The Boussinesq system with mixed nonsmooth boundary data

The Boussinesq system with mixed nonsmooth boundary data � Le syst`eme de Boussinesq `a donn´ees limites mixtes peu reguli`eres E. J. Villamizar-Roa aM. A. Rodr´ıguez-Bellido bM. A. Rojas-Medar c aEscuela de Matem´aticas, Universidad Industrial de Santander, A.A. 678, Bucaramanga-Santander, Colombia bDpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apto. 1160, 41080 Sevilla, Spain cIMECC-UNICAMP, CP 6065, 13083-970, Campinas-SP, Brazil Abstract We treat the stationary Boussinesq system with non-smooth mixed boundary conditions for the temperature, and non-smooth Dirichlet boundary condition for the velocity. We prove the existence, the continuous dependence of the solution with respect to the data and the uniqueness of the very weak solution. To cite this article: E. J. Villamizar-Roa, M. A. Rodr´ıguez-Bellido & M. A. Rojas-Medar, C. R. Acad. Sci. Paris, Ser. I 336 (2003). R´esum´e On traite le syst`eme de Boussinesq stationnaire `a donn´ees au bord mixtes peu r´eguli`eres pour la temp´erature, et donn´ee au bord Dirichlet peu r´eguli`ere pour la vitesse. On montre l’existence, la d´ependance continue de la solution par rapport aux donn´ees et l’unicit´e de solution tr`es faible pour ce syst`eme. Pour citer cet article : E. J. Villamizar-Roa, M. A. Rodr´ıguez-Bellido & M. A. Rojas-Medar, C. R. Acad. Sci. Paris, Ser. I 336 (2003). Version fran¸caise abr´eg´ee On consid`ere le syst`eme de Boussinesq avec conditions aux limites peu r´eguli`eres, mixtes DirichletNeumnann pour la temp´erature et Dirichlet pour la vitesse. On introduit les solutions tr`es faibles, et on montre l’existence et d´ependance continue de ce type de solution par rapport aux donn´ees ext´erieures et les conditions aux limites. Concretement, les r´esultats qu’on prouve sont les suivants : �The first author is supported by Universidad Industrial de Santander and COLCIENCIAS-Colombia, Project COLCIENCIAS-BID III etapa. The second and third authors have been partially supported by D.G.E.S. & M.C. y T.(Spain), Projet BFM2003-06446-C02-01. The third author has been partially supported by CNPq-Brazil, grant No 301354/03-0 Email addresses: [email protected] (E. J. Villamizar-Roa), [email protected] (M. A. Rodr´ıguez-Bellido), [email protected] (M. A. Rojas-Medar). Th´eor`eme 0.1 (Existence) Si la viscosit´e cin´ematique νest assez grande par rapport `a |f|3,|ξ|L2(Γ1) et |ζ|H−1(Γ2), alors il existe une solution tr`es faible du probl`eme (2). Th´eor`eme 0.2 (D´ependance continue) Soient (ui,θi),i=1,2deux solutions tr`es faibles du probl`eme (2) pour les forces externes f=fi∈L3(Ω),h=hi∈L2(Ω),i=1,2et les donn´ees au bord g=gi∈ L2(Γ),ζ=ζi∈H−1/2(Γ2),ξ=ξi∈L2(Γ1),i=1,2, respectivement. Alors, il existe une constante ν∗>0telle que pour toute ν≥ν∗, |u1−u2|3+|θ1−θ2|2≤c�|f1−f2|3+|h1−h2|2+|ξ1−ξ2|L2(Γ1)+|ζ1−ζ2|H−1/2(Γ2) +|g� 1−g� 2|H1(Γ)+|(g1−g� 1)−(g2−g� 2)|L2(Γ)�, (1) o`u g� isont des fonctions suffisamment proches de gi,i=1,2,danslanormeL2(Γ)et la constante cne d´epend que des donn´ees du probl`eme et de Ω.En particulier, pour ν≥ν∗, la solution de (2) est unique. 