1. Introduction Since the pioneering work by Leray [3] in 1934, there have been several studies on solutions of Navier-Stokes equations
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1 Math. Res. Lett. 13 (6, no. 3, c International Press 6 NAVIER-STOKES EQUATIONS IN ARBITRARY DOMAINS : THE FUJITA-KATO SCHEME Sylvie Monniaux Abstract. Navier-Stokes equations are investigated in a functional setting in 3D open sets Ω, bounded or not, without assuming any regularity of the boundary Ω. The main idea is to find a correct definition of the Stokes operator in a suitable Hilbert space of divergence-free vectors and apply the Fujita-Kato method, a fixed point procedure, to get a local strong solution. 1. Introduction Since the pioneering work by Leray [3] in 1934, there have been several studies on solutions of Navier-Stokes equations u t Δu + π +(u u = in ],T[ Ω, div u = in ],T[ Ω, (NS u = on ],T[ Ω, u( = u in Ω. Fujita and Kato [] in 1964 gave a method to construct so called mild solutions in smooth domains Ω, producing local (in time smooth solutions of (NS in a Hilbert space setting. These solutions are global in time if the initial value u is small enough in a certain sense. The case of non smooth domains has been studied by Deuring and von Wahl [1] in 1995 where they considered domains Ω R 3 with Lipschitz boundary Ω. They found local smooth solutions using results contained in Shen s PhD thesis [4]. Their method does not cover the critical space case as in []. One of the difficulty there was to understand the Stokes operator, and in particular its domain of definition. In Section, we give a universal definition of the Stokes operator, for any domain Ω R 3 (Defintion.4. In Section 3, we construct a mild solution of (NSwitha method similar to Fujita-Kato s [] (Theorem 3.5 for initial values u in the critical space D(A 1 4. We show in Section 4 that this mild solution is a strong solution, i.e. (NS is satisfied almost everywhere. Let Ω be an open set in R 3. The space. The Stokes operator L (Ω 3 = {u =(u 1,u,u 3 ; u i L (Ω, i =1,, 3} Received by the editors November 15, 5. Mathematics Subject Classification. Primary 35Q1, 76D5 ; Secondary 35A
2 456 SYLVIE MONNIAUX endowed with the scalar product u, v = is a Hilbert space. Define Ω u v = Ω u i v i G = { p; p L loc(ω with p L (Ω 3 }; the set G is a closed subspace of L (Ω 3.Let H = G = { u L (Ω 3 ; u, p =, p L loc(ω with p L (Ω 3}. The space H, endowed with the scalar product, is a Hilbert space. We have the following Hodge decomposition L (Ω 3 = H G. We denote by P the projection from L (Ω 3 onto H : P is the usual Helmoltz projection. We denote by J the canonical injection H L (Ω 3 : J = P (J beeing the adjoint of J andpj is the identity on H. LetnowD(Ω 3 = Cc (Ω 3 and D = {u D(Ω 3 ;divu =}. It is clear that D is a closed subspace of D(Ω 3. WedenotebyJ : D D(Ω 3 the canonical injection : J J. Let P 1 be the adjoint of J : P 1 = J : D (Ω 3 D. We have P P 1. The following theorem characterizes the elements in ker P 1. Theorem.1 (de Rham. Let T D (Ω 3 such that P 1 T =in D. Then there exists S (C c (Ω such that T = S. Conversely, if T = S with S (C c (Ω, then P 1 T =in D. We denote by H 1 (Ω 3 the closure of D(Ω 3 with respect to the scalar product (u, v u, v 1 = u, v + 3 iu, i v. By Sobolev embeddings, we have H 1 (Ω 3 L 6 (Ω 3. Define V = H H 1 (Ω 3. The space V is a closed subspace of H 1 (Ω 3 ; endowed with the scalar product, 1, V is a Hilbert space. Proposition.. The space V is dense in H. Proof. Let u Hbe in the orthogonal of V with respect to H, i.e. (.1 u, v = forallv V. Since D V, (.1 implies also u, v = forallv D. It means that u, viewed as an element of D, is. By Theorem.1, there exists a distribution S D(Ω such that Ju = S. Since Ju L (Ω 3,sois S and therefore, u = PJu = P S =.
