On the Stokes operator in general unbounded domains. Reinhard Farwig, Hideo Kozono and Hermann Sohr

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1 Hokkaido Mathematical Journal Vol. 38 (2009) p On the Stokes operator in general unbounded domains Reinhard Farwig, Hideo Kozono and Hermann Sohr (Received June 27, 2007; Revised December 19, 2007) Abstract. It is known that the Stokes operator is not well-defined in L -spaces for certain unbounded smooth domains unless = 2. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general unbounded smooth domains from the three-dimensional case, see [7], to the n-dimensional one, n 2, replacing the space L, 1 < <, by L where L = L L 2 for 2 and L = L + L 2 for 1 < < 2. In particular, we show that the Stokes operator is well-defined in L for every unbounded domain of uniform C 1,1 -type in R n, n 2, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity. Key words: General unbounded domains, domains of uniform C 1,1 -type, Stokes operator, Stokes resolvent, Stokes semigroup, maximal regularity 1. Introduction Let Ω R n, n 2, denote a general unbounded domain with uniform C 1,1 -boundary Ω, see Definition 1.1 below. As is well-known, the analysis of the instationary Navier-Stokes euations reuires L -estimates, 2, to prove the strong energy estimate, the localized energy estimate involving also the pressure function and Leray s Structure Theorem for weak solutions. Unfortunately, the standard approach to the Stokes euations in L -spaces, 1 < <, cannot be extended to general unbounded domains unless = 2. On the one hand, the Helmholtz decomposition fails to exist for certain unbounded smooth domains on L, 2, see [4], [14]. On the other hand, in L 2 the Helmholtz proection and the Stokes operator are well-defined for every domain, the latter is self-adoint, generates a bounded analytic semigroup and has maximal regularity. In order to work locally in L -spaces, but globally, to be more precise, near space infinity, in L 2, the authors introduced in [7] in the threedimensional case the function space L (Ω) = { L (Ω) L 2 (Ω), 2 < L (Ω) + L 2 (Ω), 1 < < Mathematics Subect Classification : Primary 76D05; Secondary 35Q30.

2 112 R. Farwig, H. Kozono and H. Sohr to define the Helmholtz decomposition and the space L (Ω) = { L (Ω) L 2 (Ω), 2 < L (Ω) + L 2 (Ω), 1 < < 2 of solenoidal vector fields in L (Ω) to define and to analyze the Stokes operator. It was proved that for every unbounded domain Ω R 3 of uniform C 2 -type the Stokes operator in L (Ω) satisfies the usual resolvent estimate, generates an analytic semigroup and has maximal regularity. Moreover, for every dimension n 2, the Helmholtz decomposition of L (Ω) exists for every unbounded domain Ω R n of uniform C 1 -type, see [8]. To describe this result, we introduce the space of gradients G (Ω) = { G (Ω) G 2 (Ω), 2 < G (Ω) + G 2 (Ω), 1 < < 2, where G (Ω) = { p L (Ω) : p L loc (Ω)}, and recall the notion of domains of uniform C k - and C k,1 -type. Definition 1.1 A domain Ω R n, n 2, is called a uniform C k -domain of type (α, β, K) where k N, α > 0, β > 0, K > 0, if for each x 0 Ω there exists a Cartesian coordinate system with origin at x 0 and coordinates y = (y, y n ), y = (y 1,..., y n 1 ), and a C k -function h(y ), y α, with C k -norm h C k K such that the neighborhood U α,β,h (x 0 ) := {y = (y, y n ) R n : y n h(y ) < β, y < α} of x 0 implies U α,β,h (x 0 ) Ω = {(y, h(y )) : y < α} and U α,β,h (x 0) := {(y, y n ) : h(y ) β < y n < h(y ), y < α} = U α,β,h (x 0 ) Ω. By analogy, a domain Ω R n, n 2, is a uniform C k,1 -domain of type (α, β, K), k N {0}, if the functions h mentioned above may be chosen in C k,1 such that the C k,1 -norm satisfies h C k,1 K. Theorem 1.2 ([8]) Let Ω R n, n 2, be a uniform C 1 domain of type (α, β, K) and let (1, ). Then each u L (Ω) has a uniue decomposition

3 On the Stokes operator in general unbounded domains 113 u = u 0 + p, u 0 L (Ω), p G (Ω), satisfying the estimate u 0 L + p L c u L, (1.1) where c = c(α, β, K, ) > 0. In particular, the Helmholtz proection P defined by P u = u 0 is a bounded linear proection on L (Ω) with range L (Ω) and kernel G (Ω). Moreover, L (Ω) is the closure in L (Ω) of the space C0,(Ω) = {u C0 (Ω) : div u = 0}, and the duality relations ( L (Ω) ) = L (Ω), ( P ) = P, where = 1, hold. Using the Helmholtz proection P, 1 < <, we define the Stokes operator à as the linear operator with domain D(Ã) = { D (Ω) D 2 (Ω), 2 < D (Ω) + D 2 (Ω), 1 < < 2, where D (Ω) = W 2, (Ω) W 1, 0 (Ω) L (Ω), by setting à u = P u, u D(Ã). By analogy, we define the Sobolev space W 2, (Ω) with norm u W 2, = u L + u L + 2 u L, see also (2.3) below. Let I be the identity and S ε = {0 λ C; arg λ < π 2 + ε}, 0 < ε < π 2. Then our first main result on the Stokes operator reads as follows: Theorem 1.3 Let Ω R n, n 2, be a uniform C 1,1 -domain of type (α, β, K), and let 1 < <, δ > 0, 0 < ε < π 2. (i) The operator à = P : D(Ã) L (Ω), D(Ã) L (Ω),

