Very weak solutions for the stationary Oseen and Navier–Stokes equations

Very weak solutions for the stationary Oseen and Navier-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 We consider the stationary Oseen and Navier-Stokes equations in a bounded domain of class C1,1of R3. Here we give a new and simpler proof of the existence of very weak solutions (u,q)∈Lp(Ω)×W−1,p(Ω) corresponding to boundary data in W−1/p,p(Γ). These solutions are obtained without imposing smallness assumptions on the exterior forces. We also obtain regularity results in fractional Sobolev spaces. 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 d’Oseen et de Navier-Stokes. Nous consid´erons les ´equations stationnaires d’Oseen et de Navier-Stokes dans un ouvert born´e connexe et de classe C1,1de R3. Nous donnons ici une nouvelle preuve plus simple de l’existence de solutions tr`es faibles (u,q)∈Lp(Ω)×W−1,p(Ω) correspondant `a des donn´ees au bord dans W−1/p,p(Γ). Ces solutions sont obtenues sans hypoth`ese de petitesse des forces ext´erieures. On obtient aussi des r´esultats de r´egularit´e dans des espaces de Sobolev fractionnaires. Pour citer cet article : C. Amrouche, M. A. Rodr´ıguez-Bellido, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 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) pour les ´equations d’Oseen (O) et de Navier-Stokes (NS). L’une des difficult´es consiste Email addresses: [email protected] (Ch´erif Amrouche), ang[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 28 octobre 2009 `a donner un sens aux conditions aux limites de Dirichlet. Le terme de convection rend les choses plus difficiles pour le probl`eme (O) et complique s´erieusement la situation pour l’´etude du probl`eme non lin´eaire (NS). Les r´esultats concernant l’existence de solutions tr`es faibles sont donn´es dans le th´eor`eme 2.4 pour (O) et les th´eor`ems 3.1 et 3.2 pour (NS). Les autres r´esultats concernent la r´egularit´e de telles solutions sous des hypoth`eses ad´equates. Nons consid´erons en particulier le cas o`u les donn´ees et donc les solutions appartiennent `a des espaces de Sobolev fractionnaires. 1. Introduction Let Ωbe a bounded connected open set of R3of class C1,1with boundary Γ. We are interested in some questions concerning the stationary Oseen and Navier-Stokes equations, that generally can be written as: (O)−∆u+v·∇u+∇q=fand ∇·u=hin Ω,u=gon Γ, (NS)−∆u+u·∇u+∇q=fand ∇·u=hin Ω,u=gon Γ, where udenotes the velocity field and qthe pressure, both being unknown, and f,h,gand vare given. In the case of incompressible fluids, h= 0, it has been well-known since Leray [6] that if f∈W−1,p(Ω) and g∈W1−1/p,p(Γ) with p≥2, Γiare the connected components of the boundary Γ,i=0,...,I, and �Γi g·ndσ=0,∀i=0, ..., I, (1) then there exists a solution (u,q)∈W1,p(Ω)×Lp(Ω) satisfying (NS). Serre proved [8] the existence of weak solution (u,q)∈W1,p(Ω)×Lp(Ω) for any 3 2<p<2whenh= 0 and gsatisfies the above conditions. Recently, Kim [5] improves Serre’s