Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains

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1 Various bounary conitions for Navier-Stokes equations in boune Lipschitz omains Sylvie Monniaux To cite this version: Sylvie Monniaux. Various bounary conitions for Navier-Stokes equations in boune Lipschitz omains. Discrete an Continuous Dynamical Systems - Series S, American Institute of Mathematical Sciences, 3, 6 5, pp <hal-6539> HAL I: hal Submitte on 3 Dec HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not. The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers. L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés.

2 Manuscript submitte to AIMS Journals Volume X, Number X, XX X Website: pp. X XX VARIOUS BOUNDARY CONDITIONS FOR NAVIER-STOKES EQUATIONS IN BOUNDED LIPSCHITZ DOMAINS Sylvie Monniaux LATP - UMR 663, Faculté es Sciences e Saint-Jérôme Case Cour A, Université Paul Cézanne 3397 Marseille Ceex, France Abstract. We present here ifferent bounary conitions for the Navier- Stokes equations in boune Lipschitz omains in R 3, such as Dirichlet, Neumann or Hoge bounary conitions. We first stuy the linear Stokes operator associate to the bounary conitions. Then we show how the properties of the operator lea to local solutions or global solutions for small initial ata.. Introuction. The aim of this paper is to escribe how to fin solutions of the Navier-Stokes equations t u u + π + u u = in, T, iv u = in, T, u = u in, in a boune Lipschitz omain R 3, an a time interval, T T, for initial ata u in a critical space, with one of the following bounary conitions on : Dirichlet bounary conitions: Neumann bounary conitions: Hoge bounary conitions: u =, [ u + u ]ν πν =,, ], 3 ν u =, an ν curl u =, 4 where νx enotes the unit exterior normal vector on a point x efine almost everywhere when is a Lipschitz bounary. The strategy is to fin a functional setting in which the Fujita-Kato scheme applies, such as in their funamental paper [4]. The paper is organize as follows. In Section, we efine the Dirichlet-Stokes operator an then show the existence of a local solution of the system {, } for initial values in a critical space in the L -Stokes scale. In Section 3, we aapt the previous proofs in the case of Neumann bounary conitions, i.e., for the system {, 3 }. In Section 4, we stuy a slightly moifie version of the system {, 4 } for initial conitions in the critical space Mathematics Subject Classification. Primary: 35Q3, 35Q35, 35J5. Key wors an phrases. Stokes operator, bounary conitions, Dirichlet, Neumann, Hoge, Lipschitz omains, Navier-Stokes equations.

3 SYLVIE MONNIAUX { u L 3 ; R 3 ; iv u = in, ν u = on }. Finally, in Section 5, we iscuss some open problems relate to this subject.. Dirichlet bounary conitions. For a more complete exposition of the results in this section, as well as an extension to more general omains, the reaer can refer to [6] an []. The case where is smooth was solve by Fujita an Kato in [4]. In [], the case of boune Lipschitz omains was stuie for initial ata not in a critical space... The linear Dirichlet-Stokes operator. We start with a remark about L vector fiels on. Remark. For R 3 a boune Lipschitz omain, let u L ; R 3 such that iv u L ; R. Then we can efine ν u on in the following weak sense in H ; R: for ϕ H ; R, u, ϕ + iv u, ϕ = ν u, φ 5 where φ = Tr ϕ, the right han-sie of 5 epens only on φ on an not on the choice of ϕ, its extension to. The notation, E is for the L -scalar prouct on E. The space L ; R 3 is equal to the orthogonal irect sum H G where H = { u L ; R 3 ; iv u = in, ν u = on } 6 an G = H ; R. This follows from the following theorem. Theorem. e Rham. Let T be a istribution in C c ; R 3 such that T, ϕ = for all ϕ C c ; R 3 with iv ϕ = in. Then there exists a istribution S C c ; R such that T = S. Conversely, if T = S with S C c ; R, then T, ϕ = for all ϕ C c ; R 3 with iv ϕ = in. Remark. In the case of a boune Lipschitz omain R 3, the space H coincies with the closure in L ; R 3 of the space of vector fiels u C c ; R 3 with iv u = in. We enote by J : H L ; R 3 the canonical embeing an P : L ; R 3 H the orthogonal projection. It is clear that PJ = I H. We efine now the space V = H ; R 3 H. The embeing J restricte to V maps V to H ; R 3 : we enote it by J : V H ; R 3. Its ajoint J = P : H ; R 3 V is then an extension of the orthogonal projection P. We are now in the situation to efine the Dirichlet-Stokes operator. Definition.. The Dirichlet-Stokes operator is efine as being the associate operator of the bilinear form 3 a : V V R, au, v = i J u, i J v. Proposition. The Dirichlet-Stokes operator A is the part in H of the boune operator A, : V V efine by A,u : V R, A, uv = au, v, an satisfies i= DA = { u V ; P DJ u H }, A u = P DJ u u DA,

