Remark on regularity of weak solutions to the Navier-Stokes equations

Transcription

1 Comment.Math.Univ.Carolin. 42,1(21) Remark on regularity of weak solutions to the Navier-Stokes equations Zdeněk Skalák, Petr Kučera Abstract. Some results on regularity of weak solutions to the Navier-Stokes equations published recently in[3] follow easily from a classical theorem on compact operators. Further,weaksolutionsoftheNavier-Stokesequationsinthespace L 2 (, T, W 1,3 (Ω) 3 ) are regular. Keywords: Navier-Stokes equations, weak solution, regularity Classification: 35Q1, 76D5, 76F99 Introduction Let Ω beaboundeddomainin R 3 with C 2 -boundary Ω, let T >and Q T = Ω (, T).WeconsidertheNavier-Stokesinitial-boundaryvalueproblem describingtheevolutionofthevelocity u(x, t)andthepressure p(x, t)in Q T : (1) u t ν u+u u+ p=f, (2) u=, (3) (4) u= on Ω (, T), u t= = u, where ν >istheviscositycoefficientand fistheexternalbodyforce.theinitial data u shouldsatisfythecompatibilityconditions u Ω =and u =. The definition and the proof of the existence of weak solutions of the equations (1) (4)canbefoundforexamplein[3]or[6].Ingeneral,itisunknownwhether weaksolutionsareregularornot. Serrin([5])provedthatifaweaksolution u of(1) (4)belongsto L α (, T, L q (Ω))for2/α+3/q=1and q (3, ]then uis regular. Kozono([3]) generalized this result to a certain class of functions characterizedbymeansoflocalsingularitiesintheweak-l 3 space.hefurthershowed thatthereexistsanabsoluteconstant ε >suchthatif uisaweaksolutionof (1) (4)in L (, T, L 3 (Ω) 3 )andlimsup t t u(t) L 3 (Ω) < u(t ) L 3 (Ω) + ε, then uisnecessarilyregularin Ω (t σ, t +σ)forsome σ >.Letusmention here that the Kozono s results were applied in[4] where partial regularity of weak solutionstothenavier-stokesequationsintheclass L (, T, L 3 (Ω))wasshown. Themaingoalofthispaperistoshowthattheresultsstatedabovecanbe easily derived from the following well known theorem on compact operators([2]):

2 112 Z. Skalák, P. Kučera TheoremA. Let X, Y bebanachspaces. Let Sbeaonetoonecontinuous linearoperatorfrom Xonto Y and Kalinearcompactoperatorfrom Xto Y.If Ker(S+ K)=othen(S+ K)(X)=Y. Let p >1. L p (Ω)istheLebesguespacewiththenorm p. C (Ω)denotes the set of all infinitely differentiable vector-functions defined in Ω, with a compact supportin Ω. C,σ (Ω)isasubsetof C (Ω)whichcontainsonlythedivergencefreevectorfunctions. H istheclosureof C,σ (Ω)in L2 (Ω) 3 withthescalar product(, )andthenorm 2. W m,p (Ω)and W m,p (Ω)(m N)aretheusual Sobolevspaces. V denotesthecompletionof C,σ (Ω)inthenormof W1,2 (Ω) 3 withthescalarproduct((u, v))= Ω u i v i x j x j dxandthenorm. P H isthe projectionoperatorfrom L 2 (Ω) 3 onto H. L p w(ω)denotestheweaklebesguespaceover Ωwiththequasi-norm p,w definedby φ p,w =sup R> Rµ{x Ω; φ(x, t) > R} 1/p,where µisthelebesgue measure. Thereisanotherequivalentnormtotheabove p,w (see[3]),sowe mayunderstand L p w(ω)asabanachspace.letusnotethat L p (Ω) L p w(ω)and φ p,w φ p forevery φ L p (Ω). Let D(A)={u V; f H;((u, v))=(f, v) v V }. AistheStokesoperatorfrom D(A)onto Hdefinedforevery u D(A)bytheequation((u, v))= (Au, v) v V. D(A) is endowed with the norm u D(A) = Au 2 and D(A) V. Since Ω C 2, D(A)=W 2,2 (Ω) 3 V andthenorm u D(A) on D(A)isequivalenttothenorminducedby W 2,2 (Ω) 3 (see[6,lemma3.7]). Weoftenusethisfactthroughoutthepaper. LetusdefinetheBanachspaces X = {u L 2 (, T, D(A)), u t L 2 (, T, H)}and Y = L 2 (, T, H) V with u X = u L 2 (,T,D(A)) + u t L 2 (,T,H) and (f, v ) Y = f L 2 (,T,H) + v V. Throughoutthepaper,wesupposethatin(1) (4) f L 2 (, T, H)and u H. Forsimplicity,weusethefollowingnotation:If Fisaspaceofrealfunctionsthen u F meansthateverycomponentof uisfrom F,e.g. u W 1,2 (Ω)meansin factthat u W 1,2 (Ω) 3.Similarly, u F means u F 3. Proof of regularity results At first, we prove two basic propositions. The results mentioned in Introduction will then be their straightforward consequences. Proposition1. Let u L α (, T, L q (Ω))for2/α+3/q 1and q (3, ].Then theoperator w P H (u w)iscompactfrom Xto L 2 (, T, H). Proof: Firstly,supposethat2/α+3/q <1and α, q <. UsingtheHőlder inequalitywehaveforalmostevery t (, T)andevery v H: u w v dx v 2 u q w 2q/(q 2). Ω

