On the very weak solution for the Oseen and Navier-Stokes equations

On the very weak solution for the Oseen and Navier-Stokes equations∗ Ch´erif Amrouche & Mar´ıa ´ Angeles Rodr´ıguez-Bellido Laboratoire de Math´ematiques Appliqu´ees, CNRS UMR 5142 Universit´e de Pau et des Pays de l’Adour IPRA, Avenue de l’Universit´e- 64000 Pau, France and Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. [email protected], [email protected] Abstract We study the existence of very weak solutions regularity for the Stokes, Oseen and Navier- Stokes system when non-smooth Dirichlet boundary data for the velocity are considered in domains of class C1,1. In the Navier-Stokes case, the results will be valid for external forces non necessarily small. Regularity results for more regular data will be also discussed. Keywords: Stokes equations, Oseen equations, Navier-Stokes equations, Very weak solutions, Stationary Solutions. AMS Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D05, 76D07 1 Introduction and notations In this work, we are interested in some questions concerning the Navier-Stokes equations, defined in Ωa bounded open set of R3with boundary Γ: (NS)           −∆u+u·∇u+∇q=fin Ω, ∇·u=hin Ω, u=gon Γ, ∗The second author has been partially supported by M.E.C. (Spain), Project MTM2006-07932, and by Junta de Andaluc´ıa, Project P06-FQM-02373. 1 where udenotes the velocity and qthe pressure and both are unknown. The external force f,the compressibility condition hand the boundary condition for the velocity gare given functions. The vector fields and matrix fields (and the corresponding spaces) defined over Ωor over R3are respectively denoted by boldface Roman and special Roman. In the homogeneous case, h= 0, it has been well-known since Leray [18] (see also [19]) that if f∈W−1,p(Ω) and g∈W1−1/p,p(Γ) with p≥2 and for any i=0,...,I, �Γi g·ndσ=0,(1.1) where Γidenote the connected components of the boundary Γof the open set Ω, then there exists a solution (u,q)∈W1,p(Ω)×Lp(Ω) satisfying (NS). In [25], Serre proved the existence of weak solution (u,q)∈W1,p(Ω)×Lp(Ω) for any 3 2<p<2whenh= 0 and gsatisfies the above conditions. More recently, Kim [17] improves Serre’s existence and regularity results on weak solutions of (NS) for any 3 2≤p<2 (including the case p=3 2), when the boundary of Ωis connected (I= 0) provided his small in an appropriate norm (due to the compatibility condition between hand g, then gis also small in the corresponding appropriate norm). On the other hand, the notion of very weak solutions (u,q)∈Lp(Ω)×W−1,p(Ω) for Stokes or Navier-Stokes equations, corresponding to very irregular data, has been developed in the last years by Giga [15] (in a domain Ωof class C∞), Amrouche & Girault [3] (in a domain Ωof class C1,1) and more recently by Galdi et al. [14], Farwig et al. [11] (in a domain Ωof class C2,1, see also Schumacher [24]). In this context, the boundary condition is chosen in Lp(Γ) (see Brown & Shen [7], Conca [9], Fabes et al. [10], Moussaoui & Zine [21], Shen [26], Savar´e [23], Marusiˇc-Paloka [20]) or more generally in W−1/p,p(Γ). The purpose of our work is to develop a unified theory of very weak solutions of the Dirichlet problem for Stokes, Oseen and Navier-Stokes equations (and also for the Laplace equation), see Theorem 4.10 and Theorem 5.4. One important question is to define rigorously the traces of the vector functions which are living in subspaces of Lp(Ω) (see Lemma 2.10 and Lemma 2.11). We prove existence and regularity of very weak solutions (u,q)∈Lp(Ω)×W−1,p(Ω) of Stokes and Oseen equations for any 1 <p<∞with arbitrary large data belonging to some Sobolev spaces of negative order. In the case of Navier-Stokes equations the existence of very weak solution is proved for arbitrary large external forces, but with a smallness condition for both hand g. Uniqueness of very weak solutions is also proved for small enough data. Existence of very weak solution u∈L3(Ω), for arbitrary large external forces f∈H−1(Ω), h= 0 and arbitrary large boundary condition g∈L2(Γ) and without assuming condition (1.1), was proved first by Marusiˇc-Paloka in [20] (see Theorem 5) with Ωa bounded simply-connected open set of class C1,1. But the proof of Theorem 5 becomes correct only if either condition (1.1) or condition (5.56) holds. The