INITIAL BOUNDARY VALUE PROBLEM FOR TWO-DIMENSIONAL VISCOUS BOUSSINESQ EQUATIONS FOR MHD CONVECTION

. This paper is concerned with the initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. We show that the system has a unique classical solution for H 3 initial data, and the non-slip boundary condition for velocity ﬁeld and the perfectly conducting wall condition for magnetic ﬁeld. In addition, we show that the kinetic energy is uniformly bounded in time.

1. Introduction. We consider in this paper the following 2-D incompressible Boussinesq equations for magnetohydrodynamics (MHD) convection          u t + u · ∇u − µ∆u + ∇Π = g θe 2 + J B ⊥ , θ t + u · ∇θ = 0, B t + u · ∇B − γ∆B = B · ∇u, ∇ · u = 0, ∇ · B = 0, (1) in a bounded domain Ω with prescribed initial conditions: (θ, u, B) (x, 0) = (θ 0 , u 0 , B 0 ), x ∈ Ω, (2) and boundary conditions: where n denotes the unit outward normal on ∂Ω. The condition (3) is the socalled non-slip boundary condition, and the condition (4) is known as the perfectly conducting wall condition which describes the case where the wall of container is of strong solutions for the Cauchy problem of the 2-D inviscid Boussinesq system with temperature-dependent diffusion with large initial data in Sobolev spaces. The global regularity/singularity question for the Boussinesq system with zero viscosity and zero diffusion still remains an outstanding open problem in mathematical fluid mechanics. For Boussinese-MHD system (1) with its viscosity µ, the diffusion κ , and the electrical resistivity γ being smooth, positive functions of the temperature µ = µ(θ), κ = κ(θ), γ = γ(θ), Bian-Guo-Gui [3] and Bian-Guo-Gui-Xin [4] rigorously justified the stability and instability in a fully nonlinear, dynamical setting from mathematical point of view. In this paper, we are interested in the initial boundary vulue prolem for the system (1) with µ, γ two positive constants. For the global existence of smooth solutions, we assume the following compatibility conditions where Π 0 (x) = Π(x, 0) is the solution to the Neumann boundary problem Our main results are stated in the following theorem.
Remark 1. Because of the perfectly conducting wall condition for magnetic field, we need the Poincaré inequality revised in Lemma 2.2 to deal with the term ∇B L 2 (Ω) in (62). We also use several techniques to manipulate the nonlinear terms from the strong coupling of the velocity u, magnetic B and temperatrue θ.
Remark 2. To our knowledge, the question of global regularity/finite time singularity for the cases of partial viscosity, such as (1) with either γ = 0 or µ = 0 is still open. In a separate paper, we will consider these cases.
The proof of Theorem 1.1 consists of three main parts. First, we show the global existence of weak solutions to (1)-(4), that is, solutions satisfying the following definition:  2 satisfying Ψ(x, T ) = 0 and ∇ · Ψ = 0, and for any ζ ∈ C ∞ 0 (Ω × [0, T ]) satisfying ζ(x, T ) = 0. We then establish the regularity and uniqueness of the solution by energy estimate under the initial and boundary conditions (2)-(4).
The present paper is structured as follows. Section 2 is devoted to give some basic facts which will be used in the following sections and the global existence of weak solutions to (1)-(4). In Section 3, we improve the regularity of the solution obtained in Section 2 by energy estimate. The last Section is dedicated to establish the uniqueness of the solution, and then we prove Theorem 1.1.

Preliminaries and weak solutions.
In this section, we first list several facts which will be used in the proof of Theorem 1.1, and then give the global existence of weak solutions of (1)-(4). Lemma 2.1. ( [1,22,23]) Let Ω ∈ R 2 be any bounded domain with C 1 smooth boundary. Then the following embeddings and inequalities hold: . We also need the Poincaré inequality revised.
Let Ω be any open bounded domain in R 2 with smooth boundary ∂Ω. Assume that u satisfies u ∈ W 1,p (Ω) and u · n = 0 on ∂Ω. Then there exists a constant C = C(Ω, p) such that Proof. Assume that u ∈ W 1,p (Ω) and u · n = 0 on ∂Ω. By contradiction suppose (8) is not true. This means that for ∀ j ∈ N, ∃ u j satisfies the same hypotheses such that u j L p (Ω) ≥ j ∇u j L p (Ω) . (9) Normalize u j in L p (Ω) by setting Then, from (9), Thus, {w j } is bounded in W 1,p (Ω) and by Rellich's theorem there exists a sequence w jk and w also satisfies the same hypotheses such that The continuity of the norm gives On the other hand, the weak semicontinuity of the norm yields Since Ω is connected, w is constant and since w satisfies w · n = 0 on ∂Ω, we infer w = 0, in contradiction to w L p (Ω) = 1. Now, we establish the global existence of weak solutions of (1)-(4). Proof. Following [23,28], we can show easily the proposition by a fixed point argument. This proposition can also be proved by the method in [5,13,14]. For the sake of completeness, we give the proof using the method of [23,28] here.
