Remarks on regularities for the 3D MHD equations

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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

Abstract
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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.

Keywords: regularity criterion, MHD equations, a priori estimates..

Mathematics Subject Classification: 35Q35, 35B65, 76D0.

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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