4 NAVIER-STOKES EQUATIONS IN LIPSCHITZ DOMAINS 3 where D enotes the weak vector-value Dirichlet-Laplacian in L ; R 3. The operator A is self-ajoint, invertible, A generates an analytic semigroup of contractions on H, DA = V an for all u DA, there exists π L ; R such that JA u = J u + π 7 an DA amits the following escription DA = { u V ; π L ; R : J u + π H }. Proof. By efinition, for u DA, we have, for all v V, n A u, v = au, v = j J u, j J v = n j= j= H j J u, J v H = H J u, J v H = V P J u, v V. The thir equality comes from the efinition of weak erivatives in L, the fourth equality comes from the fact that n j= j =. The last equality is ue to the fact that J = P. Therefore, A u an P J u are two linear forms which coincie on V, they are then equal. So we prove here that A, = P J : V V. Moreover, the fact that u DA implies that A u is a linear form on H, so that the linear form P J u, originally efine on V, extens to a linear form on H since V is ense in H by e Rham s theorem. The fact that A is self-ajoint an A generates an analytic semigroup of contractions comes from the properties of the form a: a is bilinear, symmetric, sectorial of angle, coercive on V V. The property that DA = V is ue to the fact that A is self-ajoint, applying a result by J.L. Lions [8, Théorème 5.3]. To prove the last assertions of this proposition, let u DA. Then A u H an P JA u = PJA u = u. Moreover, if u DA, u belongs, in particular, to V. Therefore, J u H ; R 3 an J u H ; R 3. We have then, the equalities taking place in V, P JA u J u = P JA u P J u = A u A u =. By e Rham s theorem, this implies that there exists p Cc ; R such that JA u Ju = p: p H ; R 3, which implies that p L ; R. The relations between the spaces an the operators are summarize in the following commutative iagram: V J H A, J H L V P=J H P =J D

5 4 SYLVIE MONNIAUX In the case of a boune Lipschitz omain R 3, we also have the following property of DA 3 4 ; see [, Corollary 5.5]. Proposition. The omain of A 3 4 is continuously embee into W,3 ; R 3... The nonlinear Dirichlet-Navier-Stokes equations. The system {, } is invariant uner the scaling u t, x = u t, x, t, x, T > : if u is a solution of {, } in, T for the initial value u, then u is a solution of {, } in, T for the initial value x u x. We are intereste in fining mil solutions of the system {, } for initial values u in a critical space, in the same spirit as in [4]. Lemma.3. The space DA 4 is a critical space for the Navier-Stokes equations. Proof. We have to prove that DA 4 is invariant uner the scaling u x = u x for x, >. It suffices to check that u = u an u = u an apply the fact that DA 4 is the interpolation space with coefficient between H an V = DA. For T >, efine the space E T by { E T = u C b [, T ]; DA 4 ; ut DA 3 4, u t DA 4 for all t, T ] } an t 3 A 4 ut + ta 4 u t < t,t enowe with the norm u ET = A 4 ut + t,t t,t t,t t 3 A 4 ut + t,t ta 4 u t. The fact that E T is a Banach space is straightforwar. Assume now that u E T, an that J u, p with p L ; R satisfy {, } in H ; R 3 : inee, every term p, t J u, J u an J u J u inepenently belong to H ; R 3. We can then apply P to the equations an obtain u t + A ut = P J u J u { since P } p = an P J u = A, u. We have then reuce the problem, into the abstract Cauchy problem u t + A, ut = P J u J u u = u, u E T, for which a mil solution is given by the Duhamel formula: u = α + ϕu, u, where αt = e ta u an t ϕu, vt = e t sa P J us J vs + J vs J us s. The strategy to fin u E T satisfying u = α + ϕu, u is to apply a fixe point theorem. We have then to make sure that E T is a goo space for the problem, i.e., α E T an ϕu, u E T. The fact that α E T follows irectly from the properties of the Stokes operator A an the semigroup e ta t. 8

