Very weak solutions for the stationary Stokes equations

Very weak solutions for the stationary Stokes equations. Ch´erif Amrouche a, Mar´ıa ´ Angeles Rodr´ıguez-Bellido b,1 aLaboratoire 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) bDpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. de Correos 1160 - 41080 Sevilla (Spain) Received *****; accepted after revision +++++ Presented by Abstract The concept of very weak solution introduced by Giga [9] for the stationary Stokes equations has been intensively studied in the last years for the stationary Navier-Stokes equations. We give here a new and simpler proof of the existence of very weak solution for the stationary Navier-Stokes equations, based on density arguments and an adequate functional framework in order to define more rigourously the traces of non regular vector fields. We also obtain regularity results in fractional Sobolev spaces. All these results are obtained in the case of a bounded open set, connected of class C1,1of R3and can be extended to the Laplace’s equation and to other dimensions. To cite this article: C. Amrouche, M.A. Rodr´ıguez-Bellido, C. R. Acad. Sci. Paris, Ser. I 340 (2005). R´esum´e Solutions tr`es faibles pour les ´equations stationnaires de Stokes. Le concept de solution tr`es faible introduit par Giga [9] pour les ´equations stationnaires de Stokes a ´et´e beaucoup ´etudi´e ces derni`eres ann´ees pour les ´equations stationnaires de Navier-Stokes. Nous donnons ici une nouvelle preuve plus simple de l’existence de solution tr`es faible pour les ´equations stationnaires de Navier-Stokes, qui s’appuie sur des arguments de densit´e et un cadre fonctionnel appropri´e pour d´efinir de mani`ere plus rigoureuse les traces des champs de vecteurs peu r´eguliers. On obtient aussi r´esultats de r´egularit´e dans des espaces de Sobolev fractionnaires. Tous les r´esultats sont obtenus dans le cas d’un ouvert connexe de classe C1,1de R3et peuvent ˆetre ´etendus `a l’´equation de Laplace ainsi qu’aux autres dimensions. Pour citer cet article : C. Amrouche, M.A. Rodr´ıguez-Bellido, C. R. Acad. Sci. Paris, Ser. I 340 (2005). Email addresses: [email protected] (Ch´erif Amrouche), [email protected] (Mar´ıa ´ Angeles Rodr´ıguez-Bellido). 1Partially supported by M.E.C. (Spain), Project MTM2006-07932, and by Junta de Andaluc´ıa, Project P06-FQM-02373. Preprint submitted to the Acad´emie des sciences 15th January 2010 Version fran¸caise abr´eg´ee L’objet de cette note consiste essentiellement `a ´etudier l’existence de solutions tr`es faibles (u,q)∈ Lp(Ω)×W−1,p(Ω) du probl`eme de Stokes (S) (voir Definition 3.2). L’une des difficult´es pour prouver l’existence de telles solutions consiste `a donner un sens `a la condition aux limites de Dirichlet. Utilisant un argument d’interpolation, cela nous permet d’en d´eduire l’existence de solutions appartenant `a des espaces de Sobolev fractionnaires. Les principaux r´esultats d’existence sont donn´es dans la Section 3. 