1. Introduction We consider the following Boussinesq System [1], with nonsmooth boundary data:    −ν∆u+(u·∇)u+∇p=βfθin Ω,∇·u=0inΩ,u=gon Γ, −χ∆θ+(u·∇)θ=hin Ω,∂nθ=ζon Γ2,θ=ξon Γ1; (2) where the unknowns of the problem are u(x)∈R3the fluid velocity, p(x)∈Rthe pressure and θ(x)∈R the temperature. The data are Γ=Γ1∪Γ2the boundary of Ω,h(x)∈Rthe reference temperature, f(x)∈ R3the gravitational field at x, and ν,β,χ>0 that represent the kinematic viscosity, the coefficient of volume expansion and thermal conductance, respectively. Without loss of generality, we have taken the density of fluid to be constant and equal to one. We consider the following two types of domains Ω⊆R3: CASE 1. As in [3], we consider a bounded subset Ω={(y,z)∈R×R2:z∈P, Φ0(z)<y<Φ1(z)}, where Pis a plane curvilinear polygon with C2sides, without cusp ends, Φ0(z) and Φ1(z) are C2(P) functions, such that for all z∈P, Φ0(z)<Φ1(z).We denote by Γthe boundary of Ωand we consider the following partition of Γ: Γ2={(y,z)∈R×R2:z∈∂P, Φ0(z)<y<Φ1(z)},Γ1=∪i=0,1{(y,z)∈R×R2:z∈P, y =Φi(z)}. We denote by ωthe largest inner angle of Pand φthe supremum of the dihedral angles between Γ2(side of the cylinder) and Γ0or Γ1(bases of the cylinder). We suppose 0 <φ≤π/2 and ω<π. CASE 2. The boundary Γof Ωis of C2class and it is divided into two parts, Γ=Γ1∪Γ2with measure of Γ1non zero and Γ1∩Γ2=∅. A typical example is Ω=Q1\Q2,¯ Q2⊂Q1being Q1,Q2balls of radius r1,r2(r1>r 2) respectively, which can represent the case of an obstacle within a fluid. As usual (Lp(Ω),|·|p), with 1 ≤p≤+∞and (Wk,p,�·�k,p) are the usual Sobolev spaces. In particular Hk(Ω)=Wk,2(Ω) with the norm �·�k=�·�k,2.By(·,·), we represent the inner product in L2(Ω). We denote H1 Γ1(Ω):={u∈H1(Ω):u≡0 on Γ1}and Vthe closure of {u∈C∞ 0:∇·u=0inΩ}in the norm of H1(Ω), being ((·,·)) and �·�the corresponding inner product and norm. We consider the system (2) with nonhomogeneous boundary conditions for uand θ; we only assume ζ∈H−1/2(Γ2), ξ∈L2(Γ1) and g∈L2(Γ)3with �Γg·n= 0, g·n= 0 on Γ2,wherendenotes the unit outward normal of Γ. Some authors (see for instance [11]) considered mixed boundary conditions for θand boundary Dirichlet conditions for ubut their works were done for regular functions g,ζand ξ, so these functions can be extended to the interior of the domain Ω. This allows one to use standard arguments in order to obtain 2 the existence of weak solutions. However, when the boundary data are not regular, that is, when the usual trace theorems cannot be used, the solvability of (2) has not been investigated, as far as we know. In this work, we only assume g∈L2(Γ), ξ∈L2(Γ1) and ζ∈H−1/2(Γ2), and prove the existence of a very weak solution (see Definition 1.1 below), that is a natural extension of the weak solution and it was used by Conca [2] for the Stokes problem. These results were extended in [9] to the Navier-Stokes equations, and in [12] to the Boussinesq equations with Dirichlet boundary conditions for θand u. In order to have a well-posed elliptic problem with mixed boundary data for θ, we need to impose some hypothesis over the