THE STOKES OPERATOR WITH NEUMANN BOUNDARY CONDITIONS IN LIPSCHITZ DOMAINS

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1 THE STOKES OPERATOR WITH NEUMANN BOUNDARY CONDITIONS IN LIPSCHITZ DOMAINS MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT Abstract. In the first art of the aer we give a satisfactory definition of the Stokes oerator in Lischitz domains in R n when boundary conditions of Neumann tye are considered. We then roceed to establish otimal global Sobolev regularity results for vector fields in the domains of fractional owers of this Neumann-Stokes oerator. Finally, we study existence, regularity and uniqueness of mild solutions of the Navier-Stokes system with Neumann boundary conditions. Contents 1. Introduction 1. Preliminaries 6 3. Conormal derivative in Besov-Triebel-Lizorkin saces 1 4. Sesquilinear forms and their associated oerators Fractional owers and semigrou theory The definition of the Neumann-Stokes oerator 1 7. The Stokes scale adated to Neumann boundary conditions 4 8. The Poisson roblem for the Stokes oerator with Neumann conditions 8 9. Domains of fractional owers of the Neumann Stokes oerator: I 9 1. Domains of fractional owers of the Neumann Stokes oerator: II Navier-Stokes equations Existence Regularity Uniqueness 4 References Introduction Let be a domain in R n, n, and fix a finite number T >. The Navier-Stokes equations are the standard system of PDE s governing the flow of continuum matter in fluid form, such as liquid or gas, occuying the domain. These equations describe the change with resect to time t [, T ] of the velocity and ressure of the fluid. A widely used version of the Navier-Stokes initial boundary roblem, equied with a Dirichlet Suorted in art by NSF and a UMC Miller Scholar Award Mathematics Subject Classification. Primary: 35Q3, 76D7, Secondary: 35A15, 35Q1, 76D3 Key words and hrases. Stokes oerator, Neumann boundary conditions, Lischitz domains, domain of fractional ower, regularity, Sobolev saces, Navier-Stokes equations. 1

2 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT boundary condition, reads u t x u + x π + ( u x ) u = in (, T ], (1.1) div x u = in [, T ], Tr x u = on [, T ], u() = u in, where u is the velocity field and π denotes the ressure of the fluid. One of the strategies for dealing with (1.1), brought to rominence by the ioneering work of H. Fujita, and T. Kato in the 6 s, consists of recasting (1.1) in the form of an abstract initial value roblem (1.) u (t) + (A u)(t) = f(t) t (, T ), f(t) := P D [ ( u(t) x ) u(t) ], u() = u, which is then converted into the integral equation t ] (1.3) u(t) = e ta u e (t s)a P D [( u(s) x ) u(s) ds, < t < T, then finally solving (1.3) via fixed oint methods (tyically, a Picard iterative scheme). In this scenario, the oerator P D is the Leray (orthogonal) rojection of L () n onto the sace H D := u L () n : div u = in, ν u = on }, where ν is the outward unit normal to, and A is the Stokes oerator, i.e. the Friedrichs extension of the symmetric oerator P D ( D ), where D is the Dirichlet Lalacian, to an unbounded self-adjoint oerator on the sace H D. By relying on the theory of analytic semigrous generated by self-adjoint oerators, Fujita and Kato have roved in [11] short time existence of strong solutions for (1.1) when R 3 is bounded and sufficiently smooth. Somewhat more secifically, they have shown that if is a bounded domain in R 3 with boundary of class C 3, and if the initial datum u belongs to D(A 1 4 ), then a strong solution can be found for which u(t) D(A 3 4 ) for t (, T ), granted that T is small. Hereafter, D(A α ), α >, stands for the domain of the fractional ower A α of A. An imortant asect of this analysis is the ability to describe the size/smoothness of vector fields belonging to D(A α ) in terms of more familiar saces. For examle, the estimates (1.18) and (.17) in [11] amount to (1.4) D(A γ ) C α () 3 if 3 4 < γ < 1 and < α < (γ 3 4 ), which lays a key role in [11]. Although Fujita and Kato have roved (1.4) via ad hoc methods, it was later realized that a more resourceful and elegant aroach to such regularity results is to view them as corollaries of otimal embeddings for D(A α ), α >, into the scale of vector-valued Sobolev (otential) saces of fractional order, L s() 3, 1 < <, s R. This latter issue turned out to be intimately linked to the smoothness assumtions made on the boundary of the domain. For examle, Fujita and Morimoto have roved in [1] that (1.5) C = D(A α ) L α() 3, α 1, whereas the resence of a single conical singularity on may result in the failure of D(A) to be included in L () 3.

3 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 3 The issue of extending the Fujita-Kato aroach to the class of Lischitz domains has been recently resolved in [9]. In the rocess, several useful global Sobolev regularity results for the vector fields in the fractional owers of the Stokes oerator have been established. For examle, it has been roved in [9] that for any Lischitz domain in R 3, (1.6) (1.7) (1.8) D(A 3 4 ) L () 3 >, 3 α > 3 > 3 such that D(A α ) L 4 1() 3, D(A γ ) = L γ,z() 3 H D, < γ < 3, 4 where, if s >, L s,z() is the subsace of L s() consisting of functions whose extension by zero outside belongs to L s(r n ). Also, it was shown in [9] that for any Lischitz domain in R 3 there exists ε = ε() > such that (1.9) 3 4 < γ < ε = D(Aγ ) C γ 3/ () 3, in agreement with the Fujita-Kato regularity result (1.4). For related work, as well as further ertinent references, the reader is referred to, e.g., O.A. Ladyzhenskaya [], R. Temam [41], M.E. Taylor [39], Y. Giga and T. Miyakawa [13], and W. von Wahl [43]. The aim of this aer is to derive analogous results in the case when Neumann-tye boundary conditions are considered in lace of the Dirichlet boundary condition. Dictated by secific ractical considerations (such as henomena translating into free boundary roblems), several scenarios are ossible. For examle, the Neumann condition (1.1) ( u + u )ν πν = on (, T ), (recall that ν stands for the outward unit normal to ) has been frequently used in the literature. From a hysical oint of view, it is convenient to view (1.1) as T ( u, π)ν = on (, T ), where T ( u, π) := u+( u) π denotes the stress tensor. In other words, in the case of a free boundary, (1.1) is exressing the absence of stress on the interface searating the two media. A more detailed account in this regard can be found in, e.g., D.D. Joseh s monograh [17]. See also the articles [38], [14], [15], and the references therein. Another Neumann-tye condition of interest is (1.11) ν u πν = on (, T ). This has been emloyed in [8] (in the stationary case). Here we shall work with a onearameter family of Neumann-tye boundary conditions, (1.1) [( u) + ( u)]ν πν = on (, T ), indexed by ( 1, 1] (in this context, (1.1), (1.11) corresond to choosing = 1 and =, resectively). Much as in the case of the Fujita-Kato aroach for (1.1), a basic ingredient in the treatment of the initial Navier-Stokes boundary roblem with Neumann boundary conditions, i.e., (1.13) u t x u + x π + ( u x ) u = in (, T ], div x u = in [, T ], [( x u) + ( x u)]ν πν = on [, T ], u() = u in,

4 4 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT is a suitable analogue of the Stokes oerator A = P D ( D ) discussed earlier. As a definition for this, we roose taking the unbounded oerator (1.14) B : D(B ) H N H N, where we have set H N := u L () n : div u = in }, with domain D(B ) := u L 1() n H N : there exists π L () so that u + π H N } (1.15) and such that [( u) + ( u)]ν πν = on, (with a suitable interretation of the boundary condition) and acting according to (1.16) B u := u + π, u D(B ), In order to be able to differentiate this from the much more commonly used Stokes oerator A = P D ( D ), we shall call the latter the Dirichlet-Stokes oerator and refer to (1.15)-(1.16) as the Neumann-Stokes oerator. Let us now comment on the suitability of the Neumann-Stokes oerator B vis-a-vis to the solvability of the initial Navier-Stokes system with Neumann boundary conditions (1.13). To this end, denote by P N the orthogonal rojection of L () n onto the sace H N = u L () n : div u = in }. In articular, (1.17) P N ( q) = for every q L 1() with Tr q = on. Proceed formally and assume that u, π solve (1.13) and that q solves the inhomogeneous Dirichlet roblem q = π in, (1.18) q =. Then π q is divergence-free. Based on this and (1.17) we may then comute (1.19) P N ( π) = P N ( π q) = π q = (π q). Since π q has the same boundary trace as π, it follows that [( u) + ( u)]ν (π q)ν = on. Consequently, (1.) B ( u) = u + (π q) = P N ( u + π). Thus, when P N is formally alied to the first line in (1.13) we arrive at the abstract evolution roblem u (t) + (B u)(t) = f(t) t (, T ), [ (1.1) f(t) := P N ( u(t) x ) u(t) ], u() = u, which is the natural analogue of (1.) in the current setting. This oens the door for solving (1.13) by considering the integral equation t ] (1.) u(t) = e tb u e (t s)b P N [( u(s) x ) u(s) ds, < t < T. In fact, in analogy with [4] where a similar issue is raised for the Dirichlet-Stokes oerator, we make the following:

5 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 5 Conjecture. For a given bounded Lischitz domain R 3 there exists ε = ε() > such that the Neumann-Stokes oerator associated generates an analytic semigrou on u L () 3 : div u = } rovided 3/ ε < < 3 + ε. The range of s in the above conjecture is naturally dictated by the maing roerties of the Neumann-Leray rojection which haens to extend to a well-defined bounded oerator (1.3) P N : L () 3 u L () 3 : div u = } recisely for 3/ ε < < 3 + ε where ε = ε() >. Indeed, in a recent aer, [3], M. Mitrea and S. Monniaux have roved the version of the above conjecture corresonding to the Stokes system equied with boundary conditions which, in the case of C domains, coincides with the standard Navier s sli boundary conditions ν u = on (, T ) (1.4) [( u + u )ν] tan = on (, T ), if one neglects surface tension effects (resonsible for a zero-order term, involving the curvature of the boundary). In summary, the interest in the functional analytic roerties of the Neumann-Stokes oerator B in (1.15)-(1.16) is justified. We establish shar global Sobolev regularity results for vector fields in D(B α ), the domain of fractional owers of B. Our main results in this regard arallel those for the Dirichlet-Stokes oerator which have been reviewed in the first art of the introduction. For the sake of this introduction, we wish to single out several such results. Concretely, for a Lischitz domain in R n we show that } (1.5) D(B s ) = u L s() n : div u = in if s 1, and (1.6) Also, when n = 3, (1.7) D(B α ) > n n 1 L 1() n if α > 3 4. D(B α ) C α 3/ ( ) 3 if 3 4 < α < ε, (1.8) and when n =, D(B 3 4 ) L 3 1() 3, (1.9) D(B α ) C α 1 ( ) 3 if < α < 3 + ε, 4 4 for some small ε = ε() >. It should be noted that, in the case when C, the initial boundary value roblem (1.13) has been treated (when = 1) by G. Grubb and V. Solonnikov in [14], [15], [38] (cf. also the references therein for relevant, earlier work). In this scenario, the tyical dearture oint is the regularity result D(B 1 ) L () n, which nonetheless is irreconcilably false in the class of Lischitz domains considered here. Most imortantly, the seudodifferential methods used in these references are no longer alicable in the non-smooth setting we treat. We wish to emhasize that overcoming the novel, significant difficulties caused by allowing domains with irregular boundaries reresents the main technical achievement of the current aer.