1. Introduction Let Ωbe a bounded connected open set of R3of class C1,1with boundary Γ. We study the stationary Stokes problem (S): −∆u+∇q=fand ∇·u=hin Ω,u=gon Γ, where udenotes the velocity field and qthe pressure, and both are unknown. The external force f,the compressibility condition hand the boundary condition for gare the data. The vector fields and matrix fields (and the corresponding spaces) are respectively denoted by boldface Roman and special Roman. The notion of very weak solutions (u,q)∈Lp(Ω)×W−1,p(Ω) for the stationary Stokes or Navier-Stokes equations, corresponding to very irregular data, has been developed in the last years by Giga [9] (and also by Lions-Magenes [11] for the Laplace’s equation, in a domain Ωof class C∞), Amrouche-Girault [1] (in a domain Ωof class C1,1) and more recently by Galdi-Simader-Sohr [8], Farwig-Galdi-Sohr [7] (in a domain Ωof class C2,1, see also Schumacher [14]) and finally by Kim [10] (in a domain Ωof class C2with connected boundary). In this context, the boundary condition is chosen in Lp(Γ) (see Brown-Shen [3], Conca [5], Fabes-Kenig-Verchota [6], Moussaoui [12], Shen [15], Savar´e [13]) or more generally in W−1/p,p(Γ). The purpose of this work is to develop a unified theory of very weak solutions for the Dirichlet problem associated to the stationary Stokes system. One important question is to define rigorously the traces of the vector functions which are living in subspaces of Lp(Ω) (see Lemma 2.3). We prove existence and uniqueness of very weak solutions (u,q)∈Lp(Ω)×W−1,p(Ω) for the Stokes problem for any 1 <p<∞ (see Definition 3.2). Using an interpolation argument, we deduce the existence of solutions belonging to fractional Sobolev spaces Ws,p(Ω), with 0 ≤s≤2 (see Corollary 3.6 and Theorem 3.7). Observe that the study of the Stokes problem is fundamental for the study of the Oseen and Navier-Stokes equations. The detailed proofs of the results announced in this Note are given in [2]. 2. Density and trace results We introduce the spaces: Dσ(Ω) = {ϕ∈D(Ω); ∇·ϕ=0},Dσ(Ω) = {ψ∈D(Ω)3;∇·ψ=0}, Lp σ(Ω) = {v∈Lp(Ω); ∇·v=0},Xr,p(Ω) = {ϕ∈W1,r 0(Ω); ∇·ϕ∈W1,p 0(Ω)},1<r,p<∞, and we set Xp,p(Ω) = Xp(Ω). Lemma 2.1 i) The space Dσ(Ω)is dense in Lp σ(Ω). ii) 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).(1) It is then easy to prove the following characterization: (Xr,p(Ω))�=�f=∇·F0+∇f1;F0∈Lr�(Ω),f 1∈W−1,p�(Ω),with F0=(f ij)1≤i,j≤3�.(2) 2 As a consequence of Lemma 2.1 ii) and the Sobolev embeddings, we have the embeddings W−1,r(Ω)�→ (Xr�,p�(Ω))��→W−2,p(Ω),where the second embedding holds if 1 r≤1 p+1 3. Giving a meaning to the trace of a very weak solution of a Stokes problem is not trivial. Remember that we are not in the classical variational framework. In this way, we need to introduce some spaces. First, we consider the space Yp�(Ω) = {ψ∈W2,p�(Ω); ψ|Γ=0,(∇·ψ)|Γ=0}that can also be described (see [1]) as: Yp�(Ω) = {ψ∈W2,p�(Ω); ψ|Γ=0,∂ψ ∂n·n���Γ=0}.