4 114 R. Farwig, H. Kozono and H. Sohr is a densely defined closed operator. 1 (ii) For all λ S ε, its resolvent (λi + Ã) : L (Ω) L (Ω) is well-defined. Moreover, for every f L (Ω) the solution u L (Ω) of the resolvent problem (λi + Ã)u = f satisfies the estimate λu L + 2 u L C f L, λ δ, (1.2) where C = C(, ε, δ, α, β, K) > 0. (iii) Given f L (Ω) n, λ S ε, the Stokes resolvent euation λu u + p = f, div u = 0 in Ω, u = 0 on Ω has a uniue solution (u, p) D(Ã) G (Ω) defined by u = (λi + à ) 1 P f and p = (I P )(f + u) and satisfying λu L + 2 u L + p L C f L, λ δ, (1.3) with a constant C = C(, ε, δ, α, β, K) > 0. (iv) The Stokes operator à satisfies the duality relation (Ã) = Ã, in particular, Ãu, v = u, à v for all u D(Ã), v D(à ) and generates an analytic semigroup e tã, t 0, in L (Ω) with bound e tã f L Me δt f L, f L, t 0, (1.4) where M = M(, δ, α, β, K) > 0. Note that the bound δ > 0 in Theorem 1.3 may be chosen arbitrarily small, but that it is not clear whether δ = 0 is allowed for a general unbounded domain and whether the semigroup e tã is uniformly bounded in L (Ω) for 0 t <. Our second main result concerns the instationary Stokes system u t u + p = f, div u = 0 in Ω (0, T ) u(0) = u 0, u Ω = 0. (1.5) Theorem 1.4 Let Ω R n, n 2, be a uniform C 1,1 -domain of type (α, β, K), and let 0 < T <, 1 <, s <. Then for each f L s (0, T ; L (Ω)) and each u 0 D(Ã) there exists a

5 On the Stokes operator in general unbounded domains 115 uniue solution u L s (0, T ; D(Ã)) with u t L s (0, T ; L (Ω)) of the system (1.5) satisfying the estimates u t L s (0,T ; L ) + u L s (0,T ; L ) + Ãu L s (0,T ; L ) C ( u 0 D( à ) + f ) L s (0,T ; L ) and u t L s (0,T ; L ) + u L s (0,T ; W 2, ) C( u 0 D( à ) + f ) L s (0,T ; L ) (1.6) (1.7) with C = C(, s, T, α, β, K) > 0. Remark 1.5 (i) The assumption u 0 D(Ã) in Theorem 1.4 is used for simplicity and is not optimal. Actually, it may be replaced by the weaker properties u 0 L (Ω) and T 0 Ãe tã u 0 s L dt <. Then the term u 0 D( à ) in (1.6), (1.7) can be substituted by the weaker norm ( T 0 ) 1 s à e tã u 0 s dt L, 1 < <. (1.8) (ii) Let f L s (0, T ; L (Ω)) in Theorem 1.4 be replaced by f L s (0, T ; L (Ω)). Then u L s (0, T ; D(Ã)), defined by u t + Ãu = P f, u(0) = u 0, and p, defined by p(t) = (I P ) ( f + u ) (t), is a uniue solution pair of the system satisfying u t u + p = f, u(0) = u 0, u t Ls (0,T ; L ) + u L s (0,T ; W 2, ) + p L s (0,T ; L ) C ( u 0 D( à ) + f ) L s (0,T ; L ) (1.9) with C = C(, s, T, α, β, K) > 0. Using (2.1) below we see that in the case 1 < < 2 the solution pair u, p possesses a decomposition u = u (1) + u (2), p = p (1) + p (2) such that

6 116 R. Farwig, H. Kozono and H. Sohr u (1) L s (0, T ; W 2,2 (Ω)), u (1) t L s (0, T ; L 2 (Ω)), u (2) L s (0, T ; W 2, (Ω)), u (2) t L s (0, T ; L (Ω)), (1.10) p (1) L s (0, T ; L 2 (Ω)), p (2) L s (0, T ; L (Ω)), and u t Ls (0,T ; L ) + u L s (0,T ; L ) + 2 u L s (0,T ; L ) + p L s (0,T ; L ) = u (1) t L s,2 + u (1) L s,2 + 2 u (1) L s,2 + p (1) L s,2 + u (2) t L s, + u (2) L s, + 2 u (2) L s, + p (2) L s, where L s,2 = L s (0, T ; L 2 (Ω)), L s, = L s (0, T ; L (Ω)). (iii) Note that the constant C in (1.6), (1.7), (1.9) could depend on the given interval (0, T ]. We do not know whether C can be chosen independently of T as in the usual L -theory in bounded and exterior domains, see [12]. 2. Preliminaries Let us recall some properties of sum and intersection spaces known from interpolation theory, cf. [3], [18]. Consider two (complex) Banach spaces X 1, X 2 with norms X1, X2, respectively, and assume that both X 1 and X 2 are subspaces of a topological vector space V with continuous embeddings. Further, we assume that X 1 X 2 is a dense subspace of both X 1 and X 2. Then the intersection space X 1 X 2 is a Banach space with norm The sum space u X1 X 2 = max ( u X1, u X2 ). X 1 + X 2 := {u 1 + u 2 ; u 1 X 1, u 2 X 2 } V is a well-defined Banach space with the norm u X1 +X 2 := inf { u 1 X1 + u 2 X2 ; u = u 1 + u 2, u 1 X 1, u 2 X 2 }.

7 On the Stokes operator in general unbounded domains 117 If X 1 and X 2 are reflexive Banach spaces, an argument using weakly convergent subseuences yields the following property: u X 1 + X 2 u 1 X 1, u 2 X 2 : u X1 +X 2 = u 1 X1 + u 2 X2. Concerning dual spaces we have (2.1) (X 1 X 2 ) = X 1 + X 2 with the natural pairing u, f 1 + f 2 = u, f 1 + u, f 2 for u X 1 X 2 and f = f 1 + f 2 X 1 + X 2, and (X 1 + X 2 ) = X 1 X 2 with the natural pairing u, f = u 1, f + u 2, f for all u = u 1 + u 2 X 1 + X 2, f X 1 X 2. Thus it holds u X1 +X 2 { } u1, f + u 2, f = sup ; 0 f X 1 X 2 f X 1 X 2 and f X 1 X 2 { } = sup u1, f + u 2, f ; 0 u = u 1 + u 2 X 1 + X 2 ; u X1 +X 2 see [3], [18]. Consider closed subspaces L 1 X 1, L 2 X 2 with norms L1 = X1, L2 = X2 and assume that L 1 L 2 is dense in both L 1 and L 2. Then u L1 L 2 = u X1 X 2, u L 1 L 2, and an elementary argument using the Hahn-Banach theorem shows that also u L1 +L 2 = u X1 +X 2, u L 1 + L 2. (2.2) In particular, we need the following special case. Let B 1 : D(B 1 ) X 1, B 2 : D(B 2 ) X 2 be closed linear operators with dense domains D(B 1 ) X 1, D(B 2 ) X 2 euipped with graph norms u D(B1 ) = u X1 + B 1 u X1, u D(B2 ) = u X2 + B 2 u X2,

