Regularity criteria for the solutions to the 3D MHD equations in the multiplier space

Abstract

In this paper, some improved regularity criteria for the 3D viscous MHD equations are established in multiplier spaces. It is proved that if the velocity field satisfies

$$u \in L^{\frac{2}{1-r}}\left( 0,T,\overset{.}{X}_{r}(\mathbb{R}^{3}) \right) \quad {\rm with}\,r\in [0,1[,$$

or the gradient field of velocity satisfies

$$\nabla u\in L^{\frac{2}{2-\gamma}}\left(0,T,\overset{.}{X}_{\gamma}(\mathbb{R}^{3}) \right) \quad {\rm with}\,\gamma \in \left[ 0,1\right],$$

then the solution remains smooth on [0, T].