(3) Note also that if ψ∈Yp�(Ω), then div ψ∈W1,p� 0(Ω) and 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}.Secondly, we shall use the spaces: Tp,r(Ω) = {v∈Lp(Ω); ∆v∈(Xr�,p�(Ω))�},Tp,r,σ(Ω) = {v∈Tp,r(Ω); ∇·v=0}, endowed with the norm �v�Tp,r (Ω)=�v�Lp(Ω)+�∆v�[Xr�,p�(Ω)]�.When p=r, these spaces are denoted as Tp(Ω) and Tp,σ(Ω), respectively. We also introduce the space Hp,r(div; Ω) = {v∈Lp(Ω); ∇·v∈Lr(Ω)},whichisendowedwiththe graph norm. The following lemma will help us to prove a trace result: Lemma 2.2 i) The space D(Ω)is dense in Tp,r(Ω)and in Tp,r(Ω)∩Hp,r(div; Ω)respectively. ii) The space Dσ(Ω)is dense in Tp,r,σ(Ω). The following two lemmas 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 }.Besides, we recall that we can decompose vinto its tangential, vτ, and normal parts, that is: v=vτ+(v·n)n. Lemma 2.3 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)3can be extended by continuity to a linear and continuous mapping, still denoted by γτ,fromTp,r(Ω)into W−1/p,p(Γ), and the following Green formula holds �∆v,ψ�[Xr�,p�(Ω)]�×Xr�,p�(Ω)=�Ω v·∆ψdx−�vτ,∂ψ ∂n�W−1/p,p(Γ)×W1/p,p�(Γ) ,(4) for any v∈Tp,r(Ω)and ψ∈Yp�(Ω). Proof: Consider v∈D(Ω)3and ψ∈Yp�(Ω). Then (4) holds. Observe that Yp�(Ω) is included in Xr�,p�(Ω). Suppose that µ∈W1/p,p�(Γ). Then, µ=µτ+(µ·n)n. Since Ωis of class C1,1, we know that there exists ψ∈W2,p�(Ω) such that ψ=0and ∂ψ ∂n=µτon Γ, and verifying: �ψ�W2,p�(Ω)≤C�µτ�W1/p,p�(Γ)≤C�µ�W1/p,p�(Γ). Moreover, ψ∈Yp�(Ω). Therefore, we can estimate the boundary term as follows for such functions ψ: ����vτ,µ�W−1/p,p(Γ)×W1/p,p�(Γ)���≤�v�Lp(Ω)�ψ�W2,p�(Ω)+�∆v�[Xr�,p�]��ψ�Xr�,p�. Thus, �vτ�W−1/p,p(Γ)≤C�v�Tp,r(Ω).Therefore, the linear continuous mapping v�→ vτ|Γdefined on D(Ω) is continuous for the norm of Tp,r(Ω). Since D(Ω) is dense in Tp,r(Ω), then we can extend this mapping from Tp,r(Ω) into W−1/p,p(Γ), that is, the tangential trace of functions of Tp,r(Ω) belongs to W−1/p,p(Γ) and the relation (4) holds. We can also prove that D(Ω) is dense in Hp,r(div; Ω), the mapping γn:v�→ v·n|Γis continuous from Hp,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). 3 3. Very weak solutions and regularity We focus on the study of the stationary Stokes problem (S) with the compatibility condition: �Ω h(x)dx=�g·n,1�W−1/p,p(Γ)×W1/p,p�(Γ).(5) Basic results on weak and strong solutions of problem (S) may be summarized in the following theorem (see [1], [4]). Theorem 3.1 i) For every f∈W−1,p(Ω),h∈Lp(Ω),g∈W1−1/p,p(Γ)satisfying the compatibility condition (5), the Stokes problem (S)has exactly one solution u∈W1,p(Ω)and q∈Lp(Ω)/R,and there exists a constant C>0, depending only on pand Ω, such that: �u�W1,p(Ω)+�q�Lp(Ω)/R≤C(�f�W−1,p(Ω)+�h�Lp(Ω)+�g�W1−1/p,p(Γ)).