3 NAVIER-STOKES IN ARBITRARY DOMAINS 457 The canonical injection J : V H 1 (Ω 3 is the restriction of J to V. We denote by P the adjoint of J :since J is the restriction of J to V, P is an extension of P to V.OnV V we define now the form a by a(u, v = i Ju, i Jv : a is a bilinear, symmetric, δ + a is a coercive form on V V for all δ>, then defines a bounded self-adjoint operator A : V V by (A u(v =a(u, v withδ + A invertible for all δ>. Proposition.3. For all u V, A u = P( Δ Ω D Ju,whereΔ Ω D denotes the Dirichlet- Laplacian on H 1 (Ω 3. Proof. For all u, v V,wehave (A u(v (1 = a(u, v ( = i Ju, i Jv (3 = ( Δ Ω D Ju, Jv H 1,H 1 (4 = P( Δ Ω D Ju,v V,V. ThefirsttwoequalitiescomefromthedefinitionofA and a. The third equality comes from the definition of the Dirichlet-Laplacian on H 1 (Ω 3 and the fact that for v V, Jv = v. The last equality is due to J ϕ = Pϕ in V for all ϕ H 1 (Ω 3.This shows that A u and P( Δ Ω D Ju are two continuous linear forms on V which coïncide on V, theyarethenequal. Definition.4. The operator A defined on its domain D(A ={u V; A u H} by Au = A u is called the Stokes operator. Theorem.5. The Stokes operator is self-adjoint in H, generates an analytic semigroup (e ta t, D(A 1 =V and satisfies D(A = {u V ; π (Cc (Ω : π H 1 (Ω and Δu + π H} Au = Δu + π. Remark.6. Since H 1 (Ω 3 L 6 (Ω 3, it is clear by interpolation and dualization that P maps L p (Ω 3 to D(A s for 6 5 p, s 1 and s = p. Since A is self-adjoint, one has (δ + A s D(A s = {(δ + A s u; u D(A s } = H. In particular, (δ + A 1 4 P 1 maps L 3 (Ω 3 into H. 3. Mild solution to the Navier-Stokes system Let T>. Define the following Banach space { E T = u C ([,T]; D(A 1 4 C 1 (],T]; D(A 1 4 such that sup s A u(s H + sup sa 1 4 u (s H < <s<t <s<t }
4 458 SYLVIE MONNIAUX endowed with the norm u ET = sup <s<t A 1 4 u(s H + sup <s<t s A u(s H + sup sa 1 4 u (s H. <s<t Let α be defined by α(t =e ta u where u D(A 1 4. Then α E T. Indeed, it is clear that α C ([,T]; D(A 1 4. We also have that t 1 4 A 1 α(t =t 1 4 A 1 4 e ta A 1 4 u is bounded on (,Tsince(e ta t is an analytic semigroup. Moreover, one has α (t = Ae ta u which yields to ta 1 4 α (t = tae ta A 1 4 u continuous on ],T], bounded in H. Foru, v E T,wedefinenow Φ(u, v(t = e (t sa ( 1 P((u(s v(s+(v(s u(sds, <t<t. Notation 3.1. Let X, Y be Banach spaces. For a bounded linear operator S : X Y,wedenoteby S L (X;Y the norm of S, i.e. S L (X;Y =sup{ Sx Y ; x X with x X 1}. If X = Y, we adopt the notation S L (X instead of S L (X;Y. For a bilinear operator B : X X Y, we denote by B L (X X;Y the norm of B, i.e. B L (X X;Y =sup{ B(x, x Y ; x, x X with x X 1and x X 1}. Notation 3.. For u, v L (Ω 3,wedenotebyu v the matrix defined by (u v i,j = u i v j, 1 i, j 3. Remark 3.3. If u, v are sufficiently smooth vector fields such that divu =,then div(u v := i (u i v= u i i v =(u v. Proposition 3.4. The transform Φ is bilinear, symmetric, continuous from E T E T to E T and the norm of Φ is independent of T. Proof. The fact that Φ is bilinear and symmetric is clear. Moreover, Φ(u, v =e A f, where f is defined by f(s =( 1 P((u(s v(s+(v(s u(s, s [,T]. For u, v E T, it is clear that (u(s v(s+(v(s u(s L 3 (Ω 3 and therefore (δ +A 1 4 f(s Hwith sup <s<t s 1 (δ + A 1 4 f(s H c u ET v ET. Wehavethen Φ(u, v =e A f =(δ + A 1 4 e A ((δ + A 1 4 f and therefore A 1 4 Φ(u, v(t H A (δ + A 4 e (t sa L (H (δ + A 1 4 f(s H ds ( 1 1 c s ds u ET v ET t s ( c σ dσ u ET v ET 1 σ c u ET v ET.
