summary: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, weak solutions of the Navier-Stokes equations in the space $L^2(0,T,W^{1,3}(\varOmega)^3)$ are regular

Kučera, Petr

Skalák, Zdeněk

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeZdeněk Skalák; Petr KučeraRemark on regularity of weak solutions to the Navier-Stokes equationsCommentationes Mathematicae Universitatis Carolinae, Vol. 42 (2001), No. 1, 111--117Persistent URL: http://dml.cz/dmlcz/119227Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 2001Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 42,1 (2001)111–117 111Remark on regularity of weak solutionsto the Navier-Stokes equationsZdeněk Skalák, Petr KučeraAbstract. Some results on regularity of weak solutions to the Navier-Stokes equationspublished recently in [3] follow easily from a classical theorem on compact operators.Further, weak solutions of the Navier-Stokes equations in the space L2(0, T, W 1,3(Ω)3)are regular.Keywords: Navier-Stokes equations, weak solution, regularityClassification: 35Q10, 76D05, 76F99IntroductionLet Ω be a bounded domain in R3 with C2-boundary ∂Ω, let T > 0 andQT = Ω × (0, T ). We consider the Navier-Stokes initial-boundary value problemdescribing the evolution of the velocity u(x, t) and the pressure p(x, t) in QT :∂u∂t− ν∆u + u · ∇u+∇p = f ,(1)∇ · u = 0,(2)u = 0 on ∂Ω × (0, T ),(3)u|t=0 = u0,(4)where ν > 0 is the viscosity coefficient and f is the external body force. The initialdata u0 should satisfy the compatibility conditions u0|∂Ω = 0 and ∇ · u0 = 0.The definition and the proof of the existence of weak solutions of the equations(1)–(4) can be found for example in [3] or [6]. In general, it is unknown whetherweak solutions are regular or not. Serrin ([5]) proved that if a weak solution uof (1)–(4) belongs to Lα(0, T, Lq(Ω)) for 2/α+ 3/q = 1 and q ∈ (3,∞] then u isregular. Kozono ([3]) generalized this result to a certain class of functions char-acterized by means of local singularities in the weak-L3 space. He further showedthat there exists an absolute constant ε > 0 such that if u is a weak solution of(1)–(4) in L∞(0, T, L3(Ω)3) and lim supt→t∗− ‖u(t)‖L3(Ω) < ‖u(t∗)‖L3(Ω) + ε,then u is necessarily regular in Ω× (t∗−σ, t∗+σ) for some σ > 0. Let us mentionhere that the Kozono’s results were applied in [4] where partial regularity of weaksolutions to the Navier-Stokes equations in the class L∞(0, T, L3(Ω)) was shown.The main goal of this paper is to show that the results stated above can beeasily derived from the following well known theorem on compact operators ([2]):112 Z. Skalák, P.KučeraTheorem A. Let X , Y be Banach spaces. Let S be a one to one continuouslinear operator from X onto Y and K a linear compact operator from X to Y . IfKer (S +K) = o then (S +K)(X) = Y .Let p > 1. Lp(Ω) is the Lebesgue space with the norm ‖ · ‖p. C∞0 (Ω) denotesthe set of all infinitely differentiable vector-functions defined in Ω, with a compactsupport in Ω. C∞0,σ(Ω) is a subset of C∞0 (Ω) which contains only the divergence-free vector functions. H is the closure of C∞0,σ(Ω) in L2(Ω)3 with the scalarproduct (·, ·) and the norm ‖ · ‖2. Wm,p(Ω) and Wm,p0 (Ω) (m ∈ N) are the usualSobolev spaces. V denotes the completion of C∞0,σ(Ω) in the