same result was proved by Kim [17] for arbitrary large external forces f∈[W1,3/2 0(Ω)∩W2,3(Ω)]�, for small h∈[W1,3/2(Ω)]�and g∈W−1/3,3(Γ) and 2 where the boundary of Ωis supposed connected (I= 0). Remark that the space chosen for the divergence condition his not correct, because D(Ω) is not dense in W−1/3,3(Γ) and his dual is not a subspace of distributions. Similar argument can be done for the space chosen for the external forces f. The origin of this mistake (also present everywhere in the same paper [17]) is due to the fact that when we want to solve a boundary value problem, it is necessary to have an adequate Green formula and corresponding density lemmas. In a close context, we also consider the case where the data, and then the solutions, belong to fractionary Sobolev spaces Ws,p(Ω) with sa real number possibly not integer (see Theorem 4.12) The work is organized as follows: In the remains of this section, we recall the definitions of some spaces and their respective norms. In §2, some preliminary results are stated, including density lemmas, general trace’s results, characterization of dual spaces and trace’s result for very weak solutions. In §3, we present Stokes’ results related to the very weak, weak and strong solution. Some of them generalized those appearing in [3] in order to be extended to the Oseen and Navier-Stokes systems. In §4, we extend the results of §3 for the Oseen system. The first two main results in this paper are presented here: one about existence and uniqueness of very weak solution for the Oseen equations in Lp(Ω)×W−1,p(Ω) with 1 <p<∞(see Theorem 4.10), and another one related to the regularity of solutions for the Oseen equations (see Theorem 4.12). We consider in particular the case where the external forces fand the divergence condition hare not regular, more precisely f∈Wσ−2,p(Ω) and h∈Wσ−1,p(Ω) with 1 p<σ≤2. In §5, existence of very weak solution for the Navier-Stokes system is obtained, using a fixed point technique over the Oseen system, first for the case of small data and then for arbitrary large external forces fbut sufficiently small hand gin a domain possibly multiply-connected. The results is stated in Theorem 5.4. Regularity results for this system are obtained in Theorem 5.5. The complete proofs of results can be seen in [5]. In all this work, if we do not say anything else, Ωwill be considered as a Lipschitz open bounded set of R3. When Ωis connected, we will say Ωis a domain. We will only specify the regularity of Ωwhen it to be different from the regularity presented above. 1.1 Functional framework In what follows, for any s∈R,pdenotes a real number such that 1 <p<∞and p�stands for its conjugate: 1/p +1/p�= 1. We shall denote by mthe integer part of sand by σits fractional part: s=m+σwith 0 ≤σ<1. We denote by Ws,p(R3) the space of all distributions vdefined in R3such that: •Dαv∈Lp(R3), for all |α|≤m,whens=mis a nonnegative integer 3 •v∈Wm,p(R3) and �R3×R3 |Dαv(x)−Dαv(y)|p |x−y|3+σpdxdy<∞, for all |α|=m,whens=m+σis nonnegative and is not an integer. The space Ws,p(R3) is a reflexive Banach space equipped by the norm: �v�Wm, p(R3)=�� |α|≤m�R3 |Dαv(x)|pdx�1/p in the first case, and by the norm �v�Ws, p(R3)=��v�p Wm,p(R3)+� |α|=m�R3×R3 |Dαv(x)−Dαv(y)|p |x−y|3+σpdxdy�1/p , in the second case. For s<0, we denote by Ws, p(R3) the dual space of W−s, p�(R3). In the special case of p= 2, we shall use the notation Hs(R3) instead of Ws, 2(R3). Now, we introduce the Sobolev space Hs,p(R3)={v∈Lp(R3); (I−∆)s/2v∈Lp(R3)}. It is known that Hs,p(R3)=Ws,p(R3)ifsis an integer or if p= 2. Furthermore, for any real number s, we have the following embeddings: Ws,p(R3)�→Hs,p(R3)ifp≤2 and Hs,p(R3)�→Ws,p(R3)ifp≥2. The definition of the space Ws,p(Ω) is exactly the same as in the case of the whole space. Because D(Ω) is not dense in Ws,p(Ω), the dual space of Ws,p(Ω) cannot be identified to a space of distributions in Ω. For this reason, we define Ws,p 0(Ω) as the closure of D(Ω) in Ws,p(Ω) and we denote by W−s, p�(Ω) its dual space. For every s>0, we denote by Ws,p(Ω) the space of all distributions in Ωwhich are restrictions of elements of Ws,p(R3) and by � Ws,p(Ω) the space of functions u∈Ws,p(Ω) such that the extension �uby zero outside of Ωbelongs to Ws,p(R3). 