For that, denoteH 1 0 (Ω) = {u ∈ H 1 (Ω) : u| ∂Ω = 0 or u · n = 0, (∇ × u) × n = 0}. Let A be the closed convex set in C([0, T ]; L 2 (Ω)) ∩ L 2 ([0, T ];H 1 0 (Ω)) defined by where v = (v 1 , v 2 ), H = (H 1 , H 2 ) and R 0 will be determined later. For fixed ε ∈ (0, 1) and any (v, H) ∈ A, we first molllify v and H by the standard procedure (see [23,28] wherev ε andH ε are the truncation of v and H in Ω ε = {x ∈ Ω|dist(x, ∂Ω) > ε} (extended by 0 to Ω), and ηε 2 is the standard mollifier. Then v ε and H ε satisfy for some constant C > 0 which is independent of ε. Similarly, we regularize the initial data to obtain the smooth approximation θ ε 0 for θ 0 , u ε 0 for u 0 and B ε 0 for B 0 1596 DONGFEN BIAN respectively, such that Then we solve the transport equation with smooth initial data and we denote the solution by θ ε . Next, we solve the nonhomogeneous (Linearized) MHD equations with smooth initial data and denote the solution by (u ε , B ε ) and the corresponding pressure by p ε . We then define the mapping F ε (v ε , H ε ) = (u ε , B ε ). The solvabilities of (12) and (13) follow easily from [23,28]. Next, using the energy method to prove that F ε satisfies the conditions of the schauder fixed point theorem, that is, F ε : A → A is continuous and compact. For any 2 ≤ p < ∞, multiplying the first equation of (12) by θ p−1 and integrating the resulting equation over Ω by parts, one gets where c(Ω, p) is a constant depending only on Ω and p. We then estimate (u ε , B ε ) 2 .
Recall the boundary condition (13) 4 and the identity ∆B = ∇(∇ · B) − ∇ × (∇ × B), it follows from integrating by parts that Taking the L 2 inner product of (13) 2 with u and (13) 3 with B respectively, integrating the resulting equations over Ω by parts, one gets By Young's inequality, the right-hand side of (16) can be estimated as which together with the result in [31] and (16), implies that where δ > 0 is a constant to be determined. By Lemma 2.2, one gets u L 2 ≤ C ∇u L 2 for some constant C depending only on Ω. Choosing δ = µ 2C in (16), one obtains Next, we prove the compactness of F ε . For that, we continue to find estimates of ∇u ε 2 C([0,T ;L 2 (Ω)]) and u ε t 2 L 2 (0,T ;L 2 (Ω)) . Taking L 2 inner product of (13) 2 and (13) 3 with u t and B t respectively, one gets We estimate the right-hand side of (18) as follows which together with (18), gives It follows from Gronwall's inequality and (11) that Using H 2 estimates to Stokes equations and elliptic equations, one obtains from (13) 2 and (13) which together with (20) yields Obviously, (20) and (22) imply that F ε is compact by the Sobolev embedding theorem. Now, we prove the continuity of Multiplying the first three equations of the above system with θ ε , χ ε and Z ε respectively, one has 1 2 Since which implies that Combining (25) with (26), one arrives at where T 0 C(t)dt ≤ C. Applying Gronwall's inequality to (27), one gets Integrating (27) over [0, T ] and using (28), one obtains This estimate together with (28), gives . By the definition, we know that A , which implies that F ε : A → A is continuous. Therefore, the Schauder theorem implies that for any fixed ε ∈ (0, 1), there exist u ε ∈ A and B ε ∈ A such that F ε (u ε , B ε ) = (u ε , B ε ), namely, , where u ε and B ε are the regularization of u ε and B ε , respectively. By a bootstrap argument (c.f. [23,28]) we know that (θ ε , u ε , B ε ) ∈ C ∞ (Ω × [0, T ]). Then, it is obvious that (θ ε , u ε , B ε ) satisfy the following integral identities: for any ε > 0, test In view of (14), (17) and the definition of u ε and B ε , we know that there exist functions u ∈ A, B ∈ A and θ ∈ C([0, T ]; L p (Ω)), ∀ 2 ≤ p < ∞, such that as ε → 0 + , Since u · ∇ζ belongs to C([0, T ]; L 2 (Ω)), one has Thus, using the above relations and letting ε → 0 + in (29), we verify that (θ, u, B) is a weak solution to (1)- (4) in Ω × [0, T ]. We conclude the argument by noticing that T is arbitrary, and then complete the proof of Proposition 1 by combining with (30).