6 6 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT Key ingredients in the roof of the regularity results (1.5)-(1.8) are the shar results for the well-osedness of the inhomogeneous roblem for the Stokes oerator equied with Neumann boundary conditions in a Lischitz domain in R n, with data from Besov and Triebel-Lizorkin saces from [3]. This yields a clear icture of the nature of D(B ). On the other hand, known abstract functional analytic results allow us to identify D(B 1/ ). Starting from these, other intermediate fractional owers can then be treated by relying on certain (non-standard) interolation techniques. The organization of the aer is as follows. In Section we collect a number of reliminary results of function theoretic nature. Section 3 is devoted to a discussion of the meaning and roerties of the conormal derivative [( u) + ( u)]ν πν on when R n is a Lischitz domain and u, π belong to certain Besov-Triebel-Lizorkin saces. Section 4 is reserved for a review of the definitions and roerties of linear oerators associated with sesquilinear forms. Next, in Section 5, we collect some basic abstract results about semigrous and fractional owers of self-adjoint oerators. The rigorous definition of the Neumann-Stokes oerator B is given in Section 6. Among other things, here we show that B is self-adjoint on H N and identify D(B 1/ ). The scale V,s () := u L s() n : div u = } is investigated in Section 7 where we show that, for certain ranges of indices, this is stable under comlex interolation and duality. In Section 8 we record an otimal, well-osedness result for the Poisson roblem for the Stokes system with Neumann-tye boundary conditions in Lischitz domains, with data from Besov-Triebel-Lizorkin saces, recently established in [3]. The global Sobolev regularity of vector fields belonging to D(B α ) for α [, 1] is discussed in Section 9 and Section 1, when the underlying domain is Lischitz. Finally, in the last section, we treat the solvability of (1.), thus comlementing results obtained in [9] for the Stokes oerator equied with Dirichlet boundary conditions. Acknowledgments. This work has been comleted while the authors had been visiting Université Aix-Marseille 3 and the University of Missouri-Columbia, whose hositality they wish to gratefully acknowledge.. Preliminaries We shall call an oen, bounded, nonemty set, with connected boundary R n a Lischitz domain if for every oint x there is a rotation of the Euclidean coordinates in R n, a neighborhood O of x and a Lischitz function φ : R n 1 R such that (.1) O = x = (x, x n ) R n : x n > φ(x )} O. In this scenario, we let dσ stand for the surface measure on, and denote by ν the outward unit normal to. Next, for k N and (1, ), we recall the classical Sobolev sace (.) L k () := f L () : f W k, () := } γ f L () <, (throughout the aer, all derivatives are taken in the sense of distributions) and set (.3) L k,z γ k () := the closure of Cc () in L k (). Then for every k N and 1 <, < with 1/ + 1/ = 1, we have L k () := } ( ). (.4) γ f γ : f γ L () = L k,z () γ k

7 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 7 Moving on, for s (, 1), 1, denote by ( (.5) Bs, ( ) := f L f(x) f(y) ) 1 ( ) : x y dσ xdσ n 1+s y the Besov class on. We equi this with the natural norm ( f(x) f(y) ) 1/ (.6) f B, s ( ) := f L ( ) + x y dσ xdσ d 1+s y. For s (, 1) and 1 <, < with 1/ + 1/ = 1, we also set ( B s( ), := B,. (.7) s ( )) } < +, In the sequel, we shall occasionally write L s( ) in lace of B, s ( ) for s ( 1, 1). Recall that if (1, ) and R n is a Lischitz domain, then the trace oerator (.8) Tr : L 1() B, 1 1/ ( ) is well-defined, linear and bounded (cf. [16]). Next, introduce (.9) H := u L () n : div u = in } which is a closed subsace of L () n (hence, a Hilbert sace when equied with the norm inherited from L () n ). Also, set (.1) V := L 1() n H which is a closed subsace of L 1() n hence, a reflexive Banach sace when equied with the norm inherited from L 1() n. Lemma.1. If R n is a Lischitz domain then (.11) V H continuously and densely. Proof. The continuity of the inclusion maing in (.11) is obvious. To rove that this has a dense range, fix u H. Then it has been roved in [1] that there exists a smooth domain O and w L (O) n with the following roerties: (.1) In analogy with (.9), (.1), set (.13) O, div w = in O, w = u. H(O) := v L (O) n : div v = in O}, V(O) := L 1(O) n H(O). Then the following Hodge-Helmholtz-Weyl decomositions are valid [ ] (.14) L 1(O) n = V(O) L (O) L 1,z(O), [ ] (.15) L (O) n = H(O) L 1,z(O). These can be obtained constructively as follows. Granted that O is a smooth domain (here, it suffices to have O C 1,r for some r > 1/), the Poisson roblem with homogeneous Dirichlet boundary condition q = f L (O), (.16) is well-osed, and we denote by (.17) q L (O) L 1,z(O), G : L (O) L (O) L 1,z(O), Gf = q,

8 8 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT the solution oerator associated with (.16). By the Lax-Milgram lemma, the latter further extends to a bounded, self-adjoint oerator (.18) G : L 1(O) L 1,z(O). With I denoting the identity oerator, if we now consider (.19) then in each instance below (.) P : L 1(O) n V(O), P := I G div, P : L (O) n H(O), P is a well-defined, linear and bounded oerator. Furthermore, in the second case in (.), P actually acts as the orthogonal rojection. Indeed, this is readily verified using the fact that P = P in L (O) n (.1) and P = I, the identity oerator. H(O) The Hodge-Helmholtz-Weyl decomositions (.14)-(.15) are then naturally induced by decomosing the identity oerator according to (.) I = P + G div, both on L 1(O) n and on L (O) n. After this reamble, we now turn to the task of establishing (.11). Choose a sequence w j L 1(O) n, j N, such that w j w in L (O) n as j. Then w = P w = lim j P w j in L (O) n and u j := [P w j ] V for every j N. Since these considerations imly that u = w = lim j u j in L () n, (.11) follows. Remark.. An insection of the above roof shows that, via a similar argument, we have that (.3) Thus, ultimately, (.4) P : C () H C () u C () n : div u = in } H boundedly. densely. Next, recall that ν stands for the outward unit normal to, and introduce the following closed subsace of L 1/ ( )n : (.5) L 1/,ν( ) := φ L 1/( ) n : } ν φ dσ =. Our goal is to show that the trace oerator from (.8) extends to a bounded maing (.6) Tr : V L 1/,ν( ) which is onto. In fact, it is useful to rove the following more general result. Lemma.3. Assume that R n is a Lischitz domain, with outward unit normal ν and surface measure dσ. Also, fix 1 < < and s (1/, 1 + 1/). Then the trace oerator from (.8) extends to a bounded maing (.7) Tr : which is onto. } u L s() n : div u = φ B, s 1/ ( )n : } ν φ dσ =,

9 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 9 Proof. The fact that (.7) is well-defined, linear and bounded is clear from the roerties of (.8) and the fact that (.8) ν Tr u dσ = div u dx =, whenever u L s() n is divergence-free. To see that (.7) is also onto, consider φ B, s 1/ ( )n satisfying (.9) ν φ dσ = and solve the divergence equation (.3) div u = in, u L s() n, Tr u = φ on. For a roof of the fact that this is solvable for any φ B, s 1/ ( )n satisfying (.9) see [7]. This shows that the oerator (.7) is indeed onto. Moving on, for R fixed, let (.31) a αβ jk () := δ jkδ αβ + δ jβ δ kα, 1 j, k, α, β n, and, adoting the summation convention over reeated indices, consider the differential oerator L given by (.3) (L u) α := j (a αβ jk () ku β ) = u α + α (div u), 1 α n. Next, assuming that R and u, π are sufficiently nice functions in a Lischitz domain R n with outward unit normal ν, define the conormal derivative ( ) ν ( u, π) := ν j a αβ jk () ku β ν α π 1 α n [ ] (.33) = ( u) + ( u) ν πν on, where u = ( k u j ) 1 j,k n denotes the Jacobian matrix of the vector-valued function u, and stands for transosition of matrices. Introducing the bilinear form (.34) A (ξ, ζ) := a αβ jk ()ξα j ζ β k, ξ, ζ n n matrices, we then have the following useful integration by arts formula: } (.35) L u π, w dx = ν ( u, π), w dσ A ( u, w) π(div w) dx. In turn, this readily imlies that L u π, w dx L w ρ, u dx = } ν ( u, π), w ν ( w, ρ), u dσ (.36) + } π(div w) ρ(div u) dx. Above, it is imlicitly assumed that the functions involved are reasonably behaved near the boundary. Such considerations are going to be aid aroriate attention to in each secific alication of these integration by arts formulas.

10 1 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT 3. Conormal derivative in Besov-Triebel-Lizorkin saces For <, q and s R, we denote the Besov and Triebel-Lizorkin scales in R n by Bs,q (R n ) and Fs,q (R n ), resectively (cf., e.g., [4]). Next, given R n Lischitz domain and <, q, α R, we set (3.1) A,q α () := u D () : v A,q α (R n ) with v = u}, A,q α,() := u A,q α (R n ) with su u }, A,q α,z() := u : u A,q α,()}, where A B, F }. Finally, we let Bs,q ( ) stand for the Besov class on the Lischitz manifold, obtained by transorting (via a artition of unity and ull-back) the standard scale Bs,q (R n 1 ). We shall frequently use the abbreviation (3.) L s() := F, s (), 1 < <, s R. As is well-known, this is consistent with (.) and (.4). The existence of a universal linear extension oerator, from Lischitz domains to the entire Euclidean sace, which reserves smoothness both on the Besov and the Triebel- Lizorkin scales has been established by V. Rychkov. In [37], he roved the following: Theorem 3.1. Let R n be a Lischitz domain and denote by R u := u the oerator of restriction to. Then there exists a linear, continuous oerator E, maing distributions in into temered distributions in R n, such that whenever <, q +, s R, then (3.3) E : A,q s () A,q s for A = B or A = F, in the latter case assuming <. (R n ) boundedly, satisfying R E f = f, f A,q s (), Let us also record here a useful lifting result for fractional order Sobolev saces on Lischitz domains, which has been roved in [7]. Proosition 3.. Let 1 < < and s R. Then for any distribution u in the Lischitz domain R n, the following imlication holds: (3.4) u L s 1() n = u L s(). The following useful consequence of Proosition 3. (cf. [3] for a direct roof) will be used frequently in this aer. Corollary 3.3. Let R n, n, be a Lischitz domain and suose that 1 < <. Then there exists a finite constant C > deending only on n,, and the Lischitz character of such that every distribution u L 1() with u L 1() n has the roerty that u L () and (3.5) u L () C u L 1 ()n + C diam () u L 1 () holds. Concerning R, the restriction to, let us oint out that (3.6) R : L s,() L s,z(), 1 < <, s R, is a linear, bounded, onto oerator. This ermits the factorization (3.7) L s,() r L s,() u L s(r n ) : su u } R L s,z(), 1 < <, s R,

11 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 11 where the first arrow is the canonical rojection onto the factor sace, and the second arrow is an isomorhism. Moreover, since (3.8) 1 < <, 1 + 1/ < s = u L s(r n ) : su u } = then (3.9) R : L s,() L s,z() isomorhically if 1 < <, s > 1 + 1/. In this latter case, its inverse is the oerator of extension by zero outside, denoted by tilde, i.e., (3.1) L s,z() u ũ L s,(), 1 < <, 1 + 1/ < s. In articular, this allows the identification (3.11) L s,() L s,z(), (1, ), s > 1 + 1/. Let us also oint out that, if 1 < < and s R, we have the continuous embedding (3.1) L s,z() L s() and, in fact, (3.13) L s() = L s,z() if s < 1 and 1 s / N. Moreover, for every j = 1,..., n} (3.14) j : L s() L s 1(), j : L s,() L s 1,(), j : L s,z() L s 1,z(), are well-defined, linear, bounded oerators. Later on, we shall need duality results for the scales introduced at the beginning of this section. Throughout, all duality airings on are extensions of the natural airing between test functions and distributions on. As far as the nature of the dual of L s() is concerned, when 1 <, <, 1/ + 1/ = 1 and s R we have that (3.15) C c () φ φ Cc () extends to (. an isomorhism Ψ : L s,() L s()) Above, tilde denotes the extension by zero outside, and the extension in question is achieved via density, as the inclusions (3.16) (3.17) C c () L s,(), 1 < <, s R, (, Ls()) 1 < <, s R, Cc () have dense ranges. In what follows, we shall frequently identify the saces L s,() and ( L s()) without making any secific reference to the isomorhism Ψ in (3.15). For examle, we shall write that u, v R u, v L (3.18) L s(r n ) L s (Rn ) = L s() u L s(r n ), v L s,(). s, (),