norm of W1,20 (Ω)3with the scalar product ((u, v)) =∫Ω∂ui∂xj∂vi∂xjdx and the norm ‖ · ‖. PH is theprojection operator from L2(Ω)3 onto H .Lpw(Ω) denotes the weak Lebesgue space over Ω with the quasi-norm ‖·‖p,w de-fined by ‖φ‖p,w = supR>0Rµ{x ∈ Ω; |φ(x, t)| > R}1/p, where µ is the Lebesguemeasure. There is another equivalent norm to the above ‖ · ‖p,w (see [3]), so wemay understand Lpw(Ω) as a Banach space. Let us note that Lp(Ω) ⊆ Lpw(Ω) and‖φ‖p,w ≤ ‖φ‖p for every φ ∈ Lp(Ω).Let D(A) = {u ∈ V ; ∃f ∈ H ; ((u, v)) = (f , v) ∀v ∈ V }. A is the Stokes op-erator from D(A) onto H defined for every u ∈ D(A) by the equation ((u, v)) =(Au, v) ∀v ∈ V . D(A) is endowed with the norm ‖u‖D(A) = ‖Au‖2 andD(A) →֒→֒ V . Since Ω ∈ C2, D(A) = W 2,2(Ω)3 ∩ V and the norm ‖u‖D(A)on D(A) is equivalent to the norm induced by W 2,2(Ω)3 (see [6, Lemma 3.7]).We often use this fact throughout the paper. Let us define the Banach spacesX = {u ∈ L2(0, T, D(A)), ut ∈ L2(0, T, H)} and Y = L2(0, T, H) × V with‖u‖X = ‖u‖L2(0,T,D(A))+‖ut‖L2(0,T,H) and ‖(f , v0)‖Y = ‖f‖L2(0,T,H)+‖v0‖V .Throughout the paper, we suppose that in (1)–(4) f ∈ L2(0, T, H) and u0 ∈ H .For simplicity, we use the following notation: If F is a space of real functions thenu ∈ F means that every component of u is from F , e.g. u ∈ W 1,2(Ω) means infact that u ∈ W 1,2(Ω)3. Similarly, ‖u‖F means ‖u‖F 3.Proof of regularity resultsAt first, we prove two basic propositions. The results mentioned in Introductionwill then be their straightforward consequences.Proposition 1. Let u ∈ Lα(0, T, Lq(Ω)) for 2/α+3/q ≤ 1 and q ∈ (3,∞]. Thenthe operator w 7−→ PH (u · ∇w) is compact from X to L2(0, T, H).Proof: Firstly, suppose that 2/α + 3/q < 1 and α, q < ∞. Using the Hőlderinequality we have for almost every t ∈ (0, T ) and every v ∈ H :|∫Ωu · ∇w · v dx| ≤ ‖v‖2‖u‖q‖∇w‖2q/(q−2).Regularity of weak solutions 113It follows further that∫ T0‖u‖2q‖∇w‖22q/(q−2) dt ≤ ‖u‖2Lα(0,T,Lq(Ω))(∫ T0‖∇w‖2α/(α−2)2q/(q−2)dt)(α−2)/α ≤‖u‖2Lα(0,T,Lq(Ω))(∫ T0[‖∇w‖2/α2 ‖∇w‖(α−2)/α(2αq−4q)/(αq−2α−2q)]2α/(α−2) dt)(α−2)/α ≤‖u‖2Lα(0,T,Lq(Ω))‖w‖4/αL∞(0,T,W 1,2(Ω))(∫ T0‖∇w‖2(2αq−4q)(αq−2α−2q) dt)(α−2)/α ≤‖u‖2Lα(0,T,Lq(Ω))‖w‖4/αL∞(0,T,W 1,2(Ω))‖w‖2(α−2)/αL2(0,T,W 1,(2αq−4q)/(αq−2α−2q)(Ω))and, therefore,(5) ‖PH(u · ∇w)‖L2(0,T,H) ≤‖u‖Lα(0,T,Lq(Ω))‖w‖2/αX ‖w‖(α−2)/αL2(0,T,W 1,(2αq−4q)/(αq−2α−2q)(Ω)),where we used the fact that X is embedded continuously into L∞(0, T, W 1,2(Ω)).Since (2αq−4q)/(αq−2α−2q) < 6 it follows e.g. from [5, Theorem 2.1, Chapter III]that the injection ofX into L2(0, T, W 1,(2αq−4q)/(αq−2α−2q)(Ω)) is compact. Theproof now follows immediately from (5) and the definition of compact operators.Secondly, let u ∈ Lα(0, T, L∞(Ω)), α > 2. Then |∫Ω u · ∇w · v dx| ≤‖v‖2‖u‖∞‖w‖W 1,2 for almost every t ∈ (0, T ) and every v ∈ H and∫ T0‖u‖2∞‖w‖2W 1,2 dt ≤ ‖u‖2Lα(0,T,L∞(Ω))(∫ T0‖w‖2α/(α−2)W 1,2(Ω)dt)(α−2)/α =‖u‖2Lα(0,T,L∞(Ω))(∫ T0‖w‖4/(α−2)W 1,2(Ω)‖w‖2W 1,2(Ω) dt)(α−2)/α ≤‖u‖2Lα(0,T,L∞(Ω))‖w‖4/αL∞(0,T,W 1,2(Ω))(∫ T0‖w‖2W 1,2 dt)(α−2)/α =‖u‖2Lα(0,T,L∞(Ω))‖w‖4/αL∞(0,T,W 1,2(Ω)‖w‖2(α−2)/αL2(0,T,W 1,2(Ω)).Therefore,(6) ‖PH(u · ∇w)‖L2(0,T,H) ≤ ‖u‖Lα(0,T,L∞(Ω))‖w‖2/αX ‖w‖(α−2)/αL2(0,T,W 1,2(Ω)).The injection of X into L2(0, T, W 1,2(Ω)) is compact and the proof follows im-mediately from (6) and the