometheboundarylayer.IfthenonhomogeneousEulerequationmatchtheboundaryconditionsω0·n=0inmathematicalstructure,itcancoincidewiththatofnonhomogeneousNavier-Stokesequationinthetangentialdirections.Hence,werestricttheinitialdataconditionofthedensitysatisfy∇ρ0=0.Next,weprovethemainTheorem2.First,wedenotethata=ρ−ρ0,v=u−u0,w=ω−ω0,duetothetransportequations,itfollowsthatd0a+u·∇a=−v·∇ρ.(31)dtApplyingtheoperateD2andtakingtheinnerproductwith(31)byD2a,wehaved202220222∥a(t)∥2+(u·∇Da,Da)+([D,u·∇]a,Da)=−(D(v·∇ρ),Da).dtHence,bytheYoung’sinequality,itiseasytoobtaind22−1422∥a(t)∥2≤cδν∥∆w∥+ν∥a∥2+∥a∥2+∥v∥2+cν.(32)dtSecond,weestimatethedifferencesystembetweenthevorticityequationof(1)andthevorticityequationof(6):00000awt+(ρv+au)·∇w+ρwt+ρu·∇w+Φ−ν∆w=ν∆ω,(33)withtheboundaryconditionsu·n=0,w·n=0on∂Ω,(34)whereΦ=A+B,000000A=awt+av·∇ω+ρv·∇ω+au·∇ω+aw·∇v000000000+aw·∇u+aω·∇v+aω·∇u+ρw·∇v+ρw·∇u+ρω·∇v00000+∇a×(∂tv+∂tu+v·∇v+v·∇u+u·∇v+u·∇u)000+∇ρ×(∂tv+v·∇v+v·∇u+u·∇v),-9- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnand00000B=∇a×(∂tv+∂tu+v·∇v+v·∇u+u·∇v+u·∇u)000+∇ρ×(∂tv+v·∇v+v·∇u+u·∇v).Takingtheinnerproduct(33)with−∆w,itfollowsthat1d√√(∥a∇×w∥2+∥ρ0∇×w∥2)ds+ν∥∆w∥2−(Φ,∆w)2dt∫0=((aw)t+(ρ0w)t)·(n×(∇×w))ds+ν(∆ω,∆w).(35)∂ΩIntegratingbypartyields∫1d√√(∥a∇×w+∥ρ0∇×w∥2+2(aw+ρw)·(n×(∇×ω0)))ds+ν∥∆w∥202dt∂Ω∫00=(Φ,∆w)+(aw+ρ0w)·(n×∂t(∇×ω))ds+ν(∆ω,∆w).(36)∂ΩHereweusethepropertythatn×(∇×ω)=0,v·n=0,w·n=0on∂Ω,itfollowsthat∫0(Φ,−∆w)=Φ·n×(∇×ω)−(∇×Φ,∇×w)∂Ω00=(Φ,−∆ω)−(∇×Φ,∇×ω)−(∇×Φ,∇×w).(37)Next,welistsomebasicfactstobeusedlater.Theunitoutnormalvectornhasbeenextendedasfollows:∇φ(r(x))n(x)=,x∈Ω|∇φ(r(x))|andr(x)=mind(x,y)=d(x,y0),y0∈∂Ω,y∈∂Ωwhichisuniquewhenr(x)≤σforsomeσ>0,andthefunctionissmoothandcompactsupportedin[0,σ)suchthat′φ(0)=1,φ(0)=1.Firstweestimateon(∇×Φ,∇×w)from(37),recallthat(∇×Φ,∇×w)=(∇×(A+B),∇×w).ItfollowsfromthedefinitionofAthat000|(∇×(awt),∇×w)|=|(∇a×wt+a∇×wt),∇×w)|022≤c∥a∥2∥∇×wt∥∥∇×w∥≤c(∥∇×w∥+∥a∥2),-10- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnand0|(∇×(aw·∇v+ρw·∇v),∇×w)|⊥0⊥0=|(∇(aw)·∇v+aw·∇w+∇(ρw)·∇v+ρw·∇w,∇×w)|1515≤c(∥a∥∥∇×w∥2∥∇×w∥2+∥∇×w∥2∥∇×w∥2)2112−1102−110≤cδν∥∆w∥+c(ν3∥∇×w∥3+∥a∥2+ν∥∇×w∥)+cν.similarly,itobtainsthat000000|(∇×(av·∇ω+ρv·∇ω+au·∇ω+aw·∇u0000000+aω·∇v+aω·∇u+ρw·∇u+ρω·∇v,∇×w)|224≤c(∥a∥2+∥∇×w∥+∥∇×w∥).Next,wecalculatethetermB,notethat00|(∇×B,∇×w)|=|(∇×(∇a×(∂tv+∂tu+v·∇v+v·∇u000000+u·∇v+u·∇u)+∇ρ×(∂tv+v·∇v+v·∇u+u·∇v)),∇×w)|.itfollowsthat|(∇×(∇a×(∂tv+v·∇v)),∇×w)|=|(∇a·∇(∂tv+v·∇v)−(∂tv+v·∇v)·∇(∇a)+∇a∇·(v·∇v)−(∂tv+v·∇v)∆a,∇×w)|1511≤c(∥a∥∥∇×w∥2∥∇×w∥2+∥a∥∥∂w∥∥∇×w∥2∥∇×w∥2)212t1222−38−110−14≤cδ(ν∥∆w∥+ϵ∥∂tw∥+∥a∥2+ν2∥a∥2+ν∥∇×w∥+ν2∥∇×w∥+ν).Similarly,wecanget00000|(∇×(∇a×(∂tu+v·∇u+u·∇v+u·∇u)000+∇ρ×(∂tv+v·∇v+v·∇u+u·∇v)),∇×w)|2≤c(∥a∥2∥∇×w∥+∥∇×w∥2+cν),and000|(∇ρ×(∂tv+v·∇v+v·∇u+u·∇v)),∇×w)|224≤c(∥∂tv∥+∥∇×w∥+∥∇×w∥).Hence,itfollowsthat22−18(∇×Φ,∇×w)≤c(δν∥∆w∥+ϵ∥∂tw∥+ν∥a∥2−110−110−14+ν3∥∇×w∥3+ν∥∇×w∥+ν∥∇×w∥2242+∥a∥2+∥∇×w∥+∥∇×w∥)+ϵ∥∂tw∥+cν.(38)-11- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnSecond,weestimateontheterm(∇×Φ,∇×ω0):00(∇×Φ,∇×ω)=(∇×(A+B),∇×ω).Recallthat00000000000(∇×(ρv·∇ω+au·∇ω+aω·∇u+ρw·∇u+ρω·∇v),∇×ω)∫00000000000=n×(ρv·∇ω+au·∇ω+aω·∇u+ρw·∇u+ρω·∇v)∇×ωds∂Ω000000000000−(ρv·∇ω+au·∇ω+aω·∇u+ρw·∇u+ρω·∇v),∇×ω,−∆ω),then,itfollowsfromthetracetheoremthat∫0000000νn×(awt+ρv·∇ω+au·∇ω+aω·∇u∂Ω000000+ρw·∇u+ρω·∇v),∇×ω)∇×ωds0≤cν(∥∇v∥s+∥w∥s+∥a∥s)∥∇×ω∥11−ss1−ss≤cν(∥ω∥∥∇×ω∥+∥a∥∥∇a∥)222−s≤cν(∥∇×ω∥+∥∇a∥1+ν).Atthesametime,theremandtermofAisestimatedthat00000(∇×(av·∇ω+aw·∇v+aw·∇u+aω·∇v+ρw·∇v,∇×ω)224≤c(∥a∥2+∥∇×ω∥+∥∇×ω∥),BythedefinitionofB,itfollowsthat∫000(∇×B,∇×ω)=n×B∇×ωds−(B,∆ω)Ω2221−s≤c(∥∇×ω∥+∥a∥2+ϵ∥∂tw∥+ν).Therefore,wecandeducethat02221−s|(∇×Φ,∇×ω)|≤c(∥∇×ω∥+∥a∥2+ϵ∥∂tw∥+ν).(39)Finally,weestimateon(Φ,−∆ω0):000000000000|(awt+ρv·∇ω+au·∇ω+aω·∇u+ρω·∇v+ρw·∇u,−∆ω)|221−s≤c(∥∇×ω∥+∥a∥2+ν),00000|(av·∇ω+aw·∇v+aw·∇u+aω·∇v+ρw·∇v,−∆ω)|24≤c(∥a∥2+∥∇×ω∥+ν),-12- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnand02221−s|(B,−∆ω)|≤c(∥∇×ω∥+∥a∥2+ϵ∥∂tw∥+ν).So024221−s|(Φ,−∆ω)|≤c(∥∇×ω∥+∥∇×ω∥+∥a∥2+ϵ∥∂tw∥+ν).