Local interior regularity for the 3D MHD equations in nonendpoint borderline Lorentz space

: We prove local regularity condition for a suitable weak solution to 3D MHD equations. Precisely, if a solution satisﬁes u , b ∈ L ∞ ( − ( 43 ) 2 , 0; L 3 , q ( B 34 )) , q ∈ (3 , ∞ ) in Lorentz space, then ( u , b ) is H¨older continuous in the closure of the set Q 12


Introduction
We study the three-dimensional incompressible magnetohydrodynamic (3D MHD) equations (see e.g. [5]): Here u is the flow velocity vector, b is the magnetic vector and π = p + |b| 2 2 is the scalar pressure. By suitable weak solutions we mean solutions that solves MHD in the sense of distribution and satisfy the local energy inequality (see Definition 2.1 in section 2 for details). For a point z = (0, 0) ∈ R 3 × (0, T ) by translation, we denote B r (x) := B r = {y ∈ R 3 : |y − x| < r}, Q r (z) := Q r = B r × (−r 2 , 0), r < √ T .
We say that solutions u and b are regular at z ∈ R 3 × (0, T ) if u and b are bounded for some Q r , r > 0. Otherwise, it is said that u and b are singular at z. The original paper where the weak solvability of the various boundary value problems was proved is Ladyženskaja and Solonnikov [9]. As in the Navier-Stokes equations, regularity problem remains open in dimension three. On the other hand, He and Xin proved in [8] a suitable weak solution to this equations using the construction arguments of a solution in [4]. Furthermore, they show that a suitable weak solution, (u, b) become regular in the presence of a certain type of scaling invariant local integral conditions for velocity and magnetic fields. Recently, in [14], Phuc give a new regularity condition, that is, u ∈ L ∞ (−1, 0; L 3,q (B 1 )), a weak solution to the 3D Navier-Stokes equations are regular for q ∞ (cf [2]). In this paper, we give a criterion of local interior regularity as like Phuc's result for a suitable weak solution to the 3D MHD equations in Lorentz space which is still unknown (see e.g. [20,12] for the Naiver-Stokes equations). For proofs, we prove the -regularity criteria for this solution in Lorentz space (below Proposition 2.3)based on the -regularity criteria in Sobolev space. After that, using the standard blow-up argument(or contraction argument) and the unique continuation for parabolic equation, we show a solution is regular (see e.g. [1,3,6,7,13]). In summary, overall, our proof is followed the arguments in [14,2] which is mainly contained the arguments for the Naiver-Stokes equations. Now we are ready to state the first part of our main result.
Theorem 1.1. Let a pair of functions u, b and π have the following differentiability properties: Suppose that (u, b, π) satisfy the 3D MHD equations in Q 2 in the sense of distributions. Assume, in addition, that there exists 3 < q < ∞ such that u, b ∈ L ∞ (−4, 0; L 3,q (B 2 )).
Then (u, b) is Hölder continuous in the closure of the set Q 1 2 .

Preliminaries
In this section we introduce some scaling invariant functionals and suitable weak solutions, and recall an estimation of the Stokes system.
We first start with some notations. Let Ω be an open domain in R 3 and I be a finite time interval. We denote by L p,q (R 3 ) with 1 ≤ p, q ≤ ∞ the Lorentz space with the norm [21] where m(ϕ, t) is the Lebesgue measure of the set {x ∈ R 3 : |ϕ(x)| > t}, i.e. m(ϕ, t) := m{x ∈ R 3 : |ϕ(x)| > t}.
In particular, when q = ∞, The Lorentz space L p,∞ is also called weak L p space. The norm is equivalent to the norm For a function f (x, t), we denote f L p,q x,t (Ω×I) = f L q t (I;L p x (Ω)) = f L p x (Ω) L q t (I) and vector fields u, v we write (u i v j ) i, j=1,2,3 as u ⊗ v. We denote by C = C(α, β, ...) a constant depending on the prescribed quantities α, β, ..., which may change from line to line. Next we recall suitable weak solutions for the MHD equations (1.1) in three dimensions.
for all nonnegative function φ ∈ C ∞ 0 (R 3 × R). The crucial regularity result in [8] and [23] ensures that Lemma 2.1. There exists > 0 such that if (u, b, π) is a suitable weak solution of the 3D MHD equations and for r > 0, then z is a regular point.
Before a proof, we know some necessary results, which is crucial role for our analysis (see [2] and [14]). After then, using these result, we prove Theorem 1.1.
In addition, the inequalities hold for all t ∈ (−( 3 4 ) 2 , 0), and the function Here, it is clear that q q−1 = 1 in the case q = ∞. Proof. By Sobolev's inequality, we know u ∈ L 2 (−1, 0; L 6 (B 1 )). And also by the assumptions and interpolative inequality, we have which implies u ∈ L 4 (Q 1 ). Similarly, we get b ∈ L 4 (Q 1 ). Thus by Hölder's inequality, we obtain Decompose the pressure so that π = π 1 + π 2 , Here R i is Riesz operator and we adopt summation convention. It is not difficult to notice that in B ρ : By Calderón-Zygmund estimate we have and thus (2.4), it holds Estimates (2.4) and (2.4) imply that the pressure π ∈ L 2 (Q 5 6 ). With the energy class, estimate (2.2), (2.3) and (2.5), and the local interior regularity of Stokes systems , we have It then follows that ) and thus the function g ϕ (t) := . This yields Thus by the density of Then it can be seen, again by density, that the function g ϕ above is actually continuous on . Finally, using u ∈ L 4 (B 1 ) and a standard mollification in R 3+1 combined with a truncation in time of test functions, we obtain the local generalized energy equality in Q 5 6 .