(6) ii) Moreover, if f∈Lp(Ω),h∈W1,p(Ω),g∈W2−1/p,p(Γ), then u∈W2,p(Ω),q∈W1,p(Ω)satisfy an analogous estimate to (6) with the corresponding norms. We wonder about minimal necessary assumptions on f,hand g, in order that a very weak solution, that is, (u,q)∈Lp(Ω)×W−1,p(Ω)/Rexists. We are interested here in the case of singular data satisfying the following assumptions: f∈(Xr�,p�(Ω))�,h∈Lr(Ω),g∈W−1/p,p(Γ),with 1 r≤1 p+1 3and r≤p. (7) Observe that the space (Xr�,p�(Ω))�is an intermediate space between W−1,r(Ω) and W−2,p(Ω). Definition 3.2 (Very weak solution for the Stokes problem) Apair(u,q)∈Lp(Ω)×W−1,p(Ω)is avery weak solution of (S)if the following equalities hold: For any ϕ∈Yp�(Ω)and π∈W1,p�(Ω), −�Ω u·∆ϕdx−�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, (8) with �·,·�Ω=�·,·�[Xr�,p�(Ω)]�×Xr�,p�(Ω)and �·,·�Γ=�·,·�W−1/p,p(Γ)×W1/p,p�(Γ). Note that W1,p�(Ω)�→Lr�(Ω) and Yp�(Ω)�→Xr�,p�(Ω) if 1 r≤1 p+1 3, which means that all the brackets and integrals have a sense. We can then prove that, if f,hand gsatisfy (7), then (u,q)∈Lp(Ω)×W−1,p(Ω) is a very weak solution of (S) if and only if (u,q) satisfies the system (S) in the sense of distributions. Proposition 3.1 Let f∈(Xp�(Ω))�,h∈Lp(Ω)and g∈W−1/p,p(Γ)satisfy the compatibility condition (5). Then, the Stokes problem (S)has exactly one solution u∈Tp(Ω)and q∈W−1,p(Ω)/R. Moreover, there exists a constant C>0, depending only on pand Ω, such that: �u�Tp(Ω)+�q�W−1,p(Ω)/R≤C��f�[Xp�(Ω)]�+�h�Lp(Ω)+�g�W−1/p,p(Γ)�.(9) Proof: The case f=0and h= 0 is considered in [1]. Here, we generalize the result as follows: Step 1: We suppose g·n= 0 on Γand �Ω h(x)dx= 0. It remains to consider the equivalent problem: Find (u,q)∈Lp(Ω)×W−1,p(Ω)/Rsuch that: for any w∈Yp�(Ω) and any π∈W1,p�(Ω) it holds �Ω u·(−∆w+∇π)dx−�q,∇·w�W−1,p(Ω)×W1,p� 0(Ω)=�f,w�[Xp�(Ω)]�×Xp�(Ω)−�gτ,∂w ∂n�Γ−�Ω hπdx. We can prove (as in [1]) that for any pair (F,ϕ)∈Lp�(Ω)×(W1,p� 0(Ω)∩Lp� 0(Ω)): 4 ����f,w�[Xp�(Ω)]�×Xp�(Ω)−�gτ,∂w ∂n�Γ −�Ω hπdx��� ≤C��f�[Xp�(Ω)]�+�g�W−1/p,p(Ω)+�h�Lp(Ω)���F�Lp�(Ω)+�ϕ�W1,p�(Ω)�, being (w,π)∈Yp�(Ω)×W1,p�(Ω)/Rthe unique solution of the Stokes (dual) problem: −∆w+∇π=Fand ∇·w=ϕin Ω,w=0on Γ. Note that for any k∈R,�����Ω hπdx����=�����Ω h(π+k)dx����≤�h�Lp(Ω)�π�Lp�(Ω)/Rand �w�W2,p�(Ω)+�π�W1,p�(Ω)/R≤C��F�Lp�(Ω)+�ϕ�W1,p�(Ω)�.From this bound, we deduce that the mapping (F,ϕ)→�f,w�Ω−�gτ,∂w ∂n�Γ−�Ω hπdxdefines an element of the dual space of Lp�(Ω)×(W1,p� 0(Ω)∩ Lp� 0(Ω)) with norm bounded by C(�f�[Xp�(Ω)]�+�h�Lp(Ω)+�g�W−1/p,p(Ω)). That means that there exists aunique(u,q)∈Lp(Ω)×W−1,p(Ω)/Rsolution of (S) satisfying the estimate (9). Step 2: Now, we suppose that �Ω h(x)dx=�g·n,1�Γ.Defineu0=∇θwith θ∈W1,p(Ω) the solution of the Neumann problem: ∆θ=hin Ωand ∂θ ∂n=g·non Γ.By Step 1, there exists a unique (z,q)∈Lp(Ω)×W−1,p(Ω)/Rsatisfying: −∆z+∇q=f+∇hand ∇·z= 0 in Ωand z=g−u0|Γon Γ, where ∇h∈(Xp�(Ω))�and g−u0|Γsatisfies the hypothesis of Step 1. Thus, the pair of functions (u,q)=(z+u0,q) is the required solution. The following result is a generalization of Proposition 4.11 in [1], where f=0and h= 0. Theorem 3.3 Let f,h,gbe given satisfying (5) and (7). Then, the Stokes problem (S)has exactly one solution (u,q)∈Tp,r(Ω)×W−1,p(Ω)/R. Moreover, there exists a constant C>0, only depending on p and Ω, such that: �u�Tp,r(Ω)+�q�W−1,p(Ω)/R≤C��f�[Xr�,p�(Ω)]�+�h�Lr(Ω)+�g�W−1/p,p(Γ)�(10) Remark 1 i) Observe