8 118 R. Farwig, H. Kozono and H. Sohr respectively. Obviously each functional F D(B i ), i = 1, 2, is given by some pair f, g X i in the form u, F = u, f + B iu, g. We assume that the intersection D(B 1 ) D(B 2 ) is dense in both D(B 1 ) and D(B 2 ) in the corresponding graph norms. Then (2.2) with L i = {(u, B i u); u D(B i )} X i X i, i = 1, 2, and the euality of norms (X1 X 1 )+(X 2 X 2 ) and (X1 +X 2 ) (X 1 +X 2 ) on (X 1 X 1 )+(X 2 X 2 ) yield the following result: For each u D(B 1 ) + D(B 2 ) with decomposition u = u 1 + u 2, u 1 D(B 1 ), u 2 D(B 2 ), to be more precise, for an element (u 1, B 1 u 1 ) + (u 2, B 2 u 2 ) L 1 L 2 (X 1 X 1 ) + (X 2 X 2 ), u D(B1 )+D(B 2 ) = u 1 + u 2 X1 +X 2 + B 1 u 1 + B 2 u 2 X1 +X 2. (2.3) For instationary problems we need, given a Banach space X, the usual Banach space L s (0, T ; X), 0 < T, of measurable X valued (classes of) functions u with norm ( T u L s (0,T ;X) = If X is reflexive and 1 < s <, then 0 ) 1 u(t) s X dt s, 1 s <. L s (0, T ; X) = L s (0, T ; X ), s = s s 1, with the natural pairing u, f T = T u(t), f(t) dt, where, denotes the 0 pairing between X and its dual X. Let X = L (Ω), 1 < <. Then we use the notation ( T 1/s L s, := L s (L (Ω)) = L s (0, T ; L (Ω)), u L s, = u s dt). The pairing of L s (0, T ; L (Ω)) with its dual L s (0, T ; L (Ω)) is given by u, f T = u, f Ω,T = T ( 0 Ω u f dx) dt. Moreover, we see that since L s, L s,2 = L s (0, T ; L L 2 ) and L s, + L s,2 = L s (0, T ; L + L 2 ) 0

9 On the Stokes operator in general unbounded domains 119 (L s, + L s,2 ) = (L s, ) (L s,2 ) = L s (0, T ; L L 2 ) = L s (0, T ; L + L 2 ) ; the pairing between L s, +L s,2 and (L s, ) (L s,2 ) is given by u 1 +u 2, f T = u 1, f T + u 2, f T for u 1 L s,, u 2 L s,2, f (L s, ) (L s,2 ). Furthermore, we can choose the decomposition u = u 1 + u 2 L s (0, T ; L + L 2 ) in such a way that u L s, +L s,2 = u 1 L s, + u 2 L s,2. We conclude that u 1 + u 2 L s, +L s,2 { } = sup u1 + u 2, f T ; 0 f L s (0, T ; L L 2 ). f (L s, ) (L s,2 ) Let us introduce the short notation { L s, L s,2, 2 < L s, = L s, + L s,2, 1 < < 2, and note the duality relation ( Ls, ) = Ls,. Concerning domains of uniform C 1,1 -type (α, β, K), see Definition 1.1, we have to introduce further notations. Obviously, the axes e i, i = 1,..., n, of the new coordinate system (y, y n ) may be chosen in such a way that e 1,..., e n 1 are tangential to Ω at x 0. Hence at y = 0 the function h C 1,1 satisfies h(y ) = 0 and h(y ) = ( h/ y 1,..., h/ y n 1 )(y ) = 0. By a continuity argument, for any given constant M 0 > 0, we may choose α > 0 sufficiently small such that h C 1 M 0 is satisfied. It is easily shown that there exists a covering of Ω by open balls B = B r (x ) of fixed radius r > 0 with centers x Ω, such that with suitable functions h C 1,1 of type (α, β, K) B U α,β,h (x ) if x Ω, B Ω if x Ω. (2.4) Here runs from 1 to a finite number N = N(Ω) N if Ω is bounded, and N if Ω is unbounded. Moreover, as an important conseuence, the covering {B } of Ω may be constructed in such a way that not more than a fixed number N 0 = N 0 (α, β, K) N of these balls have a nonempty intersection:

10 120 R. Farwig, H. Kozono and H. Sohr If 1 1 < 2 < < N and N > N 0, then N B k =. (2.5) k=1 Related to the covering {B }, there exists a partition of unity {ϕ }, ϕ C 0 (R n ), such that 0 ϕ 1, supp ϕ B, and N ϕ = 1 or =1 ϕ = 1 on Ω. (2.6) =1 The functions ϕ may be chosen so that ϕ (x) + 2 ϕ (x) C uniformly in and x Ω with C = C(α, β, K). If Ω is unbounded, then Ω can be represented as the union of an increasing seuence of bounded uniform C 1,1 -domains Ω k Ω, k N, Ω 1 Ω k Ω k+1, Ω = Ω k, (2.7) where each Ω k is of the same type (α, β, K ), see [13, p. 652]. Without loss of generality we assume that α = α, β = β, K = K. Using the partition of unity {ϕ } we will perform the analysis of the Stokes operator by starting from well-known results for certain bounded and unbounded domains. For this reason, given h C 1,1 (R n 1 ) satisfying h(0) = 0, h(0) = 0 and with compact support contained in the (n 1)- dimensional ball of radius r, 0 < r = r(α, β, K) < α, and center 0, we introduce the bounded domain k=1 H = H α,β,h;r = { y = (y, y n ) R n : h(y ) β < y n < h(y ), y < α } B r (0); here we assume that B r (0) {y R n : y n h(y ) < β, y < α}. On H we consider the classical Sobolev spaces W k, (H) and W k, 0 (H), k N, the dual space W 1, (H) = ( W 1, 0 (H) ) and the space L 0 {u (H) = L (H) : H } u dx = 0