Some estimates
For simplicity, we write Also, we introduce following the scale invariant functional : for 0 < r < 1, ds.
Now, we begin with stating a well known algebraic Lemma, whose proof is omitted but found in [4].
with A, B, C ≥ 0, α > β > 0 and θ ∈ [0, 1). Then there holds Lemma 2.3. Let (u, b, π) be a suitable weak solution to 3D MHD equations. Then for 0 < r the following holds Φ( Proof. Without loss of generally, consider z 0 to be the origin. Let 0 < r 2 ≤ s < ρ ≤ r < 1. Let η 1 ∈ C ∞ 0 (B(ρ)) such that 0 ≤ η 1 ≤ 1 in R 3 and η 1 = 1 on B(s). Furthermore for |α| ≤ 2: From the local energy inequality, we are known for all t ∈ I = (−1, 0) and for all non-negative functions φ ∈ C ∞ 0 (R 3 × R). Let us treat the term E 1 first. By O'Neil's inequality in space, the property of φ, and then Hölder in time, we have Lorentz spaces is characterization as interpolation space between L 2 and L 6 as follows: Before the term E 3 is estimated, we note that where we use the interpolation (2.8), Sobolev embedding and the property of φ. Set I(ρ) = ρΦ(ρ).
Using O'Neil inequality and the estimate (2.9), the term E 3 is estimated as follows: for ρ ≤ r, Similarly, we are obtained the following estimate as like E 3 : So thus, with the estimates (2.11) and (2.12), the term E 2 + E 4 is estimated by We combine with the estimate (2.7), (2.10) and (2.13) and Young's inequality to get Since r 2 ≤ s < ρ ≤ r and by Lemma 2.2, we obtain

Proof of main theorem
Following the notation in [14], we suppose that z 0 := (x 0 , t 0 ) ∈ Q 1 2 (0, 0) is a singular point. It means that there exists no neighborhood N of z 0 such that (u, b) has a Hölder continuous representative on N ∩ [B 1 (0) × (−1, 0]). By Theorem 3.2 [13], there exist c 0 > 0 and a sequence of numbers k ∈ (0, 1) such that k → 0 as k → ∞ and (2.14) for any k ∈ N. Moreover, by Proposition 2.1, we have in particular The following proposition is a key in the proof of Theorem 1.1, which says the properties in the limit.
Before proving the main statement we introduce some notation Proposition 2.3. Let (u, b, π) be a suitable weak solution to 3D MHD equations. Then there exists a universal constants c 0 and c 0k ( 0 ) (with k = 1, 2, · · · ) with the following property. Assume then for any natural number k, ∇ k−1 u is Hölder continuous inQ 1/8 and the following bound is valid:

Proof. From Lemma 2.3 and assumptions (2.31), it follows that
(2.32) By interpolation and Sobolev embedding theorem one can show that For similar reasons it is not so difficult to see that Thus, ∇ · (u × u) On the other hand, by Hölder's inequality, it is obvious that Using O'Neil's inequality, we have |π(x, t)| .