12 1 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT Other duality results of interest are ( ) (3.19) L s,z() = L s() if 1 < < and s > 1 + 1, and (3.) ( ) L s() = L s,z() if 1 < < and s < 1. In articular, if 1 < <, ( ) (3.1) L s() = L s(), s ( 1 + 1/, 1/). The duality in (3.19) is related to the duality in (3.15) via u, v = R u, R v (3.) s > 1 + 1/ = L s, () L s (Rn ) L s,z() u L s,(), v L s(r n ). As a consequence, (3.3) and, hence, (3.4) s > 1 + 1/ = L s, () 1 + 1/ < s < 1/ = ũ, w L s (Rn ) = L s,z() u, R w u L s,z(), w L s(r n ), L s, () ũ, w L s (Rn ) = L s() L s (), u, R w u L s(), w L s(r n ). L s (), L s (), See the discussion in [9]. Moving on, we shall need a refinement of (.8) in the context of of Besov and Triebel- Lizorkin saces (cf. [16], [5]). To state this result, let (a) + := max a, }. Proosition 3.4. Let be a Lischitz domain in R n and assume that the indices, s satisfy n 1 < and (n 1)( 1 1) n + < s < 1. Then the following hold: (i) The restriction to the boundary extends to a linear, bounded oerator (3.5) Tr : B,q () B s+ 1 s,q ( ) for < q. For this range of indices, Tr is onto and has a bounded right inverse (3.6) Ex : Bs,q ( ) B,q (). s+ 1 As far as the null-sace of (3.5) is concerned, if n 1 < <, (n 1)(1/ 1) n + < s < 1 and < q <, then (3.7) B,q s+1/,z () = u B,q s+1/ }, () : Tr u = and (3.8) Cc () B,q s+1/,z () densely. (ii) Similar considerations hold for (3.9) Tr : F,q () B s+ 1 s, ( )

13 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 13 with the convention that q = if =. More secifically, Tr in (3.9) is a linear, bounded, oerator which has a linear, bounded right inverse (3.3) Ex : Bs, ( ) F,q (). s+ 1 Also, if n 1 < <, (n 1)(1/ 1) n + < s < 1 and min 1, } q <, then (3.31) F,q s+1/,z () = u F,q s+1/ }, () : Tr u = and (3.3) Cc () F,q s+1/,z () densely. Let X be a Banach sace with dual X. For every n n matrix F = (F α j ) α,j with entries from X, every n n matrix G = (G β k ) β,k with entries from X, and each R, we set (3.33) A X (F, G) := a αβ jk () X F α j, G β k X, where X, X is the duality airing between X and X, and a αβ jk () are as in (.31). In the sequel, our notation will not emhasize the deendence of, and A on X; however, the articular nature of X should be clear from the context in each case. Let R n, n, be a bounded Lischitz domain and assume that 1 <, q <, < s < 1. If u B,q s+ 1 u π = f in, then as suggested by (.36), it is natural to define (3.34) by setting (3.35) ν ( u, π) f, ψ () n, π B,q s+ 1 1() and f B,q s+ 1,()n are such that ν ( u, π) f B,q s 1( ) n = ( B,q 1 s ( ) n ), 1/ + 1/ = 1, 1/q + 1/q = 1, R, := f, Ex( ψ) ( + A u, Ex( ψ) ) π, div Ex( ψ), ψ B,q 1 s ( ) n, where Ex is the extension oerator introduced in Proosition 3.4. The conditions on the indices, q, s ensure that all duality airings in the right-hand side of (3.35) are well-defined. Similar considerations aly to the case when u, π, f belong to aroriate Triebel-Lizorkin saces (in which case the conormal ν ( u, π) f belongs to a suitable diagonal boundary Besov sace). Remark 3.5. Since the conormal ν ( u, π) f has been defined for a class of (trilets of) functions u, π, f [ ] for which the exression ( u) +( u) ν πν is, in the standard sense of the trace theory, utterly ill-defined on, it is aroriate [ to remark ] that ( u, π, f) ν ( u, π) f is not an extension of the oeration ( u, π) Tr ( u) +( u) ν Tr π ν in an ordinary sense. In fact, it is more aroriate to regard the former as a re-normalization of the latter trace, in a fashion that deends strongly on the choice of f A,q s+1/, ()n as an extension of u π A,q s+1/,z ()n. To further shed light on this issue, recall that, for u L 1() n, π L (), u π is naturally defined as a linear functional in (L 1,() n ). The choice of f is the choice of an extension of this linear functional to a functional in (L 1() n ) = L 1,() n. As

14 14 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT an examle, consider [ u L 1() n,] π L (), and suose that actually u L () n, π L 1() so Tr ( u) + ( u) ν Tr π ν is well defined in L ( ) n. In this case, u π L () n has a natural extension f L 1,() n (i.e., f is the extension of u π to R n by setting this equal zero outside ). Any other extension f 1 L 1,() n differs from f by a distribution η L 1(R n ) n suorted on. As is well-known, the sace of such distributions is nontrivial. In fact, we have [ ] (3.36) ν ( u, π) f = Tr ( u) + ( u) ν Tr π ν in L ( ) n, but if η then ν ( u, π) f is not equal to ν ( u, π) f1. Indeed, by linearity we have that ν ( u, π) f1 = ν ( u, π) f + ν (, ) η and (3.35) shows that ν (, ) η, ψ = η, Ex( ψ) (3.37) for every ψ L 1/ ( )n. Consequently, ν (, ) η if η. We continue by registering an natural integration by arts formula, which builds on the definition of the renormalized conormal (3.35). Proosition 3.6. Assume that R n is a Lischitz domain. Fix s (, 1), as well as 1 <, q <, and denote by, q the Hölder conjugate exonents of and q, resectively. Next, suose that w A q 1 s+1/ () n, u A,q () n, π A,q s+ 1 s+ 1 1() and f A,q are such that u π = f s+ 1,()n in (where, as usual, A B, F }). Then, for every R, the following integration by arts formula holds: ( ) (3.38) ν ( u, π) f, Tr w = f, w + A u, w Proof. By linearity, it suffices to show that ( ) (3.39) f, w + A u, w π, div w = π, div w if w, u, π, f are as in the statement of the roosition and, in addition, Tr w =. Note that the latter condition entails that w A,q 1 s,z() n by (3.7), (3.31). Thus, by (3.8), (3.3), w can be aroximated in A,q 1 s,z() n by a sequence of vector fields w j Cc () n. Since, thanks to the fact that u π = f as distributions in, we have ) (3.4) f, w j + A ( u, w j π, div w j =, j N, we can obtain (3.39) by letting j. In order to continue, we introduce the following adatations of the Besov and Triebel- Lizorkin scales to the Stokes oerator Bs,q () := ( u, π, f) B,q () n B,q s+ 1 s+ 1 1() B,q : s+ 1,()n. (3.41) and (3.4) F,q s () := u π = f }, ( u, π, f) F,q () n F,q,q s+ 1 s+ 1 1() F : s+ 1,()n u π = f }.

15 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 15 Corollary 3.7. Suose that R n is a Lischitz domain, and assume that s (, 1), 1 <, q <, 1/ + 1/ = 1/q + 1/q = 1. Then f, w g, u = ν ( u, π) f, Tr w ν ( w, ρ) g, Tr u (3.43) + π, div w ρ, div u rovided either (3.44) or (3.45) ( u, π, f) B,q s (), ( w, ρ, g) B,q 1 s (), ( u, π, f) F,q s (), ( w, ρ, g) F,q 1 s (). Proof. This follows from (3.38) much as (.36) follows from (.35). 4. Sesquilinear forms and their associated oerators In this section we describe a few basic facts on sesquilinear forms and linear oerators associated with them. Throughout, given two Banach saces X, Y, we denote by B ( X, Y ) the sace of linear, bounded oerators from X into Y, equied with the strong oerator norm. Also, we let I X stand for the identity oerator on X. Finally, we adot the convention that if X is a Banach sace then X denotes the adjoint sace of continuous conjugate linear functionals on X, also known as the conjugate dual of X. In this scenario, we let X, X denote the duality airing between X and X. Let H be a comlex searable Hilbert sace with scalar roduct (, ) H (antilinear in the first and linear in the second argument), V a reflexive Banach sace continuously and densely embedded into H. Then also H embeds continuously and densely into V, i.e., (4.1) V H V continuously and densely. Here the continuous embedding H V is accomlished via the identification (4.) H u (, u) H V. In articular, if the sesquilinear form (4.3) V, V : V V C denotes the duality airing between V and V, then (4.4) V u, v V = (u, v) H, u V, v H V, that is, the V, V airing V, V is comatible with the scalar roduct (, ) H in H. Let T B(V, V ). Since V is reflexive, i.e. (V ) = V, one has (4.5) T : V V, T : V V and (4.6) V u, T v V = V T u, v (V ) = V T u, v V = V v, T u V. Self-adjointness of T is then defined as the roerty that T = T, that is, (4.7) V u, T v V = V T u, v V = V v, T u V, u, v V, nonnegativity of T is defined as the demand that (4.8) V u, T u V, u V,

16 16 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT and boundedness from below of T by c R is defined as the roerty that (4.9) V u, T u V c u H, u V. (Note that, by (4.4), this is equivalent to V u, T u V c V u, u V for all u V.) Next, let the sesquilinear form a(, ): V V C (antilinear in the first and linear in the second argument) be V-bounded. That is, there exists a c a > such that (4.1) a(u, v) c a u V v V, u, v V. Then à defined by (4.11) Ã: satisfies V V, v Ãv = a(, v), (4.1) à B(V, V ) and V u, Ãv V = a(u, v), u, v V. In the sequel, we shall refer to à as the oerator induced by the form a(, ). Assuming further that a(, ) is symmetric, that is, (4.13) a(u, v) = a(v, u), u, v V, and that a is V-coercive, that is, there exists a constant C > such that (4.14) a(u, u) C u V, u V, resectively, then, (4.15) Ã: V V is bounded, self-adjoint, and boundedly invertible. Moreover, denoting by A the art of à in H, defined by (4.16) D(A) := u V : Ãu H } H, A := à D(A) : D(A) H, then A is a (ossibly unbounded) self-adjoint oerator in H satisfying (4.17) (4.18) In articular, A C I H, D ( A 1/) = V. (4.19) A 1 B(H). The facts (4.1) (4.19) are a consequence of the Lax Milgram theorem and the second reresentation theorem for symmetric sesquilinear forms. Details can be found, for instance, in [3, VI.3, VII.1], [7, Ch. IV], and [3]. Next, consider a symmetric form b(, ): V V C and assume that b is bounded from below by c b R, that is, (4.) b(u, u) c b u H, u V. Introducing the scalar roduct (, ) V(b) : V V C (with associated norm V(b) ) by (4.1) (u, v) V(b) := b(u, v) + (1 c b )(u, v) H, u, v V, turns V into a re-hilbert sace (V; (, ) V(b) ), which we denote by V(b). The form b is called closed if V(b) is actually comlete, and hence a Hilbert sace. The form b is called closable if it has a closed extension. If b is closed, then (4.) b(u, v) + (1 c b )(u, v) H u V(b) v V(b), u, v V,

17 and NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 17 (4.3) b(u, u) + (1 c b ) u H = u V(b), u V, show that the form b(, ) + (1 c b )(, ) H is a symmetric, V-bounded, and V-coercive sesquilinear form. Hence, by (4.11) and (4.1), there exists a linear ma V(b) V(b), (4.4) Bcb : with v B cb v := b(, v) + (1 c b )(, v) H, (4.5) B cb B(V(b), V(b) ) and V(b) u, Bcb v V(b) = b(u, v) + (1 c b )(u, v) H, u, v V. Introducing the linear ma (4.6) B := Bcb + (c b 1)Ĩ : V(b) V(b), where Ĩ : V(b) V(b) denotes the continuous inclusion (embedding) ma of V(b) into V(b), one obtains a self-adjoint oerator B in H by restricting B to H, (4.7) D(B) = u V : Bu H } H, B = B D(B) : D(B) H, satisfying the following roerties: (4.8) B c b I H, (4.9) (4.3) (4.31) (4.3) D ( B 1/) = D ( (B c b I H ) 1/) = V, b(u, v) = ( B 1/ u, U B B 1/ v ) H = ( (B c b I H ) 1/ u, (B c b I H ) 1/ v ) + c H b(u, v) H = V(b) u, Bv, u, v V, V(b) (4.33) (4.34) (4.35) b(u, v) = (u, Bv) H, u V, v D(B), D(B) = v V : there exists f v H such that Bu = f u, b(w, v) = (w, f v ) H for all w V}, u D(B), D(B) is dense in H and in V(b). Proerties (4.34) and (4.35) uniquely determine B. Here U B in (4.31) is the artial isometry in the olar decomosition of B, that is, (4.36) B = U B B, B = (B B) 1/. Definition 4.1. The oerator B is called the oerator associated with the form b(, ). The norm in the Hilbert sace V(b) is given by (4.37) l V(b) = su V(b) u, l V(b) : u V(b) 1}, l V(b), with associated scalar roduct, (4.38) (l 1, l ) V(b) = V(b) ( B + (1 cb )Ĩ) 1 l1, l V(b), l 1, l V(b).