6 NAVIER-STOKES EQUATIONS IN LIPSCHITZ DOMAINS 5 Proposition 3. The application ϕ : E T E T E T is bilinear, continuous an symmetric. Proof. The fact that ϕ is bilinear an symmetric is immeiate, once we have prove that it is well-efine. For u, v E T, let ft = P J ut J vt + J vt J ut, t, T. 9 By the efinition of E T an Sobolev embeings, it is easy to see that J ut J vt + J vt J ut L ; R 3 an J ut J vt + J vt J ut C t 3 4 u ET v ET where C is a constant inepenent from t. Inee, by Proposition, if u, v E T, then u, v L 3, R 3 3 with the estimates ut 3 t u ET an vt 3 t v ET for all t >. Moreover, since DA L 6 ; R 3, we also have ut 6 t 4 u ET an vt 6 t 4 v ET for all t >. This, combine with the fact that L 3 L 6 L, gives the following estimate ft C t 3 4 u ET v ET for all t >. Therefore, we have A 4 ϕu, vt t A 4 e t sa L H C s 3 4 u ET v ET s t C t s 4 s 3 4 s u ET v ET, an since t t s 4 s 3 4 s = s 4 s 3 4 s, we finally obtain the estimate A 4 ϕu, vt C u ET v ET. The proof of the continuity of t A 4 ϕu, vt on H is straightforwar once we have the estimate. The proof of the fact that ta 3 4 ϕu, vt C u ET v ET is prove the same way, replacing A 4 by A 3 4 an using the fact that an t A 3 4 e t sa L H C t s 3 4 t s 3 4 s 3 4 s = t s 3 4 s 3 4 s. It remains to prove the estimate on the erivative with respect to t of ϕu, vt. Let us rewrite f as efine in 9 as follows: fs = P J us J vs + J vs J us where u v enotes the matrix u i v j i,j 3 an acts on matrices M = m i,j i,j 3 the following way: 3 M = i m i,j j 3. i=

7 6 SYLVIE MONNIAUX For u, v E T an s, T, we have f s = P Ju s J vs + J us Jv s + Jv s J us + J vs Ju s For all s, T we have s 5 4 Ju s J vs sju s 3 s 4 J vs 6 sa 4 u s s 4 A vs u ET v ET, where the first inequality comes from the fact that L 3 L 6 L, the secon comes from the Sobolev embeings DA 4 L 3 ; R 3 an DA L 6 ; R 3 an the thir inequality follows irectly from the efinition of the space E T. Of course the same occurs for the other three terms J us Jv s, Jv s J us an J vs Ju s. Therefore, since A maps V to H, we obtain We have an therefore which yiels ϕu, vt = s 5 4 A f s c u ET v ET. 3 <s<t t t e sa ft ss + e t sa fss t, T, ϕu, v t = e t A f t + t A 4 ϕu, v t c t 4 t + A e t sa fss, + c c t t f t + c t + t s 5 4 σ A e sa A, f t ss s 3 4 t s 5 4 s u ET v ET s 3 4 σ 5 4 σ 3 4 u ET v ET, s u ET v ET where we use the estimates, 3, an the fact that A generates a boune analytic semigroup, so that A α e ta L H C t α. This last inequality together with an ensure that ϕu, v E T whenever u, v E T. We conclue this section by applying Picar s fixe point theorem see e.g. [7, Theorem A.] to obtain the following existence result for the system {, }. Theorem.4. Let R 3 be a boune Lipschitz omain an let u DA 4. Let α an ϕ be efine as above. i If A 4 u is small enough, then there exists a unique u E solution of u = α + ϕu, u.