domain. This is a quite interesting question which has been studied by M. Dauge in [3,4]. The elliptic problem for the temperature is coupled with a Navier-Stokes system for the velocity, which must be solved in a nonsmooth domain. We recall that the domains considered in [9] were of C1,1 class. Our main results are: Definition 1.1 Atriple(u,θ,p)in L3(Ω)×L2(Ω)×W−1,3(Ω)is a very weak solution of problem (2) if � Ω�−νu·∆Φ −(u·∇)Φ·u�dx−�p, ∇·Φ�W−1,3(Ω),W 1,3/2(Ω)=β� Ω θf·Φdx−ν� Γ g·∂nΦds, � Ω u·∇τdx=� Γ (g·n)τds, −χ(θ,∆ψ)−(u·∇ψ,θ)=(h, ψ)+χ�ζ,ψ�H−1/2(Γ2),H1/2(Γ2)−χ� Γ1 ξ∂ nψds, (3) ∀Φ∈W2,3/2(Ω)∩W1,3/2 0(Ω),∀τ∈W1,3/2(Ω)with �Ωτdx=0and ∀ψ∈H2(Ω)∩H1 Γ1(Ω)with ∂nψ|Γ2≡0. Theorem 1.2 (Existence) If νis large enough with respect to the norms |f|3,|ξ|L2(Γ2)and |ζ|H−1/2(Γ2), then there exists a very weak solution of the problem (2) in the sense of the above definition. Theorem 1.3 (Continuous Dependence) Let (ui,θi),i =1,2be very weak solutions of problem (2) corresponding to the external forces f=fi∈L3(Ω),h=hi∈L2(Ω),i=1,2and boundary data g=gi∈L2(Γ),ζ=ζi∈H−1/2(Γ2),ξ=ξi∈L2(Γ1),i=1,2,respectively. Then, there exists a constant ν∗>0such that for all ν≥ν∗, (1) is verified. In particular, for ν≥ν∗, the solution of (2) is unique. 2. Problem in θ Problem 1. Consider h∈L2(Ω), ξ∈L2(Γ1), ζ∈H−1/2(Γ2) and choose (see §3) u∈L3(Ω)with ∇·u= 0 (in the weak sense), splitting uinto u�+v�,beingu�∈H1(Ω) the regular part and v�∈L3(Ω) the irregular part with |v�|3small enough. First, we want to find θin L2(Ω) such that (3)3holds. Lemma 2.1 There exists a unique solution of Problem 1. Outline of the proof: For the data b∈L2(Ω), we consider the weak solution ψ∈H1 Γ1(Ω) of −χ∆ψ−(u·∇)ψ=bin Ω,ψ= 0 on Γ1,∂nψ= 0 on Γ2.(4) There exists a constant c=c(Ω) such that χ|∇ψ|2≤c|b|2(weak bound for θ). Taking a smooth family of mollifiers {ρη},η>0, we obtain that ψ∈H1 Γ1(Ω)∩H2(Ω). Indeed, let uη:= u� η+v� η,where u� η=u�∗ρη,v� η=v�∗ρη, and let ψηbe the solution in H1 Γ1(Ω) of the regularized system −χ∆ψη=b+(u� η·∇)ψη+(v� η·∇)ψη:= Fη,ψη|Γ1=0,∂nψη|Γ2=0.(5) Since u� η,v� η∈C(¯ Ω) and ∇ψη∈L2(Ω), we have Fη∈L2(Ω). Now, we use Lemma 2.2 below in order to obtain ψη∈H2(Ω) such that χ�ψη�2≤c|Fη|2, for some constant c=c(Ω). Using interpolation 3 inequalities, the weak bound for θ, and the Poincar´e’s inequality, we get �ψη�2≤cχ−1(1+χ−2�u� η�2 1)|b|2. Taking the limit for a subsequence of {η}and using the uniqueness of solution of (4) in H1 Γ1(Ω), we get �ψ�2≤cχ−1(1 + χ−2�u��2 1)|b|2.(6) Finally, we consider the map that transforms b∈L2(Ω) into the unique solution ψof (4) which is in H2(Ω). This mapping is linear. By (6), it is continuous from L2(Ω)intoH2(Ω). Thus L(b)=(h, ψ)+χ<ζ,ψ>H−1/2(Γ2),H1/2(Γ2)−χ� Γ1 ξ∂ nψds, (7) defines a continuous linear function of bacting on L2. We conclude from the Riesz representation that there exists a unique θ∈L2(Ω) such that L(b)=(θ,b) for all b∈L2(Ω). This proves the Lemma. Lemma 2.2 Let Ωbe either CASE 1 or CASE 2, and let ψ∈H1 Γ1(Ω)be the weak solution of the problem −∆ψ=fin Ωwith ψ=0on Γ1and ∂nψ=0on Γ2,for some data f∈L2(Ω).Then, ψ∈H2(Ω). The result for domains of CASE 1 is a particular case of Theorem 1 in [3,4]. The result for domains of CASE 2 follows using analogous arguments to those in Theorem 4, IV-II of [10]. An estimate for θ.Settingb=θin the equation (θ,b)=L(b), we obtain |θ|2 2=(h, ψ)+χ<ζ,ψ>H−1/2(Γ2),H1/2(Γ2)−χ� Γ1 ξ∂ nψds, (8) where ψ∈H2∩H1 Γ1satisfying ∂nψ|Γ2≡0 is the (unique) solution of (4) with b=θ. Using the weak bound of θfor b=θ, and (6), we obtain |θ|2≤cχ−1{(χ−1|h|2+(1+χ−1�u��1)(|ξ|L2(Γ1)+|ζ|H−1/2(Γ2))}.