2 Preliminary results We present here some trace results, density results, De Rham’s theorems and characterizations of some spaces, either known or designed specially for the Stokes, Oseen and Navier-Stokes problems, that will be used in the following sections. Recall now some density results ([1, 16]): i) The space D(Ω) is dense in Ws,p(Ω) for any real s. 4 ii) The space D(R3) is dense in Ws,p(R3) and in Hs,p(R3) for any real s. iii) The space D(Ω) is dense in � Ws,p(Ω) for all s>0. iv) The space D(Ω) is dense in Ws,p(Ω) for all 0 <s≤1/p, that means that Ws,p(Ω) = Ws,p 0(Ω). Next result gives some properties of traces of functions living in Ws, p(Ω) ([1, 16]). Theorem 2.1 Let Ωbe a bounded open set of class Ck,1, for some integer k≥0. Let sbe real number such that s≤k+1,s−1/p =m+σ, where m≥0is an integer and 0<σ<1. i) The following mapping γ0:u�→ u|Γ Ws, p(Ω)→Ws−1/p, p(Γ) is continuous and surjective. When 1/p < s < 1+1/p, we have Ker γ0=Ws, p 0(Ω). ii) For m≥1,thefollowingmapping (γ0,γ 1):u�→ (u|Γ,∂u ∂n|Γ) Ws, p(Ω)→(Ws−1/p, p(Γ)×Ws−1−1/p, p(Γ)) is continuous and surjective. When 1+1/p < s < 2+1/p, we have Ker (γ0,γ 1)=Ws, p 0(Ω). We recall also the following embeddings: Ws, p(Ω)�→Wt, q(Ω) for t≤s, p ≤qsuch that s−3/p =t−3/q and Ws, p(Ω)�→C k, α(Ω) for k<s−3/p < k +1,α=s−k−3/p, where kis a non negative integer. Then, we introduce the following spaces: Dσ(Ω) = {ϕ∈D(Ω); ∇·ϕ=0},Dσ(Ω) = {ψ∈D(Ω)3;∇·ψ=0}. Recall now two versions of De Rham’s Theorem, the first one proved by G. de Rham [22] and the second by C. Amrouche & V. Girault [3]: Lemma 2.2 (De Rham’s Theorem for distributions) Let Ωbe any open subset of R3and let fbe a distribution of D�(Ω)that satisfies: ∀v∈Dσ(Ω),�f,v�=0. Then, there exists a distribution πin D�(Ω)such that f=∇π. 5 Lemma 2.3 (De Rham’s Theorem in W−m,p(Ω))Let mbe any integer, pany real number with 1<p<∞. Let f∈W−m,p(Ω)satisfy: ϕ∈Dσ(Ω),�f,ϕ�=0. Then, there exists π∈W−m+1,p(Ω)such that f=∇π. If in addition the set Ωis connected, then πis defined uniquely, up to an additive constant, and there exists a positive constant C, independent of f, such that: inf K∈R�π+K�W−m+1,p(Ω)/R≤C�f�W−m,p(Ω). The two next lemmas are density results: Lemma 2.4 The space Dσ(Ω)is dense in Hp(Ω) = {v∈Lp(Ω); ∇·v=0}. Proof.[Sketch of the proof] Let �be a linear and continuous mapping in Hp(Ω) such that ��,v�= 0 for any v∈Dσ(Ω). We want to prove that �=0. Since Hpis a subspace of Lp(Ω), we can extend �to L∈Lp�(Ω). We will suppose that Ωis bounded, connected but eventually multiply-connected (when Ωis not connected, we can repeat the procedure above in each connected component of Ω), being �1≤i≤Iωiits wholes, and its boundary Γis Lipschitzcontinuous. We denote by ω0the exterior of Ω, by Γ0the exterior boundary of Ωand by Γi, 1≤i≤I, the other components of Γ. The duality between W−1/p,p�(Γi) and W1/p,p(Γi), and W−1/p,p�(Γ0) and W1/p,p(Γ0), will be denoted by �·,·�Γiand �·,·�Γ0, respectively. By De Rham’s Lemma 2.3, there exists a unique q∈W1,p�(Ω)∩Lp� 0(Ω) such that L=∇qand Lp� 0(Ω) = �ϕ∈Lp�(Ω); �Ω ϕ(x)dx=0 �. Moreover, ∀v∈Dσ(Ω),��,v�=�q,v·n�Γ=0. We extend Lby zero out of Ωand denote the extension by � L. Then, for any ϕ∈D(R3)such that ∇·ϕ=0inR3,�R3� L·ϕdx=�Ω L·ϕdx=0. From that, we deduce that, thanks to De Rham’s Lemma 2.2, there exists h∈D �(R3) verifying ∇h∈Lp�(R3) such that � L=∇h(see Lemma 2.1 in [4]). It is clear that h∈W1,p� loc (R3). As h is unique up to an additive constant and ∇h=0inω0, we can choose this constant in such a way that h=0inω0. Therefore, we deduce that: h=0 inω0,h=ciin each ωi,h= q+c0in Ω, and thus: q=−c0on Γ0,q=ci−c0on Γi,1≤i≤I. 6 Let j∈{1,...,I}be a fixed index, choosing vj∈Dσ(Ω) such that �vj·n,1�Γk=δjk for 1≤k≤Iand �vj·n,1�Γ0=−1, we can deduce that cj= 0. In consequence, for every v∈Hp(Ω), we have: ��,v�=�Ω ∇h·vdx=0. Thus, we deduce that �=0in H� p(Ω). In the sequel, we will use the following space Xr,p(Ω) = {ϕ∈W1,r 0(Ω); ∇·ϕ∈W1,p 0(Ω)},1<r,p<∞,(2.2) and we set Xp,p(Ω) = Xp(Ω). Observe that the space Xp,p(Ω) were used in [3] in order to define very weak solution for the Stokes problem. In the case of Navier-Stokes problem, the generalization to the space Xr,p(Ω) is necessary. In this sense, the proof of the next result follows from an argument appearing in [2]. Lemma 2.5 The space D(Ω)is dense in Xr,p(Ω)and for all q∈W−1,p(Ω)and ϕ∈Xr�,p�(Ω), we have �∇q, ϕ�[Xr�,p�(Ω)]�×Xr�,p�(Ω)=−�q, ∇·ϕ�W−1,p(Ω)×W1,p� 0(Ω).