8 NAVIER-STOKES EQUATIONS IN LIPSCHITZ DOMAINS 7 ii For all u DA 4, there exists T > an a unique u E T u = α + ϕu, u. solution of Uniqueness in the larger space C b [, T ; DA 4 can be obtaine, applying [5, Theorem.]. 3. Neumann bounary conitions. In this section, we stuy the system {, 3 }. We will only survey the results prove in [4], the metho to prove existence of solutions being similar to what we one in Section. 3.. The linear Neumann-Stokes operator. The bounary conitions 3 are inexe by, ]: if =, 3 becomes T u, πν = on, T, 4 where T u, π = u + u πi enotes the stress tensor; if =, 3 becomes ν u πν = on, T. 5 Before efining the Neumann-Stokes operator, we nee the following integration by parts formula. Lemma 3.. Let R, u, w : R 3, π, ρ : R sufficiently nice functions efine on the Lipschitz omain R 3. Let L u = u + iv u an efine the conormal erivative ν u, π = u + u ν πν on. 6 Then the following integration by parts formula hol [ L u π w x = I u, w π iv w ] x where I ξ, ζ = = + ν u, π w σ 7 [ ] L w ρ u x + π iv w ρ iv u x + [ ν u, π w ν w, ρ u ] σ, 8 3 ξ i,j ζ i,j + ξ i,j ζ j,i, for ξ = ξ i,j i,j 3 an ζ = ζ i,j i,j 3. i,j= Recall that u = i u j i,j 3. The space L ; R 3 amits the following orthogonal ecomposition: H n G, where G = { π; π H ; R } an H n = { u L ; R 3 ; iv u = }. 9 Following the steps of the previous section, we efine V n = H ; R 3 H n an J n : H n L ; R 3 the canonical embeing, P n = J n : L ; R 3 H n the orthogonal projection, Jn : V n H ; R 3 the restriction of J n on V n an J n = P n : H ; R 3 V n, extension of P n to H ; R 3. We can now efine the Neumann-Stokes operator.

9 8 SYLVIE MONNIAUX Definition 3.. Let R. The Neumann-Stokes operator is efine as being the associate operator of the bilinear form a : V n V n R, a u, v = I J n u, J n v x In the case where, ], the bilinear form a is continuous, symmetric, coercive an sectorial. So its associate operator is self-ajoint, invertible an the negative generator of an analytic semigroup of contractions on H n. The following proposition is a consequence of the integration by parts formula 7, [4, Theorem 6.8] an [8, Théorème 5.3]. Proposition 4. Let, ]. The Neumann-Stokes operator A is the part in H n of the boune operator A, : V n V n efine by A, uv = a u, v. The operator A is self-ajoint, invertible, A generates an analytic semigroup of contractions on H n, DA = V n an for all u DA, there exists π L ; R such that J n A u = J n u + π an DA amits the following escription DA = { u V n ; π L ; R : f = J n u + π H n an ν u, π f = }, where ν u, π f is efine in a weak sense for all f H ; R 3 by ν u, π f, ψ = H f, Ψ H + I J n u, Ψ x L π, iv Ψ L for Ψ H an ψ = Tr Ψ. Remark 3. If f H ; R 3, the quantity ν u, π f exists in the Besov space ; R 3 = H, R 3 accoring to [4, Proposition 3.6]. B, Thanks to [4, Sections 9 & ], we have a goo escription of the omain of fractional powers of the Neumann-Stokes operator A. In particular, in [4, Corollary.6] it was establishe that DA 3 4 is continuously embee into W,3 ; R The nonlinear Neumann-Navier-Stokes equations. The results in 3. allow us to prove a result similar to Theorem.4 for the system {, 3 }. As in the previous section, it is not ifficult to see that DA 4 L 3 ; R 3 is a critical space for the system. For T, ], following the efinition of E T in Section, we efine { F T = u C b [, T ]; DA 4 ; ut DA 3 4, u t DA 4 for all t, T ] an t,t enowe with the norm t A 3 4 ut + u FT = A 4 ut + t,t t,t t,t } ta 4 u t < t 3 A 4 ut + t,t ta 4 u t. The same tools as in. apply, so we can prove the following result see [4, Theorem.3].

