A scaling invariant regularity criterion for the 3D incompressible magneto-hydrodynamics equations

Abstract

In this paper, we are concerned with the regularity criterion for weak solutions to the 3D incompressible MHD equations. We show that if any two groups functions of $$(\partial _{1}u_{1}, \partial _{1}b_{1})$$ , $$(\partial _{2}u_{2}, \partial _{2}b_{2})$$ and $$(\partial _{3}u_{3}, \partial _{3}b_{3})$$ belong to the space $$L^{\theta }([0,T);L^{r}(\mathbb {R}^3)), \frac{2}{\theta }+\frac{3}{r}=2, \frac{3}{2}<r\le \infty $$ , then the solution (u, b) to the MHD equations actually is smooth on (0, T).