(2.3) Next lemmas characterize the space (Xr,p(Ω))�and give a density result. Lemma 2.6 Let f∈(Xr,p(Ω))�. Then, there exist F0=(fij)1≤i,j≤3such that F0∈Lr�(Ω)and f1∈W−1,p�(Ω)such that f=∇·F0+∇f1.(2.4) Moreover, �f�[Xr,p(Ω)]�= max{�fij�Lr�(Ω),1≤i, j ≤3,�f1�W−1,p�(Ω)}. Conversely, if fsatisfies (2.4), then f∈(Xr,p(Ω))�. As a consequence of Lemma 2.5, we have the following embeddings: W−1,r(Ω)�→(Xr�,p�(Ω))��→W−2,p(Ω),(2.5) where the second embedding holds if 1 r≤1 p+1 3. Lemma 2.7 Let Ωbe a Lipschitz bounded open set. Then, the space D(Ω)is dense in (Xr,p(Ω))�. One of the main difficulties for the definition of a very weak solution for Stokes, Oseen and Navier-Stokes problems is to give a meaning to the trace, because we are not in the classical variational framework. We shall use the spaces1: Tp,r(Ω) = {v∈Lp(Ω); ∆v∈(Xr�,p�(Ω))�},Tp,r,σ(Ω) = {v∈Tp,r(Ω); ∇·v=0}, 1When p=r, these spaces are denoted as Tp(Ω) and Tp,σ(Ω), respectively. Observe that these spaces were introduced in [2, 3] 7 endowed with the topology given by the norm: �v�Tp,r(Ω)=�v�Lp(Ω)+�∆v�[Xr�,p�(Ω)]�, and Hp,r(div; Ω) = {v∈Lp(Ω); ∇·v∈Lr(Ω)}, which is equipped with the graph norm. Next density lemmas will be necessary: Lemma 2.8 i) The space D(Ω)is dense in Tp,r(Ω). ii) The space D(Ω)is dense in Tp,r(Ω)∩Hp,r(div; Ω). Lemma 2.9 The space Dσ(Ω)is dense in Tp,r,σ(Ω). For the following two lemmas, we will need to introduce the space: Yp�(Ω) = {ψ∈W2,p�(Ω); ψ|Γ=0,(∇·ψ)|Γ=0} that can also be described (see [3]) as: Yp�(Ω) = {ψ∈W2,p�(Ω); ψ|Γ=0,∂ψ ∂n·n���Γ=0}.(2.6) Observe that the range space of the normal derivative γ1:Yp�(Ω)→W1/p,p�(Γ) is Zp�(Γ) = {z∈W1/p,p�(Γ); z·n=0}. In these lemmas, we prove that the tangential trace of functions vof Tp,r,σ(Ω) belongs to the dual space of Zp�(Γ), which is: (Zp�(Γ))�={µ∈W−1/p,p(Γ); µ·n=0}.(2.7) Recall that we can decompose vinto its tangential, vτ, and normal parts, that is: v=vτ+(v· n)n. Lemma 2.10 Let Ωbe a bounded open set of R3of class C1,1. Let 1<p<∞and r>1be such that 1 r≤1 p+1 3.Themappingγτ:v�→ vτ|Γon the space D(Ω)can be extended by continuity to a linear and continuous mapping, still denoted by γτ,fromTp,r(Ω)into W−1/p,p(Γ). The Green formula reads: for any v∈Tp,r(Ω)and ψ∈Yp�(Ω), �∆v,ψ�[Xr�,p�(Ω)]�×Xr�,p�(Ω)=�Ω v·∆ψdx−�vτ,∂ψ ∂n�W−1/p,p(Γ)×W1/p,p�(Γ) . Lemma 2.11 i) The space D(Ω)is dense in Hp,r(div; Ω). 8 ii) Let 1<p<∞and r>1such that 1 r≤1 p+1 3.Themappingγn:v�→ v·n|Γon the space D(Ω)can be extended by continuity to a linear and continuous mapping, still denoted by γn,fromHp,r(div; Ω)into W−1/p,p(Γ), and we have the Green formula: for any v∈Hp,r(div; Ω)and ϕ∈W1,p�(Ω), �Ω v·∇ϕdx+�Ω ϕdiv vdx=�v·n,ϕ�W−1/p,p(Γ)×W1/p,p�(Γ). Lemma 2.12 Let Ωbe a Lipschitz bounded open set. Let h∈Lr(Ω)and g∈W−1/p,p(Γ)be given such that the condition (3.11) holds. For every ε>0, there exist sequences (hε)⊂D(Ω) and (gε)⊂C∞(Γ)such that �Ω hε(x)dx=�Γ gε·ndσ(2.8) and verifying �h−hε�Lr(Ω)≤εand �g−gε�W−1/p,p(Γ)≤ε(2.9) �hε�Lr(Ω)+ i=I � i=0 |�gε·n,1�Γi|≤2��h�Lr(Ω)+ i=I � i=0 |�g·n,1�Γi|�.(2.10) In all the rest of this work , if we do not say anything else, we assume that Ωis a bounded connected open set of class C1,1. 3 The Stokes problem Before starting the study of the Oseen and Navier-Stokes problems, we focus on the study of the Stokes problem in order to make an appointment about all the knowing results about this system. Recall that the Stokes problem is: (S)−∆u+∇q=fand ∇·u=hin Ω,u=gon Γ, with the compatibility condition: �Ω h(x)dx=�g·n,1�W−1/p,p(Γ)×W1/p,p�(Γ).(3.11) Basic results on weak and strong solutions of problem (S) in Lp(Ω) Sobolev spaces may be summarized in the following theorem (see [3], [8], [12]). Theorem 3.1 i) For every f,h,gwith f∈W−1,p(Ω),h∈Lp(Ω),g∈W1−1/p,p(Γ),and satisfying the compatibility condition (3.11), the Stokes problem (S)has exactly one solution u∈W1,p(Ω)and q∈Lp(Ω)/R. Moreover, there exists a constant C>0depending only on pand Ωsuch that: �u�W1,p(Ω)+�q�Lp(Ω)/R≤C(�f�W−1,p(Ω)+�h�Lp(Ω)+�g�W1−1/p,p(Γ)).