10 NAVIER-STOKES EQUATIONS IN LIPSCHITZ DOMAINS 9 Theorem 3.3. Let R 3 be a boune Lipschitz omain an let u DA 4. Let β an ψ be efine by an for u, v F T an t, T, ψu, vt = t βt = e ta u, t, e t sa P n J n us J n vs + J n vs J n us s. i If A 4 u is small enough, then there exists a unique u F solution of u = β + ψu, u. ii For all u DA 4, there exists T > an a unique u F T solution of u = β + ψu, u. A comment here may be necessary to link the solution u obtaine in Theorem 3.3 an a solution of the system {, 3 }. If u F T, then u H n an J n u J n u L ; R n. Moreover, if u satisfies the equation u = β + ψu, u, then u is a mil solution of A u = u P n Jn u J n u H n. Going further, we may write where q H ; R satisfies J n P n Jn u J n u = J n u J n u q q = iv J n u J n u H ; R n. Therefore, we have by efinition of A, there exists π L, R such that J n u + π = J n A u = J n u J n u J n u + q an at the bounary, u, π satisfies 3 in the weak sense as in Proposition 4. Since q H ; R, u, π q satisfies also 3. This proves that u, π q is a solution of the system {, 3 }. The uniqueness is true in a larger space than F T : for each u DA 4, there is at most one u C b [, T ; DA 4, mil solution of the system {, 3 }. For a more precise statement, see [4, Theorem.8]. 4. Hoge bounary conitions. Most of the results presente here are prove thoroughly in [] for the linear theory an [3] for the nonlinear system. We start with the stuy of the linear Hoge-Laplacian on L p -spaces an then move to the Hoge-Stokes operator before applying the properties of this operator to prove the existence of mil solutions of the Hoge-Navier-Stokes system in L The Hoge-Laplacian. Let H = L ; R 3 an V = { u H; curl u H, iv u L ; R an ν u = on }. We start by efining on V V the following form b : V V R, bu, v = curl u, curl v + iv u, iv v, where, enotes either the scalar or the vector-value L -pairing.

11 SYLVIE MONNIAUX Remark 4. Contrary to the case of smooth boune omains with a C, bounary, the space V is not containe in H ; R 3. The Sobolev embeing associate to the space V is as follows: V H ; R 3 with the estimate see for instance []. u H C [ u + curl u + iv u ], u V ; Proposition 5. The Hoge-Laplacian operator B, efine as the associate operator in H of the form b, satisfies DB = { u V ; iv u H, curl curl u H an ν curl u = on } Bu = u, u DB. Since the form b is continuous, bilinear, symmetric, coercive an sectorial, the operator B generates an analytic semigroup of contractions on H, B is self-ajoint an DB = V. Remark 5. As in Remark for a boune Lipschitz omain an a vector fiel w H satisfying curl w H, we can efine ν w on in the following weak sense in H ; R 3 : for ϕ H ; R 3, curl w, ϕ w, curlϕ = ν w, ϕ 3 where φ = Tr ϕ, the right han-sie of 3 epens only on φ on an not on the choice of ϕ, its extension to. To prove that B extens to L p -spaces, we prove that its resolvent amits L L off-iagonal estimates. This was prove in [, Section 6] Proposition 6. There exist two constants C, c > such that for any open sets E, F such that ist E, F > an for all t >, f H an we have u = I + t B χ F f, ist E,F c χ E u + t χ E iv u + t χ E curl u Ce t χ F f. 4 Proof. We start by choosing a smooth cut-off function ξ : R 3 R satisfying ξ = k on E, ξ = on F an ξ ist E,F. We then efine η = eαξ where α > is to be chosen later. Next, we take the scalar prouct of the equation u t u = χ F f, u DB with the function v = η u. Since η = on F an u χ F f, it is easy to check then that ηu + t ηiv u + t ηcurl u χ F f + α ξ t ηu ηiv u + ηcurl u ist E,F an therefore, using the estimate on ξ an choosing α = 4kt, we obtain ηu + t ηiv u + t ηcurl u χ F f. Using now the fact that η = e α on E, we finally get χ E u + t χ E iv u + t χ E curl u ist E,F e 4kt χ F f, which gives 4 with C = an c = 4k.