(9) 3. Problem in u and a related problem Problem 2. Consider θ∈L2(Ω),g∈L2(Γ) and f∈L3(Ω). Find u∈L3(Ω) such that (3)1,2are satisfied. Now, we consider solenoidal test functions in (3)1. Lemma 3.1 Let Ωsatisfy the hypothesis of Lemma 2.2. Consider the following boundary value problem for the Navier-Stokes equations with data g∈L2(Γ)satisfying �Γg·nds =0: −ν∆z+(z·∇)z+∇p=0,∇·z=0in Ω,z=gon Γ.(10) If |g|L2(Γ)is small enough, then there exists a unique weak solution z∈L3(Ω)of problem (10). Moreover, there is a constant c1depending only on νsuch that |z|3<c 1ν|g|L2(Γ)(ν−c1|g|L2(Γ))−1. PROOF. [Lemma 3.1] If Ωis a domain of CASE 2, the proof follows from Theorem 4 in [9]. If Ω⊂R3 is a lipschitz bounded domain of CASE 1, we need to modify Lemma 2 in [9] (cf. remark 6 in [9]) in order to guarantee the strong regularity results for the Stokes problem in irregular domains. To this aim, we use Theorem 9.20 due to M. Dauge in [5]. In order to find u, we split it into a large regular part u�and a small irregular part v�,i.e.,u=u�+v�, where u�∈H1(Ω), ∇·u�= 0 and u�|Γ=g�, and v�∈L3(Ω) is the very weak solution of problem (10) such that v�|Γ=g−g�. We choose g�(for instance in H1/2(Γ)) as an approximation of gin L2(Γ)such 4 that |g−g�|L2�1). By Lemma 3.1, we know that if |g−g�|L2(Γ)is small enough, then there exists a unique solution v�∈L3(Ω) such that |v�|3<c 1ν|g−g�|L2(Γ)(ν−c1|g−g�|L2(Γ))−1where c1=c1(ν). On the search for u�∈H1(Ω), we first lift the boundary data g�using a suitable extension � g�satisfying |B(w,� g�,w)|≤1 2ν�w�2for all w∈V(see for example [6] or [7] for Lipschitz domains, or [8,13] for the regular case). Thus, for some v∈V,we rewrite u�=v+� g�, and u=u�+v�=v+� g�+v�=v+V� where V�:= � g�+v�∈L3(Ω) and V�|Γ=g. In this way, we need to solve the following problem in v: ν((v,Φ)) = B(v,Φ,v)+β(θf,Φ)+L(v,Φ)+ <F,Φ>, (11) for all Φ∈W2,3/2(Ω)∩W1,3/2 0(Ω)with∇·Φ= 0. The bilinear form L(·,·)isdefinedbyL(v,Φ)= B(V�,Φ,v)+B(v,Φ,V�), and the linear form Fby <F,Φ>=−ν(( � g�,Φ))+B(v�,Φ,� g�)+B(� g�,Φ,V�). If vis a solution of problem (11), then it is also a variational solution, i.e., equation (11) is also valid for all Φ∈V. One can check that L(v,Φ), <F,Φ>and B(v,Φ,v) are continuous linear mappings in Φwith respect to the topology of H1(Ω). Therefore, if we prove that v∈Vis a solution of (11), then it follows that u=v+V�is a very weak solution of Problem 2. 