3 Regularity of weak solutions 113 It follows further that u 2 q w 2 2q/(q 2) dt u 2 L α (,T,L q (Ω)) ( u 2 L α (,T,L q (Ω)) ( w 2α/(α 2) 2q/(q 2) dt) (α 2)/α [ w 2/α 2 w (α 2)/α (2αq 4q)/(αq 2α 2q) ]2α/(α 2) dt) (α 2)/α u 2 L α (,T,L q (Ω)) w 4/α L (,T,W 1,2 (Ω)) ( w 2 (2αq 4q)(αq 2α 2q) dt)(α 2)/α u 2 L α (,T,L q (Ω)) w 4/α L (,T,W 1,2 (Ω)) w 2(α 2)/α L 2 (,T,W 1,(2αq 4q)/(αq 2α 2q) (Ω)) and, therefore, (5) P H (u w) L 2 (,T,H) u L α (,T,L q (Ω)) w 2/α X w (α 2)/α L 2 (,T,W 1,(2αq 4q)/(αq 2α 2q) (Ω)), whereweusedthefactthat Xisembeddedcontinuouslyinto L (, T, W 1,2 (Ω)). Since(2αq 4q)/(αq 2α 2q) < 6 it follows e.g. from[5, Theorem 2.1, Chapter III] thattheinjectionof Xinto L 2 (, T, W 1,(2αq 4q)/(αq 2α 2q) (Ω))iscompact.The proof now follows immediately from(5) and the definition of compact operators. Secondly, let u L α (, T, L (Ω)), α > 2. Then Ω u w v dx v 2 u w W 1,2foralmostevery t (, T)andevery v Hand u 2 w 2 W 1,2 dt u 2 L α (,T,L (Ω)) ( u 2 L α (,T,L (Ω)) ( Therefore, w 2α/(α 2) W 1,2 (Ω) dt) (α 2)/α = w 4/(α 2) W 1,2 (Ω) w 2 W 1,2 (Ω) dt)(α 2)/α u 2 L α (,T,L (Ω)) w 4/α L (,T,W 1,2 (Ω)) ( w 2 W 1,2 dt) (α 2)/α = u 2 L α (,T,L (Ω)) w 4/α L (,T,W 1,2 (Ω) w 2(α 2)/α L 2 (,T,W 1,2 (Ω)). (6) P H (u w) L 2 (,T,H) u L α (,T,L (Ω)) w 2/α X w (α 2)/α L 2 (,T,W 1,2 (Ω)). Theinjectionof Xinto L 2 (, T, W 1,2 (Ω))iscompactandtheprooffollowsimmediately from(6) and the definition of compact operators. If u L (, T, L q (Ω))and q >3thentheproofproceedsinthesamewayas inthepreviousparagraphsandwewillskipit.

4 114 Z. Skalák, P. Kučera Finally,let u L α (, T, L q (Ω))for2/α+3/q=1, q (3, ].Let M n = {t (, T); u(t) q > n}, n Nanddefine u n on(, T)as: u n (t)=u(t) if t / M n, u n (t)= if t M n. Obviously, u n L (, T, L q (Ω))andaccordingtothepreviousparagraphsthe operators w P H (u n w)arecompactfrom Xto L 2 (, T, H).Further,the Lebesguemeasureof M n goestozerofor n sothat u u n L α (,T,L q (Ω)) = ( M n u α q dt) 1/α. Therefore,theoperator w P H (u w)iscompact from Xto L 2 (, T, H)asalimitofcompactoperators w P H (u n w)inthe usualnormofthespaceofalllinearboundedoperatorsfrom Xto L 2 (, T, H). Let us consider the following Stokes equations with the perturbed convection term P H (u w): (7) (8) w t + νaw+ P H (u w)=f, w()=w. Proposition2. Let2/α+3/q=1with q (3, ].Thenthereexists ε >with thefollowingproperty:if u=u + u 1 in(, T), u(t) V foralmostevery t (, T), u L (, T, L 3 w (Ω)), u 1 L α (, T, L q (Ω))andsup <t<t u (t) 3,w < ε,thenforevery w V and f L 2 (, T, H)thereexistsauniquesolution wof (7),(8)in X. Proof:Theoperator w (w t + νaw, w())isaonetoonecontinuouslinear operatorfrom Xonto Y. Itispossibletoprove(seealso[3,Lemma2.7])that theoperator w P H (u w)islinearandboundedfrom Xto L 2 (, T, H) withthenormlessthan C u L (,T,L 3 w (Ω)).Sincethesetoflinearboundedone tooneoperatorsisopeninthespaceofalllinearboundedoperators(usingthe usualtopology)wegetthattheoperator w (w t +νaw+p H (u w), w()) isaonetooneoperatorfrom Xonto Y for εbeingsufficientlysmall.finally,it followsfromproposition1thattheoperator w P H (u 1 w)iscompactfrom Xto L 2 (, T, H).Moreover,theoperator w (w t +νaw+p H (u w), w()) isonetoonefrom Xto Y andtheprooffollowsimmediatelyfromtheorema. Now, we present proofs of the results stated in Introduction. The proofs are basedonpropositions1and2. Theorem3isageneralizationofthefamous Serrin s result([5]) on regularity of weak solutions in the subcritical case and was provedin[3]. Theorem4whichisdealingwiththepartialregularityofweak solutionsinthesupercriticalcase L (, T, L 3 (Ω))wasalsoprovedin[3]. We present these theorems in a little more general way.