existence and regularity results on weak solutions of (NS) for any 3 2≤p<2, when Γis connected (I= 0) provided his small in an appropriate norm (due to (2), see below, gis also small in the corresponding appropriate norm). Existence of very weak solutions (u,q)∈L3(Ω)×W−1,3(Ω), for h= 0, arbitrary large f∈H−1(Ω) and large g∈L2(Γ), without assuming condition (1), was proved first by Marusic-Paloka in [7] (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) or condition (12) hold. The same result was proved by Kim [5] 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), with Γ supposed connected (I= 0). Observe that the space chosen for hand for fare not correct either and led us to some erros (in particular, the equivalence given in Theorem 5 there does not work). The purpose of our work is to generalize the theory of very weak solutions of the Dirichlet problem from the Stokes equations to the Oseen and Navier-Stokes equations, defining rigorously the traces of the vector functions which are living in subspaces of Lp(Ω) (see [1], [2]), and the spaces for the data. We prove existence and regularity of very weak solutions (u,q)∈Lp(Ω)×W−1,p(Ω) of Oseen equations for any p∈(1,+∞) with arbitrary large data in Sobolev spaces of negative order. In the Navier-Stokes case, 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. The detailed proofs of the results announced in this Note are given in [2]. 2. Oseen Equations For any 1 <r,p<∞, we define the spaces: Hp(Ω) = {v∈Lp(Ω); ∇·v=0},Xr,p(Ω) = {ϕ∈ W1,r 0(Ω); ∇·ϕ∈W1,p 0(Ω)}, and Tp,r(Ω) = {v∈Lp(Ω); ∆v∈(Xr�,p�(Ω))�}, endowed with the topology 2 given by the norm �v�Tp,r (Ω)=�v�Lp(Ω)+�∆v�[Xr�,p�(Ω)]�. As for the Navier-Stokes system, we can prove that if f∈H−1(Ω), v∈H3(Ω), h∈L2(Ω) and g∈H1/2(Γ) with hand gverifying the compatibility condition �Ω h(x)dx=�Γ g·ndσ,(2) then the problem (O) has a unique solution (u,q)∈H1(Ω)×L2(Ω)/Rverifying the following estimate: �u�H1(Ω)≤C��f�H−1(Ω)+�1+�v�L3(Ω)�(�h�L2(Ω)+�g�H1/2(Γ))�. Theorem 2.1 (Strong solutions) Consider p≥6 5,f∈Lp(Ω),h∈W1,p(Ω),v∈Hs(Ω)and g∈ W2−1/p,p(Γ),withs=3if p<3,s=pif p>3, or s=3+εif p=3, for some arbitrary ε>0, and satisfying the compatibility condition (2). Then, the unique solution of (O) 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)��. Proof: First, let (u,q)∈H1(Ω)×L2(Ω)/Rbe the unique solution of Problem (O). For a given vλ∈D(Ω) (λ>0) such that ∇·vλ= 0 and �vλ−v�Ls(Ω)≤λ,let(uλ,q λ)∈W2,p(Ω)×W1,p(Ω) be the unique solution of the problem (Oλ): −∆uλ−vλ·∇uλ+∇qλ=fand ∇·uλ=hin Ω,uλ=gon Γ(use the Stokes regularity and a bootstrap argument). Secondly, we focus on the obtention of a strong estimate for (uλ,q λ). If � vis the extension by zero of vto R3and ρεthe classical mollifier, we consider vλ=vε 1+vε λ,2where vε 1=� v�ρ ε/2,vε λ,2=vλ−� v�ρ ε/2for ε>0,and 0 <λ<ε/2.(3) 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) In order to estimate the term �vλ·∇uλ�Lp(Ω), we use (3) and Sobolev embeddings. First: �vε λ,2·∇uλ�Lp(Ω)≤�vε λ,2�Ls(Ω)�∇uλ�Lk(Ω)≤Cε�uλ�W2,p(Ω),with 1 k=1 p−1 s.