(3.12) 9 with �q�L2(Ω)/R≤C�∇q�H−1(Ω). Finally, estimate (4.21) follows from the previous equation and estimate for z. As a consequence of Theorem 4.1, Theorem 3.1 and the inequality �v·∇u�L6/5(Ω)≤�v�L3(Ω)�∇u�L2(Ω), we can deduce the following result: Corollary 4.2 Let us assume f∈L6/5(Ω),v∈H3(Ω),h∈W1,6/5(Ω) and g∈W7/6,6/5(Γ) be given verifying the compatibility condition (3.11). Then, the solution (u,q)given by Theorem 4.1 belongs to W2,6/5(Ω)×W1,6/5(Ω)and verifies the following estimate: �u�W2,6/5(Ω)+�q�W1,6/5(Ω)/R ≤C�1+�v�L3(Ω)���f�L6/5(Ω)+�1+�v�L3(Ω)���h�W1,6/5(Ω)+�g�W7/6,6/5(Γ)�� Theorem 4.3 (Strong regularity for p≥6/5)Let p≥6 5, f∈Lp(Ω),h∈W1,p(Ω),v∈Hs(Ω) and g∈W2−1/p,p(Γ), be given with s=3 if p<3,s=pif p>3,s=3+εif p=3,(4.23) for some arbitrary ε>0, and satisfying the compatibility condition: �Ω h(x)dx=�Γ g·ndσ. Then, the unique solution of (O) given by Theorem 4.1 verifies (u,q)∈W2,p(Ω)×W1,p(Ω). Moreover, there exists a constant C>0such that: �u�W2,p(Ω)+�q�W1,p(Ω)/R≤C�1+�v�Ls(Ω)�× ×��f�Lp(Ω)+�1+�v�Ls(Ω)���h�W1,p(Ω)+�g�W2−1/p,p(Γ)�� (4.24) Proof. First, by Corollary 4.2, we can suppose p≥6/5 and then we have the following embeddings: Lp(Ω)�→H−1(Ω),W 1,p(Ω)�→L2(Ω),and W2−1/p,p(Γ)�→H1/2(Γ). Thanks to the regularity of fand Theorem 4.1 there exists a unique solution (u,q)∈H1(Ω)× L2(Ω)/Rverifying the following estimates: �u�H1(Ω)≤C��f�H−1(Ω)+�1+�v�L3(Ω)�(�h�L2(Ω)+�g�H1/2(Γ))�(4.25) 16 and �u�H1(Ω)+�q�L2(Ω)/R≤C�1+�v�L3(Ω)�× ×��f�H−1(Ω)+�1+�v�L3(Ω)�(�h�L2(Ω)+�g�H1/2(Γ))�.(4.26) Observe that, apriori, the regularity for the Oseen problem cannot be deduced from the Stokes one. This follows from the fact that v·∇u=∇·(v⊗u)∈H−1(Ω). In order to obtain the strong solution in W2,p(Ω)×W1,p(Ω), first we apply Lemma 2.4 to function v, and we take for any λ>0,vλas the velocity of the convection term, where vλ∈D(Ω) such that ∇·vλ= 0 and �vλ−v�Ls(Ω)≤λ. Therefore, we search for (uλ,q λ)∈ W2,p(Ω)×W1,p(Ω) solution of the problem: (Oλ)           −∆uλ−vλ·∇uλ+∇qλ=fin Ω, ∇·uλ=hin Ω, uλ=gon Γ. From above we can obtain a unique solution (uλ,q λ) bounded in H1 0(Ω)×L2(Ω)/Rindependently from λ. Then, we obtain again estimates (4.25) and (4.26). As vλ·∇uλ∈L2(Ω), if fand hare regular enough, then using the Stokes regularity we deduce that (uλ,q λ)∈H2(Ω)×H1(Ω) if 2≤pand (uλ,q λ)∈W2,p(Ω)×W1,p(Ω) if 6/5<p≤2. A bootstrap argument moreover shows that (uλ,q λ)∈W2,p(Ω)×W1,p(Ω) if 2 <p. Thus, we focus on the getting of a strong estimate for (uλ,q λ). Let ε>0with0<λ<ε/2. We consider vλ=vε 1+vε λ,2where vε 1=� v�ρ ε/2,and vε λ,2=vλ−� v�ρ ε/2.(4.27) where � vis the extension of vby zero to R3and ρεis the classical mollifier. By regularity estimates for the Stokes problem, we have �uλ�W2,p(Ω)+�qλ�W1,p(Ω)/R≤C(�f�Lp(Ω) +�h�W1,p(Ω)+�g�W2−1/p,p(Γ)+�vλ·∇uλ�Lp(Ω)). (4.28) Now, we use the decomposition (4.27) in order to bound the term �vλ·∇uλ�Lp(Ω). We observe first that �vε λ,2�Ls(Ω)≤�vλ−v�Ls(Ω)+�v−� v�ρ ε/2�Ls(Ω)≤λ+ε/2<ε. Recall that W2,p(Ω)�→W1,k(Ω) (4.29) for any k∈[1,p∗], with 1 p∗=1 p−1 3,ifp<3, for any k≥1ifp= 3 and for any k∈[1,∞]if p>3. Moreover the embedding W2,p(Ω)�→W1,q(Ω) (4.30) 17 is compact for any q∈[1,p∗[ifp<3, for any q∈[1,∞[ifp= 3 and for q∈[1,∞]ifp>3. Then, using the H¨older inequality and the Sobolev embedding, we obtain �vε λ,2·∇uλ�Lp(Ω)≤�vε λ,2�Ls(Ω)�∇uλ�Lk(Ω)≤Cε�uλ�W2,p(Ω)(4.31) where 1 k=1 p−1 s, which is well defined (see the defintion of the real number s). For the second estimate, we consider two cases. i) Case p≤2. Let r∈]3,∞] be such that 1 p=1 r+1 2and t≥1 such that 1 + 1 r=1 3+1 tsatisfy �vε 1·∇uλ�Lp(Ω)≤�vε 1�Lr(Ω)�∇uλ�L2(Ω) ≤�v�L3(Ω)�ρε/2�Lt(R3)�∇uλ�L2(Ω). Using the estimate (4.25), we have �vε 1·∇uλ�Lp(Ω)≤Cε�v�L3(Ω)��f�Lp(Ω)+�1+�v�L3(Ω)� ×(�h�W1,p(Ω)+�g�W2−1/p,p(Γ))�. (4.32) From (4.32) and (4.31), we deduce that �uλ�W2,p(Ω)+�qλ�W1,p(Ω)/R≤C�1+�v�L3(Ω)�× ×��f�Lp(Ω)+�1+�v�L3(Ω)�(�h�W1,p(Ω)+�g�W2−1/p,p(Γ))�(4.33) ii) Case p>2. First, we choose the exponent qgiven in (4.30) such that q>2. For any ε�,we known that there exists Cε�>0 such that �∇uλ�Lq(Ω)≤ε��uλ�W2,p(Ω)+Cε��uλ�H1(Ω). Let first consider p<3 and choose q<p∗and close of p∗. Then, there exist r>3 such that 1 p=1 r+1 qand t>1 such that 1 + 1 r=1 3+1 tsatisfying �vε 1·∇uλ�Lp(Ω)≤�vε 1�Lr(Ω)�∇uλ�Lq(Ω) ≤�v�L3(Ω)�ρε/2�Lt(R3)�∇uλ�Lq(Ω). If p≥3, �vε 1·∇uλ�Lp(Ω)≤�vε 1�Ls(Ω)�∇uλ�Lq(Ω) ≤�v�Ls(Ω)�ρε/2�L1(R3)�∇uλ�Lq(Ω), where we choose q=∞if p>3 and qlarge enough if p= 3. In the both cases, in order to control the first term on the right hand side of (4.28) with the term on the left hand side, we fix εand ε�small enough to obtain �uλ�W2,p(Ω)+�qλ�W1,p(Ω)/R≤C��f�Lp(Ω)+�h�W1,p(Ω)+�g�W2−1/p,p(Γ) +Cε��v�Ls(Ω)�ρε/2�Lt(Ω) ×��f�Lp(Ω)+�1+�v�Ls(Ω)�(�h�W1,p(Ω)+�g�W2−1/p,p(Γ))�. (4.34) 18 Thus, we deduce that (uλ,q λ) satisfies (4.33), where we replace �v�L3by �v�Ls. The estimate (4.33) is uniform with respect to λ, and therefore we can extract subsequences, that we still call {uλ}λand {qλ}λ, such that if λ→0, uλ−→ uweakly in W2,p(Ω), and for the pressure, there exists a sequence of real numbers kλsuch that qλ+kλ→qweakly in W1,p(Ω). It is easy to verify that (u,q) is solution of (O) satisfying estimate (4.24) and this solution is unique. Thanks to the strong regularity, we can deduce the following regularity: Theorem 4.4 (Regularity in W1,p(Ω)×Lp(Ω),p>1)Let us consider f∈W−1,p(Ω),v∈H3(Ω),h∈Lp(Ω) and g∈W1−1/p,p(Γ) be given verifying the compatibility condition (3.11). Then, the problem (O)has a unique solution (u,q)∈W1,p(Ω)×Lp(Ω)/R. Moreover, there exists some constant C>0such that: i) if p≥2, then �u�W1,p(Ω)+�q�Lp(Ω)/R≤ ≤C�1+�v�L3(Ω)�2��f�W−1,p(Ω)+�h�Lp(Ω)+�g�W1−1/p,p(Γ)�(4.35) holds, ii) if p<2, then �u�W1,p(Ω)+�q�Lp(Ω)/R≤C�1+�v�L3(Ω)�2× ×��f�W−1,p(Ω)+�1+�v�L3(Ω)��h�Lp(Ω)+�g�W1−1/p,p(Γ)�(4.36) holds. Proof.[Sketch of the proof] i) First case: p≥2. Let (u0,q 0)∈W1,p(Ω)×Lp(Ω) be the solution of: −∆u0+∇q0=fand ∇·u0=hin Ω,u0=gon Γ. verifying the estimate: �u0�W1,p(Ω)+�q0�Lp(Ω)/R≤C��f�W−1,p(Ω)+�h�Lp(Ω)+�g�W1−1/p,p(Γ)�(4.37) 19 and (z,θ)∈W2,t(Ω)×W1,t(Ω) verifying: −∆z+v·∇z+∇θ=−v·∇u0and ∇·z=0 inΩ,z=0on Γ, with 1 t=1 3+1 pand satisfying the estimate �z�W2,t(Ω)+�θ�W1,t(Ω)/R≤C�1+�v�L3(Ω)��v·∇u0�Lt(Ω) ≤C�1+�v�L3(Ω)��v�L3(Ω)(�f�W−1,p(Ω)+�h�Lp(Ω)+�g�W1−1/p,p(Γ)). (4.38) Here, we have applied Theorem 4.3 because of v·∇u0∈Lt(Ω). Observe that 6 5≤t<3, if and only if p≥2. Thanks to the embedding W2,t(Ω)�→W1,p(Ω), the pair (u,q)=(z+u0,θ+q0)∈W1,p(Ω)× Lp(Ω) verifies the problem (O). Estimate (4.35) follows from (4.37) and (4.38). ii) Second case: p<2. We use duality argument. Using quickly the reasoning given in Theorem 4.3, we can improve estimates (4.35) and (4.36) for some values of p: Proposition 4.5 Under the assumptions of Theorem 4.4 and supposing that 6 5≤p≤6,the solution (u,q)satisfies the estimate: �u�W1,p(Ω)+�q�Lp(Ω)/R≤C�1+�v�L3(Ω)�× ×��f�W−1,p(Ω)+�1+�v�L3(Ω)���h�Lp(Ω)+�g�W1−1/p,p(Γ)�� (4.39) Moreover assuming v·n=0on Γ, then the estimate (4.39) holds for any 1<p<∞. Remark 4.6 If we suppose that v∈Hp(Ω), then estimate (4.39), where we replace the norm �v�L3(Ω)by �v�Lp(Ω), holds when p>6(and then also, by duality argument, when p<6/5and v∈Hp�(Ω)). Corollary 4.7 (Strong regularity for 1<p<6/5)Let 1<p<6/5and let us f∈Lp(Ω),v∈H3(Ω),h∈W1,p(Ω) and g∈W2−1/p,p(Γ) be satisfied the compatibility condition (3.11). Then, the solution given by Theorem 4.4 satisfies (u,q)∈W2,p(Ω)×W1,p(Ω)and the estimate �u�W2,p(Ω)+�q�W1,p(Ω)/R≤C�1+�v�L3(Ω)� ×��f�Lp(Ω)+�1+�v�L3(Ω)���h�W1,p(Ω)+�g�W2−1/p,p(Γ)�� (4.40) holds. 20 Proof.[Sketch of the proof] Let us consider 1 <p<6/5 and (u,q)∈W1,p(Ω)×Lp(Ω) be the solution given by Theorem 4.4. Then Lp(Ω)�→W−1,r(Ω),W 1,p(Ω)�→Lr(Ω),W2−1/p,p(Γ)�→W1−1/r,(Γ) where r∈]3 2,2[ satisfies 1 r=1 p−1 3. From Theorem 4.4, we deduce that (u,q)∈W1,r(Ω)×Lr(Ω) and then v·∇u∈Lp(Ω). By Stokes regularity let us to conclude that (u,q)∈W2,p(Ω)× W1,p(Ω). To obtain estimate (4.40), we proceed similarly to the proof of Theorem 4.3. We can summarize Theorem 4.3 and Corollary 4.7 by the following theorem: Theorem 4.8 (Strong regularity) Let f,h,gbe such that f∈Lp(Ω),h∈W1,p(Ω) and g∈W2−1/p,p(Γ) verifying the compatibility condition (3.11) and v∈Hs(Ω)be with sdefined by (4.23). Then, the solution given by Theorem 4.4 satisfies (u,q)∈W2,p(Ω)×W1,p(Ω)and satisfies estimate (4.24). The concepts of weak and strong solutions are known for the Oseen equations. Now, we define and prove the existence of a very weak solution for the Oseen equations. Definition 4.9 (Very weak solution for the Oseen problem) Let f,h,gbe given satisfying (3.14) and (3.11) and v∈Hs(Ω)for sas (4.42). We say that (u,q)∈Lp(Ω)×W−1,p(Ω) is a very weak solution of (O)if the following equalities hold: For any ϕ∈Yp�(Ω)and π∈W1,p�(Ω), �Ω u·(−∆ϕ−v·∇ϕ)dx−�q,∇·ϕ�W−1,p(Ω)×W1,p� 0(Ω) =�f,ϕ�Ω−�gτ,∂ϕ ∂n�Γ, �Ω u·∇πdx=−�Ω hπdx+�g·n,π�Γ, (4.41) where the dualities on Ωand Γare defined by (3.16). As for the Stokes problem, the previous duality have sense. Moreover, note that W1,p�(Ω)�→ Lp�∗(Ω) and then the integral �Ω u·(v·∇)ϕdxis well defined. Theorem 4.10 (Very weak solution for Oseen equations) Let us f,h,gsatisfy (3.11), f∈(Xr�,p�(Ω))�,h∈Lr(Ω),g∈W−1/p,p(Γ),with 1 r=1 p+1 s 21 and v∈Hs(Ω)with s=3 if p>3/2,s=p�if p<3/2,s=3+εif p=3/2.