On Regularity for the 3D MHD Equations via One Directional Derivative of the Pressure

Abstract

This work establishes a new regularity criterion for the 3D incompressible MHD equations in term of one directional derivative of the pressure (i.e., \(\partial _{3}P\)) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that for \(T>0\), if \(\partial _{3}P\in L^{\beta }(0,T; L^{\alpha }(\mathbb {R}^{2}_{x_{1}x_{2}};L^{\gamma }(\mathbb {R}_{x_{3}})))\) with \(\frac{2}{\beta }+\frac{1}{\gamma }+\frac{2}{\alpha }=k\in [2,3)\) and \(\frac{3}{k}\le \gamma \le \alpha \le \frac{1}{k-2},\) then the corresponding solution (u, b) to the 3D MHD equations is regular on [0, T].