18 18 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT Since (4.39) the Riesz reresentation theorem yields ( B + (1 cb )Ĩ) v V(b) = v V(b), v V, (4.4) ( B + (1 cb )Ĩ) B(V(b), V(b) ) and ( B + (1 cb )Ĩ) : V(b) V(b) is unitary. In addition, ( u, B + (1 cb )Ĩ) v = (( ) 1/u, ( ) 1/v ) B + (1 c V(b) b )I H B + (1 cb )I H (4.41) V(b) In articular, (4.4) and hence = (u, v) V(b), u, v V(b). (B + (1 cb )I H ) 1/ u H = u V(b), u V(b), (4.43) (B + (1 c b )I H ) 1/ B(V(b), H) and (B + (1 c b )I H ) 1/ : V(b) H is unitary. The facts (4.) (4.43) comrise the second reresentation theorem of sesquilinear forms (cf. [7, Sect. IV.], [1, Sects ], [, Sect. VI..6], [34]). 5. Fractional owers and semigrou theory Assume that H is a (ossibly comlex) searable Hilbert sace with scalar roduct (, ) H and that V a reflexive Banach sace continuously and densely embedded into H. Also, fix a sesquilinear form b(, ) : V V C, which is assumed to be symmetric, nonnegative, bounded, and which satisfies the following coercivity condition: There exist C R and C 1 > such that (5.1) b(u, u) + C u H C 1 u V, u V. As a consequence, V(b) V. Thus V(b) = V and, hence, b(, ) is also closed. Let B : D(B) H H be the (ossibly unbounded) oerator associated with the form b(, ) as in Definition 4.1. In articular, B is self-adjoint and nonnegative. Also, ti H +B is invertible on H for every t >, and t(ti H +B) 1 B(H,H) C for t > (cf., e.g., Proosition 1. on. 13 in [34]). In fact, there exist θ (, π/) and a finite constant C > such that Σ θ := z C : arg (z 1) θ + π/} is contained in C \ Sec (B) (where Sec (B) denotes the sectrum of B as an oerator on H) and (5.) (zi H + B) 1 B(H,H) C 1 + z, z Σ θ, i.e., B is sectorial. See, e.g., [4, Theorem 3,. 374 and Proosition 3,. 38]. In articular, the oerator B generates an analytic semigrou on H according to the formula (5.3) e zb u := 1 e ζz (ζi H + B) 1 u dζ, arg (z) < π/ θ, u H, πi Γ θ where θ (θ, π/) and Γ θ := ± re iθ : r > }. Cf. [4] and [35] for a more detailed discussion in this regard. Moving on, we denote by E B (µ)} µ R the family of sectral rojections associated with B, and for each u H introduce the function ρ u by (5.4) ρ u : R [, ), ρ u (µ) := (E B (µ)u, u) H = E B (µ)u H. H

19 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 19 Clearly, ρ u is bounded, non-decreasing, right-continuous, and (5.5) lim µ ρ u(µ) =, lim ρ u (µ) = u µ H, u H. Hence, ρ u generates a measure, denoted by dρ u, in a canonical manner. A function f : R C is then called de B -measurable if it is dρ u -measurable for each u H. As is well-known, all Borel measurable functions are de B -measurable functions. For a Borel measurable function f : R C we then define the (ossibly) unbounded oerator by setting D(f(B)) := u H : } R f dρ u < + (5.6) f(b)u := f(µ) de R B(µ)u, u D(f(B)). In articular, for each α [, 1], the fractional ower B α of B is a self-adjoint oerator (5.7) B α : D(B α ) H H. Since in our case B is maximally accretive, then so is B α if α (, 1) and for every u D(B) D(B α ) we have the reresentation (5.8) B α u = sin (π z) π t α B(tI H + B) 1 u dt t. See [18], []. Other roerties are discussed in, e.g., A. Pazy s book [35] and the survey article [1] by W. Arendt, to which we refer the interested reader. Here we only wish to summarize some well-known results of T. Kato and J.-L. Lions (see [19], [3]) which are relevant for our work. Secifically, if B is as above, then (5.9) D(B 1/ ) = V and, with [, ] θ denoting the comlex interolation bracket, (5.1) D(B θ ) = [H, D(B)] θ, θ 1. Hence, by the reiteration theorem for the comlex method, the family } (5.11) D(B s ) : s is a comlex interolation scale. In articular, (5.1) D(B θ/ ) = [H, V] θ, θ 1. We wish to further elaborate on this toic by shedding some light on the nature of D(B α ) when α (1/, 1). This requires some rearations. To get started, denote by B B(V, V ) the oerator induced by the form b(, ) (so that B is the art of B in H), and let Ĩ stand for the inclusion of V into V. It then follows from (5.1) that (5.13) (Ĩ + B) B(V, V ) is an isomorhism. The idea is to find another suitable context in which the oerator Ĩ + B is an isomorhism, and then interolate between this and (5.13). However, in contrast to what goes on for boundedness, invertibility is not, generally seaking, reserved under interolation. There are, nonetheless, certain secific settings in which this is true. To discuss such a case recall that, if (X, X 1 ) are a coule of comatible Banach saces, X X 1 and X + X 1 are equied, resectively, with the norms (5.14) x X X 1 := max x X, x X1 }, and z X +X 1 = inf x X + x 1 X1 : z = x + x 1, x i X i, i =, 1}.

20 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT We have: Lemma 5.1. Let (X, X 1 ) and (Y, Y 1 ) be two coules of comatible Banach saces and assume that T : X + X 1 Y + Y 1 is a linear oerator with the roerty that (5.15) T : X i Y i is an isomorhism, i =, 1. In addition, assume that there exist Banach saces X, Y such that the inclusions (5.16) are continuous with dense range, and that (5.17) Then the oerator (5.18) X X X 1, Y Y Y 1, T : X Y is an isomorhism for each θ 1. is an isomorhism. T : [X, X 1 ] θ [Y, Y 1 ] θ Proof. Denote by R i B(Y i, X i ), i =, 1, the inverses of T in (5.15). Since the oerators R and R 1 coincide as maings in B(Y, X ), by density they also agree as maings in B(Y Y 1, X X 1 ). It is therefore meaningful to define (5.19) R : Y + Y 1 X + X 1, by R(y + y 1 ) := R (y ) + R 1 (y 1 ), y i Y i, i =, 1. Then R is a linear oerator which belongs to B(Y, X ) B(Y 1, X 1 ). Thus, by the interolation roerty, R mas [Y, Y 1 ] θ boundedly into [X, X 1 ] θ for every θ [, 1]. In this latter context, R rovides an inverse for T : [X, X 1 ] θ [Y, Y 1 ] θ, since RT = I X X 1 on X X 1, which is a dense subsace of [X, X 1 ] θ, and T R = I Y Y 1 on Y Y 1, which is a dense subsace of [Y, Y 1 ] θ. This roves that the oerator in (5.18) is indeed an isomorhism for every θ [, 1]. After this reamble, we are ready to resent the following. Proosition 5.. With the above assumtions and notation, (5.) D(B 1+θ ) = ( Ĩ + B) ( ) 1 D(B 1 θ ) for every θ 1. Proof. As already remarked above, the oerator Ĩ + B : V V is boundedly invertible. We claim that (5.1) Ĩ + B : D(B) H is invertible as well, when D(B) is equied with the grah norm u u H + Bu H. Indeed, this oerator is clearly well-defined, linear and bounded, since B coincides with B on D(B). Also, the fact that the oerator in (5.13) is one-to-one readily entails that so is (5.1). To see that the oerator (5.1) is onto, ick an arbitrary w H V. It follows from (5.13) that there exists u V H such that (Ĩ + B)u = w. In turn, this imlies that Bu = w u H and, hence, u D(B). This shows that the oerator (5.1) is onto, hence ultimately invertible. Interolating between (5.13) and (5.1) then roves (with the hel of Lemma 5.1, (5.9)- (5.1), and the duality theorem for the comlex method) that the oerator (5.) Ĩ + B : D(B 1+θ ) = [V, D(B)]θ [V, H] θ = ([H, V] 1 θ ) = ( D(B 1 θ ) )

21 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 1 is an isomorhism, for every θ 1. From this, (5.) readily follows. 6. The definition of the Neumann-Stokes oerator In this section we define the Stokes oerator when equied with Neumann boundary conditions in Lischitz domains in R n. Subsequently, in Theorem 6.8, we study the functional analytic roerties of this oerator. We begin by making the following: Definition 6.1. Let R n be a Lischitz domain and assume that R is fixed. Define the Stokes oerator with Neumann boundary condition as the unbounded oerator (6.1) with domain D(B ) := (6.) B : D(B ) H H u V : there exists π L () so that f := u + π H and such that ν ( u, π) f = in L 1/( ) n }, and acting according to (6.3) B u := u + π, u D(B ), assuming that the air ( u, π) satisfies the requirements in the definition of D(B ) and where we have identified f with its extension by zero outside according to (3.11) (for = and s = ) in the exression ν ( u, π) f. As it stands, it is not entirely obvious that the above definition is indeed coherent and our first order of business is to clarify this issue. We do so in a series of lemmas, starting with: Lemma 6.. If the air ( u, π) satisfies the requirements in the definition of D(B ), then π = in. Proof. Since the vector fields u and f := u + π are both divergence-free, it follows that π = div ( u + π) = div f =. Lemma 6.3. If u D(B ), then there exists a unique scalar function π L () such that f := u + π H and ν ( u, π) f = in L 1/ ( )n. Proof. Fix a vector field u D(B ) and assume that π j L (), j = 1,, are such that (6.4) f j := u + π j H and ν ( u, π j ) fj = in L 1/( ) n, for j = 1,. Set π := π 1 π L (), and note that (6.5) π = f 1 f H L 1() n. As a consequence, (6.6) π L 1().

22 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT Next, we emloy (3.35) and (6.4) in order to write = ν ( u, π 1 ) f1 ν ( u, π ) f, ψ = f1, Ex( ψ) ( + A u, Ex( ψ) ) π 1, div Ex( ψ) f, Ex( ψ) ( A u, Ex( ψ) ) + π, div Ex( ψ) = f1 f, Ex( ψ) π, div Ex( ψ) (6.7), for every ψ L 1/ ( )n. At this stage, we recall (6.5)-(6.6) in order to transform the last exression in (6.7) into π, Ex( ψ) π, div Ex( ψ) = Tr π, ν ψ (6.8). In concert with (6.7), this shows that (Tr π) ν, ψ (6.9) = for every ψ L 1/( ) n, from which we may conclude that (6.1) Tr π = in L 1/( ) n. This, (6.6) and Lemma 6. amount to saying that π L 1() is harmonic and satisfies Tr π =. Thus, π = in, by the uniqueness for the Dirichlet roblem. Hence, π 1 = π in, as desired. Remark 6.4. In articular, Lemma 6.3 imlies that there is no ambiguity in defining B u as in (6.3). Recall now the bilinear form (.34), and consider (6.11) b (, ) : V V R, b ( u, v) := A ( u, v) dx. Our goal is to study this sesquilinear form. This requires some rerequisites which we now disense with. First, the following Korn tye estimate has been roved in [3]. Proosition 6.5. Let be a Lischitz domain and assume that 1 < <. Then there exists a finite constant C > which deends only on and the Lischitz character of such that (6.1) u L C u + u 1 ()n L () + C diam n () 1 u L () }, n uniformly for u L 1() n. We shall also need the the following algebraic result from [3] regarding the bilinear form A (, ) from (.34). Proosition 6.6. For every ( 1, 1] there exists κ > such that for every n n- matrix ξ (6.13) A (ξ, ξ) κ ξ for < 1 and A 1 (ξ, ξ) κ 1 ξ + ξ. The following well-known result (cf. [5]) is also going to be useful shortly. Lemma 6.7. Let be an oen subset of R n, and assume that v [D() ] n is a vectorvalued distribution which annihilates w Cc () n : div w = in }. Then there exists a scalar distribution q D() with the roerty that v = q in.