11 On the Stokes operator in general unbounded domains 121 of L -functions with vanishing mean on H. Lemma 2.1 Let 1 < < and H = H α,β,h;r. (i) There exists a bounded linear operator R : L 1, 0 (H) W0 (H) such that div R = I on L 0 (H) and R( L 1, 0 (H) W0 (H) ) W 2, 0 (H). Moreover, there exists a constant C = C(α, β, K, ) > 0 such that Rf W 1, C f L for all f L 0 (H) Rf W 2, C f W 1, for all f L 1, 0 (H) W0 (H). (2.8) (ii) There exists C = C(α, β, K, ) > 0 such that for every p L 0 (H) p C p W 1, { } p, div v = C sup : 0 v W 1, 0 (H). (2.9) v (iii) For given f L (H) let u L (H) W 1, 0 (H) W 2, (H), p W 1, (H) satisfy the Stokes resolvent euation λu u+ p = f with λ S ε, 0 < ε < π 2. Moreover, assume that supp u supp p B r(0). Then there are constants λ 0 = λ 0 (, α, β, K) > 0, C = C(, ε, α, β, K) > 0 such that if λ λ 0. λu L (H) + u W 2, (H) + p L (H) C f L (H) (2.10) Proof. (i) It is well-known that there exists a bounded linear operator R : L 1, 0 (H) W0 (H) such that u = Rf solves the divergence problem div u = f. Moreover, the estimate (2.8) 1 holds with C = C(α, β, K, ) > 0, see [10, III, Theorem 3.1]. The second part follows from [10, III, Theorem 3.2]. (ii) A duality argument and (i) yield (ii), see [8], [16, II.2.1]. (iii) We extend u, p by zero so that (u, p) may be considered as a solution of the Stokes resolvent system in a bent half space; then we refer to [6, Theorem 3.1, (i)]. The next lemma concerns the instationary Stokes systems

12 122 R. Farwig, H. Kozono and H. Sohr u t u + p = f, u(0) = u 0 or u t u + p = f, u(t ) = u 0, (2.11) in the domain H. To describe this crucial result we define the Stokes operator as usual by A = P with domain D(A ) = L (H) W 1, 0 (H) W 2, (H). Lemma 2.2 Let 0 < T <, u 0 D(A ) and f L (0, T ; L (H)) be given. Assume that u L (0, T, D(A )), p L (0, T ; W 1, (H)) solve one of the systems in (2.11) and satisfy supp u 0 supp u(t) supp p(t) B r (0) for a.a. t [0, T ]. Then there is a constant C = C(, α, β, K, T ) > 0 such that u t L (0,T ;L (H)) + u L (0,T ;W 2, (H)) + p L (0,T ;L (H)) C ( u 0 W 2, (H) + f L (0,T ;L (H))). (2.12) Proof. In the case u(0) = u 0 this estimate follows from [17, Theorem 4.1, (4.2) and (4.21 )], see also [15]. A careful inspection of the proofs shows that the constant C in (2.12) depends only on the type (α, β, K) and on, T ; actually, it suffices to assume the boundary regularity C 1,1 since only the boundedness of second order derivatives of functions locally describing the boundary is used. The second case u t u + p = f, u(t ) = u 0, can be reduced to the first one by the transformation ũ(t) = u(t t), f(t) = f(t t), p(t) = p(t t). We note that the assumption u 0 D(A ) is used for simplicity and can be weakened as in Remark 1.5 (i). Since u t L (0, T ; L ), the conditions u(0) = u 0 or u(t ) = u 0, resp., are well defined. Next we collect several results on Sobolev embedding estimates and on the Stokes operator A, 1 < <, on bounded C 1,1 -domains. Lemma 2.3 Let Ω R n be a bounded C 1,1 -domain of type (α, β, K). (i) Let 1 < <. Then for every M (0, 1) there exists some constant C = C(, M, α, β, K) > 0 such that u L M 2 u L + C u L, u W 2, (Ω). (2.13)

13 On the Stokes operator in general unbounded domains 123 (ii) Let 2 <. Then for every M (0, 1) there exists a constant C = C(, M, α, β, K) > 0 such that u L M 2 u L + C ( 2 u L 2 + u L 2), u W 2, (Ω). (2.14) Proof. The proofs of (i), (ii) are easily reduced to the case u W 2, (R n ), using an extension operator on Sobolev spaces the norm of which is shown to depend only on and (α, β, K). In (ii) we choose an r [2, ) such that u L M 2 u L r + C u L r and use the interpolation ineuality v L r γ ( ) 1/γ 1 v L 2 + (1 γ)ε 1/(1 γ) v L, (2.15) ε with γ (0, 1), 1 r = γ γ, for v = u and v = 2 u for suitable ε > 0 to get (2.14). For basic details see [1, IV, Theorem 4.28], [9] and [16, II.1.3]. Lemma 2.4 Let 1 < < and let Ω R n be a bounded C 1,1 -domain. (i) The Stokes operator A = P : D(A ) L (Ω), where D(A ) = L (Ω) W 1, 0 (Ω) W 2, (Ω), satisfies the resolvent estimate λu L + A u L C f L, C = C(ε,, Ω) > 0, (2.16) where u D(A ), λu + A u = f L (Ω), λ S ε, 0 < ε < π 2, and it holds the estimate u W 2, C A u L, C = C(, Ω). Moreover, A u, v = u, A v for all u D(A ), v D(A ) and A = A. (ii) If = 2, then the resolvent problem λu + A 2 u = f L 2 (Ω), λ S ε, has a uniue solution u D(A 2 ) satisfying the estimate λu L 2 + A 2 u L 2 C f L 2 (2.17) with the constant C = 1 + 2/ cos ε independent of Ω. Moreover, A 2 is selfadoint and A 2 u, u = A u 2 L 2 = u 2 L 2 for all u D(A 2 ). Proof. For (i) see [6], [11], [17]. For (ii) including even general unbounded domains we refer to [16].