(4.42) Then, the Oseen problem (O)has a unique solution (u,q)∈Tp,r(Ω)×W−1,p(Ω)/Rverifying the following estimates: �u�Tp,r(Ω)≤C�1+�v�Ls(Ω)���f�[Xr�,p�(Ω)]�+�h�Lr(Ω)+�g�W−1/p,p(Γ)�,(4.43) �q�W−1,p(Ω)/R≤C�1+�v�Ls(Ω)�2 ×��f�[Xr�,p�(Ω)]�+�h�Lr(Ω)+�g�W−1/p,p(Γ)�.(4.44) Proof. First, we shall prove that if the pair (u,q)∈Lp(Ω)×W−1,p(Ω)/Rsatisfies the two first equations of (O), then ubelongs to Tp,r(Ω) and thus the boundary condition u=gon Γmakes sense. Hence, if a pair (u,q)∈Lp(Ω)×W−1,p(Ω) satisfies the two first equations of (O), because of v∈Hs(Ω) with ∇·v= 0 and thanks (again) to Lemma 2.6, then ∆u= ∇·(v⊗u)+∇q−f∈(Xr�,p�(Ω))�. Therefore, u∈Tp,r,σ(Ω) and its tangential trace belongs to W−1/p,p(Γ). Moreover, as u∈Lp(Ω) and ∇·u∈Lr(Ω), then u·n|Γ∈W−1/p,p(Γ), and the whole trace u|Γ∈W−1/p,p(Γ) can be identified with u|Γ=g. It suffices to consider the case where g·n|Γ= 0 and �Ω h(x)dx= 0, the general case is similar to the proof given in the end of Proposition 3.4. The result can be deduced (see [5]) applying the Riesz’s Lemma. Similarly to Corollary 3.9, we can prove: Corollary 4.11 i) Let σbe a real number such that 0<σ<1. Let f=∇·F0+∇f1,hand gsatisfy the compatibility condition (3.11) with F0∈Wσ,r(Ω),f 1∈Wσ−1,p(Ω),g∈Wσ−1/p,p(Γ),h∈Wσ,r(Ω), with 1 r=1 p+1 sand r≤p. Let us consider v∈Hs(Ω)with s=3 if p>3/2,s=p�if p<3/2,s=3+εif p=3/2. Then, the Oseen problem (O) has a unique solution (u,q)belonging to Wσ,p(Ω)×Wσ−1,p(Ω)/Rand satisfying the estimate �u�Wσ,p(Ω)+�q�Wσ−1/p,p(Ω)/R≤C(1 + �v�Ls(Ω)) ×��F0�Wσ,r(Ω)+�f1�Wσ−1,p(Ω)+(1+�v�Ls(Ω))(�h�Wσ,r (Ω)+�g�Wσ−1/p,p(Γ))�. 22 ii) If moreover F0,f 1,g,h satisfy that F0∈Wσ+1,r(Ω),f 1∈Wσ,p(Ω),g∈Wσ+1−1/p,p(Γ),h∈Wσ+1,r(Ω), with 1 r≤1 p+1 sand v∈Hs(Ω), where s=3 if p<3,s=pif p>3,s=3+εif p=3, then (u,q)∈Wσ+1,p 0(Ω)×Wσ,p(Ω)and satisfies �u�Wσ+1,p(Ω)+�q�Wσ,p(Ω)/R ≤C(1 + �v�Ls(Ω))×��F0�Wσ+1,r(Ω)+�f1�Wσ,p(Ω) +(1+�v�Ls(Ω))(�h�Wσ+1,r(Ω)+�g�Wσ+1−1/p,p(Γ))�. Theorem 4.12 (Regularity for Oseen equations) Let σbe a real number such that 1 p< σ≤2. Let f,hand gsatisfy the compatibility condition (3.11) with f∈Wσ−2,p(Ω),h∈Wσ−1,p(Ω),g∈Wσ−1/p,p(Γ). Let v∈Hs(Ω)satisfy (4.42). Then, the Oseen problem (O) has exactly one solution (u,q)∈ Wσ,p(Ω)×Wσ−1,p(Ω)/Rsatisfying the estimate �u�Wσ,p(Ω)+�q�Wσ−1,p(Ω)/R≤C(�f�Wσ−2,p(Ω))+�h�Wσ−1,p(Ω)+�g�Wσ−1/p,p(Ω)). Proof. The proof is similar to proof of Theorem 3.11. It suffices to study the new term containing the function v. Remark 4.13 i) When f∈W1/p−2,p(Ω), we can conjecture that u/∈W1/p,p(Ω). ii) If 1/p < σ<1,f∈Wσ−2,p(Ω),g∈Wσ−1/p,p(Γ), then the solution (u,q)of (O) belongs to Wσ,p(Ω)×Wσ−1,p(Ω). This assumptions are weaker than those of Corollary 4.11 point i). Moreover, they are optimal for this case. iii) If 0≤σ≤1/p, Theorem 4.12 cannot be applied. Indeed, the trace mapping is not continuous (and not surjective) from Wσ,p(Ω)into Wσ−1/p,p(Γ). If we like to solve Problem (O) with boundary condition g∈Wσ−1/p,p(Γ), it is necessary to suppose that fand hare more regular, precisely we must assume f=∇·F0+∇f1with F0∈Wσ,r(Ω),f 1∈Wσ−1,p(Ω), and h∈Wσ,r(Ω), where 1 r≤1 p+1 3and r≤p. The solution is then obtained by Corollary 4.11 point i). 23 5 The Navier-Stokes problem First of all, we give the definition of a very weak solution for the Navier-Stokes equations. Definition 5.1 (Very weak solution for the Navier-Stokes problem) Let f∈(Xr�,p�(Ω))�, h∈Lr(Ω)and g∈W−1/p,p(Γ)satisfy the compatibility condition (3.11).Wesaythat(u,q)∈ Lp(Ω)×W−1,p(Ω)is a very weak solution of (NS)if the following equalities hold: For any ϕ∈Yp�(Ω)and π∈W1,p�(Ω), �Ω u·(−∆ϕ−u·∇ϕ)dx−�q,∇·ϕ�W−1,p(Ω)×W1,p� 0(Ω) =�f,ϕ�Ω−�gτ,∂ϕ ∂n�Γ, �Ω u·∇πdx=−�Ω hπdx+�(g·n),π�Γ, (5.45) where the dualities on Ωand Γare defined in (3.16). In the stationary Navier-Stokes equations, the data hand gplay an special role, making possible or not the existence of a very weak solution. If hand gare small enough, then the result is true. Until we now, we think that it is not possible to eliminate this latest condition. Therefore, we present first three results related to the existence of very weak solution: the two first