12 NAVIER-STOKES EQUATIONS IN LIPSCHITZ DOMAINS With a slight moification of the proof, we can show that for all θ, π there exist two constants C, c > such that for any open sets E, F such that ist E, F > an for all z Σ π θ = { ω C \ {}; arg z < π θ }, f H an we have u = zi + B χ F f, z χ E u + z χe iv u + z χe curl u C e c iste,f z χ F f. 5 With that in han an the Sobolev embeing, together with the rescale Sobolev inequality R χe u 3 C χ E u + R χ E iv u + R χ E curl u 6 where R = iam E, we can prove that, choosing E = Bx, z an F j = Bx, j+ z \ Bx, j z for x an j N: where f j = χ Fj f. z χ E u 3 C z 4 e c j f j 7 Proposition 7. There exists a constant C > such that for all f L ; R 3 L 3 ; R 3, z Σ π θ, the following estimate hols: z zi + B f 3 C f 3. 8 Proof. For x an r >, enote by B x, r the ball centere in x with raius r intersecte with. Let u = zi + B f. For x, let f j = χ Fj f for F j = B x, j+ z \ B x, j z an u j = zi + B f j. From 7 an Fubini s theorem, keeping in min that a Lipschitz omain in R n is a n-set in the terminology of [7] which means that balls centere in with raius r intersecte with have a volume equivalent to r n, we have [ ] z u 3 C z B x, t uy 3 3 y x B x,t [ C z [ B ] x, t uy ] 3 y x B x,t [ B x, t [ C z [ C j= j= B x,t Ce cj 3j B x, j t C e cj 3j M f L 3 ;R j= C f 3 ] u j y ] 3 y x B x, j t fy 3 ] 3 y x where we use the notation t = z an M enotes the Hary-Littlewoo maximal operator which is boune on L p for all p,.

13 SYLVIE MONNIAUX Corollary. The semigroup e tb t extens to a boune analytic semigroup on L p ; R 3 for p [ 3, 3]. Proof. For p = 3, this comes irectly from Proposition 7. We obtain the result for all p [, 3] by interpolation an for all p [ 3, ] by uality since the operator B is self-ajoint. We can actually prove that the semigroup e tb t extens to a boune analytic semigroup on L p ; R 3 for p in an interval containing [ 6 5, 6]. In an open interval p, q containing [ 3, 3], the negative generator B p of this semigroup satisfies DB p = { u L p ; R 3 ; iv u W,p ; R 3, curl u L p ; R 3, curl curl u L p ; R 3, ν u = an ν curl u = on } B p u = u, u DB p. To obtain estimates in L p for p > 3, the metho is in the same spirit as what we have just one, combine with a bootstrap argument an regularity results for B. For a complete proof, the reaer may refer to [, Section 5]. 4.. The nonlinear Hoge-Navier-Stokes equations. Grante that u is a sufficiently smooth vector fiel, we have the following ientification u u = u + u curl u. That is, replacing π in by π + u, the system {, 4 } reas t u u + π + u curl u = in, T, iv u = in, T, u = u in, ν u = on, T, ν curl u = on, T. Before trying to solve this system, we nee some facts about the Hoge-Stokes operator. In [3], it was prove that the orthogonal projection P efine in Section on L ; R 3 extens to a boune projection on L p ; R 3 for p in an open interval p, q containing [ 3, 3] ; enote it by P p. In [, Lemma 3.7], it was prove that P p an B p, the Hoge-Laplacian in L p ; R 3 commute on DB p. This allows us to efine the Hoge-Stokes operator A p on H p = { u L p ; R 3 ; iv u = in, an ν u = on }. The results we prove for the Hoge-Laplacian naturally exten to the Hoge-Stokes operator as state in the following theorem. Theorem 4.. Let p p, q. The Hoge-Stokes operator A p efine on H p by DA p = { u H p ; curl u L p ; R 3, curl curl u L p ; R 3 an ν curl u = on } A p u = P p u = u, u DA p is the negative generator of a boune analytic semigroup on H p efine by e ta p u = P p e tb p u = e tb p P p u = e tb p u, u H p. 9