14 124 R. Farwig, H. Kozono and H. Sohr Finally we return to the instationary Stokes system for a bounded C 1,1 - domain Ω R n, written in the form of the abstract evolution problem u t + A u = f, u(0) = u 0, (2.18) with initial value u 0 D(A ) and f L s (0, T ; L (Ω)), 1 <, s <. In view of the variation of constants formula we define the operators J s,, J s, by J s, f(t) = t 0 e (t τ)a f(τ)dτ, J s,f(t) = T t e (τ t)a f(τ)dτ. (2.19) Lemma 2.5 Let Ω R n be a bounded C 1,1 -domain. (i) Let 1 <, s < and 0 < T <. Then for every initial value u 0 D(A ) and external force f L s (0, T ; L (Ω)) the nonstationary Stokes system (2.18) has a uniue solution u L s (0, T ; D(A )) given by u(t) = e ta u 0 + J s, f(t) satisfying the estimate u t L s, + u L s, + A u L s, C ( ) u 0 D(A ) + f L s, (2.20) with a constant C = C(, s, T, Ω). Analogously, the nonstationary Stokes system u t + A u = f, u(t ) = u 0, has a uniue solution u L s (0, T ; D(A )), namely, u(t) = e (T t)a u 0 + (J s,f)(t); this solution satisfies (2.20) with the same constant C. Moreover, there holds the duality relation (J s, ) = J s,. (ii) In the case = 2 the constant C = C(2, s, T, Ω) = C(s, T ) in (2.20) does not depend on the domain Ω. Proof. For (i) see [12], [17]. The assertions on J s, follow from the transformation ũ(t) = u(t t), f(t) = f(t t) and by duality arguments. For (ii) including even general unbounded domains we refer to [16, IV.1.6]. Note that in (2.16) and (2.20) it is not clear up to now how the constant C will depend on the underlying bounded domain Ω except for = 2.

15 On the Stokes operator in general unbounded domains Proofs After a preliminary result on the norm u W 2, and the graph norm u D(A ) = u L + A u L, u D(A ), for bounded domains Ω R n we turn to the proofs of Theorem 1.3, see Subsection 3.1, and of Theorem 1.4, see Subsection 3.2. In both cases we consider first of all bounded domains for > 2, then for 1 < < 2, and finally unbounded domains. Lemma 3.1 Let Ω R n be a bounded C 1,1 -domain of type (α, β, K). Then there exists a constant C = C(, α, β, K) > 0 such that C u W 2, u D(A ), u D(A ). (3.1) Proof. We use the system of functions {h }, 1 N, parametrizing Ω, the covering of Ω by balls {B }, and the partition of unity {ϕ } as described in Section 2. Let U = U α,β,h (x ) B if x Ω and U = B if x Ω, 1 N. (3.2) Given f L (Ω) and u D(A ) satisfying A u = f, i.e. u+ p = f, div u = 0 in Ω, let w = R ( ( ϕ ) u ) W 2, 0 (U ) be the solution of the divergence euation div w = div (ϕ u) = ( ϕ ) u in U, 1 N. Moreover, let M = M (p) be the constant such that p M L 0 (U ). By Lemma 2.1 (i), (ii) and the euation p = f + u we conclude that w W 1, (U ) C u L (U ), w W 2, (U ) C u W 1, (U ) as well as p M L (U ) C ( ) f L (U ) + u L (U ) with C = C(, α, β, K) > 0 independent of. Finally, let λ 0 > 0 denote the constant in Lemma 2.1 (iii). Then ϕ u w satisfies the local resolvent euation λ 0 (ϕ u w ) (ϕ u w ) + ( ϕ (p M ) ) = ϕ f + w 2 ϕ u ( ϕ )u + ( ϕ )(p M ) + λ 0 (ϕ u w ) in U. By (2.10) with λ = λ 0 and the previous a priori estimates we get the local ineualities

16 126 R. Farwig, H. Kozono and H. Sohr ϕ 2 u L (U ) + ϕ p L (U ) C( f L (U ) + u W 1, (U )), (3.3) 1 N. Taking the sum over = 1,..., N and exploiting the crucial property of the number N 0, see (2.5), we are led to the estimate 2 u L (Ω) + p L (Ω) = Ω Ω CN (( N 0 0 ( ) ) ϕ u ) 2 + ϕ p dx ( ϕ 2 u + ) ϕ p dx ( f L (U ) + ) u W 1, (U ). (3.4) Next we use (2.13) for the term u W 1, (U ). Choosing M > 0 sufficiently small in (2.13), exploiting the absorption principle and again the property of the number N 0, (3.4) may be simplified to the estimate 2 u L (Ω) C ( ) f L (Ω) + u L (Ω) (3.5) where C = C(, α, β, K) > 0. Since f = A u, the proof is complete Proof of Theorem The Stokes resolvent in a bounded domain Ω when 2 We consider for λ S ε, 0 < ε < π 2, the resolvent euation λu + A u = λu u + p = f in Ω with f L (Ω), where 2 <. Our aim is to prove for its solution u D(A ) and p = (I P ) u the estimate λu L L u L L 2 + p L L 2 C f L L2, λ δ > 0 (3.6) with a constant C = C(, ε, δ, α, β, K) > 0. Note that this estimate is wellknown for bounded domains with a constant C = C(, ε, δ, Ω) > 0. As in Subsection 3.1 let w = R ( ( ϕ ) u ) W 2, 0 (U ) and choose a constant M = M (p) such that p M L 0 (U ). Then we obtain the local euation