(5) For the estimate on vε 1, we consider two cases: If p≤2, let r∈]3,∞] be such that 1 p=1 r+1 2, and t≥1 such that 1+1 r=1 3+1 tsatisfying: �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 (5), we deduce from (4) 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))). If p>2, using the compact embedding W2,p(Ω)�→W1,q(Ω), with q<p ∗, for any ε�>0, we known that there exists Cε�>0 such that �∇uλ�Lq(Ω)≤ε��uλ�W2,p(Ω)+Cε��uλ�H1(Ω).Considering the case p<3 and then the case p≥3, we can choose the exponent qand fix ε>0 and ε�>0 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))��. Thus, we deduce that there exists a sequence of real numbers kλsuch that (uλ,q λ+kλ) converges weakly in W2,p(Ω)×W1,p(Ω) to (u,q), solution of Problem (O) with the corresponding estimate. Theorem 2.2 Let f∈W−1,p(Ω),v∈H3(Ω),h∈Lp(Ω)and g∈W1−1/p,p(Γ)verify the compatibility condition: �Ω h(x)dx=�g·n,1�W−1/p,p(Γ)×W1/p,p�(Γ).(6) 3 Then, the problem (O)has a unique solution (u,q)∈W1,p(Ω)×Lp(Ω)/R. Moreover, there exists some constant C>0such that, for α=1if p≥2and α=1+�v�L3(Ω)if p<2, we have �u�W1,p(Ω)+�q�Lp(Ω)/R≤C�1+�v�L3(Ω)�2��f�W−1,p(Ω)+α�h�Lp(Ω)+�g�W1−1/p,p(Γ)�.(7) Sketch of the proof: We split it in two cases. If p≥2, we decompose the solution (u,q) as (z,θ)+(u0,q 0), being (u0,q 0)∈W1,p(Ω)×Lp(Ω) satisfying −∆u0+∇q0=fand ∇·u0=hin Ω,u0=gon Γ, and (z,θ)∈W2,t(Ω)×W1,t(Ω) satisfying −∆z+v·∇z+∇θ=−v·∇u0and ∇·z= 0 in Ω,z=0on Γ, where 1 t=1 3+1 p. The corresponding estimates (see Theorem 2.1) and the embedding W2,t(Ω)�→W1,p(Ω) conclude the proof in this case. Secondly, if p<2, we are able to conclude by a duality argument. Remark 1 Estimate (7) can be improved for p∈[6 5,6], and for any p>1 if v·n= 0 on Γas: �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)��. Corollary 2.3 Consider 1<p<6/5and f∈Lp(Ω),v∈H3(Ω),h∈W1,p(Ω)and g∈W2−1/p,p(Γ) verifying the compatibility condition (6). Then, the solution given by Theorem 2.2 satisfies (u,q)∈ W2,p(Ω)×W1,p(Ω)and the following estimate holds: �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)��. Using the previous results and following arguments in [2], we obtain: Theorem 2.4 (Very weak solution of Oseen equations) Let f∈(Xr�,p�(Ω))�,h∈Lr(Ω),g∈ W−1/p,p(Γ),with1 r=1 p+1 s, be given, satisfying the compatibility condition (6), and v∈Hs(Ω)with s=3if p>3/2,s=p�if p<3/2, or s=3+εif p=3/2.Then, the Oseen problem (O)has a unique solution (u,q)∈Tp,r(Ω)×W−1,p(Ω)/Rverifying the estimates �u�Tp,r(Ω)≤C�1+�v�Ls(Ω)���f�[Xr�,p�(Ω)]�+�h�Lr(Ω)+�g�W−1/p,p(Γ)�,(8) �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)�. Concerning the regularity of solutions for the Oseen equations in fractional Sobolev spaces, we obtain: Theorem 2.5 (Regularity for Oseen equations) Consider σ∈(1/p, 2]. Let f∈Wσ−2,p(Ω),h∈ Wσ−1,p(Ω),g∈Wσ−1/p,p(Γ)be given satisfying the compatibility condition (6), and v∈Hs(Ω)with sas in Theorem 2.4. Then, the Oseen problem (O) has a unique solution (u,q)∈Wσ,p(Ω)×Wσ−1,p(Ω)/R satisfying �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)). 