14 NAVIER-STOKES EQUATIONS IN LIPSCHITZ DOMAINS 3 Moreover, this semigroup satisfies the uniform estimate e tap L Hp + t curl e tap L Hp,L p 3 t> + t curl curl e ta p L Hp,L p <. We now rewrite the nonlinear Hoge-Navier-Stokes system for initial ata in the ciritical space H 3 in the abstract form u t + A p ut + P p ut curl ut =, u H 3, 3 for p to be etermine. The iea to solve 3 is to apply the same metho as in Sections & 3. To o so, we nee a regularizing property of the Hoge-Stokes semigroup, which was prove in [3, Theorem 3. an Theorem 4.]: the Hoge- Stokes semigroup satisfies the estimate t α e ta p L Hp,L q + t +α curl e ta p L Hp,L q < 3 t> whenever p p, q, q p, q with p α 3 = q for some α,. The proof of this results relies on the possibility to fin an inverse of the curl moulo graient vectors an uses results prove in [9]. With these properties of the Hoge-Stokes semigroup in han, the following existence result for 3 is almost immeiate. For T, ], we efine the space G T by { G T = u C b [, T, H 3 C, T, H 3+ε ; curl u C, T, L 3, R 3 with ε s +ε us 3+ε + } s curl us 3 < <t<t enowe with the norm u GT = <t<t ε us 3 s +ε us 3+ε + s curl us 3, where ε > is such that 3 + ε < q. Theorem 4.. Let R 3 be a boune Lipschitz omain an let u H 3. Let γ an Φ be efine by γt = e tap u, t, an for u, v G T, an t, T, Φu, vt = t e t sa 3 P 3 us curl vs + vs curl us s. i If u 3 is small enough, then there exists a unique u G solution of u = γ + Φu, u. ii For all u H 3, there exists T > an a unique u G T solution of u = γ + Φu, u. For a complete proof of this theorem, we refer to [3, Section 5]. 5. Remarks an open problems.

15 4 SYLVIE MONNIAUX 5.. Comparison between the bounary conitions. The bounary conitions, 3 an 4 can be ecompose, for sufficiently regular vector fiels u, into their normal part an their tangential part as follows i becomes ii 3 becomes ν u = an ν u = on, 33 ν [ u + u ν] = π an [ u + u ν] tan = on if =, 34 iii the Navier s slip bounary conitions rea ν u = an [ u + u ν] tan = on, 35 iv 4 is alreay ecompose into its normal part ν u = an its tangential part ν curlu = on. It is common to ientify the Navier s slip bounary conitions 35 with the Hoge bounary conitions 4. This is true only on flat parts of the bounary. In the case of a C omain, it can be prove that 35 an 4 iffer only by a zero-orer term. For more informations on this subject, the intereste reaer coul refer to [3, Section ]. 5.. Open problems. In the case of a smooth boune omain in R n, it was prove by Y. Giga an T. Miyakawa in [6] that the Dirichlet-Navier-Stokes system amits a local mil solution for initial values in L n critical space for the system in imension n. Their metho relies on the fact that the Dirichlet-Stokes operator, as efine in Section, extens to all L p spaces an is the negative generator of an analytic semigroup there, which was prove in [5]. The situation in Lipschitz omains is ifferent. For instance, P. Deuring provie in [] an example of a omain with one conical singularity such that the Dirichlet-Stokes semigroup oes not exten to an analytic semigroup in L p for p large or p small, away from. As alreay mentione, E. Fabes, O. Menez an M. Mitrea prove in [3] that the orthogonal projection P efine in Section on L ; R 3 extens to a boune projection on L p ; R 3 for p in an open interval containing [ 3, 3] if is C, then this interval is,. This le M. Taylor in [8] to formulate the conjecture that the Dirichlet-Stokes semigroup efine originally on H extens to an analytic semigroup on L p for p in the same interval as in [3]. Remark 6. This conjecture is actually true when, instea of consiering Dirichlet bounary conitions, we consier Hoge bounary conitions, as prove in Section 4. In the same paper [3], the authors prove that the orthogonal projection P n efine in Section 3 on L ; R 3 also extens to a boune projection on L p ; R 3 for p in the same open interval containing [ 3, 3]. This leas to the conjecture similar to Taylor s that the Neumann-Stokes semigroup efine originally on H n extens to an analytic semigroup on L p for p in the same interval. As for now, no positive result is known in L p for p for these two conjectures. To apply the Fujita-Kato scheme as in Sections & 3, proving that the Stokes semigroup extens to an analytic semigroup in L 3 seems to be the first step to obtain mil solutions of the Navier-Stokes system with either Dirichlet or Neumann bounary conitions.