17 On the Stokes operator in general unbounded domains 127 λ(ϕ u w ) (ϕ u w ) + ( ϕ (p M ) ) = ϕ f + w 2 ϕ u ( ϕ )u λw + ( ϕ )(p M ) (3.7) Concerning the term λw, we apply the embedding W 1,r (U ) L (U ) for some r [2, ), then Lemma 2.1(i) and use the interpolation estimate (2.15) for v = u to get for M (0, 1) that w L (U ) C 1 w W 1,r (U ) M u L (U ) + C 2 u L 2 (U ); here C i = C i (M,, r, α, β, K) > 0 (i = 1, 2). Moreover, 2 w L (U ) C u L (U ). For p M we use (2.9) and the euation p = λu+ u+f to see that p M L (U ) ( { λu, v C f L (U ) + u L (U ) + sup v }) : 0 v W 1, 0 (U ), where C = C(, α, β, K) > 0. Again we choose r [2, ), use the embedding W 1, (U ) L r (U ), then (2.15) for v = λu to get that p M L (U ) C ( f L (U ) + u L (U ) + λu L2 (U )) + M λu L (U ). Finally, we apply to the local resolvent euation (3.7) the estimate (2.10) with λ replaced by λ + λ 0 where λ 0 0 is sufficiently large such that λ + λ 0 λ 0 for λ δ, λ 0 as in (2.10). Now we combine these estimates and are led to the local ineuality λϕ u L (U ) + ϕ 2 u L (U ) + ϕ p L (U ) C ( f L (U ) + u L (U ) + u L (U ) + λu L2 (U )) + M λu L (U ) (3.8) with C = C(M,, δ, ε, α, β, K) > 0. Raising each term in (3.8) to the th power, taking the sum over = 1,..., N in the same way as in (3.3) (3.5) and using the crucial property (2.5) of the integer N 0 we get the ineuality

18 128 R. Farwig, H. Kozono and H. Sohr λu L (Ω) + 2 u L (Ω) + p L (Ω) C ( f L (Ω) + u L (Ω) + u L (Ω) + λu L 2 (Ω)) + M λu L (Ω) (3.9) with C = C(M,, δ, ε, α, β, K) > 0, λ δ. For the proof of (3.9) we also used the reverse Hölder ineuality ( ) 1/ ( 1/2 a ) a2 for the real numbers a = λu L 2 (U ) valid for 2. Applying (2.13) and choosing M sufficiently small we remove the terms u L (Ω) and λu L (Ω) from the right-hand side in (3.9) by the absorption principle. The term u L (Ω) is removed with the help of (2.14). Hence we get that λu + 2 u + p C ( f + λu 2 + u u 2 ). Now we combine this ineuality with the estimate (2.17) for λ δ and we apply (3.1) with = 2. This proves the desired estimate (3.6) for 2 < The case Ω bounded, 1 < < 2 We consider for f L 2 + L = L and λ S ε, λ δ, the euation λu u + p = f and its uniue solution u D(A ) + D(A 2 ) = D(A ), p = (I P ) u. Note that A = Ã, P = P and that C0,(Ω) is dense in L (Ω) L 2 (Ω) = L (Ω). Using f = λu P u, the density of D(A ) D(A 2 ) = D(A ) in L L 2, (3.6) with replaced by > 2, and setting g = λv + à v for v D(A ) D(A 2) we obtain that f L 2 +L { } λu + = sup à u, v ; 0 v D(A ) D(A 2 ) v L L 2 { u, λv + à v = sup { = sup v L L 2 u, g (λi P ) 1 g L L 2 } ; 0 v D(A ) D(A 2 ) ; 0 g L L 2 { u, g λ C 1 sup ; 0 g L L 2 g L L 2 } }. (3.10) By Section 2 the last term sup{...} in (3.10) defines a norm on L +L 2 which

19 On the Stokes operator in general unbounded domains 129 is euivalent to the norm L +L ; the constants in this norm euivalence 2 are related to the norm of P and depend only on and (α, β, K), cf. Theorem 1.2. Hence we proved the estimate λu L +L C f 2 L +L and 2 even λu L +L 2 + u L +L 2 + A u L +L 2 C f L +L 2, λ S ε, λ δ. (3.11) By virtue of Lemma 3.1 and (2.3) with B 1 = A, B 2 = A 2, we conclude that u W 2, +W 2,2 c( u L +L 2 + A u L +L 2 ) with a constant c > 0 depending only on and (α, β, K). Then (3.11) and the identity p = f λu + u lead to the estimate λu L +L 2 + u W 2, +W 2,2 + p L +L 2 C f L +L 2 with C = C(, δ, ε, α, β, K) > 0. Hence we proved for every (1, ) the ineuality λu L + u W 2, + p L C f L, u D(Ã), (3.12) with C = C(, δ, ε, α, β, K) > 0 when λ δ > 0. Now the proof of Theorem 1.3 (i) (iii) is complete for bounded domains The case Ω unbounded Consider the seuence of bounded subdomains Ω Ω, N, of uniform C 1,1 -type as in (2.7), let f L (Ω) and f := P f Ω. Then consider the solution (u, p ) of the Stokes resolvent euation λu P u = λu u + p = f, p = (I P ) u in Ω. From (3.12) we obtain the uniform estimate λu L (Ω ) + u W 2, (Ω ) + p L (Ω ) C f L (Ω) (3.13) with λ δ > 0, C = C(, δ, ε, α, β, K) > 0. Extending u and p by 0 to vector fields on Ω we find, suppressing subseuences, weak limits u = w lim u in L (Ω), p = w lim p in L (Ω) n

20 130 R. Farwig, H. Kozono and H. Sohr satisfying u D(Ã), λu u + p = λu P u = f in Ω and the a priori estimates (1.2), (1.3). Note that each p when extended by 0 need not be a gradient field on Ω; however, by de Rham s argument, the weak limit of the seuence { p } is a gradient field on Ω. Hence we solved the Stokes resolvent problem λu + Ãu = λu u + p = f in Ω. Finally, to prove uniueness of u we assume that there is some v D(Ã) and λ S ε satisfying λv P v = 0. Given f L (Ω) n let u D(Ã ) be a solution of λu P u = P f. Then 0 = λv P v, u = v, (λ P )u = v, P f = v, f for all f L (Ω) n ; hence, v = 0. Now Theorem 1.3 (i) (iii) is proved. The assertions (iv) of this Theorem are proved by standard duality arguments and semigroup theory Proof of Theorem 1.4 Let 0 < T <, 1 < s, <, and consider a domain Ω R n, n 2, of uniform C 1,1 -type (α, β, K). Then we define the subspace L s, := L s (0, T ; L (Ω)) of L s, := L s (0, T ; L (Ω)) with norm = Ls, L s (0,T ; L (Ω) ). In addition to the operators J s,, J s, for bounded domains, see Lemma 2.4, we define J s,, J s, by J s, f(t) = t 0 e (t τ)ã f(τ) dτ, T J s, f(t) = e (τ t)ã f(τ) dτ, t for f g L s, L s, and 0 t T. Since (Ã) = that J s, f, g T = f, J s, g T. Ã, we obtain for all f L s,, Maximal regularity in a bounded domain Ω when s = 2 First we consider the case u 0 = 0 and s =. Then u = J, f solves the euation u t + Ãu = f, u(0) = 0, and u = J,f is the solution of the system u t + Ãu = f, u(t ) = 0. Our aim is to prove in both cases the estimate (1.7) with a constant C = C(T,, α, β, K) > 0. Obviously it suffices to consider the case u = J, f since the other case follows using the transformation ũ(t) = u(t t), f(t) = f(t t). By Lemma 2.5 we know

