EXISTENCE AND UNIQUENESS OF VERY WEAK SOLUTION OF THE MHD TYPE SYSTEM

. This paper studies the very weak solution to the steady MHD type system in a bounded domain. We prove the existence of very weak solutions to the MHD type system for arbitrary large external forces ( f , g ) in L r (Ω) × [ X θ (cid:48) ,q (cid:48) (Ω)] (cid:48) and suitable boundary data ( B 0 , U 0 ) in W − 1 /p,p ( ∂ Ω) × W − 1 /q,q ( ∂ Ω), under certain assumptions on p,q,r,θ . The uniqueness of very weak solution for small data ( f , g , B 0 , U 0 ) is also studied.

1. Introduction. In this paper, we consider a system of Magnetohydrodynamic type (MHD type) which has been used in the study of magnetic properties of electrically conducting fluids with applications in the study of Geophysics and Astrophysics. In stationary state and where there is free motion of heavy ions, this model can be reduce to the form (see for instance [23,32]) in Ω, − η ρ ∆u + (u · ∇)u + ∇π − µ ρ (B · ∇)B = g in Ω, div B = 0, div u = 0 in Ω, where ρ, σ, η, µ > 0 are constants and B is the magnetic field, u is the fluid velocity, p is the hydrostatic pressure and w is a function related to the motion of heavy ions. The given vector fields curl f and g are external forces on the magnetically charged fluid flows.
There are many studies on the time-dependent system of the above system (1). For example, existence and uniqueness of strong solution for small data in bounded or unbounded domains are studied in [29,35,27]. The temporal and spatial decay of the solution are studied in [26,27]. For more studies on the time-dependent system of (1), we refer to [2,25] and references therein.
When there is no potential term ∇w in the first equation, this model reduce to the usual MHD system. The time-dependent MHD system has been extensively studied by many mathematicians. For the existence of weak and strong solution with smooth enough data, we refer to [13,15,31] and references therein. For 5618 YONG ZENG the regularity criteria see for example [11,22] and references therein. The timeindependent MHD system is also well studied. For instance, existence of weak solutions to the steady MHD system has been studied in [1,21] by applying the Lax-Milgram theorem to some continuous forms on the Sobolev spaces. Regularity of the weak solutions follows from a standard bootstrap argument.
In this paper, we study the existence and uniqueness of very weak solution to the MHD type system (1) with singular datum. This is motivated by the works on the existence and uniqueness of very weak solutions to the stationary Navier-Stokes system, especially by the seminal papers of Kim [24] and Amrouche et.al [8]. For more studies of very weak solution to Navier-Stokes system, see for instance [4,5,6,7,17,20,28,33] and references therein. See [10] for the study of non-linear Stokes equations with singular forcing.
We mention here that Villamzar-Roa et. al. [32] studied this model in two dimensional space. They considered the Dirichlet boundary condition (u, B) = (u 0 , B 0 ) on ∂Ω with u 0 , B 0 ∈ L 2 (∂Ω) and proved the existence of very weak solution for small µσ and the uniqueness for large η of small ρ. Different from their result, we consider three dimensional space and more general boundary datum B 0 ∈ W −1/p,p (∂Ω) and U 0 ∈ W −1/q,q (∂Ω) for some 3 ≤ p ≤ q < ∞. Moreover, we shall show the existence of very weak solution for arbitrary external forces and suitable boundary datum.
To formulate our result, we introduce some notations and functional spaces for vector fields. For any functional space X, we denote X by its dual space. The notation X(div 0, Ω) denotes the space of divergence free vector fields in X(Ω). For any constant p ∈ (1, ∞), we set p = p p−1 . Mainly, we follow the notations of Amrouche et.al. [8]. For any 1 < r, p < +∞, we define Note that Y p (Ω) can also be characterized [3] by where ν denotes the unit outer normal vector. Moreover, Amrouche et.al. [8,Lemma 12] showed that the map γ τ : u → u T | ∂Ω = u − (u · ν)ν can be extended to a linear continuous map from T r,p (Ω) to W −1/p,p (∂Ω), and we have the Green formula: for all φ ∈ Y p (Ω). With the help of the above Green formula, we can now give the definition of very weak solutions to the MHD type system (1) with datum For simplicity, throughout this paper, we assume that ρ = µ = σ = η = 1.
Note that the Sobolev embedding gives W 1,p (Ω) → Lp(Ω) and W 1,q (Ω) → Lq(Ω) with 1 p = 2 3 − 1 p , and 1 q = 2 3 − 1 q . On the other hand, we have 1 p By the Hölder's inequality we see that Ω B × u · curl φ dx is well-defined. Similarly, the other integrations in the above definition are also well-defined.
Throughout this paper, Ω ⊂ R 3 always denotes a bounded connected C 2 domain, which is needed when applying the L q theory for the Stokes equations (see [12] or [9, Theorem IV.6.6, p.303]). For convenience, we set 2. Main results.
Note that the last condition in (6) implies that θ ≤ q 2 . This proposition shows only the uniqueness of "small solution". Generally, if we have another very weak solution, we do not know whether it coincides with the small solution or not. However, under some additional assumptions, we can show that any weak solution coincides with the small solution.