definition of compact operators.If u ∈ L∞(0, T, Lq(Ω)) and q > 3 then the proof proceeds in the same way asin the previous paragraphs and we will skip it.114 Z. Skalák, P.KučeraFinally, let u ∈ Lα(0, T, Lq(Ω)) for 2/α+ 3/q = 1, q ∈ (3,∞]. Let Mn = {t ∈(0, T ); ‖u(t)‖q > n}, n ∈ N and define un on (0, T ) as:un(t) = u(t) if t /∈ Mn,un(t) = 0 if t ∈ Mn.Obviously, un ∈ L∞(0, T, Lq(Ω)) and according to the previous paragraphs theoperators w 7−→ PH (un · ∇w) are compact from X to L2(0, T, H). Further, theLebesgue measure ofMn goes to zero for n → ∞ so that ‖u−un‖Lα(0,T,Lq(Ω)) =(∫Mn‖u‖αq dt)1/α 7−→ 0. Therefore, the operator w 7−→ PH(u · ∇w) is compactfrom X to L2(0, T, H) as a limit of compact operators w 7−→ PH (un ·∇w) in theusual norm of the space of all linear bounded operators from X to L2(0, T, H).Let us consider the following Stokes equations with the perturbed convectionterm PH(u · ∇w):wt + νAw + PH (u · ∇w) = f ,(7)w(0) = w0.(8)Proposition 2. Let 2/α+3/q = 1 with q ∈ (3,∞]. Then there exists ε > 0 withthe following property: if u = u0 + u1 in (0, T ), u(t) ∈ V for almost every t ∈(0, T ), u0 ∈ L∞(0, T, L3w(Ω)), u1 ∈ Lα(0, T, Lq(Ω)) and sup0<t<T ‖u0(t)‖3,w <ε, then for every w0 ∈ V and f ∈ L2(0, T, H) there exists a unique solution w of(7), (8) in X .Proof: The operator w 7−→ (wt + νAw, w(0)) is a one to one continuous linearoperator from X onto Y . It is possible to prove (see also [3, Lemma 2.7]) thatthe operator w 7−→ PH (u0 · ∇w) is linear and bounded from X to L2(0, T, H)with the norm less than C‖u0‖L∞(0,T,L3w(Ω)). Since the set of linear bounded oneto one operators is open in the space of all linear bounded operators (using theusual topology) we get that the operator w 7−→ (wt+νAw+PH(u0 ·∇w), w(0))is a one to one operator from X onto Y for ε being sufficiently small. Finally, itfollows from Proposition 1 that the operator w 7−→ PH (u1 ·∇w) is compact fromX to L2(0, T, H). Moreover, the operator w 7−→ (wt+νAw+PH (u ·∇w), w(0))is one to one from X to Y and the proof follows immediately from Theorem A.Now, we present proofs of the results stated in Introduction. The proofs arebased on Propositions 1 and 2. Theorem 3 is a generalization of the famousSerrin’s result ([5]) on regularity of weak solutions in the subcritical case and wasproved in [3]. Theorem 4 which is dealing with the partial regularity of weaksolutions in the supercritical case L∞(0, T, L3(Ω)) was also proved in [3]. Wepresent these theorems in a little more general way.Regularity of weak solutions 115Theorem 3. There exists a constant ε with the following property. If u is aweak solution of (1)–(4) and there exists a non-negative L2-function M = M(t)on (0, T ) such that(9) supR≥M(t)Rµ{x ∈ Ω; |u(x, t)| > R}1/3 ≤ εfor almost every t ∈ (0, T ), then u is regular, that is ∂u∂t , Dαxu ∈ C(Ω × (0, T ))for every multi-index α with |α| ≤ 2.Proof: Due to the condition (9) u can be easily decomposed as u = u0 + u1,where u0 ∈ L∞(0, T, L3w(Ω)), u1 ∈ L2(0, T, L∞(Ω)) and sup0<t<T ‖u0(t)‖3,w <ε (see [3]). Let σ ∈ (0, T ) be an arbitrary number. Since the weak solutionu ∈ L2(0, T, V ), there exists a t0 ∈ (0, σ) such that u(t0) ∈ V . If ε is sufficientlysmall it follows from Proposition 2 that there exists a unique solution w ∈ X of(7), (8) on (t0, T ) with w(t0) = u(t0). It is easy to show that u = w on (t0, T )and therefore u ∈ X on (t0, T ). Since σ was chosen arbitrarily the theorem followsimmediately 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 uis a weak solution of (1)–(4) and there existsw ∈ L3(Ω) such that ‖u(t)−w‖3,w <ε for almost every t ∈ (a, b) ⊂ (0, T ), then ∂u∂t , Dαxu ∈ C(Ω × (a, b)) for everymulti-index α with |α| ≤ 2.Proof: There exists w1 ∈ L4(Ω) such that ‖w − w1‖3 < ε. If we put u0 =u − w1 and u1 = w1, then u = u0 + u1 on (a, b), u0 ∈ L∞(a, b, L3w(Ω)),u1 ∈ L∞(a, b, L4(Ω)) and supa<t<b ‖u0(t)‖3,w < 2ε. Now, applying again Propo-sitions 1 and 2 on (a, b) and using the same arguments as in Theorem 3, Theorem 4follows immediately. It was proved in [1] and [3] that if u is a weak solution of (1)–(4) and u ∈C([0, T ), L3(Ω)) or u ∈ BV ([0, T ), L3(Ω)) — the set of all functions of boundedvariation on [0, T ) with values in L3(Ω) — then u is regular. These results areconsequences of Theorem 4.The following theorem is another example of the use of Theorem A in theregularity theory of the Navier-Stokes equations. Let us note here that the spaceL2(0, T, W 1,3(Ω)) is not imbedded into any Lα(0, T, Lq(Ω)) with 2/α+ 3/q = 1and q ∈ (3,∞].Theorem 5. Let u be a weak solution of (1)–(4) and u ∈ L2(0, T, W 1,3(Ω)).Then ∂u∂t , Dαxu ∈ C(Ω × (0, T )) for every multi-index α with |α| ≤ 2.Proof: Firstly, let us show that the operatorw 7−→ PH(w ·∇u) is compact fromX to L2(0, T, H). Using the Hőlder inequality we have for almost every t ∈ (0, T )and every v ∈ H :|∫Ωw · ∇u · v dx| ≤ c‖v‖2‖w‖W 1,2(Ω)‖u‖W 1,3(Ω).116 Z. Skalák, P.KučeraIt follows easily as in the first paragraph of Proposition 1 that‖PH(w · ∇u)‖L2(0,T,H) ≤ c‖w‖X‖u‖L2(0,T,W 1,3(Ω)) so that w 7−→ PH(w · ∇u)is a linear bounded operator from X to L2(0, T, H). As in the last paragraph ofProposition 1 it is possible to construct un ∈ L∞(0, T, W 1,3(Ω)) such that ‖u −un‖L2(0,T,W 1,3(Ω)) 7−→ 0 and the compactness of the operator w 7−→ PH(w ·∇u)follows now from this and from the fact that the operators w 7−→ PH(w · ∇un)are compact.It follows from the standard estimates in Sobolev spaces, the Gronwall lemmaand Theorem A that for everyw0 ∈ V and f ∈ L2(0, T, H), the following problemhas a unique solution w ∈ X :wt + νAw + PH (w · ∇u) = f ,(12)w(0) = w0.(13)The proof is concluded using the same arguments as in the proof of Theorem 3.Remark 6. If e.g. f ∈ H (f independent of time) then in Theorem 3 and The-orem 5, resp. Theorem 4 u is analytic in time, in a neighborhood of the in-terval (0, T ), resp. (a, b), as a D(A)-valued function (see [7]). It follows thatu ∈ C∞(0, T, C(Ω)), resp. u ∈ C∞(a, b, C(Ω)). Therefore, u has no singularpoints in Ω × (0, T ), resp. Ω × (a, b). Also, u(x, ·) is an infinitely differentiablefunction in (0, T ), resp. (a, b), for every x ∈ Ω.Remark 7. If Ω ∈ C0,1 then the information from the Introduction — D(A) =W 2,2(Ω)3 ∩V and the norm ‖u‖D(A) on D(A) is equivalent to the norm inducedby W 2,2(Ω)3 — cannot be used. We do not even know in this case whetherD(A) →֒ W 1,2+ε(Ω)3 for a positive ε or not. What we only have here is thatD(A) →֒→֒ V and also X →֒ L∞(0, T, V ). As a consequence, Propositions 1and 2 can be proved only if u ∈ L2(0, T, L∞(Ω)) and