that in [8] Theorem 3, the domain was of class C2,1(here it is of class C1,1), and the divergence term was h∈Lp(Ω) (here of h∈Lr(Ω)). Moreover, our solution is obtained in the space Tp,r(Ω), which has been clearly characterized, contrary to the space � W1,p(Ω) appearing in [8], which was not characterized, completely abstract and obtained as the closure of W1,p(Ω) for the norm �u�� W1,p(Ω)=�u�Lp(Ω)+�A−1/2 rPr∆u�Lr(Ω),where Aris the Stokes operator with domain equal to W2,p(Ω)∩W1,p 0(Ω)∩Lp σ(Ω) and Pris the Helmholtz projection operator from Lr(Ω) onto Lr σ(Ω). ii) Existence of very weak solution u∈Lp(Ω) was proved by Kim [10] for f∈[W1,q� 0(Ω)∩W2,q�(Ω)]�, for h∈[W1,q�(Ω)]�and g∈W−1/q,q(Γ), but the spaces chosen for hand fare not correct either and the equivalence in Theorem 5 of [10] is not valid. Corollary 3.4 Let f,h,gbe given satisfying (5) and f=∇·F0+∇f1with F0∈Lr(Ω),f1∈W−1,p(Ω), h∈Lr(Ω),g∈W1−1/r,r(Γ). Then the solution ugiven by Theorem 3.3 belongs to W1,r(Ω). If moreover f1∈Lr(Ω), then qbelongs to Lr(Ω). In both cases, we have analogous estimates to (10). Remark 2 It is clear that W1,r(Ω)�→Tp,r(Ω) when 1 r≤1 p+1 3,i.e.,Tp,r(Ω) is an intermediate space between W1,r(Ω) and Lp(Ω). Corollary 3.5 Let h∈Lr(Ω)and g∈W−1/p,p(Γ)be given, satisfying (5), with 1 r≤1 p+1 3and r≤p. Then, there exists at least one solution u∈Tp,r(Ω)verifying ∇·u=hin Ωand u=gon Γ. Moreover, there exists a constant C=C(Ω,p,r)such that �u�Tp,r(Ω)≤C��h�Lr(Ω)+�g�W−1/p,p(Γ)�. The following corollary gives Stokes solutions (u,q) in fractionary Sobolev spaces of type Wσ,p(Ω)× Wσ−1,p(Ω), with 0 <σ<2. 5 Corollary 3.6 Let sbe a real number such that 0≤s≤1. i) Let f=∇·F0+∇f1,hand gsatisfy the compatibility condition (5) with F0∈Ws,r(Ω),f1∈Ws−1,p(Ω), g∈Ws−1/p,p(Γ)and h∈Ws,r(Ω),with1 r≤1 p+1 3and r≤p. Then, the Stokes problem (S) has exactly one solution (u,q)∈Ws,p(Ω)×Ws−1,p(Ω)/Rsatisfying the estimate �u�Ws,p(Ω)+�q�Ws−1,p(Ω)/R≤C��F0�Ws,r (Ω)+�f1�Ws−1,p(Ω)+�h�Ws,r(Ω)+�g�Ws−1/p,p(Γ)� ii) Assume that f∈Ws−1,p(Ω),g∈Ws+1−1/p,p(Γ)and h∈Ws,p(Ω), fulfill the compatibility condition (5). Then, the Stokes problem (S) has exactly one solution (u,q)∈Ws+1,p(Ω)×Ws,p(Ω)/Rwith �u�Ws+1,p(Ω)+�q�Ws,p(Ω)/R≤C��f�Ws−1,p(Ω)+�h�Ws,p(Ω)+�g�Ws+1−1/p,p(Γ)�. The following theorem provides solutions for external forces f∈Ws−2,p(Ω) and divergence condition h∈Ws−1,p(Ω) with 1/p < s < 2. In particular, if p=2,we obtain solutions in H1/2+ε(Ω)×H1/2+ε(Ω), for any 0 <ε≤3/2. Theorem 3.7 Let sbe a real number such that 1 p<s≤2. Let f,hand gsatisfy the compatibility condition (5) with f∈Ws−2,p(Ω),h∈Ws−1,p(Ω)and g∈Ws−1/p,p(Γ). Then, the Stokes problem (S) has exactly one solution (u,q)∈Ws,p(Ω)×Ws−1,p(Ω)/Rsatisfying the estimate �u�Ws,p(Ω+�q�Ws−1,p(Ω/R≤C��f�Ws−2,p(Ω)+�h�Ws−1,p(Ω+�g�Ws−1/p,p(Γ)�.(11) References [1] C. Amrouche and V. Girault,Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Mathematical Journal, 44, 119 (1994), 109–140. [2] C. Amrouche and M. A. 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