Prodi–Serrin condition for 3D MHD equations via one directional derivative of velocity and magnetic fields

Abstract

In this paper, we consider regularity conditions on weak solutions of 3D MHD equations with viscosity coefficient μ and resistivity coefficient ν being not equal. The main contribution of the present result is to establish the Prodi-Serrin regularity criterion in the case of $\mu \ne \nu$. More precisely, it shows that the weak solution of the 3D MHD equations is regular if ${\partial }_{3}u\in L^{p_0,1}(0,T;L^{q_0}(R^3\left)\right)$ and ${\partial }_{3}b\in L^{p_1,1}(0,T;L^{q_1}(R^3\left)\right)$ with $\frac{2}{p_0}+\frac{3}{q_0}=\frac{2}{p_1}+\frac{3}{q_1}=2$, $q_0\in (\frac{3}{2},2],q_1\in (\frac{3}{2},\infty ]$ or $q_0\in (\frac{3}{2},\infty ],q_1\in (\frac{3}{2},2]$. Moreover, if $\mu =\nu$, then the range of $q_0,q_1$ can be improved to $\frac{3}{2}<q_0,q_1\le \infty$. This is an alternative new Prodi-Serrin regularity criterion for the 3D MHD equations by comparing with the result of the Chen et al. (2022) [11].