5 NAVIER-STOKES IN ARBITRARY DOMAINS 459 Continuity with respect to t [,T]oft A 1 4 Φ(u, v(t is clear once we have proved the boundedness. We also have A 1 Φ(u, v(t H A 1 1 (δ + A 4 e (t sa L (H (δ + A 1 4 f(s H ds ( 1 1 c ds u (t s 3 ET v ET 4 s ( 1 ct dσ u (1 σ 3 ET v ET 4 σ ct 1 4 u ET v ET. Continuity with respect to t ],T] is clear once we have proved the boundedness. To prove the last part of the norm of Φ(u, v ine T,wefirstwritef, using Notation 3. and Remark 3.3, in the following form f(s =( 1 P div(u(s v(s+v(s u(s, s [,T]. We have then for s ],T[ f (s =( 1 P div(u (s v(s+u(s v (s+v (s u(s+v(s u (s. For all s ],T]wehave s 5 4 u (1 (s v(s su (s 3 s 1 4 v(s 6 ( sa 1 4 u (s H s A v(s H (3 u ET v ET, where the first inequality comes from the fact that L 3 L 6 L, the second comes from the inclusions D(A 1 4 L 3 (Ω 3 and D(A 1 L 6 (Ω 3 and the third inequality follows directly from the definition of the space E T. Of course the same occurs for the other three terms u(s v (s, v (s u(s andv(s u (s. Therefore, since A 1 maps V to H, we obtain We have and therefore Φ(u, v(t = sup s 5 4 (δ + A 1 f (s H c u ET v ET. <s<t e sa f(t sds + e (t sa f(sds t ],T[, Φ(u, v (t = e t A f( t + (δ + A 1 e sa (δ + A 1 f (t sds + A(δ + A 1 4 e (t sa (δ + A 1 4 f(sds,
6 46 SYLVIE MONNIAUX which yields ( A 1 4 Φ(u, v (t H c (δ + A 1 4 f( t t H c ds u s 1 (t s 5 ET v ET 4 ( 1 1 +c ds u (t s 5 4 s 1 ET v ET ( c 1 dσ u t (1 σ 5 4 σ 1 ET v ET. This last inequality ensures that Φ(u, v E T whenever u, v E T. Theorem 3.5. For all u D(A 1 4, thereexistst> such that there exists a unique u E T solution of u = α+φ(u, u on [,T]. This function u is called the mild solution to the Navier-Stokes system. Proof. Let T>. Since Φ : E T E T E T is bilinear continuous, it suffices to apply Picard fixed point theorem, as in []. The sequence in E T (v n n N defined by v = α as first term and v n+1 = α +Φ(v n,v n, n N converges to the unique solution u E T of u = α +Φ(u, u provided A 1 4 u H is 1 small enough ( α ET < 4 Φ. In the case where A 1 4 L (ET u E T ;E T H is not small 1 (that is, if α ET 4 Φ thenforε>, there exists u L (ET,ε D(A such E T ;E T that A 1 4 (u u,ε H ε. If we take as initial value u,ε D(A, we have α ε ET ct 3 4 Au,ε H. T 1 Therefore, we can find T>such that α ET < 4 Φ. L (ET E T ;E T 4. Strong solutions Let u be the mild solution to the Navier-Stokes system. We show in this section that u in fact satisfies the equations of the Navier-Stokes system in an L p sense (for a suitable p. To begin with, we know that u E T and satisfies u = α +Φ(u, u =α + e A ϕ(u, where ϕ(u = P((u u and we have t 1 (u(t u(t 3 c u E T. Therefore, we get (4.1 u( = α( = u, (4. divu(t =inthel sense for t ],T[, and u + Au = f in C (],T[; V, which means that for all t ],T[, P(u (t Δ Ω Du(t+(u(t u(t =.
7 NAVIER-STOKES IN ARBITRARY DOMAINS 461 Then, by Theorem.1, there exists ( π(t (Cc (Ω such that π(t H 1 (Ω 3 and (4.3 ( π(t =u (t Δ Ω Du(t+(u(t u(t and we have for <t<t Δ Ω Du(t+ π(t = u (t (u(t u(t L 3 (Ω 3 + L 3 (Ω 3. The equation (4.3, together with (4.1 and (4., give the usual Navier-Stokes equations which are fulfilled in a strong sense (a.e. where we consider the expression Δu + π undecoupled. References [1] P. Deuring, W. von Wahl. Strong solutions of the Navier-Stokes system in Lipschitz bounded domains. Math. Nachr. 171 ( [] H. Fujita, T. Kato. On the Navier-Stokes initial value problem I. Arch. Rational Mech. Anal. 16 ( [3] J. Leray. Sur le mouvement d un liquide visqueux emplissant l espace. ActaMath. J. 63 ( [4] Z. Shen. Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders. Amer. J. Math. 113 ( LATP UMR 663, Case cour A, Faculté des sciences de Saint-Jérôme, Université Paul Cézanne (Aix-Marseille 3, Marseille Cédex, France address: sylvie.monniaux@univ.u-3mrs.fr
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