A regularity criterion for the 3D MHD equations in terms of the gradient of the pressure in the multiplier spaces

In this paper, we consider the regularity criterion for the 3D MHD equations and prove that if the gradient of the pressure belongs to L22-r(0,T;X˙r(R3))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^\frac{2}{2-r}(0,T;\dot X_r(\mathbb{R}^{3}))}$$\end{document} with 0≤r≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0\leq r\leq 1}$$\end{document} , then the solution is smooth. Notice that we extend the result given by Gala (Appl Anal 92:96–103, 2013).

We are interested in the regularity condition in terms of the pressure, the pressure gradient or its partial components. Let us now list some finest results up to date.
• In [6], the author improved [25], and established the fundamental Serrin-type regularity criterion in terms of the pressure, or the pressure gradient, that is, if one of the above two conditions holds on (0, T ) with 0 < T < ∞, then the solution is smooth on (0, T ). • Jia and Zhou [13] used intricate decomposition technique and delicate inequalities to obtain the following regularity criterion: Applying a more subtle decomposition technique (see [23,Remark 3]), Zhang et al. [23] consider the range 3/2 ≤ q ≤ 3.
In this paper, we would like to make a further contribution in this direction. We shall extend the smoothness condition for the Navier-Stokes equations (see [8]) to the MHD equations (1), whereẊ r is the multiplier spaces (see Sect. 2 below). However, due to strong coupling of the velocity field with the magnetic field, we could only be able to prove the following regularity condition for (1), Before stating the main result, let us recall the weak formulation of the MHD equations (1). (u, b) is called a weak solution to (1) with initial data (u 0 , b 0 ), provided that the following three conditions hold: (2) (1) 1,2,3,4 are satisfied in the distributional sense; (3) the energy inequality holds for almost every t ≥ 0. Now, we are ready to announce the main result of the paper. that (u, b) is a given weak solution pair of the MHD system (1) with initial data (u 0 , b 0 ) on (0, T ). If

then, the solution pair (u, b) is smooth on (0, T ).
Noticing thatẊ 0 ∼ = B M O (see Sect. 2 for the definition, and [10,19] for the equivalence relation), we have the following corollary. that (u, b) is a given weak solution pair of the MHD system (1) with initial data (u 0 , b 0 ) on (0, T ). If then the solution pair (u, b) is smooth on (0, T ).
In the rest of the paper, we make some preliminaries in Sect. 2 and prove Theorem 1.2 in Sect. 3.

Preliminaries
In this section, we recall the definition and fine properties of the multiplier spacesẊ r (see e.g., [10,19]).

Definition 2.1
For 0 ≤ r < 3/2, the homogeneous spaceẊ r is defined as the space of f ∈ L 2 loc (R 3 ) such that We have the following scaling properties: When r = 0, we haveẊ where B M O is the homogeneous space of bounded mean oscillations associated with semi-norm (see [17]) Furthermore, for 0 < r < 3 2 , we have the following strict imbeddings: which could be justified simply as In this section, we are ready to prove Theorem 1.2.
The classical Serrin-type regularity criterion, as in [11], then concludes the Proof of Theorem 1.2.