LOCAL REGULARITY OF THE MAGNETOHYDRODYNAMICS EQUATIONS NEAR THE CURVED BOUNDARY

. We study a local regularity condition for a suitable weak solutions of the magnetohydrodynamics equations near the curved boundary.

1. Introduction. We study the local regularity problem for suitable weak solutions (u, b, π) : Q T → R 3 × R 3 × R to the magnetohydrodynamics equations (MHD) in dimensions three (1) where Ω is a bounded domain in R 3 . Here u is the flow velocity vector, b is the magnetic vector and π = p+ |b| 2 2 is the magnetic pressure. The boundary conditions of u and b are given as no-slip and slip conditions, respectively, namely where ν is the outward unit normal vector along boundary Ω. The MHD equations describe the macroscopic dynamics of the interaction of moving highly conducting fluids with electro-magnetic fields such as plasma liquid metals, two-phase mixtures (see e.g. [2]).
For the existence of weak solutions for MHD equiations, it is well known that it is globally in time. Moreover, in the two-dimensional case, it become regular in [3]. On the other hand, the existence of weak solution for MHD equations with boundary condition (2) in dimension three is proved in [8] and it is shown in [10] that if weak solutions become regular under some condition. However, a regularity question remains open in dimension three not yet as in Navier-Stokes equations.
I will briefly list known results for MHD equations (1) relevant to the regularity criteria in terms of the scaled invariant quantities.

JAE-MYOUNG KIM
Numerous works of a regularity criteria for suitable weak solutions have been also studied in terms of the scaled norms. In particular, in [6], the following regularity criteria for a velocity vector for a half space was proved following as: Other types of conditions in terms of scaled invariant norms near boundary are also found in [15] and [16] (compare to [4,14,5] and [18] in the interior cases) Also, refer to papers [11,12,1] and [9] for Navier-Stokes equations. This paper is to establish the regularity criteria for the domain near the curved boundary (cf [7] for Navier-Stokes equations). To be more precise, our main result is that Hölder continuity of suitable weak solution u is ensured near sufficiently regular curved boundary provided that the scaled mixed L p,q −norm of the velocity field u is small (see Theorem 1.1 for the details).
For notational convenience, we denote for a point For x ∈Ω, we use the notation Ω x,r = Ω ∩ B x,r for some r > 0. If x = 0, we drop x in the above notations, for instance Ω 0,r is abbreviated to Ω r . A solution u and b to magnetohydrodynamics equations (1) is said to be regular at for some r > 0. In such case, z is called a regular point. Otherwise we say that u is singular at z and z is a singular point. Next, we give the assumption and remark on the boundary of Ω (see [7], [17]).
Assumption 1. Suppose that Ω be a class of C 2 ∩ W 3,∞ −boundary (which is its second derivatives are Lipschitz continuous) such that the following is satisfied: For each point x = (x , x 3 ) ∈ ∂Ω there exist absolute positive constants L, µ and r 0 independent of x such that we can find a Cartesian coordinate system {y i } 3 i=1 with the origin at x and a function ϕ : D r0 → R satisfying Remark 1. The main condition on the Assumption 1 is the uniform estimate of the C 2 −norms of the function ϕ for each x ∈ ∂Ω. More precisely, there exists a sufficiently small r 1 with r 1 < r 0 , where r 0 is the number in the Assumption 1 such that for any r < r 1 sup This can be easily shown by the Taylor formula. Now we are ready to state the main part of our main results which is local regularity criteria for a suitable weak solution for MHD equations. Theorem 1.1. Let (u, b, π) be a suitable weak solution of the MHD equations (1) according to Definition 2.1. Suppose that for a one pair p, q satisfying 3 p + 2 q ≤ 2, 2 < q ≤ ∞ and (p, q) = ( 3 2 , ∞), there exists > 0 depending only on p, q such that for some point z = (x, t) ∈ ∂Ω x,r × (0, T ) u is locally in L p,q x,t near z and lim sup r→0

LOCAL REGULARITY OF THE MAGNETOHYDRODYNAMICS EQUATIONS 509
Then, u and b are regular at z.
This paper is organized as follows. In Section 2 we introduce the definition of suitable weak solutions and give some known results for our proof of Theorems. In Section 3 we present the proofs of Theorem 1.1.