for the small external forces case (following the scheme used by Marusiˇc-Paloka [20]) and the third one for the general Navier-Stokes case, always supposing that hand gare small enough in their respective norms. Last result involves the regularity for the Navier-Stokes equations. Theorem 5.2 (Very weak solution for Navier-Stokes, small data case) Let us consider f∈(X3,3/2(Ω))�,h∈L3/2(Ω)and g∈W−1/3,3(Γ)be given verifying (3.11). i) There exists a constant α1>0such that, if �f�[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ)≤α1,(5.46) then, there exists a very weak solution (u,q)∈L3(Ω)×W−1,3(Ω)to the problem (NS) verifying the following estimates: �u�L3(Ω)≤C��f�[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ)�(5.47) �q�W−1,3/R≤C1�f�[X3,3/2)]� + 2(1 + C2)C��f�[X3,3/2]�+�h�L3/2+�g�W−1/3,3�(5.48) where C>0is the constant given by (4.43),α1=min�(2C)−1,(2C2)−1�,C1and C2are constants of Sobolev embeddings. 24 ii) Moreover there exists a constant α2∈]0,α 1]such that this solution is unique, up to an additive constant for q,if �f�[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ)≤α2.(5.49) Proof. i) Existence. The existence of a very weak solution is made through the application of the Banach’s fixed point theorem. We do this fixed point over the Oseen equations, written in an adequate manner. We are searching for a fixed point for the application T,    T:H3(Ω)→H3(Ω) v�→ Tv=u (5.50) where given v∈H3(Ω), Tv=uis the unique solution of (O) given by Theorem 4.10. We also need to define a neighborhood Br, in the form: Br={v∈H3(Ω); �v�L3(Ω)≤r}.(5.51) In order to prove the contraction of the operator, we must prove that: there exists θ∈]0,1[ such that �Tv1−Tv2�L3(Ω)=�u1−u2�L3(Ω)≤θ�v1−v2�L3(Ω).(5.52) Searching for an estimate of �u1−u2�L3(Ω), we observe that for each i=1,2, we have −∆ui+vi·∇ui+∇qi=fin Ω, ∇·ui=hin Ω, ui=gon Γ, with the estimates �ui�L3(Ω)≤C�1+�vi�L3(Ω)� ×��f�[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ)�,(5.53) where C>0 is the constant given by (4.43). Moreover, for estimating the difference u1−u2, we look for the problem verified by (u,q)=(u1−u2,q 1−q2), which is: −∆u+v1·∇u+∇q=−v·∇u2and ∇·u=0 inΩ,u=0on Γ, where u1=Tv1,u2=Tv2and v=v1−v2. Using the very weak estimates (4.43) made for the Oseen problem successively for uand for u2, we obtain that: �u�L3(Ω)≤C�1+�v1�L3(Ω)��(v·∇)u2�[X3,3/2(Ω)]� ≤C2β�1+�v1�L3(Ω)��1+�v2�L3(Ω)��v�L3(Ω), 25 [13] (MR1284206) G. P. Galdi, “An Introduction to the Matematical Theory of the Navier- Stokes Equations,” Vol 2: Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, vol. 39, Springer, New York, 1994. [14] (MR2107439) G. P. Galdi, C. G. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W−1/q,q,Math. Ann., 331 (2005), 41–74. [15] (MR635201) Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lp-spaces, Math. Z., 178 (1981), 287–329. [16] (MR775683) P. Grisvard, “Elliptic Problems in Nonsmooth Domains,” Pitman, Boston, 1985. [17] H. Kim, Existence and regularity of very weak solutions of the stationary Navier-Stokes equations, Arch. Rational Mech. Anal., 193 (2009), 117–152. [18] J. Leray, ´ Etude de diverses ´equations int´egrales non lin´eaires et de quelques probl`emes que pose l’hydrodynamique, J. Math. Pures Appl., 12(1933), 1–82. [19] (MR0259693) J. L. Lions, “Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin´eaires,” Dunod, Paris, 1969. [20] (MR1739399) E. Marusiˇc-Paloka, Solvability of the Navier-Stokes system with L2boundary data, Appl. Math. Optim., 41 (2000), 365–375. [21] (MR1663460) M. Moussaoui and A. M. Zine, Existence and regularity results for the Stokes system with non-smooth boundary data in a polygon, Math. Mod. Meth. Appl. Sc., 8(1998), 1307–1315. [22] (MR0068889) G. de Rham, “Vari´et´es Diff´erentiables,” Hermann, Paris, 1960. [23] (MR1600081) G. Savar´e, Regularity results for elliptic equations in Lipschitz domains,J. Funct. Anal., 152 (1998), 176–201. [24] (MR2403378) K. Schumacher, Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces, Ann. Univ. Ferrara Sez. VII Sci. Mat., 54 (2008), 123–144. [25] (MR753154) D. Serre, ´ Equations de Navier-Stokes stationnaires avec donn´ees peu r´eguli`eres, Ann. Sc. Norm. Sup. Pisa, 10 (1983), 543–559. [26] (MR1223521) Z. Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proc. AMS, 123 (1995), 801–811. 32