23 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 3 We are now ready to state and rove the main result of this section. Recall the saces V, H from (.1), (.9), along with the form b (, ) from (6.11). Theorem 6.8. Let R n be a Lischitz domain and assume that ( 1, 1] is fixed. Then the sesquilinear form b (, ) introduced in (6.11) is symmetric, bounded, non-negative, and closed. Furthermore, the Neumann-Stokes oerator B, originally introduced in (6.1)-(6.3), is (in the terminology of 4) the oerator associated with b (, ). As a consequence, (6.14) (6.15) B is self-adjoint and nonnegative on H, B generates an analytic semigrou on H, (6.16) D( B 1/ ) = D(B 1/ ) = V, (6.17) D(B ) is dense both in V and in H. Finally, Sec(B ), the sectrum of the oerator (6.1)-(6.3) is a discreet subset of [, ). Proof. Lemma.1 ensures that (4.1) holds, hence the formalism from 4 alies. That the form b (, ) in (6.11) is nonnegative, symmetric, sesquilinear and continuous is clear from its definition. In addition, this form is coercive, hence closed. Indeed, when < 1 this follows directly from Proosition 6.6, whereas when = 1 this is a consequence of the second inequality in (6.13) and Proosition 6.5. We next wish to show the coincidence between the domain D(B ) of the Neumann- Stokes oerator in (6.) and the sace (6.18) u V : there exists f H such that b ( w, u) = ( w, f) H for all w V }. In one direction, fix u V such that there exists f H for which (6.19) A ( w, u) dx = w, f dx for every w V. Secializing (6.19) to the case when w Cc of (.35) used with π =, that (6.) the distribution f + u vanishes on () n is divergence-free yields, e.g., on account } w Cc () n : div w = in. Then, by virtue of Lemma 6.7, there exists a scalar distribution π in such that (6.1) π = f + u L 1() n. Going further, (6.1) and Corollary 3.3 imly that, in fact, (6.) π L () and f = u + π in. At this oint we make the claim that there exists a constant c R with the roerty that (6.3) π := π c = ν ( u, π) f = in L 1/( ) n. To justify this, we first note that (3.38) (used with f in lace of f) and (6.19) force (6.4) ν ( u, π) f, Tr w = for every w V, hence, further, (6.5) ν ( u, π) f, φ = for every φ L 1/,ν( ),

24 4 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT by Lemma.3. To continue, fix some vector field φ o L 1/ ( )n with the roerty that ν φ o dσ = 1, and define (6.6) c := ν ( u, π) f, φ o. Now, given an arbitrary φ L 1/ ( )n, set η := ν φ dσ and comute ν ( u, π) f, φ = ν ( u, π) f, φ η φ o + η ν ( u, π) f, φ o (6.7) = + c ν, φ, by (6.5), (6.6) and the definition of. Since φ L 1/ ( ) is arbitrary, this roves that (6.8) ν ( u, π) f = c ν in L 1/( ) n. Thus, (6.9) ν ( u, π c) f = ν ( u, π) f ν (, c) = c ν c ν = in L 1/( ) n, hence (6.3) holds. Note that (6.) also ensures that π L () and f = u + π in. Together, these conditions rove that the sace in (6.18) is contained in D(B ) (defined in (6.)). Conversely, the inclusion of D(B ) into the sace in (6.18) is a direct consequence of the definition of the domain of the Neumann-Stokes oerator (in (6.)) and the integration by arts formula (3.38). Once D(B ) has been identified with the sace in (6.18), the fact that the Neumann- Stokes oerator B, in (6.1)-(6.3) is, in the terminology of 4, the oerator associated with the form b (, ) follows from (4.34). Finally, the claim made about Sec (B ) is a consequence of the fact that B is nonnegative and has a comact resolvent. 7. The Stokes scale adated to Neumann boundary conditions Given a Lischitz domain R n and 1 < <, s R, we set } (7.1) V s, () := u L s() n : div u = in. The first main result of this section is to show that the above scale is stable under comlex interolation. Theorem 7.1. For each Lischitz domain R n, the family } (7.) V s, () : 1 < <, s R is a comlex interolation scale. In other words, if [, ] θ stands for the usual comlex interolation bracket, then [ ] (7.3) V s, (), V s 1, 1 () = V s, () θ whenever 1 < i <, s i R, i =, 1, θ [, 1], 1 := 1 θ + θ 1 and s := (1 θ)s + θs 1. Before turning to the roof of Theorem 7.1, we recall a version of an abstract interolation result from [4].

25 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 5 Lemma 7.. Let X i, Y i, i =, 1, be two airs of Banach saces such that X X 1 is dense in both X and X 1, and similarly for Y, Y 1. Let D be a linear oerator such that D : X i Y i boundedly for i =, 1, and consider the following closed subsaces of X i, i =, 1: (7.4) Ker (D; X i ) := u X i : Du = }, i =, 1. Finally, suose that there exists a continuous linear maing G : Y i X i with the roerty D G = I, the identity on Y i for i =, 1. Then, for each < θ < 1, (7.5) [Ker (D; X ), Ker (D; X 1 )] θ = u [X, X 1 ] θ : Du = }, θ (, 1). Proof of Theorem 7.1. Denote by Π the harmonic Newtonian otential, i.e., the oerator of convolution with the standard fundamental solution for the Lalacian in R n. Recall the universal extension oerator E from Theorem 3.1. Without loss of generality, we may assume that E u is suorted in a fixed comact neighborhood of for every distribution u in. Assuming that this is the case, we set (7.6) Π := R Π E, where, as before, R u := u is the oerator of restriction to. Given that Π is smoothing of order two, it follows that (7.7) Π : L s() L s+(), 1 < <, s R, is a well-defined, linear and bounded oerator. Next, fix, 1,, s, s 1, s, θ as in the statement of the theorem. We shall imlement Lemma 7. in which we take (7.8) X i := L i s i () n and Y i := L i s i 1(), i =, 1, as well as (7.9) D := div and G := Π. Then since (7.1) D : X i Y i, G : Y i X i, i =, 1, are well-defined, linear and bounded, and since D G = I, the identity, the conclusion in Theorem 7.1 follows from Lemma 7.. Our next goal is to identify the duals of the saces in the Stokes scale introduced in (7.1). As a reamble, we rove the following. Proosition 7.3. Let be a Lischitz domain in R n with outward unit normal ν and assume that 1 < <, < s < 1. Define the maing (7.11) ν : V s, () B, ( ) s 1 by setting (7.1) ν u, ϕ := u, Φ ( ) for each ϕ B, ( ) = B, ( ), where Φ L s 1 s+ 1 s() is such that Tr Φ = ϕ. 1 Then the above definition is meaningful and the oerator (7.11) is bounded in the sense that (7.13) ν u B, s 1 ( ) C u L s () n,

26 6 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT for some finite C = C(, s, ) >. Finally, the range of the oerator (7.11)-(7.1) is } (7.14) f B, ( ) : f, 1 =. s 1 Proof. This follows from Proosition.7 in [9] and Proosition.1 in [8]. Theorem 7.4. Let R n be a Lischitz domain and fix 1 < <. Next, for each 1 + 1/ < s < 1/, let (7.15) J s, : V s, () L s() n be the canonical inclusion, and consider its adjoint (7.16) J s, : L s() n ( V s, ()), where 1/ + 1/ = 1. Then the maing (7.16) is onto and its kernel is recisely [L 1 s,z()]. In articular, (7.17) J s, : is an isomorhism. [ L s() n L 1 s,z() ] ( ) V s, () Proof. Since V s, () is a closed subsace of L s,z(), Hahn-Banach s theorem immediately gives that the maing (7.16) is onto. That (7.17) is an isomorhism will then follow as soon as we show that Ker Js,, the null-sace of the alication (7.16), coincides with ( ) [L 1 s,z()]. In one direction, if u L s() n = L s() n is such that J s, ( u) =, then u, v = for each v V s, (). Choosing v Cc () n such that div v = in shows, on account of Lemma 6.7, that there exists a distribution w in such [ that w ] = u. Proosition 3. then ensures that w L 1 s(), so that u = w L 1 s(). There remains to show that, after subtracting a suitable constant from w, this function can be made to have trace zero and, hence, belong to L 1 s,z(). To this end, note that for each v V s, () we have (7.18) = u, v = w, v = Tr w, ν v. Then the last claim in Proosition 7.3 shows that Tr w is a constant, as wanted. Conversely, if u = Φ L s() n for some Φ L 1 s,z() then Proosition 7.3 allows us to write (7.19) J s,( u), v = Φ, v = Tr Φ, ν v =, for every v V s, (). Thus, J s,( u) =, finishing the roof of the theorem. Theorem 7.5. For each Lischitz domain R n there exists ε = ε() (, 1] with the following significance. Assume that 1 < <, 1 + 1/ < s < 1/ and that the air (s, 1/) satisfies either of the following three conditions: (7.) (I) : (II) : < 1 < 1 ε and < s < ε; 1 ε 1 1+ε and < s < 1 ; (III) : 1+ε < 1 < 1 and + 3 ε < s < 1.

27 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 7 Then [ (7.1) L s() n = V s, () L s+1,z() where the direct sum is toological (in fact, orthogonal when s = and = ). Furthermore, if (7.) P : L s() n V s, () denotes [ the rojection ] onto the first summand in the decomosition (7.1), then its kernel is L s+1,z(). In articular, ], (7.3) P : L [ s() n L s+1,z() ] V s, () is an isomorhism. Also, the adjoint of the oerator (7.4) P,s : L s() n P V s, () Js, L s() n is the oerator P, s, where 1/ + 1/ = 1, and ( (7.5) V ()) s, = V s, (). Proof. The decomosition (7.1) corresonding to the case when s = has been established in [9] via an aroach which reduces matters to the well-osedness of the inhomogeneous Dirichlet roblem for the Lalacian in the Lischitz domain. The more general case considered here can be roved in an analogous fashion. With (7.1) in hand, the claims about the rojection (7.) are straightforward. Consider next the identification in (7.5). If u V s, () define Λ u ( V s, () ) by setting (7.6) Λ u ( v) := L s () J n s, v, J s, u, v V s, (). L s ()n Note that since 1 + 1/ < s < 1/, the dual of L s() is L s(), hence the duality bracket makes sense. Then the maing (, (7.7) Φ : V s, () V ()) s, Φ( u) := Λ u, is well-defined, linear and bounded. Our goal is to show that this is an isomorhism. To rove that Φ is onto, fix Λ ( V s, () ). Recall the oerator P from (7.) and note that Λ P ( L s() ) = L s(). That is, there exists w L s() such that (7.8) (Λ P) u = L s () w, J n s, u Then Λ P w = Φ(P w) satisfies Λ P w ( v) = L s () J n s, v, J s, P w (7.9) L s ()n, L s ()n = L s () n = L s () P n,s J s, v, w = L L s () n s ()n = (Λ P) v = Λ( v), v V s, (). u V s, (). J s, v, P, s w L s ()n J s, v, w L s ()n

28 8 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT Hence Λ = Λ P w, roving that Φ is onto. To see that Φ is also one-to-one, we note that if u V s, () is such that Λ u =, then L s() J n s, v, J s, u = v V s, () L s ()n = L s () J n s, P w, J s, u = w L s() L s ()n = L s () P n,s w, J s, u = w L s() L s ()n (7.3) = L s () w, P n, sj s, u L s ()n = = L s () w, J n s, u = L s ()n = J s, u = in L s() n = u = in V s, (). w L s() w L s() This shows that Φ in (7.7) is an isomorhism, thus finishing the roof of (7.5). The roof of the theorem is therefore comleted. 8. The Poisson roblem for the Stokes oerator with Neumann conditions For a given Lischitz domain in R n, n, the range of indices for which the Poisson roblem in for the Stokes oerator equied with Neumann boundary conditions is wellosed on Besov and Triebel-Lizorkin saces deends on the dimension n of the ambient sace and the Lischitz character of. The latter is manifested by a arameter ε (, 1] which can be thought of as measuring the degree of roughness of (thus, the larger ε the milder the Lischitz nature of, and the smaller ε, the more acute Lischitz nature of ). To best describe these regions, for each n and ε > we let R n,ε denote the following sets. For n =, R,ε is the collection of all airs of numbers s, with the roerty that either one of the following two conditions below is satisfied: (8.1) (I ) : 1 < s + 1+ε and < s 1+ε, (II ) : 1+ε < 1 s < 1+ε and 1+ε < s < 1. Corresonding to n = 3, R 3,ε is the collection of all airs s, with the roerty that either of the following two conditions holds: (8.) (I 3 ) : 1 < s + 1+ε and < s < ε, (II 3 ) : ε < 1 s < 1+ε and ε s < 1. Finally, corresonding to n 4, we let R n,ε denote the collection of all airs s, with the roerty that (8.3) n 3 (I n ) : ε < 1 s < 1 + ε and < s < 1, 1 < <. (n 1) n 1 The following well-osedness result has been recently established in [3]. Theorem 8.1. Let be a bounded Lischitz domain in R n, n, with connected comlement, and fix n 1 <, < q, and (n 1) ( 1 1) < s < 1. Also, n +

