American Institute of Mathematical Sciences

October  2005, 12(5): 881-886. doi: 10.3934/dcds.2005.12.881

Remarks on regularities for the 3D MHD equations

 1 Department of Mathematics, East China Normal University, Shanghai 200062, China

Received  January 2004 Revised  November 2004 Published  February 2005

In this paper we consider the regularity criteria for the solution to the 3D MHD equations. It is proved that if the gradient of the velocity field belongs to $L^{\alpha,\gamma}$ with $2/\alpha+3/\gamma \leq 2$ or the velocity field belongs to $L^{\alpha,\gamma}$ with $2/\alpha+3/\gamma \leq 1$ on $[0,T]$, then the solution remains smooth on $[0,T]$. The significance is that there are no restriction on the magnetic field. Moreover, the norms $||\nabla u||_{L^{\alpha,\gamma}}$ and $\|\|u\|\|_{L^{\alpha,\gamma}}$ are scaling dimension zero for $2/\alpha+3/\gamma=2$ and $2/\alpha+3/\gamma=1$ respectively.
Citation: Yong Zhou. Remarks on regularities for the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 881-886. doi: 10.3934/dcds.2005.12.881
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