2.
Preliminaries. In this section we introduce some scaling invariant functionals and the notion of the suitable weak solutions.
We first start with some notations. Let Ω be a bounded domain in R 3 . For 1 ≤ q ≤ ∞, we denote the usual Sobolev spaces by W k,q (Ω) = {u ∈ L q (Ω) : D α u ∈ L q (Ω), 0 ≤ |α| ≤ k}. As usual, W k,q 0 (Ω) is the completion of C ∞ 0 (Ω) in the W k,q (Ω) norm. We also denote by W −k,q (Ω) the dual space of W k,q 0 (Ω), where q and q are Hölder conjugates. We write the average of f on E as We denote by C = C(α, β, ...) which may change from line to line.
As defined earlier, We also denote Ω r = Ω ∩ B r and Q r = Ω r × (−r 2 , 0). Let r 0 and r 1 be the numbers in the Assumption 1 and the Remark 1, respectively. For any r < r 1 , we introduce where κ, κ * and λ are numbers satisfying 3 κ where κ, κ * and λ are numbers satisfying (6).
Next we recall suitable weak solutions for the magnetohydrodynamics equations (1) in three dimensions.
Let Ω ⊂ R 3 be a bounded domain satisfying the Assumption 1 and I = [0, T ). We denote Q T = Ω×I. A pair of (u, b, π) is a suitable weak solution to (1) if the following conditions are satisfied: (a) The functions u, b : Q T → R 3 and p : where κ, κ * and λ be numbers satisfying 3 solves the MHD equations in Q T in the sense of distributions and u and b satisfy the boundary conditions (2) in the sense of traces. (c) u, b and π satisfy the local energy inequality Ωr for all t ∈ I = (0, T ) and for all nonnegative function φ ∈ C ∞ 0 (R 3 × R). Let x 0 ∈ ∂Ω. Under the Assumption 1, we can represent Ω x0,r0 = Ω ∩ B x0,r0 = {y = (y , y 3 ) ∈ B x0,r0 : y 3 > ϕ(y )}, where ϕ is the graph of C 2 in the Assumption 1. Flatting the boundary near x 0 , we introduce new coordinates x = ψ(y) by formulas We note that the mapping Then using the change of variables (8), the equations (1) result in the following equations for v, h andπ: where∇ and∆ are differential operators with variable coefficients defined bŷ where a ij and b i are given as As mentioned in Remark 1, if we take a sufficiently small r 1 with r 1 < r 0 , then (4) holds for any r < r 1 . In addition, the followings are satisfied: for a sufficiently small r < r 0 , ). (12) From now on, we fix x 0 = 0 without loss of generality. We suppose that, as above, ψ is a coordinate transformation so that v, π satisfies (9) in ψ(Ω r0 ).

Remark 2.
Due to the suitability of u, b, π (see Definition 2.1), (v, π) solve (9) in a weak sense and satisfies the following local energy inequality: There exists r 2 with r 2 < r 0 where r 0 is the number in the Assumption 1 such that where η ∈ C ∞ 0 (B r ) with r < r 2 and η ≥ 0, and∇ and∆ are differential operators in (10).
Next lemma shows relations between scaling invariant quantities above (see [7]).

Lemma 2.2.
Let Ω be a bounded domain satisfying the Assumption 1 and x 0 ∈ ∂Ω. Suppose that (u, p) and (v, π) are suitable weak solutions of (1) in Ω × I and (9) in ψ(Ω x0 ) × I, respectively, where ψ is the mapping flatting the boundary in the Assumption 1. Let x = ψ(x 0 ). Then there exist sufficiently small r 1 and an absolute constant C such that for any 4r < r 1 the followings are satisfied: Also, Lemma 2.2 holds for the quantitiesÊ h (2r),Â h (2r),M v (2r) andK v (2r).

Proof of Theorem.
In this section, we present the proof of the Theorem 1.1. We first show a -regularity criterion for the suitable weak solution of MHD equations (1) near the boundary. Next we prove a local regularity integration condition for the velocity vector u near boundary. For simplicity, we write Ψ(r) := A v (r) + A h (r) + E v (r) + E h (r). Let z = (x, t) ∈ Γ × I and from now on, without loss of generality, we assume x = 0 by translation. We first recall that the local energy estimate.
Next we prove a local -regularity condition near boundary for MHD equations. Its proof is similar to the proof in [6, Proposition 3.1] and so we omit. Proposition 1. There exist * > 0 and r 0 > 0 such that if (u, b, π) is a suitable weak solution of MHD equations satisfying Definition 2.1, z = (x, t) ∈ Γ × I, and then z is regular point.
The proof of Proposition 1 is based on the following lemma 3.1, which shows a decay estimate of (u, b, π).  We estimate the scaled norm for suitable weak solutions.
Next, we may continue with scaled norm of L 2,2 x,t (Q + z0,r ) estimate of b. Lemma 3.3. Let z = (x, t) ∈ Γ×I. Suppose that u ∈ L p,q x,t (Q + z,r ) with 3/p+2/q = 2 and 3/2 ≤ p < 3. Then for 0 < r < ρ/4 Proof. Although the process of proof is similar as in [6, Lemma 3.3], we will give its details for the convenience of readers. For convenience, we write x = (x 1 , x 2 , x 3 ) = (x , x 3 ) and by translation, we assume that without loss of generality, z = (0, 0) ∈ Γ×I. Let ζ(x, t) be a standard cut off function supported in Q ρ such that ζ(x, t) = 1 in Q ρ/2 . We set g(x, t) : z,ρ and we then defineg(x, t), an extension of g from Q + ρ onto Q ρ , in the following way:g( . This can be done by extending tangential components of v and h as even functions and normal components of v and h as odd functions, respectively. We denote such extensions byṽ andh for simplicity. Here we also used the fact that ζ and ∇ ζ are even and ∂ x3 ζ is odd with respect to x 3 −variable, where ∇ = (∂ x1 , ∂ x2 ).
In next lemma we show an estimate of the gradient of pressure .
We remark that, via Young's inequality, (22) can be estimated as follows: Next lemma shows an estimate of a scaled norm of pressure.
We are ready to present the proof of Theorem 1.1.
Therefore, we obtain Ψ(r) + Q(r) ≤ * /2 for all r < r 1 , which implies the regularity condition in Proposition 1. This completes the proof.