Preliminaries.
To prove our main results, we need some auxiliary lemmas. The first lemma is a variant of the Gerhardt inequality [18]. See also [24,Lemma 1].
for some positive constant C = C(s, t, Ω).
(ii) For any constant ε > 0, there exists a constant C ε = C(ε, s, Ω, u L 3 (Ω) ) > 0 such that The second lemma is a result of Amrouche [8] which concerns the Stokes equation, we quote it here for convenient of the reader.
Proof. We shall first make use of the Leray-Schauder fixed point theorem to show the existence of unique solution to the problem (14), and then prove that (ξ, Π 2 ) ∈ W 2,q (div 0, Ω) × W 1,q (Ω)/R by using the L q theory for the Stokes equations. Define Therefore, the L q theory for the Stokes system then ensures a unique (H,ξ) ∈ X and (P 1 , Denotes (H,ξ) = F(H, ξ), the above shows that F maps X into itself. Step Using Lemma 3.1 again, we get and similarly Letting k → +∞ and then ε → 0 in (17) and (18), we deduce that (H k , ξ k ) → (H, ξ) strongly in X, which shows that F is compact.
Concluding the above two steps, the Leray-Schauder fixed point theorem then gives the existence of (H,ξ) ∈ X to (14). The existence of Π 1 and Π 2 then follows from the de Rham theorem. The uniqueness follows immediately from the estimate (29).
4. Proof of Proposition 1. Now we are ready to prove Proposition 1.
Proof of Proposition 1. We shall use a fixed point argument. Step by the unique very weak solution to the following problem: We shall show that when (f , g, B 0 , U 0 ) ≤ δ 0 , and δ 0 and δ 3 both small enough, the map T defined above then maps M into itself. We first show that T is well defined. Let us definef = −curl (B × u + f ). Recall that B ∈ L p (div 0, Ω), u ∈ L q (div 0, Ω) and f ∈ L r (Ω, R 3 ). On the other hand, for any φ ∈ X r ,p (Ω), we have curl φ ∈ L r (Ω, R 3 ). Thus, we have (B × u + f ) · curl φ ∈ L 1 (Ω) by noting that Therefore, for the givenf and any φ ∈ X r ,p (Ω), defines a bounded linear map on X r ,p (Ω), which implies thatf = −curl (B×u+f ) ∈ [X r ,p (Ω)] . Moreover, Using Lemma 3.2, there exists a unique (D, w 0 ) which solves the equation of D in (30) such that Similarly, we haveg . Thus, by Lemma 3.2, there exists a unique (v, π 0 ) which solves the equation of v in (30) This shows that T is well defined. Combining (31) and (32), we see that where . Thus, once we choosing

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we immediately get Thus, T maps M into itself provided that (34) holds.
Step 2. We show that T is a contraction map on M provided that δ 3 is small enough.
2 ) and (w 1 , π 1 ), (w 2 , π 2 ) be the associated pressure terms of (D 1 , v 1 ), (D 2 , v 2 ) given by Lemma 3.2, respectively. Denote Thus, we have where the last inequality used that fact that Therefore, once we choosing we then have Hence, T is a contraction map on M provided that The Banach fixed point theorem then give a unique (B, u) ∈ M such that T(B, u) = (B, u). The existence of w, π then follows from the de Rham theorem.
Step 3. Now we show that (B, u, w, π) satisfies (8). To this end, recall the first inequality of (33), we conclude that by the definition of δ 3 , we immediately get

VERY WEAK SOLUTION TO MHD TYPE SYSTEM 5629
We are done.
6. Proof of Proposition 3. We need a density result which is an analogue of Lemma 15 in [8], we give the proof of this lemma in the Appendix for completeness. Lemma 6.1. Assume that p, q, r, θ ∈ (1, +∞). There exists a sequence With the help of the above density result, we can now prove Proposition 3. The general idea is that divide the MHD system (1) into two systems: one is an MHD system with small datum which is solved by Proposition 1, the other is a system with more regular datum that can be solved by using the Galerkin approximation method.
Step 2. We shall use the Galerkin Approximation method to show the existence of weak solution to (46). This is similar to the proof of existence of weak solutions to Navier-Stokes equations (see for instance [9, Theorem V.3.1, p.392]) and the steady Hall-MHD system (see [34]), with a delicately different while dealing with the coupled terms, especially the terms includingB andû. We give the proof here for completion. Let (η i , µ i ) be the eigenpairs of the Stokes operator, where η i ∈ H 1 0 (div 0, Ω). Namely, −∆η i + ∇p i = µ i η i , div η i = 0 in Ω, For each integer N , define finite dimensional space and consider the following approximation problem: where α = (α 1 , α 2 ) ∈ R N × R N . Now we define a map P from R N × R N to itself such that P (α) = β = (β 1 , β 2 ), with components By the Brouwer's fixed point theorem, it suffices to check that α · P (α) ≥ 0 when |α| large enough. Direct computation gives We now estimate T i , i = 1, · · · , 9 term by term. Thanks to (5), we see that Recall that 3 ≤ q ≤ p < +∞, by choosing γ = δ and using (44), we find that Substitute these estimates into (50), we find that By choosing δ < 1 8C(Ω) and applying the Hölder inequality, we immediately get Thus, for each N > 0, problem (49) admits a solution ( which further implies that (see for example [14,Theorem 3,p.209]) Therefore, up to a subsequence, there exists a (D, v) ∈ H 1 (div 0, Ω) × H 1 (div 0, Ω) such that v N → v weakly in H 1 (div 0, Ω) and strongly in L r (Ω, R 3 ) for all r ∈ [1, 6); Passing to the limit N → +∞ in (49), we find that holds for all Ψ and Φ in H N . By the density of ∪ ∞ N =1 H N in H 1 0 (div 0, Ω), we immediately deduce that (D, v) with the associated pressure w 2 , π 2 is a weak solution to (46).
Remark 3. From (45) and the choosing of γ and δ in Step 2, we see that δ 2 is a constant depends only on Ω, p and q.
Let B 0 ε = v ε | ∂Ω , we see that The existence of U 0 ε follows in the same way.