3. Navier-Stokes Equations Now, we present two theorems giving existence of very weak solutions for the Navier-Stokes equations in L3(Ω)×W−1,3(Ω), first one for the small data case, and second one for arbitrary large fbut hand g small enough in a domain possibly multiply-connected. Theorem 3.1 (Very weak solution for Navier-Stokes, small data case) Let f∈(X3,3/2(Ω))�,h∈ L3/2(Ω)and g∈W−1/3,3(Γ)verify (6). Then, 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,then, there exists a very weak solution (u,q)∈L3(Ω)×W−1,3(Ω)to problem (NS)verifying the estimates �u�L3(Ω)≤C��f�[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ)�,(9) �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�,(10) 4 where C>0is the constant given in (8), α1= min �(2C)−1,(2C2)−1�, and C1and C2constants of Sobolev embeddings. ii) Moreover, there exists a constant α2∈]0,α 1]such that if �f�[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ)≤ α2, then this solution is unique, up to an additive constant for q. Proof: We prove existence of a very weak solution by applying Banach’s fixed point theorem over the Oseen equations. Indeed, let T:H3(Ω)→H3(Ω) be the application defined as v�→ Tv=u, where uis the unique solution of (O) provided by Theorem 2.4. We set Br={v∈H3(Ω); �v�L3(Ω)≤r}. We will prove that there exists θ∈]0,1[ such that �Tv1−Tv2�L3(Ω)=�u1−u2�L3(Ω)≤θ�v1−v2�L3(Ω).(11) In order to estimate �u1−u2�L3(Ω), we observe that for each i=1,2, (ui,q i) is the solution of −∆ui+ vi·∇ui+∇qi=fand ∇·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)�, being C>0 the constant given in (8). However, to estimate the difference u1−u2, we have to argue differently. Consider the problem fulfilled by (u,q)=(u1−u2,q 1−q2), which is −∆u+v1·∇u+∇q= −v·∇u2and ∇·u= 0 in Ω,u=0on Γ, where u1=Tv1,u2=Tv2and v=v1−v2. Using the very weak estimates (8) 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), where β=�f�[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ). Thus, we obtain estimate (11) if we consider C2β(1 + r)2<1, and (9)-(10) hold for C1the continuity constant of the Sobolev embedding [X3,3/2(Ω)]��→W−2,3(Ω) and C2the continuity constant of the Sobolev embedding W1,3/2 0(Ω)�→L3(Ω). The uniqueness result is a simple consequence of Sobolev embeddings and the Stokes estimates. Theorem 3.2 (Very weak solution for Navier-Stokes, arbitrary forces) Let f∈(X3,3/2(Ω))�, h∈L3/2(Ω)and g∈W−1/3,3(Γ)be given, and satisfying the compatibility condition (6). There exists a constant δ>0(depending only on Ω) such that the problem (NS)has a very weak solution (u,q)∈L3(Ω)×W−1,3(Ω)if �h�L3/2(Ω)+ i=I � i=0 |�g·n,1�Γi|≤δ.