3. Global regularity. This section is devoted to giving the regularity of the solution obtained in Proposition 1.
The proof of Proposition 2 is based on a priori estimates which are stated as a sequence of lemmas.
Proof. For any 2 ≤ p < ∞, multiplying the temperature equation of (1) by θ p−1 , integrating the resulting equation over Ω by parts, one gets d dt Ω θ p dx = 0, which implies that Furthermore, letting p → ∞ in above estimate, one shows This completes the proof of Lemma 3.1.
Proof. Multiplying the momentum and the magnetic equations of (1) by u and B respectively, integrating the resulting equations over Ω by parts, one has where we have used the boundary condition (4) and (15). By Cauchy-Schwarz inequality, the right-hand side of (34) can be estimated as which together with the result in [31], (34) and Lemma 3.1, implies that By Gronwall's inequality, one gets for any t ∈ [0, T ], which together with (35), gives This completes the proof of Lemma 3.2.
Proof. Multiplying the momentum and magnetic equations by u t and B t respectively, integrating the resulting equations over Ω by parts, one gets The right-hand side of (36) can be estimated as It follows from Lemma 2.1 (iii) and (iv) that where we have used Lemma 3.2 and δ > 0 is a small number to be determined. Hence, (36) can be rewritten as Applying H 2 estimates to Stokes equations and elliptic equations, it follows from (1) 1 , (1) 3 and (37)-(40), one obtains Choosing δ = 1 8C , combing (41)-(43), one gets µ 2 By Young inequality and Gronwall's inequality, one shows This completes the proof of Lemma 3.3.

Lemma 3.4.
Under the assumptions of Theorem1.1, it holds that Proof. Derivating the momentum and the magnetic equations of (1) with respect to t, one gets B tt + u t · ∇B + u · ∇B t = γ∆B t + B t · ∇u + B · ∇u t . (47) Multiplying (46) by u t and (47) by B t , and then integrating the resulted equations over Ω by parts, one has 1 2 . Using Gronwall's inequality, one obtains (45). Thus, the proof of Lemma 3.4 is finished.
Proof. From Lemma 2.1 (i), Lemma 3.3 and Lemma 3.4, one gets easily Thus, the proof of Lemma 3.5 is completed. Hence, which, together with Lemma 2.1 (i), gives . Therefore, applying W 2,p estimates to Stokes equations and elliptic equations, and using Lemma 3.1 and Lemma 3.5, one gets from (1) 1 and (1) 3 that This finishes the proof of Lemma 3.6.
Proof. For any p ≥ 2, applying the operator ∇ to the both sides of (1) 2 , multiplying the resulted equation with |∇θ| p−2 ∇θ, and then integrating by parts over Ω, one obtains 1 p
Lemma 3.8. Under the assumptions of Theorem 1.1, it holds that Proof. Multiplying (46) and (47) by u tt and B tt respectively, and integrating the resulting equations over Ω by parts, one obtains µ 2 For any t ∈ [0, T ], the right-hand side of the above inequality can be estimated as in the following Combining these estimates, one gets µ 2 which together with Lemma 3.4 and Lemma 3.6 gives that This completes the proof of Lemma 3.8.
Proof. Using H 3 estimates to Stokes equations and elliptic equations, it follows from (1) 1 , (1) 3 , (50), (52) and (53) that which together with Lemma 2.1 (i), gives Thus, by Lemma 2.1 (ii), one has Notice that Applying H 2 estimates to Stokes equations and elliptic equations, then from (46), (47) and Lemma 3.8, one can show that By Sobolev inequality and (55), one gets Now, it remains to give a higher order estimate on θ to complete the proof of this lemma. For that, applying the operator ∇ 2 to (1) 2 , multiplying the resulting equation by p|∇ 2 θ| p−2 ∇ 2 θ, and then integrating over Ω by parts, one obtains , where we have used (51), (56) and (57). Thus, Gronwall's inequality gives that Similarly, one can obtain that d dt (60) Applying H 4 estimates to Stokes equations and elliptic equations, one can prove from (1) 1 , (1) 3 and (58)-(60) that Thus, the proof of Lemma 3.9 is finished.
Proof. From (1) 1 and (1) 3 , one can show that for any positive δ. It follows from Lemma 2.2 that there is a constantC =C(Ω) such that Choosing δ =C −1 , it holds from (62) that Set σ = min{µ, 2γ}, then the above estimate can be rewritten as Integrating this estimate with respect to t, leads to which concludes (61). Therefore, the proof of Lemma 3.10 is completed.
4. Uniqueness. In this section, we will use the global regularity established in Lemmas 3.1-3.9 to prove the uniqueness of the solution.
Proof of Theorem 1.1. Our main result, Theorem 1.1, can be proved from Proposition 2 and Proposition 3 immediately.