16 NAVIER-STOKES EQUATIONS IN LIPSCHITZ DOMAINS 5 REFERENCES [] Paul Deuring, The Stokes resolvent in 3D omains with conical bounary points: nonregularity in L p -spaces, Av. Differential Equations 6, no., [] Paul Deuring an Wolf von Wahl, Strong solutions of the Navier-Stokes system in Lipschitz boune omains, Math. Nachr , 48. [3] Eugene Fabes, Osvalo Menez, an Marius Mitrea, Bounary layers on Sobolev-Besov spaces an Poisson s equation for the Laplacian in Lipschitz omains, J. Funct. Anal , no., [4] Hiroshi Fujita an Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal , [5] Yoshikazu Giga, Analyticity of the semigroup generate by the Stokes operator in L r spaces, Math. Z , no. 3, [6] Yoshikazu Giga an Tetsuro Miyakawa, Solutions in L r of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal , no. 3, [7] Alf Jonsson an Hans Wallin, Function spaces on subsets of R n, Math. Rep. 984, no., xiv+. [8] Jacques-Louis Lions, Espaces interpolation et omaines e puissances fractionnaires opérateurs, J. Math. Soc. Japan 4 96, [9] Dorina Mitrea, Marius Mitrea, an Sylvie Monniaux, The Poisson problem for the exterior erivative operator with Dirichlet bounary conition in nonsmooth omains, Commun. Pure Appl. Anal. 7 8, no. 6, [] Dorina Mitrea, Marius Mitrea, an Michael Taylor, Layer potentials, the Hoge Laplacian, an global bounary problems in nonsmooth Riemannian manifols, Mem. Amer. Math. Soc. 5, no. 73, x+. [] Marius Mitrea an Sylvie Monniaux, The regularity of the Stokes operator an the Fujita- Kato approach to the Navier-Stokes initial value problem in Lipschitz omains, J. Funct. Anal. 54 8, no. 6, [] Marius Mitrea an Sylvie Monniaux, On the analyticity of the semigroup generate by the Stokes operator with Neumann-type bounary conitions on Lipschitz subomains of Riemannian manifols, Trans. Amer. Math. Soc. 36 9, no. 6, [3] Marius Mitrea an Sylvie Monniaux, The nonlinear Hoge-Navier-Stokes equations in Lipschitz omains, Differential Integral Equations 9, no. 3-4, [4] Marius Mitrea, Sylvie Monniaux, an Matthew Wright, The Stokes operator with Neumann bounary conitions in Lipschitz omains, J. Math. Sci. N. Y. 76, no. 3, [5] Sylvie Monniaux, On uniqueness for the Navier-Stokes system in 3D-boune Lipschitz omains, J. Funct. Anal. 95, no.,. [6] Sylvie Monniaux, Navier-Stokes equations in arbitrary omains: the Fujita-Kato scheme, Math. Res. Lett. 3 6, no. -3, [7] Sylvie Monniaux, Maximal regularity an applications to PDEs, Analytical an numerical aspects of partial ifferential equations, Walter e Gruyter, Berlin, 9, pp [8] Michael E. Taylor, Incompressible flui flows on rough omains, Semigroups of operators: theory an applications Newport Beach, CA, 998, Progr. Nonlinear Differential Equations Appl., vol. 4, Birkhäuser, Basel,, pp Receive xxxx xx; revise xxxx xx. aress: sylvie.monniaux@univ-cezanne.fr