(12) Sketch of the proof: We decompose (NS) into two problems. One system, denoted (NS1), for small data: −∆vε+vε·∇vε+∇q1 ε=f−fε,∇·vε=h−hεin Ω,and vε=g−gεon Γ. with ε>0 and the (NS2) system: −∆zε+zε·∇zε+zε·∇vε+vε·∇zε+∇q2 ε=fε,∇·zε=hεin Ω,zε=gεon Γ. where fε∈H−1(Ω),h ε∈L2(Ω) and gε∈H1/2(Γ) satisfy �f−fε�[X3,3/2(Ω)]�+�h−hε�L3/2(Ω)+�g−gε�W−1/3,3(Γ)≤εand �hε�L3/2(Ω)+ i=I � i=0 |�gε·n,1�Γi|≤2δ, (here, we have used density arguments). Finally, we use an extension of Hopf’s lemma: (see [3], Remark VIII.4.4 for instance) for any α>0, there exists yε∈H1(Ω), depending on α, such that for C1>0 depending only on Ω,∇·yε=hεin Ω,yε=gεon Γand for any w∈H1 0(Ω), �����Ω (w·∇)yε·wdx����≤�α+�hε�L3/2(Ω)+C i=I � i=0 |�gε·n,1�Γi|��w�2 H1(Ω)≤(α+2C1δ)�w�2 H1(Ω). 5 To finish, we prove some regularity results on very weak solutions for the Navier-Stokes equations by using the regularity results for the Stokes and Oseen problems. Theorem 3.3 (Regularity for Navier-Stokes equations) Let (u,q)∈L3(Ω)×W−1,3(Ω)be the solution given by Theorem 3.2. Then, the following regularity results hold: i) If f∈(Xr�,p�(Ω))�,h∈Lr(Ω)and g∈W−1/p,p(Γ), with 1 r≤1 p+1 3and max{r, 3}≤p, then (u,q)∈ Lp(Ω)×W−1,p(Ω). ii) Consider r≥3/2,f∈W−1,r(Ω),h∈Lr(Ω)and g∈W1−1/r,r(Γ). Then (u,q)∈W1,r(Ω)×Lr(Ω). iii) For r∈(1,+∞), if f∈Lr(Ω),h∈W1,r(Ω)and g∈W2−1/r,r(Γ), then (u,q)∈W2,r(Ω)×W1,r(Ω). iv) Suppose that 3/2≤p≤3,f=∇·F0+∇f1for F0∈Wσ,r(Ω)and f1∈Wσ−1,p(Ω),h∈Wσ,r(Ω), and g∈Wσ−1/p,p(Γ), with σ=3 p−1,1 r≤1 p+1 3and r≤p. Then (u,q)∈Wσ,p(Ω)×Wσ−1,p(Ω). v) Let σbe such that 1/p<σ≤1and σ≥3/p −1. Suppose that f∈Wσ−2,p(Ω),h∈Wσ−1,p(Ω), and g∈Wσ−1/p,p(Γ). Then (u,q)∈Wσ,p(Ω)×Wσ−1,p(Ω). Remark 2 i) Point i) shows in particular that for any p≥3, if f∈W−1,r(Ω) and g∈W1−1/r,r(Γ), with 3p 3+p≤r≤p, and �Γig·n= 0 for any i=1,...,I and h= 0, then Problem (NS) has a solution (u,q)∈Lp(Ω)×W−1,p(Ω). Serre [8] proves that for any 3/2<r<2 (and then for r>3/2), if f∈W−1,r(Ω), g∈W1−1/r,r(Γ), h= 0 and (1) is verified for any i=0,...,I, then (NS) has a solution (u,q)∈W1,r(Ω)×Lr(Ω). Point ii) proves that this result holds if r=3/2 without assuming hor the flux gthrough Γito be equal to 0. Actually, it suffices to assume the smallness condition (12). ii) From relation (6), condition (12) is automatically fulfilled when the norm �h�L3/2(Ω)is small enough and I= 0, that means that the boundary Γis connected, which is the case considered by Kim [5]. iii) Marusic-Paloka [7] proves Theorem 3.2 with f∈H−1(Ω)⊆(X3,3/2(Ω))�,h= 0 and g∈L2(Γ)⊆ W−1/3,3(Γ) with �g�L2(Γ)small, in a domain Ωsimply-connected. In fact, the solution u∈L3(Ω) obtained in [7] is more regular and belongs to H1/2(Ω) by point iv) with p= 2. iv) Galdi et al. [4] prove Theorem 3.2 and Theorem 3.3 point i) with f= div F0,F0∈Lr(Ω), h∈Lp(Ω) and g∈W−1/p,p(Γ) with 1 r≤1 p+1 3and max{2r, 3}≤p, in a domain Ωof class C2,1, assuming that f, hand gare small enough in their respective norms. The smallness condition on fis in fact unnecessary. References [1] C. Amrouche and V. 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Marusiˇ c-Paloka,Solvability of the Navier-Stokes system with L2boundary data, Appl. Math. Optim. 41 (2000), pp. 365-375. [8] D. Serre,´ Equations de Navier-Stokes stationnaires avec donn´ees peu r´eguli`eres, Ann. Sc. Norm. Sup. Pisa 10-4, (1983), pp. 543-559. 6