29 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 9 assume that ( 1, 1] and µ C \ Sec (B ). Then there exists ε = ε() (, 1] such that the Poisson roblem for the Stokes system with Neumann boundary condition µ u u + π = f, f B,q, div u = in, (8.4) s+ 1,()n u B,q () n, π B,q s+ 1 s+ 1 1(), ν ( u, π) f µ u = in Bs 1( ),q n, has a unique solution if the air s, belongs to the region R n,ε, described in (8.1)-(8.3). In addition, the solution satisfies the estimate (8.5) u B,q s+ 1 () + π n B,q C f B,q s+ 1 1() s+ 1,()n, for some finite constant C = C(, n,, s,, µ) >. Moreover, an analogous well-osedness result holds for the roblem µ u u + π = f, f F,q, div u = in, (8.6) s+ 1,()n u F,q () n, π F,q s+ 1 s+ 1 1(), ν ( u, π) f µ u = in Bs 1( ), n, assuming that, q <. Strictly seaking, the above theorem has been roved in [3] when µ = (in which case the data must satisfy certain necessary comatibility conditions, and uniqueness is valid u a finite dimensional sace). The method of roof in [3] is constructive as it relies on integral reresentation formulas (involving hydrostatic otential oerators). As such, this aroach can be easily adated to the slightly more general case above, since the difference between the fundamental solutions for the original Stokes system u+ π = } }, div u = and the lower-order erturbation (µ ) u + π =, div u = is only weakly singular. We leave the straightforward details to the interested reader. 9. Domains of fractional owers of the Neumann Stokes oerator: I Here we study the global regularity, measured on the Sobolev scale, of vector fields in the domains of fractional owers of the Neumann Stokes oerator. Our first result in this regard reads as follows: Theorem 9.1. Let be a Lischitz domain in R n and fix ( 1, 1]. Then the domain of the fractional ower of the Neumann Stokes oerator B introduced in (6.)-(6.3) satisfies } (9.1) D(B s ) = u L s() n : div u = if s 1, and (9.) u D(B s ) granted that s (3/, ]. Proof. Consider the families of saces u V and there exists π L () such that f := u + π L s () n and ν ( u, π) f =, } V s, () : s R and } D(B s ) : s. From Theorem 7.1 and (5.11) we know that both are comlex interolation scales, and (9.3) D(B ) = H = V, (), D(B 1 ) = V = V 1, ().

30 3 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT Thus, by comlex interolation, (9.4) D(B s ) = V s, (), s 1, which gives the descrition of D(B s ) in (9.1). To study larger values of s, recall the form b (, ) and the oerator B induced by it. From (5.)-(9.4) we obtain (9.5) D(B s ) = (Ĩ + B ) 1 (V s, ()), 1 s. Thus, by (7.5), (9.6) u D(B s ) u V and (Ĩ + B ) u V s, (), if 3 < s. Consequently, if s (3/, ], then by taking into account the very definition of B we arrive at the conclusion that u V and f L s () n such that (9.7) u D(B s ) f, v = u v dx + A ( u, v) dx, v V. Much as before, by relying on Lemma 6.7, Corollary 3.3 and Proosition 3.6, it follows from (9.7) that u V and there exists π L () such that (9.8) u D(B s ) f := (1 ) u + π L s () n and ν ( u, π) f u =, With this in hand, (9.) follows after re-denoting f u by f. It is ossible to further extend the scoe of the above analysis. In order to facilitate the subsequent discussion, for each ε (, 1], s [ 3, ] and n, define the two dimensional region (θ, 1 ) : < 1 < θ < <, θ s, and 1 (9.9) R n,s,ε := + ε > 1 θ 1 s 3 if s < n + εn, n n n ε > 1 θ > ε n if + εn < s. n n n 1 The figures below deict the region R n,s,ε in the case when 3 s < 1/ 1 sloe 1/n sloe 1/n n + εn, n 1 1/ + ε 1/ 1/ s/n 1 3/ s θ and when n + εn < s, resectively: n 1

31 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 31 1/ 1 sloe 1/n sloe 1/n 1/ + ε 1/ n 3 (n 1) ε 1 3/ s θ Theorem 9.. For every Lischitz domain R n, n, there exists ε = ε( ) > with the roerty that for every s (3/, ] and ( 1, 1] the following imlication holds: (9.1) (θ, 1/) R n,s,ε = D(B s/ ) L θ ()n. Proof. The strategy is to combine the characterization (9.) with the well-osedness result for the Poisson roblem for the Stokes system equied with Neumann boundary conditions. In concert, these two results show that D(B α/ ) L θ ()n rovided s, belonging to the region R n,ε such that (9.11) θ = s + 1/ and L α () L θ (). Now, elementary algebra shows that, given α (3/, ], the condition (9.11) holds if and only if (θ, 1/) R n,α,ε. Clearly, this roves (9.1), after re-adjusting notation. Corollary 9.3. For a Lischitz domain in R n one has (9.1) D(B α ) L 1() n if α > 3. 4 Also, when n = 3, (9.13) and when n =, > n n 1 D(B α ) C α 3/ ( ) 3 if 3 4 < α < ε, (9.14) D(B α ) C α 1 ( ) 3 if < α < 3 + ε, 4 4 for some small ε = ε() >. Proof. These are all immediate consequences of Theorem 9. and classical embedding results. 1. Domains of fractional owers of the Neumann Stokes oerator: II The aim of this section is to augment the results in Theorem 9.1 by including a descrition of D(B s/ ) in the case when s (1, 3/]. See Theorem 1.5 below. We begin by revisiting the Neumann-Leray rojection (7.), with the goal of further extending the range of action of this oerator. Lemma 1.1. Assume that is a Lischitz domain in R n and that s R,, (1, ), 1/ + 1/ = 1. Then the oerator (1.1) P s, : L s,() n = ( L s() n) ( V s, () )

32 3 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT (where we have used the isomorhism Ψ defined in (3.15) to identify L ( L s () n) ) defined by the requirement that s,() n and (1.) V s, () v, Ps, u (V s, ()) = L s() n Js, v, u (L s() n ) v V s, (), is well-defined, linear, bounded and onto. In addition, any two such oerators act coherently, i.e., P s1, 1 = P s, on L 1 s 1,() n L s,() n for any numbers s 1, s R and 1, (1, ). Also, the null-sace of (1.1)-(1.) is ( ) ] [ ] Ker [ P : L (1.3) s,() n V s, () = L 1 s,(). Finally, if corresonding to s = 1 and = one considers (1.4) then the diagram (1.5) P 1, : L 1,() n = ( L 1() n) V, V v, P1, u V = L 1 ()n J1, v, u (L 1 ()n ) v V, L 1,() n b P1, L () n V P H in which the vertical arrows are natural inclusions, is commutative. Consequently, the Neumann-Leray rojection (7.) extends as in (1.4). Proof. That (1.1)-(1.) is well-defined and bounded is clear from the continuity of the inclusion J s, : V s, () L s() n. Using the fact that V s, () is a closed subsace of L s() n, the Hahn-Banach theorem, and (3.15), it is straightforward to show that the oerator (1.1) is onto. It is also clear from (1.) that this family of oerators acts in a mutually comatible fashion. The left-to-right inclusion in (1.3) is a direct consequence of the fact that (1.6) L s() J n s, v, π =, v V s, (), π L 1 s,(). L s, ()n In turn, (1.6) follows from a standard density argument (based on the fact that the inclusion (3.16) has dense range), and the fact that vector fields in V s, () are, in the sense of distribution, divergence-free. To rove the oosite inclusion in (1.3), assume that u L s,() n is such that (1.7) L s() n Js, v, u (L s() n ) =, Then, on account of (3.18), for every w C c (1.8) v V s, (). (R n ) n such that div w = in R n we have L s(r n ) n w, u L s (Rn ) n = L s () n R w, u L s, ()n =, thanks to (1.7), used with v := R w. With this in hand, Lemma 6.7 then shows that u = π for some distribution π in R n. In fact, since u is suorted in and R n \ is connected, after eventually subtracting a constant it can be arranged that π is also suorted in. Finally, Proosition 3. gives that π L 1 s(r n ) so that, all together, ] and comletes the roof of the right- [ π L 1 s,(). This shows that u to-left inclusion in (1.3). Thus, (1.3) holds. L 1 s,()

33 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 33 Next, to show that the diagram (1.5) is commutative, ick u L () n and use (7.1) (with s = and = ) in order to decomose it as P u + π for some π L 1,z(). Then, since by (1.6) (1.9) for every v V we have (J 1, v, π) L () n = v V, V J1, v, P 1, u V = L 1 ()n J1, v, u (L 1 ()n ) = ( J 1, v, u ) L () n (1.1) = J 1, v, P u L () n + ( J 1, v, π ) L () n = V v, P u V. This shows that P 1, u = P u in V, as desired. Remark 1.. A close insection of the above argument shows that, for each (1, ) and s R, the oerator (1.1)-(1.) factors as (1.11) P : L s,() n L s,() n [ L 1 s,() ] ( V s, ()), where the first arrow is the canonical rojection onto the quotient sace, and the second arrow is an isomorhism naturally induced by J s,, the adjoint of (7.15). Next, let R n be a Lischitz domain and assume that ( 1, 1] has been fixed. Recall the oerator B induced by the sesquilinear form b (, ), i.e., (1.1) B : V V, B u := b (, u) V, u V. Next, fix u V, so that B u : V C is a linear, bounded functional. Since V is a closed subsace of L 1() n, the Hahn-Banach theorem ensures the existence of some linear, bounded functional f : L 1 () n C with the roerty that f V = (Ĩ + B ) u. Thus, f ( L 1() n) = L 1, () n satisfies L 1 v, f = (L ()n 1 ()n ) V v, ( Ĩ + B ) u V (1.13) = u v dx + A ( u, v) dx, v V L 1() n. Secializing this to the case when v belongs to v Cc () n : div v = in } shows that the distribution f (1 ) u L 1() n annihilates this sace. Thus, by Lemma 6.7, there exists a distribution π in such that π = f (1.14) (1 ) u L 1() n. In articular, π L () by Corollary 3.3. Returning with this information back in (1.13) and invoking (3.38) then shows that, after an eventual re-normalization of π (done by subtracting a suitable constant, similar in sirit to (6.3)), matters can be arranged so that (1.15) ν ( u, π) f u = in L 1/( ) n. The stage is now set for roving the following result. Proosition 1.3. Suose that R n is a Lischitz domain and that ( 1, 1]. Then for every u V there exist (1.16) π L () and f L 1, () n