21 On the Stokes operator in general unbounded domains 131 that u = J, solves the euation u t + Ãu = u t u + p = f L (0, T ; L ), u(0) = 0, with p = (I P ) u, and that u satisfies (2.20) with a constant C = C(Ω, ) > 0; note that the norms u W 2, and u D(A ) are euivalent. Thus it remains to prove that C in (2.20) can be chosen depending only on T, and (α, β, K). For this reason, we use the system of functions {h }, 1 N, the covering of Ω by balls {B }, and the partition of unity {ϕ } as described in Section 2 as well as the bounded sets U B, cf. (3.2). On U define w = R(( ϕ ) u) L (0, T ; W 2, 0 (U )), and let M = M (p) be the constant depending on t (0, T ) such that p M L (0, T ; L 0 (U )), see Lemma 2.1. Since div w = ( ϕ ) u and div w t = ( ϕ ) u t for a.a. t (0, T ), the term (ϕ u w) solves in U the local euation (ϕ u w) t (ϕ u w) + ( ϕ (p M ) ) = ϕ f w t + w 2 ϕ u ( ϕ )u + ( ϕ )(p M ). (3.14) From (2.8), (2.9) using w t = R(( ϕ ) u t ) and p = f u t + u we will prove for all ε (0, 1) the estimates w t L (L (U )) C u t L (L 2 (U )) + ε u t L (L (U )), 2 w L (L (U )) C ( u L (L (U )) + u L (L (U ))), (3.15) p M L (L (U )) C ( ) f L (L (U )) + u t L (L 2 (U )) + u L (L (U )) + ε u t L (L (U )) with C = C(, T, ε, α, β, K) > 0. In fact, for the proof of (3.15) 1, choose r [2, ) such that the embedding W 1,r (U ) L (U ) holds with an embedding constant c = c(, r, α, β, K) > 0 independent of. Moreover, w t L (U ) c w t W 1,r (U ) c u t Lr (U ) for a.a. t (0, t). Then the interpolation ineuality (2.15) proves (3.15) 1, and (2.8) 2 implies (3.15) 2. For the proof of (3.15) 3 we use (2.9), the embed-

22 132 R. Farwig, H. Kozono and H. Sohr ding W 1, (U ) L r (U ) with an embedding constant c = c(, r, α, β, K) > 0 independent of and apply the previous interpolation argument to u t. Applying the local estimate (2.12) to (3.14) and using (3.15) we get that ϕ u t L (L (U )) + ϕ u L (L (U )) + ϕ 2 u L (L (U )) + ϕ p L (L (U )) C ( f L (L (U )) + u L (W 1, (U )) + u t L (L 2 (U ))) + ε ut L (L (U )) with C = C(T,, ε, α, β, K) > 0. Raising this ineuality to its th power, taking the sum over = 1,..., N and exploiting the crucial property of the number N 0, see (2.5), we are led to the estimate u t L + u, L + 2 u, L + p, L, T ( = ϕ u t + ϕ u + ϕ 2 u 0 T 0 CN Ω Ω 0 ) + ϕ p dx dt ( N 0 ϕ u t + ϕ u + ϕ 2 u + ) ϕ p dx dt ( f L (0,T ;L (U )) + u L (0,T ;W 1, (U )) + ) u t L (0,T ;L 2 (U )) + εn 0 u t L (0,T ;L (U )). (3.16) Choosing ε > 0 sufficiently small, exploiting the absorption principle and again the property of the number N 0, we may simplify (3.16) to the estimate u t L, + u L, + 2 u L, + p L, C ( ) f L, + u L, + u t L,2 (3.17) where C = C(, α, β, K) > 0; note that in order to deal with the sum of the terms u t L (0,T ;L 2 (U )) we also used the reverse Hölder ineuality. Now, concerning the term u L,, we use (2.14) with ε > 0 sufficiently small and exploit the absorption principle. Finally we apply Lemma 2.5 (ii), i.e., we add the estimate (2.20) with = 2 to (3.17), to prove the estimate (1.7) for

23 On the Stokes operator in general unbounded domains 133 bounded domains when s = > 2, u(0) = 0. Since the operator norm of P is bounded by a constant c = c(, α, β, K) > 0 we get (1.6) for s =, u(0) = 0. To prove (1.6) with u 0 D(Ã) we solve the system ũ t + Ãũ = f, ũ(0) = 0, with f = f Ãu 0. Then u(t) = ũ(t) + u 0 yields the desired solution with u 0 D(Ã). This proves Theorem 1.4 for bounded Ω and s = The case Ω bounded, 1 < s = < 2 In this case we consider for f L, + L,2 = L, and the initial value u 0 = 0 the Stokes system u t +Ãu = f, u(0) = 0. By Lemma 2.5 there exists a uniue solution u(t) = J, f(t) = J, f(t); here we used that P = P and à = A. For the following duality argument we need that the space C 0 (C 0,) = { v C 0 (Ω (0, T )); div v(x, t) = 0 t (0, T ) } is dense in L, L,2 = ( ). L, + L,2 Then the identity u t + Ãu, à v = u, ( t + à )à v = Ãu, ( t + à )v ( holds for u = J, f and every v à 1 C 0 (C0,) ), since ( J, ) = J,. Let g = v t + à v. Then we obtain by (1.6) with s = replaced by s = 2 and u replaced by v that { ut + f L, +L,2 = sup Ãu, à v T à v L, L,2 { à u, g T = sup ; 0 v à 1 ( C 0 (C 0,) )} ( à v ; 0 v à 1 C 0 (C0,) )} L, L,2 1 C Ãu L, +L,2, (3.18) where C = C(T,, α, β, K) > 0. Here we used that the estimate (1.6) with, s replaced by, s also holds with u, u 0, f replaced by v, v(t ) = 0, g due to the transformation in time in the proof of Lemma 2.5, and exploited the norm euivalence