the proofs of Theorems 3and 4 fail totally. On the other hand, it is interesting that Theorem 5 can bestated and proved without any change.Remark 8. If Ω is the half-space or R3 (or possibly some other special unboundeddomain) then we are able to obtain almost the same results as in the case of abounded domain. Let us discuss it briefly. V denotes the completion of C∞0,σ(Ω) inthe norm ofW 1,2(Ω)3 with the scalar product ((u, v))V =∫Ω(∂ui∂xj∂vi∂xj+uivi) dx.D(A) is then defined as {u ∈ V ; ∃f ∈ H ; ((u, v))V = (f , v) ∀v ∈ V } and usingthe cut-off method it is possible to show that D(A) →֒ W 2,2(Ω). It implies thatX →֒ L2(0, T, W 2,2(Ω)) and, consequently, X →֒→֒ L2(0, T, W 1,6−ε(Θ)) for everysmall ε > 0 and every smooth domain Θ ⊆ Ω. As a result, Proposition 1 can beproved in a similar way as in the case of a bounded domain and Proposition 2holds with only one change: the weak Lebesgue space L3w(Ω) is replaced byRegularity of weak solutions 117the Lebesgue space L3(Ω). In Theorem 3 the condition (9) is replaced by theassumption u = u0 + u1 and u0 ∈ L∞(0, T, L3(Ω)), u1 ∈ Lα(0, T, Lq(Ω)),sup0<t<T ‖u0(t)‖3 < ε and 2/α + 3/q = 1 with q ∈ (3,∞]. In Theorem 4, thespace L3(Ω) is used instead of the space L3w(Ω). Theorem 5 can be stated withoutany change.ConclusionThe results on regularity of weak solutions to the Navier-Stokes equations pre-sented in this paper have been proved recently in [3]. It is interesting, however,that an easy proof of these results can be based on a well known classical theoremon compact operators. Further, weak solutions of the Navier-Stokes equationsin the space L2(0, T, W 1,3(Ω)3) are regular (Theorem 5), which is interesting inconnection with the famous Prodi-Serrin’s conditions (see [3]).Acknowledgment. The research was supported by the Ministry of Educationof the Czech Republic (project No. VZ J04/98210000010), by the Grant Agencyof the Academy of Sciences of the Czech Republic through the grant A2060803and by the Institute of Hydrodynamics AS CR (project No. 5921).References[1] Giga Y., Solutions for semilinear parabolic equations in Lp and regularity of weak solutionsof the Navier-Stokes system, J. Differential Equations 62 (1986), 182–212.[2] Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg,New York 1980.[3] Kozono H., Uniqueness and regularity of weak solutions to the Navier-Stokes equations,Lecture Notes in Num. and Appl. Anal. 16 (1998), 161–208.[4] Neustupa J., Partial regularity of weak solutions to the Navier-Stokes Equations in theclass L∞(0, T, L3(Ω)), J. Math. Fluid Mech. 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), 187–195.[6] Temam R., Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland Pub-lishing Company, Amsterdam, New York, Oxford, 1979.[7] Temam R., Navier-Stokes Equations and Nonlinear Functional Analysis, Society for In-dustrial and Applied Mathematics, Philadelphia, Pennsylvania, second edition, 1995.Department of Mathematics, Faculty of Civil Engineering, Czech Technical Uni-versity, Thákurova 7, 166 29 Prague 6, Czech Republic(Received December 10, 1999)

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

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

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