34 34 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT such that (1.17) (1.18) (1.19) (1 ) u + π = f in L 1() n, ν ( u, π) f u = in L 1/( ) n, and (Ĩ + B ) u = P 1,f in V. Furthermore, if g L 1,() n is such that P 1, g = P 1, f, then there exists q L () with the roerty that (1.) (1 ) u + (π q) = g in L 1() n, (1.1) ν ( u, π q) g u = in L 1/( ) n. Proof. The existence of π, f as in (1.16) and for which (1.17)-(1.18) are satisfied is clear from the discussion receding the statement of the roosition. Hence, there remains to rove (1.19). This, however, is a direct consequence of Lemma 1.1 and the first equality in (1.13). There remains to take care of the claim in the second art of the statement. To this end, we first note that P 1, ( f g) = entails (1.) L 1 ()n J1, v, f g (L 1 ()n ) =, v V. Thus, based on (1.3) we may conclude that there exists some scalar function ˆq L () with the roerty that ( f g) = ˆq in L 1(). In turn, this and (1.17) yield (1.3) (1 ) u + (π ˆq) = g in L 1() n. Going further, formula (3.38) gives that for every w V L 1() n ) Tr w, ν ( u, π ˆq) g u = w u dx + A ( w, u (1.4) On the other hand, for every w V we have L 1 () n J1, w, g (L 1 ()n ). L 1 ()n J1, w, g (L 1 ()n ) = V w, P 1, g V = V w, P 1, f V (1.5) = L 1 () n J1, w, f (L 1 ()n ) ) = w u dx + A ( w, u, by hyotheses, (1.4), (3.38) and (1.18). Together, this and (1.4) then rove that (1.6) ν ( u, π ˆq) g u, Tr w =, w V L 1() n. With this in hand, and by roceeding as in (6.4)-(6.9), we may then conclude that there exists a constant c R with the roerty that if q := ˆq c then (1.)-(1.1) hold. Once again, suose that R n is a Lischitz domain and that ( 1, 1]. Also, fix (1, ) and assume that 1/ < s < 1 + 1/, 1 < <, 1/ + 1/ = 1. Then the

35 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 35 oerator B from (1.1) extends to a bounded maing (, B : V s, () V ()) s, (1.7) (, B u := A (, u) V ()) s, u V s, (). A similar line of reasoning as in the roof of Proosition 1.3 (the only significant difference is that Proosition 3. is used in lace of Corollary 3.3) then yields the following. Proosition 1.4. Retain the above notation and conventions. Also, assume that µ R. Then for every u V s, () there exist (1.8) such that (1.9) (1.3) (1.31) π L s 1() and f L P s+1/, () n (µ ) u + π = f in L s+1/ ()n, ν ( u, π) f µ u = in B, s 1( ) n, and (µĩ + B ) u = P s, f in (V s, ()). The stage has now been set for us to rove the following. Theorem 1.5. Let R n be a Lischitz domain and assume that ( 1, 1]. Then the domain of the fractional ower of the Neumann-Stokes oerator B satisfies } (1.3) D(B s ) = u L s() n : div u = in if s (1, 3). Furthermore, corresonding to s = 3/, one has that u D(B 3 4 ) if and only if u V and π L (), f L 1/, ()n L 1,() n, such that (1 ) u + π = (1.33) f in L 1/ ()n L 1() n, and for which ν ( u, π) f u = in L 1/ ( )n. Proof. Assume that s [1, ] and recall (9.5). Much as with (9.6), we have (1.34) u D(B s ) u V and (Ĩ + B ( ) u V ()) s, V. Now, given u D(B s ), Proosition 1.3 ensures that there exist f, π as in (1.16) such that (1.17)-(1.19) are satisfied. On the other hand, from Lemma 1.1 we know that the oerator (1.1) is onto. This imlies that there exists g L s,() n such that P 1, g = (Ĩ + B ) u in V. Then, according to the second art in the statement of Proosition 1.3, there exists q L () such that (1.)-(1.1) hold. As a consequence, if π := π q, then for each s [1, ], (1.35) u V and π L (), g L s,() n L 1,() n, u D(B s ) such that (1 ) u + π = g in L s () n L 1() n, and for which ν ( u, π) g u = in L 1/ ( )n. After adjusting notation, this equivalence with s = 3/ roves (1.33).

36 36 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT Assume next that s (1, 3 ). With u, π and g as in the right-hand side of (1.35), let ( w, ρ) solve u L s() n, ρ L s 1(), (1 ) w + ρ = g, (1.36) div w = in, ν ( w, ρ) g w = in L s 3/ ( )n. That this is ossible is ensured by Theorem 8.1. Then the difference ( v, η) := ( u, π) ( w, ρ) solves the homogeneous system v L s() n, η L s 1(), (1 ) v + η = in, (1.37) div v = in, ν ( v, η) v = in L 1/ ( )n. This then forces ν ( v, η) v = in L s 3/ ( )n and, hence, v =, η = in by the uniqueness art in Theorem 8.1. Thus, ultimately, u = w L s() n and π = ρ L s 1(). This roves the left-to-right inclusion in (1.3). The oosite imlication in (1.3) then follows from (1.35) and Proosition 1.4 (considered with = and µ = 1). Having established Theorem 1.5, the same argument as in the roof of Theorem 9. yields the following: Corollary 1.6. The end-oint case s = 3/ in (9.1) holds as well. As a corollary, if n = 3 then (1.38) D(B 3 4 ) L 3 1() 3. It should be noted that this is the counterart of a similar result for the Dirichlet-Stokes oerator first established in [9]. We conclude this section with the following remark, whose veracity is aarent from a close insection of earlier roofs: Remark 1.7. All regularity results established in this aer for D(B α ) are also valid in the case of D((µI + B ) α ) if µ. Furthermore, results similar in sirit hold in the case of the Stokes system with Neumann boundary conditions considered in Lischitz subdomains of Riemannian manifolds (cf. [9] for the Dirichlet-Stokes oerator in such a setting). 11. Navier-Stokes equations In this section, we make use of our earlier analysis of the fractional owers of the Stokes system in order to study issues such as existence, uniqueness and regularity for the Navier- Stokes system in bounded Lischitz subdomains of R 3, in the sense of mild solutions as in (1.). We are interested in the critical case, as far as functional saces are concerned, i.e., u D(B 1/4 ), where B has been defined in Section 6. The ideas here follow the lines develoed in [9] in the case of Dirichlet boundary conditions.

37 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS Existence. Let be a bounded Lischitz domain in R 3 and fix ( 1, 1]. Then for each T >, define the following Banach sace: F T := u C ([, T ]; D(B 1 4 )) C 1 ((, T ]; D(B 3 4 )) : (11.1) su <s<t s 1 3 B 4 u(s) H + su s 3 4 u (s) H + su <s<t <s<t where H and B are as in Section 6, endowed with the norm (11.) u FT := su B 1 4 u(s) H + su <s<t <s<t s 1 B 3 4 u(s) H + su s 3 4 u (s) H + su <s<t <s<t s 3 3 } B 4 u (s) H <, s 3 B 3 4 u (s) H. Following (5.)-(5.3), B generates an analytic semigrou. For the convenience of notation, let us denote the Neumann-Stokes semigrou by (11.3) (S u )(t) := e tb u, u H, t, Lemma If u D(B 1 4 ) then S u F T for each T > and (11.4) S u FT C B 1 4 u H where C > is a finite constant indeendent of T >. Proof. Fix some number T >, as well as a divergence-free vector field u D(B 1 4 ). Since (S u ) (t) = B e tb u for t >, it follows from (5.11) that (11.5) We also have that S u C ([, T ]; D(B 1 4 )) C 1 ((, T ]; D(B 3 4 )). t 1 3 B 4 (S u )(t) = t 1 1 B e tb B 1 4 (11.6) u is bounded from (, T ) into H. Likewise, the functions (11.7) and t 3 3 B 4 (S u ) (t) = t 3 3 B e tb B 1 4 u t 3 4 (S u ) (t) = t B 4 e tb B 1 4 (11.8) u are bounded from (, T ) into H. This roves that S u F T. Now, (11.4) is imlicit in the above analysis. Recall the oerator P from (7.) for =, s ( 1, 1) and, for each u, v F T, introduce (11.9) Φ( u, v)(t) := t Proosition 11.. The alication e (t s)b ( 1 P)(( u(s) ) v(s) + ( v(s) ) u(s)) ds, < t < T. (11.1) Φ : F T F T F T is well-defined, bilinear, symmetric and continuous. Furthermore, (11.11) Φ( u, v) FT κ u FT v FT, u, v F T, where κ = κ() > is a finite constant, indeendent of T.

38 38 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT Proof. The fact that Φ is bilinear and symmetric is clear. Moreover, Φ( u, v) = e B f, where f is defined by (11.1) f(s) := ( 1P)(( u(s) ) v(s) + ( v(s) ) u(s)), < s < T. We have D(B 3 4 ) L 3 1(, R 3 ) by (1.38) and [D(B 1 4 ), D(B 3 4 )] 1 = D(B 1 ) L 6 (, R 3 ). Thus, by Hölder s inequality, ( u(s) ) v(s)+( v(s) ) u(s) L (, R 3 ) for each u, v F T and, therefore, f(s) H for s (, T ), with s 3 4 (11.13) su s 3 4 f(s) H su <s<t <s<t ( u(s) L 3 1 (,R 3 ) v(s) L 6 (,R 3 ) )} + v(s) L 3 1 (,R ) u(s) 3 L 6 (,R 3 ) ( C su <s<t s 3 4 u(s) 3 D(B 4 ) v(s) 1/ v(s) 1/ D(B 4 1 ) D(B 4 3 ) ) } + v(s) 3 D(B 4 ) u 1/ u(s) 1/ D(B 4 1 ) D(B 4 3 ) ( C su s 3 4 B 3 4 u(s) B 1 4 v(s) 1/ H B 3 4 v(s) 1/ H <s<t )} C u FT v FT. Based on (11.13) and (5.11) we may then estimate (11.14) B 1 4 Φ( u, v)(t) H + B 3 4 v(s) H B 1 4 u(s) 1/ H B 3 4 u(s) 1/ H t B 1 4 e (t s)b L (H) f(s) H ds ( t C (t s) 1 4 s 3 4 ds ) u FT v FT ( 1 C (1 σ) 1 4 σ 3 4 dσ ) u FT v FT C u FT v FT. In order to check that the alication [, T ] t Φ( u, v)(t) D(B 1 4 ) is continuous, fix an arbitrary t o [, T ] and estimate B 1 4 Φ( u, v)(t) B 1 4 Φ( u, v)(t o ) H by distinguishing two scenarios: t t o, and t o t T. In the first case, we recall a general identity to the effect that ( to ) (11.15) e tb w e t ob w = B e τb w dτ, w H. Cf. [35, (.4),. 5]. Formula (11.15) allows us to write t (11.16) B 1 4 Φ( u, v)(t) B 1 4 Φ( u, v)(t o ) t ( = B 1 to ) 4 B e (τ s)b f(s) dτ ds + =: I 1 + I. to t t B 1 4 e (t s)b f(s) ds

39 Now, (11.17) and NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 39 ] I 1 H C su [s [ t 3 4 f(s) H <s<t t [ C u FT v FT ( to t (t o s) 1 4 (t s) 1 4 ( t o ) (11.18) I H C u FT v FT (t s) 1 4 s 3 4 ds Thus, altogether, B 1 4 Φ( u, v)(t) B 1 4 Φ( u, v)(t o ) H t to t t o and this ultimately shows that t ) ] dτ s 34 ds (τ s) 5/4 ] s 3 4 ds t to,. t t o. In fact, the same is true when (11.19) Φ( u, v) C ([, T ]; D(B 1 4 )) and su B 1 4 Φ( u, v)(t) H C u FT v FT <t<t for every u, v F T, where C > is a finite constant, indeendent of T >. Going further, we estimate (11.) B 3 4 Φ( u, v)(t) H t B 3 4 e (t s)b L (H) f(s) H ds ( t ) C (t s) 3 4 s 3 4 ds u FT v FT ( 1 C t 1 (1 σ) 3 4 σ 3 4 dσ ) u FT v FT C t 1 u FT v FT. The continuity of the ma (, T ] t B 3 4 Φ( u, v)(t) H can then be established as before. In order to estimate the derivative in time of Φ( u, v)(t), we first note that for each s (, T ) (11.1) f (s) = ( 1 P)(( u (s) ) v(s) + ( u(s) ) v (s) + ( v (s) ) u(s) + ( v(s) ) u (s)). In articular, much as in (11.13), (11.) su s 7 4 f (s) H C u FT v FT. <s<t where C > is indeendent of T. After this reamble we write (11.3) Φ( u, v)(t) = and, therefore, t t (11.4) Φ( u, v) (t) = e t B f( t ) + e sb f(t s) ds + t e sb f (t s) ds + e (t s)b f(s) ds, t ], T [, t B e (t s)b f(s) ds.

