THE ANISOTROPIC INTEGRABILITY LOGARITHMIC REGULARITY CRITERION FOR THE 3D MHD EQUATIONS

This study is devoted to investigating the regularity criterion of the 3D MHD equations in terms of pressure in the framework of anisotropic Lebesgue spaces. The result shows that if a weak solution (u, b) satisfies

1 + ln e + π(·, t) 2 then (u, b) is regular at t = T , which improve the previous results on the MHD equations

Introduction
Let us consider the following Cauchy problem of the incompressible magnetohydrodynamic (MHD) equations in three-spatial dimensions : where u is the velocity field, b the magnetic field and π the pressure, while u 0 , b 0 are given initial data with ∇ · u 0 = ∇ · b 0 = 0 in the sense of distributions.
Due to the importance of both physics and mathematics, there is large literature on the well-posedness for weak solutions of magnetohydrodynamic equations. However, similar to the Navier-Stokes equations (b = 0), the question of global regularity of the weak solutions to (2) still one of the most challenging problems in the theory of PDE's (see for example [3,4,5,6,9,10,11,12,13] and the references therein). It is interesting to study the regularity of the weak solutions of (2) by imposing some growth sufficient conditions on the velocity or the pressure. In particular, the condition via only one directional derivative of the pressure was established in [2] and showed that the weak solution becomes regular if the pressure π satisfies Later on, Jia and Zhou [7] improve (3) as Recently, some interesting logarithmical pressure regularity criteria of MHD equations are studied. In particular, Benbernou et al. [1] refined (4) by imposing the following regularity criterion The aim of this paper is to establish the logaritmical regularity criterion in terms of the partial derivative of pressure in the framework of anisotropic Lebesgue space.
Throughout the rest of this paper, we endow the usual Lebesgue space L p (R 3 ) with the norm · L p . We denote by ∂ i = ∂ ∂xi the partial derivative in the x i −direction. Recall that the anisotropic Lebesgue space consists on all the total measurable where (i, j, k) belongs to the permutation group S =span{1, 2, 3}. Before giving the main result, let us first recall the definition of weak solutions for MHD equations (2). Definition 1.1 (weak solutions). Let (u 0 , b 0 ) ∈ L 2 (R 3 ) with ∇ · u 0 = ∇ · b 0 = 0 in the sense of distribution and T > 0. A pair vector field (u(x, t), b(x, t)) is called a weak solution of (2) on (0, T ) if (u, b) satisfies the following properties : (2) in the sense of distribution.
(iv): (u, b) satisfies the energy inequality, that is, Now, our result read as follows.
is a weak solution of (2) in (0, T ). If the pressure π satisfies the condition then the weak solution (u, b) becomes a regular solution on (0, T ].
This allows us to obtain the regularity criterion of weak solutions via only one directional derivative of the pressure. This extends and improve some known regularity criterion of weak solutions in term of one directional derivative, including the notable works of Jia and Zhou [7].
As an application of Theorem 1.2, we also obtain the following regularity criterion of weak solutions.
is a weak solution of (2) in (0, T ). If the pressure satisfies the condition then the weak solution (u, b) becomes a regular solution on (0, T ]. In order to prove the main result, we need to recall the following lemma, which is proved in [9] (see also [8]). Lemma 1.4. Let us assume that r > 1 and 1 < γ ≤ α < ∞. Then for f, g, ϕ ∈ C ∞ 0 (R 3 ), we have the following estimate .
and C is a constant independent of f, g, ϕ.
2. Proof of Theorem 1.2 We are now ready to give the proof of Theorem 1.2. Clearly, in order to prove Theorem 1.2, it suffices to show that the assumption (5) ensures the following a priori estimate : Proof. We first convert the MHD system equations into a symmetric form. Adding and substracting (2) 1 and (2) 2 , we get w + and w − satisfy Next, we establish some fundamental estimates between the pressure π and w ± . Taking the divergence operator ∇· on both sides of the first and second equations of (2.1)gives where we have used the divergence free condition ∇ · w + = ∇ · w − = 0. Due to the boundedness of Riesz transform in Lebesgue space L p (1 < p < ∞), we have Similarly, acting the operator ∇div on both sides of the first and second equations of (9), one shows that −∆∇π = ∇div(w − · ∇w + ) = ∇div(w + · ∇w − ), together with Calderón-Zygmund inequality, implies that for any 1 < p < ∞, (10) ∇π L p ≤ C w − · ∇w + L p or (11) ∇π L p ≤ C w + · ∇w − L p . By means of the local existence result, (2) with (u 0 , b 0 ) ∈ L 2 (R 3 ) ∩ L 4 (R 3 ) admit a unique L 4 −strong solution (u, b) on a maximal time interval. For the notation simplicity, we may suppose that the maximal time interval is [0, T ). It is obvious that to prove regularity for u and b, it is sufficient to prove it for w + and w − . We shall show lim and thus u 2 This will lead to a contradiction to the estimates to be derived below. We now begin to follow this argument.
This completes the proof of Theorem 1.2.