Regularity criteria for the 3D MHD equations involving partial components

Abstract

In this paper, we establish two new regularity criteria for the 3D incompressible MHD equations involving partial components of the velocity and magnetic fields. It is proved that if $u_3,b\in L^{\alpha }(0,T;L^{\gamma }\left(R^3\right)),\frac{2}{\alpha }+\frac{3}{\gamma }\le \frac{3}{4}+\frac{1}{2\gamma },\gamma >\frac{10}{3}$ or $u_3,b\in L^{{\alpha }_1}(0,T;L^{{\gamma }_1}\left(R^3\right)),\text{with}\frac{2}{{\alpha }_1}+\frac{3}{{\gamma }_1}\le 1,3<{\gamma }_1\le \infty$,${\partial }_3{u}_1,{\partial }_3{u}_2\in L^{{\alpha }_2}(0,T;L^{{\gamma }_2}\left(R^3\right)),\text{with}\frac{2}{{\alpha }_2}+\frac{3}{{\gamma }_2}\le 2,\frac{3}{2}<{\gamma }_2\le \infty$, then the local strong solution $(u,b)$ remains smooth on $[0,T]$.