40 4 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT In concert with (11.13) and (11.), this allows us to estimate Φ( u, v) (t) H C f( t ) H + C (11.5) +C t t B e (t s)b L (H) f(s) H ds e sb L (H) f (t s) H ds C t 3 4 u FT v FT + C +C C t 3 4 t t (t s) 7 4 ds u FT v FT ( C t 3 4 u FT v FT, (1 σ) 7 4 dσ + 1 (t s) 1 s 3 4 ds u FT v FT (1 σ) 1 σ 3 4 dσ ) u FT v FT where C > is indeendent of T. Furthermore, by reasoning as before, one can show that the alication (, T ] t Φ( u, v) (t) D(B 3 4 ) is continuous. Finally, B 3 4 Φ( u, v) (t) H C B 3 4 e t B L (H) f( t ) H + C (11.6) +C t t B 3 4 e sb L (H) f (t s) H ds C t 3 u FT v FT + C +C C t 3 t t (t s) 7 4 s 3 4 ds u FT v FT ( C t 3 u FT v FT, (1 σ) 7 4 σ 3 4 dσ + 1 B 7 4 e (t s)b L (H) f(s) H ds (t s) 7 4 s 3 4 ds u FT v FT (1 σ) 7 4 σ 3 4 dσ ) u FT v FT where, once again, the constant C does not deendent of T. The above analysis ensures that Φ( u, v) F T whenever u, v F T. Moreover, from (11.19), (11.), (11.5) and (11.6), there exists a constant κ > indeendent of T > such that (11.11) holds. We are now ready to discuss the existence of mild solutions for the Navier-Stokes system. Theorem Given u D(A 1 4 ) and T >, the equation (11.7) u(t) = e tb u + Φ( u, u)(t), < t < T, has a unique solution u F T, if either u D(A 1 4 ) or T are sufficiently small. Proof. Let T > be given and consider the bilinear, continuous maing Φ : F T F T F T defined as in (11.9). As in [11], a solution of (11.7) will be found imlementing

41 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 41 Picard s fixed oint theorem. That is, consider the sequence in v j } j of functions in F T defined by v := S u and (11.8) v j+1 := v + Φ( v j, v j ), j N. As is well-known (cf., e.g., [6, Lemma,. 157]), this sequence converges to the unique solution u F T of (11.7) rovided (11.9) v FT < 1 4κ, where κ is the constant aearing in (11.11). In turn, since v FT C B 1 4 u H, the estimate (11.9) is satisfied granted that u 1 D(B 4 is small enough. ) To finish the roof, it suffices to show that, irresective of the size of u 1, matters D(B 4 ) can be arranged so that (11.9) holds by taking T small enough (relative to u D(B 1 4 ) ). To see this, we shall make use of the fact that for each ε > there exists u,ε D(B ) such that B 1 4 ( u u,ε ) H ε. If we now consider v,ε (t) := S u,ε for < t < T, then (11.3) v v,ε FT C B 1 4 (u u,ε ) H Cε, by (11.4) and, for each fixed ε, (11.31) v,ε FT C T 3 4 B u,ε H T +. By first choosing ε > small enough, we can therefore find T > such that (11.9) is valid. This concludes the roof of the theorem. Remark A somewhat smaller sace for which the analogues of (11.4) and (11.1) hold is as follows (11.3) F T := u F T : lim τ + u F τ = } Regularity. Here, we shall rove that the solution u F T of the fixed oint roblem (11.7) is actually a solution of the Navier-Stokes system (1.13) in the suitable sense, made recise in the theorem below. Theorem Any solution u F T of the roblem (11.7) satisfies u() = u in and, in addition, has the following roerties. For every t [, T ], the field u(t, ) is divergence free in and there exists π L (, T ; L ()) such that x u + x π L (, T ; L () 3 ), ( u, π) has vanishing conormal derivative (cf. 3) on, and for which the first equation in (1.13) is satisfied everywhere in the time variable t (, T ] and almost everywhere in the sace variable x. Furthermore, (11.33) u L 1(, T ; H) L (, T ; D(B )), 1 < < 4 3, and matters can be arranged so that (11.34) lim τ + u F τ =. Proof. Assume that u F T solves (11.7) and introduce (11.35) f(s) := P [ ( u(s) x ) u(s) ], s [, T ].

42 4 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT From (11.13) we may conclude that f L (, T ; H) whenever 1 < < 4 and, from (11.7), 3 that u = e B u + e B f. Now, the maximal regularity roerty for the Neumann- Stokes oerator in H (since B generates an analytic semigrou in the Hilbert sace H; cf. [6]) and the fact that u D(B 1 4 ) entail B u L (, T ; H) and that u solves (11.36) u (t) + (B u)(t) = f(t) for a.e. t (, T ), and u() = u. It follows from the definition of the Neumann-Stokes oerator B (cf. (6.)-(6.3)) that there exists q 1 L (, T ; L ()) such that (11.37) (11.38) L (, T ; L () 3 ) B u = x u + x q 1 L (, T ; L 1/( ) 3 ) ν (u, q 1 ) =. Moreover, by the definition of P ((7.) for = and s = ), there exists q L (, T ; L 1,z() 3 ) such that (11.39) P[( u x ) u] = ( u x ) u q. Since ( u x ) u C ((, T ]; L () 3 ), we also have (11.4) q C ((, T ]; L 1,z()). Combining (11.37)-(11.39) with (11.36) yields (1.13) with π = q 1 q. Moreover, since u C ((, T ]; H) and f C ((, T ]; H), we may finally conclude from (11.37) and (11.4) that x u + x π C ((, T ]; L () 3 ). Thus, the Navier-Stokes system (1.13)) holds as mentioned, whereas (11.34) is a consequence of the remark made at the end of Uniqueness. We have already roved that there exists a local mild solution to the Navier-Stokes system which is unique in the sace F T. Following [33], here we shall rove that, in fact, uniqueness holds in the larger sace C ([, T ]; D(B 1 4 )). Prior to formally stating this as a theorem, need to make sense of the non-linearity Φ( u, u) for fields u C ([, T ]; D(B 1 4 )). To this end, for u, v C ([, T ]; D(B 1 4 )) consider (11.41) f(s) := ( 1 P )( ) x u(s) v(s) + v(s) u(s), s (, T ), where, generally seaking, a b denotes the matrix (a i b j ) 1 i,j 3 for any a = (a 1, a, a 3 ) and b = (b 1, b, b 3 ) R 3. In this connection, let us also note that if a and b are smooth vector fields then (11.4) x ( a b) = ( a x ) b + (div a) b. This elementary identity allows us to extend the bilinear form Φ, originally defined on F T F T, to the larger sace C ([, T ]; D(B 1 4 )) C ([, T ]; D(B 1 4 )) in the following sense. First, if u, v C ([, T ]; D(B 1 4 )) are arbitrary then both u v and v u belong to C ([, T ]; L 3 () 3 3 ), since D(B 1 4 ) L 3 () 3. In articular, (11.43) x ( u v + v u) C ([, T ]; L 3 1 () 3 ). We now digress momentarily in order to establish a useful auxiliary result. Lemma The oerator P, introduced in Lemma 1.1, has the roerty that (11.44) B 3 4 P : L 3 1 () 3 H in a bounded fashion.

43 NEUMANN-STOKES OPERATOR IN LIPSCHITZ DOMAINS 43 Proof. Using (3.13) and (3.), we know from (1.1) that P mas L 3 1 () 3 boundedly into the sace (V 1,3 ()) which, in turn, embeds continuously into D(B 3 4 ) by (1.38). Since B is self-adjoint, we also have B 3 [ 3 4 D(B 4 ) ] = H, and (11.44) follows. Returning to the mainstream discussion, we note that B 3 4 f C ([, T ]; H), by (11.43) and Lemma Therefore, writing (11.45) Φ( u, v)(t) = it follows that t B 3 4 e (t s)b B 3 4 f(s) ds, t [, T ], (11.46) Φ : C ([, T ]; D(B 1 4 )) C ([, T ]; D(B 1 4 )) C ([, T ], H) in a bilinear, bounded fashion. Another useful roerty of this ma is as follows. Proosition For each (1, ) the maing (11.46) further extends to a bounded bilinear alication (11.47) Φ : L (, T ; D(B 1 4 )) L (, T ; D(B 1 4 )) L (, T ; D(B 1 4 )). Furthermore, the norm of (11.47) is bounded by a constant which deends exclusively on. Proof. For u L (, T ; D(B 1 4 )) and v L (, T ; D(B 1 4 )), the function f defined in (11.41) satisfies the estimate (11.48) B 3 4 f L (,T ;H) C B 1 4 u L (,T ;H) B 1 4 v L (,T ;H) for a finite constant C >. Then, thanks to the maximal regularity roerty for B, we have (11.49) and B 1 4 Φ( u, v) = B (e B B 3 4 f) L (, T ; H) (11.5) B 1 4 Φ( u, v) L (,T ;H) C B 1 4 u L (,T ;H) B 1 4 v L (,T ;H), as desired. We are now in a osition to discuss the uniqueness of mild solutions for the Navier- Stokes system, which is the main result of this subsection. To state it formally, for a measurable set E R and a Banach sace X, we set C b (E; X ) := C (E, ; X ) L (E; X ). Theorem For each u D(B 1 4 ), there is at most one field u C b ([, T ); D(B 1 4 )) which satisfies (11.7). Proof. Assume that for some u D(B 1 4 ) there exist two vector fields u 1, u which belong to C b ([, T ); D(B 1 4 )) and which solve (11.7). Then w := u 1 u also belongs to C b ([, T ); D(B 1 4 )) and, in addition, satisfies (11.51) w = Φ( u 1, u 1 ) Φ( u, u ) = Φ( w, u 1 + u ) = Φ( w, u 1 + u S u ) + Φ( w, S u ), where S is the Stokes semigrou (cf. (11.3)).

44 44 MARIUS MITREA, SYLVIE MONNIAUX AND MATTHEW WRIGHT The traditional strategy (cf., e.g., [33] and the references therein) is to rove that, for a fixed (1, ), there exists τ (, T ] such that w (11.5) w L (,τ;d(b 1 L (,τ;d(b 1 4 )). 4 )) Granted this estimate, we may conclude that w vanishes on [, τ) which, in turn, roves that τ (, T ] : w(t) = for t < τ } is nonemty. Let us denote its suremum by τ max. If τ max < T, the continuity of w entails w(τ max ) =. In this scenario, the above scheme can be reiterated, taking τ max as the initial time, and we eventually conclude that there exists some δ > such that w = on [, τ max + δ). This contradicts the maximality of τ max and roves that τ max = T. Thus w = on [, T ], as wanted. There remains to establish (11.5). For starters, we note that for any (1, ), Proosition 11.7 gives Φ( w, u 1 + u S u ) L (,τ;d(b 1 4 )) ( ) (11.53) C w L (,τ;d(b 4 1 u 1 S u )) L (,τ;d(b 1 + u S u 4 )) L (,τ;d(b 4 1. )) Since (11.54) u j S u L (,τ;d(a 1, j = 1,, 4 )) τ + it follows that (11.53) is useful for the urose of establishing (11.5). There remains to handle the term Φ( w, S u ). To this end, for an arbitrary ε >, to be secified later, ick u,ε D(B ) such that u u,ε 1 D(B 4 < ε and then write ) (11.55) Φ( w, S u ) L (,τ;d(b 1 4 )) C w L (,τ;d(b 1 4 )) ( ) S( u u,ε ) L (,τ;d(b S u,ε )) L (,τ;d(b 4 1. )) Next, (11.56) S( u u,ε ) L (,τ;d(b 1 4 )) u u,ε D(B 1 4 ) < ε Finally, much as with (11.31), (11.57) S u,ε L (,τ;d(b 1 4 )) C τ 3 4 B u,ε H τ +. In summary, by first choosing ε > small enough (relative to the constant C in (11.55)) it is then ossible to ensure that (11.5) holds rovided τ > is sufficiently small. This justifies (11.5) and concludes the roof of the theorem. References [1] W. Arendt, Semigrous and evolution equations: functional calculus, regularity and kernel estimates, in Evolutionary Equations, Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 4. [] R. Brown and Z. Shen, Estimates for the Stokes oerator in Lischitz domains, Indiana Univ. J., 44 (1995), [3] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume, Functional and Variational Methods, Sringer, Berlin,. [4] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 5, Evolution Problems I, Sringer, Berlin,.

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