24 134 R. Farwig, H. Kozono and H. Sohr { }, h L +L sup ; 0 h L 2 L 2 h L L 2 with constants depending only on and (α, β, K), cf. Theorem 1.2. Hence we obtain the estimate Ãu L, +L,2 C f L, +L,2, and it follows u t L, +L,2 + Ãu L, +L,2 C f L, +L,2. (3.19) Since u W 2, +W 2,2 c( u L +L 2 + Ãu L +L 2 ) with a constant c > 0 depending only on and (α, β, K), (3.19) and the identity p = f u t + u lead to the estimate u t L, +L,2 + u L (0,T ;W 2, +W 2,2 ) + p L, +L,2 C f L, +L,2 (3.20) with C = C(, ε, α, β, K) > 0. Now the proof of Theorem 1.4 is complete for bounded domains in the case s =, u(0) = 0. The case u 0 D(Ã) is treated as in The case Ω unbounded Consider the seuence of bounded subdomains Ω Ω, N, of uniform C 1,1, () () -type as in (2.7), let f L and f := P f Ω where P denotes the Helmholtz proection in L (Ω ). Then consider the solution (u, p ) of the instationary Stokes euation t u P u = t u u + p = f, p = (I P ) u in Ω (0, T ) with initial condition u (0) = 0. estimate From (1.6) with s = we obtain the t u + u L, L (0,T ; W 2, (Ω )) + p L, C f L, (3.21) on Ω with C = C(T,, α, β, K) > 0 independent of N. Extending u and p for a.a. t (0, T ) from Ω by 0 to vector fields on Ω we find, suppressing subseuences, weak limits u = w lim u in L, (Ω), p = w lim p in L, (Ω)

25 On the Stokes operator in general unbounded domains 135 satisfying u L (0, T ; L (Ω), t u u + p = t u + Ãu = f in Ω (0, T ) and the a priori estimate (1.6) with u 0 = 0; it follows (1.7) for this case. Hence we solved the instationary Stokes euation t u + Ãu = t u u + p = f, u(0) = 0, in Ω (0, T ) and proved (1.6), (1.7). Up to now we considered only the case when s =, u(0) = 0. However, an abstract extrapolation argument shows that the validity of (1.6) with s = immediately extends to all s (1, ), see [2, p. 191] and [5, (1.12)], where A has to be replaced by à δi with δ > 0 as in (1.4). The case u(0) = u 0 0 can be reduced to the case u 0 = 0 in the same way as before. Finally, to prove uniueness let v L s (0, T ; W 2, ) satisfy t v + Ãv = 0 and v(0) = 0. Given f L s, let u L s (0, T ; W 2, ) be a solution of u t + à u = P f, u(t ) = 0. Then 0 = v t + Ãv, u T = v, ( t + à )u T = v, P f T = v, f T for all f L s, ; hence, v = 0. Now Theorem 1.4 is proved. Acknowledgement The authors are grateful to the referee for several improvements of the paper, in particular in Lemma 3.1. References [ 1 ] Adams R.A., Sobolev Spaces. Academic Press, New York, [ 2 ] Amann H., Linear and Quasilinear Parabolic Problems, Vol. I: Abstract Linear Theory. Birkhäuser Verlag Basel, [ 3 ] Bergh J. and Löfström J., Interpolation Spaces. Springer-Verlag, New York, [ 4 ] Bogovski M.E., Decomposition of L p (Ω, R n ) into the direct sum of subspaces of solenoidal and potential vector fields. Soviet Math. Dokl. 33 (1986), [ 5 ] Cannarsa P. and Vespri V., On maximal L p regularity for the abstract Cauchy problem. Boll. Un. Mat. Ital. (6) 5 (1986), [ 6 ] Farwig R. and Sohr H., Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Japan 46 (1994), [ 7 ] Farwig R., Kozono H. and Sohr H., An L approach to Stokes and Navier- Stokes euations in general domains. Acta Math. 195 (2005), [ 8 ] Farwig R., Kozono H. and Sohr H., On the Helmholtz decomposition in general unbounded domains. Arch. Math. 88 (2007),

26 136 R. Farwig, H. Kozono and H. Sohr [ 9 ] Friedman A., Partial Differential Euations. Holt, Rinehart and Winston, New York, [10] Galdi G.P., An Introduction to the Mathematical Theory of the Navier- Stokes Euations, Vol. I, Linearized Steady Problems. Springer-Verlag, New York, [11] Giga Y., Analyticity of the semigroup generated by the Stokes operator in L r spaces. Math. Z. 178 (1981), [12] Giga Y. and Sohr H., Abstract L p -estimates for the Cauchy problem with applications to the Navier-Stokes euations in exterior domains. J. Funct. Anal. 102 (1992), [13] Heywood J., The Navier-Stokes euations: On the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29 (1980), [14] Maslennikova V.N. and Bogovski M.E., Elliptic boundary value problems in unbounded domains with noncompact and nonsmooth boundaries. Rend. Sem. Mat. Fis. Milano. LVI (1986), [15] Maremonti P., Solonnikov V.A., On nonstationary Stokes problem in exterior domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), [16] Sohr H., The Navier-Stokes Euations. Birkhäuser Verlag, Basel, [17] Solonnikov V.A., Estimates for solutions of nonstationary Navier-Stokes euations. J. Soviet Math. 8 (1977), [18] Triebel H., Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Reinhard Farwig Technische Universität Darmstadt Fachbereich Mathematik Darmstadt, Germany farwig@mathematik.tu-darmstadt.de Hideo Kozono Tôhoku University Mathematical Institute Sendai, Japan kozono@math.tohoku.ac.p Hermann Sohr Universität Paderborn Fakultät für Elektrotechnik Informatik und Mathematik Paderborn, Germany hsohr@math.uni-paderborn.de