, Volume 61, Issue 2, pp 193-199
Date: 17 Jul 2009

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

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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].