5 Regularity of weak solutions 115 Theorem3. Thereexistsaconstant εwiththefollowingproperty. If uisa weaksolutionof (1) (4)andthereexistsanon-negative L 2 -function M= M(t) on(, T)suchthat (9) sup Rµ{x Ω; u(x, t) > R} 1/3 ε R M(t) foralmostevery t (, T),then uisregular,thatis u t, Dα xu C(Ω (, T)) foreverymulti-index αwith α 2. Proof: Duetothecondition(9) ucanbeeasilydecomposedas u=u + u 1, where u L (, T, L 3 w(ω)), u 1 L 2 (, T, L (Ω))andsup <t<t u (t) 3,w < ε(see[3]). Let σ (, T)beanarbitrarynumber. Sincetheweaksolution u L 2 (, T, V),thereexistsat (, σ)suchthat u(t ) V.If εissufficiently smallitfollowsfromproposition2thatthereexistsauniquesolution w Xof (7),(8)on(t, T)with w(t )=u(t ). Itiseasytoshowthat u=won(t, T) andtherefore u Xon(t, T).Since σwaschosenarbitrarilythetheoremfollows immediately using the results on interior regularity of weak solutions proved in[5]. Theorem 4. There exists a positive constant ε with the following property. If u isaweaksolutionof(1) (4)andthereexists w L 3 (Ω)suchthat u(t) w 3,w < εforalmostevery t (a, b) (, T),then u t, Dα xu C(Ω (a, b))forevery multi-index α with α 2. Proof: Thereexists w 1 L 4 (Ω)suchthat w w 1 3 < ε. Ifweput u = u w 1 and u 1 = w 1, then u = u + u 1 on(a, b), u L (a, b, L 3 w (Ω)), u 1 L (a, b, L 4 (Ω))andsup a<t<b u (t) 3,w <2ε.Now,applyingagainPropositions1and2on(a, b)andusingthesameargumentsasintheorem3,theorem4 follows immediately. Itwasprovedin[1]and[3]thatif uisaweaksolutionof(1) (4)and u C([, T), L 3 (Ω))or u BV([, T), L 3 (Ω)) thesetofallfunctionsofbounded variationon[, T)withvaluesin L 3 (Ω) then uisregular. Theseresultsare consequences of Theorem 4. ThefollowingtheoremisanotherexampleoftheuseofTheoremAinthe regularity theory of the Navier-Stokes equations. Let us note here that the space L 2 (, T, W 1,3 (Ω))isnotimbeddedintoany L α (, T, L q (Ω))with2/α+3/q=1 and q (3, ]. Theorem5. Let ubeaweaksolutionof (1) (4)and u L 2 (, T, W 1,3 (Ω)). Then u t, Dα xu C(Ω (, T))foreverymulti-index αwith α 2. Proof:Firstly,letusshowthattheoperator w P H (w u)iscompactfrom Xto L 2 (, T, H).UsingtheHőlderinequalitywehaveforalmostevery t (, T) andevery v H: w u v dx c v 2 w W 1,2 (Ω) u W 1,3 (Ω). Ω