(40)Theremainingtermsin(36)canbeestimatedasfollows:∫0221−s|(aw+ρ0w)·(n×∂t(∇×ω))ds|≤∥∇×ω∥+∥a∥2+cν,(41)∂Ω021−sν|(∆ω,∆w)|≤cδν∥∆w∥+cν,(42)∫01221−s|(aw+ρ0w)·(n×(∇×ω)))ds|≤∥∇×ω∥+∥a∥2+cν.(43)∂Ω4Inordertoestimate∥∂w∥2,takingtheinnerproduct(33)with∂w,itfollowsthattt∫202d2a|wt|+ρ|wt|+ν∥∇×w∥Ωdt∫∫0000=((ρv+au)·∇w+ρu·∇w+Φ+ν∆ω)∂tw+n×(∇×w)wt,(44)Ω∂Ωduetotheboundaryconditionn×(∇×ω)=0,wehave∫∫∫d00n×(∇×w)wtds=−n×(∇×ω)wds+n×(∇×ωt)wds.(45)∂Ωdt∂Ω∂ΩItfollowsfromtheformula(44)and(45)that∫∫2d20ρ|wt|dx+ν(∥∇×w∥+n×(∇×ω)wds)Ωdt∂Ω∫∫∫0000=((ρv+au)·∇w+ρu·∇w)∂tw+Φ∂tw+n×(∇×ωt)wΩΩ∂Ω=I+II+III.Hence,2244m2I≤c(∥a∥2+∥∇×w∥+∥∇×w∥+∥a∥2)+∥∂tw∥,4and2m2II≤c∥Φ∥+∥∂tw∥42223m2≤c(∥a∥2+∥∇×w∥+∥∂tv∥)+∥∂tw∥4-13- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnItfollowsfromthetracetheoremthat1−ss121−sIII≤c∥ω∥∥∇×ω∥≤∥∇×ω∥+cν.4Itfollowsthat∫2d20m∥wt∥+ν(∥∇×w∥+n×(∇×ω)wds)dt∂Ω2223m21−s≤c(∥a∥2+∥∇×w∥+∥∂tv∥)+∥∂tw∥+cν.(46)2Throughtheestimate(32),(38)-(43),(46)wecanobtaind√222222(∥a∇×w∥+∥∇×w∥+ν∥∇×w∥+∥a∥2)+ν∥∆w∥+m∥wt∥dt2223−14−38=c((∥a∥2+∥∇×w∥+∥∂tv∥)+ν∥a∥2+ν2∥a∥2−110−110−141−s+ν3∥∇×w∥3+ν∥∇×w∥+cν2∥∇×w∥+ν+ν).(47)Ifs∈(0,1)and221−s21−s∥a∥2≤cν,∥∇×ω∥≤cν.Sowecandeducethat−34−110−110−141−sν2∥a∥2+ν3∥∇×w∥3+ν∥∇×w∥+cν2∥∇×w∥=o(ν),andthereexistsomeconstantcsuchthat−141−sν∥a∥2≤cν.Usingtheinitialdatea(0)=0,w(0)=0,bythelemma2,weobtain∫∫√222221−s∥a∥2+∥a∇×w(t)∥+∥∇×w(t)∥+∥wt∥dx+ν∥∆w(s)∥dx≤cν.(48)ΩΩontheinterval[0,T]fors∈(0,1)andν∈(0,ν)⊂(0,ν),whereT=T(ν,s)>0is121011111′independentofν∈(0,ν).Ifs≥,wecanchoseas′∈(0,)suchthatνs≤cνs.The023theorem2isproved.参考文献（References）[1]S,Itoh.Onthevanishingviscosityinthecauchyproblemfortheequationsofanonhomo-geneousincompressiblefluid[J].J.GlasgowMath.,1994,36:123-129.[2]S,Itoh,A,Tani.Solvabilityofnonstationaryproblemsfornonhomogeneousincompressiblefluidsandconvergencewithvanishingviscosity[J].Tokyo.J.Math.,1999,22:17-42.-14- 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