4. Existence theorem We shall show how to construct a map A:V−→ Vwhose fixed point gives a very weak solution of (2). Having θ(the unique solution of Problem 1), we define A(v)∈Vby the relation E(Av,Φ)=B(v,Φ,v)+β(θf,Φ)+ <F,Φ>, ∀Φ∈V(12) where E(v,Φ):=ν((v,Φ)) −L(v,Φ). The bilinear form E(·,·) is continuous and coercive under our assumptions. For each θ∈L2(Ω) and v∈V, the right-hand side of (12) defines a linear and bounded functional in Φon V. Thus, by the Lax-Milgram Lemma, the mapping Ais well defined. We observed that each fixed point v∈Vof the map Adefines a pair (u,θ)=(v+V�,θ), which is a very weak solution of (2). Using the De Rham’s Lemma we show then that there exists a p∈W−1,3(Ω) such that the triple (u,θ,p) satisfies all conditions in Definition 1.1. The existence of a fixed point of Afollows from the Schauder’s fixed point Theorem. To this aim, we prove the existence of a radius Rsuch that the ball BR(V)={v∈V:�v�1≤R}satisfies the hypothesis of this theorem. We strongly use estimate (9) and the choice of Rdepending on the data χ,ν,f,ζ,ξverifying the hypothesis of Theorem 1.2. 5. Continuous dependence In [12], the continuous dependence with respect to the boundary data for θwas considered, but not for the u-boundary data. As long as the first one does not present any additional difficulty, here we focus on the dependence result on |g1−g� 1|L2(Γ)�= 0, for fixed data for ξ,ζ. We consider ν((vi,Φ)) = B(vi,Φ,vi)+β(θif,Φ)+L(vi,Φ)+ <Fi,Φ>, i =1,2,∀Φ∈V,(13) where L(vi,Φ)=B(V� i,Φ,vi)+B(vi,Φ,V� i) and <Fi,Φ>=−ν((� g� i,Φ)) + B(� g� i,Φ,V� i)+B(v� i,Φ,� g� i), for V� i=� g� i+v� i. Taking the difference between the cases i= 1 and i= 2, setting Φ=v1−v2, and estimating in a suitable manner, we obtain ν�v1−v2�1≤c�v1�1�v1−v2�1+c|θ1−θ2|2|f|3+c(�v1�1+�� g� 1�1)|v� 1−v� 2|3 +(ν+c|V� 2|3+�v1�1+�� g� 1�1)|� g� 1−� g� 2|1+c|v� 2|3�v1−v2�1. 5 In order to bound |v� 1−v� 2|3by |(g1−g� 1)−(g2−g� 2)|L2(Γ), one first proves: |v� 1−v� 2|3≤C�1+|v� 1|3+|v� 2|3�|(g1−g� 1)−(g2−g� 2)|L2(Γ). Thus, assuming that νis large enough and using the smallness of |v� 2|3and that ν�v1�1≤Mwith M= M(|f|3,|h1|2,�� g� 1�1,|ξ1|L2(Γ1),|ζ|H−1/2(Γ2)), we obtain the following estimate of continuous dependence: ν�v1−v2�1≤c|θ1−θ2|2|f|3+c(�v1�1+�� g� 1�1)�1+|v� 1|3+|v� 2|3�|(g1−g� 1)−(g2−g� 2)|L2(Γ) +c(ν+c|V� 2|3+�v1�1+�� g� 1�1)|g� 1−g� 2|H1/2(Γ). To obtain the estimate for |θ1−θ2|2, we use that (θi,b i)=L(bi)withL(b) as (7), and bi=Lui(ψi)with Lu:= −χ∆ψ+(u·∇)ψ.First,puttingξ1=ξ2and ζ1=ζ2, and taking (θ1,b i)−(θ2,b i)withbi=θ1−θ2 |θ1−θ2|2 2=(h1,ψ1)−(h2,ψ2)+χ<ζ1,(ψ1−ψ2)>H−1/2(Γ2),H1/2(Γ2)−χ� Γ1 ξ1∂n(ψ1−ψ2).(14) Suitable estimates allow us to obtain |θ1−θ2|2≤c|h1−h2|2+c(ν)|f1−f2|3provided νlarge enough. Since |u1−u2|3=|v1−v2|3≤c�v1−v2�1,wefind|u1−u2|3+|θ1−θ2|2≤c(ν)(|h1−h2|2+|f1−f2|3). If ξ1�=ξ2and ζ1�=ζ2, then the additional terms χ�Γ1(ξ1−ξ2)∂nψ2and χ<ζ1−ζ2,ψ2>H−1/2(Γ2),H1(Γ2) appearing in (14) can be estimated by c|ξ1−ξ2|L2(Γ1)|θ1−θ2|2and c|ζ1−ζ2|H−1/2(Γ2)|θ1−θ2|2respectively. Acknowledgements The authors want to express their sincere gratitude to M. Dauge, V. Girault, C. Amrouche and F. Guill´en-Gonz´alez for useful discussions and bibliographic references about this work. References [1] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, New york: Dover. [2] C. Conca, Stokes equations with non-smooth data, Rev. Mat. 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