6 116 Z. Skalák, P. Kučera ItfollowseasilyasinthefirstparagraphofProposition1that P H (w u) L 2 (,T,H) c w X u L 2 (,T,W 1,3 (Ω)) sothat w P H(w u) isalinearboundedoperatorfrom Xto L 2 (, T, H).Asinthelastparagraphof Proposition1itispossibletoconstruct u n L (, T, W 1,3 (Ω))suchthat u u n L 2 (,T,W 1,3 (Ω)) andthecompactnessoftheoperator w P H(w u) followsnowfromthisandfromthefactthattheoperators w P H (w u n ) are compact. It follows from the standard estimates in Sobolev spaces, the Gronwall lemma andtheoremathatforevery w Vand f L 2 (, T, H),thefollowingproblem hasauniquesolution w X: (12) (13) w t + νaw+ P H (w u)=f, w()=w. TheproofisconcludedusingthesameargumentsasintheproofofTheorem3. Remark6. Ife.g. f H(f independentoftime)thenintheorem3andtheorem5, resp.theorem4uisanalyticintime, inaneighborhoodoftheinterval(, T),resp.(a, b), asad(a)-valuedfunction(see[7]). Itfollowsthat u C (, T, C(Ω)),resp. u C (a, b, C(Ω)). Therefore, uhasnosingular pointsin Ω (, T),resp. Ω (a, b). Also, u(x, )isaninfinitelydifferentiable functionin(, T),resp.(a, b),forevery x Ω. Remark7. If Ω C,1 thentheinformationfromtheintroduction D(A)= W 2,2 (Ω) 3 V andthenorm u D(A) on D(A)isequivalenttothenorminduced by W 2,2 (Ω) 3 cannotbeused. Wedonotevenknowinthiscasewhether D(A) W 1,2+ε (Ω) 3 forapositive εornot. Whatweonlyhavehereisthat D(A) V andalso X L (, T, V). Asaconsequence,Propositions1 and2canbeprovedonlyif u L 2 (, T, L (Ω))andtheproofsofTheorems3 and4failtotally. Ontheotherhand,itisinterestingthatTheorem5canbe stated and proved without any change. Remark8.If Ωisthehalf-spaceor R 3 (orpossiblysomeotherspecialunbounded domain)thenweareabletoobtainalmostthesameresultsasinthecaseofa boundeddomain.letusdiscussitbriefly. Vdenotesthecompletionof C,σ (Ω)in thenormof W 1,2 (Ω) 3 withthescalarproduct((u, v)) V = Ω ( u i x j v i x j +u i v i )dx. D(A)isthendefinedas {u V; f H;((u, v)) V =(f, v) v V }andusing thecut-offmethoditispossibletoshowthat D(A) W 2,2 (Ω).Itimpliesthat X L 2 (, T, W 2,2 (Ω))and,consequently, X L 2 (, T, W 1,6 ε (Θ))forevery small ε >andeverysmoothdomain Θ Ω.Asaresult,Proposition1canbe provedinasimilarwayasinthecaseofaboundeddomainandproposition2 holdswith onlyonechange: theweaklebesguespace L 3 w(ω) isreplacedby

7 Regularity of weak solutions 117 thelebesguespace L 3 (Ω). InTheorem3thecondition(9)isreplacedbythe assumption u = u + u 1 and u L (, T, L 3 (Ω)), u 1 L α (, T, L q (Ω)), sup <t<t u (t) 3 < εand2/α+3/q=1with q (3, ]. InTheorem4,the space L 3 (Ω)isusedinsteadofthespace L 3 w (Ω).Theorem5canbestatedwithout any change. Conclusion The results on regularity of weak solutions to the Navier-Stokes equations presented in this paper have been proved recently in[3]. It is interesting, however, thataneasyproofoftheseresultscanbebasedonawellknownclassicaltheorem on compact operators. Further, weak solutions of the Navier-Stokes equations inthespace L 2 (, T, W 1,3 (Ω) 3 )areregular(theorem5),whichisinterestingin connection with the famous Prodi-Serrin s conditions(see[3]). Acknowledgment. The research was supported by the Ministry of Education of the Czech Republic(project No. VZ J4/98211), by the Grant Agency of the Academy of Sciences of the Czech Republic through the grant A2683 and by the Institute of Hydrodynamics AS CR(project No. 5921). References [1] GigaY.,Solutionsforsemilinearparabolicequationsin L p andregularityofweaksolutions of the Navier-Stokes system, J. Differential Equations 62(1986), [2] Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, New York 198. [3] Kozono H., Uniqueness and regularity of weak solutions to the Navier-Stokes equations, Lecture Notes in Num. and Appl. Anal. 16(1998), [4] Neustupa J., Partial regularity of weak solutions to the Navier-Stokes Equations in the class L (, T, L 3 (Ω)),J.Math.FluidMech.1(1999),1 17. [5] Serrin J., On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9(1962), [6] Temam R., Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland Publishing Company, Amsterdam, New York, Oxford, [7] Temam R., Navier-Stokes Equations and Nonlinear Functional Analysis, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, second edition, Department of Mathematics, Faculty of Civil Engineering, Czech Technical University, Thákurova 7, Prague 6, Czech Republic (Received December 1, 1999)