Remarks on regularities for the 3D MHD equations
Remarks on regularities for the 3D MHD equations
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DOI：
10.3934/dcds.2005.12.881
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发表时间：
2005-02
影响因子：
1.1
通讯作者：
Yong Zhou
中科院分区：
文献类型：
--
作者：
Yong Zhou
本文考虑了三维MHD方程解的正则性准则。证明了如果速度场的梯度在$2/\alpha+3/\gamma \leq 2$上属于$L^{\alpha,\gamma}$，或者在$[0,T]$上速度场的梯度在$L^{\alpha,\gamma}$上属于$2/\alpha+3/\gamma \leq 1$，则在$[0,T]$上解保持光滑。重要的是磁场没有限制。此外，规范$||\nabla u||_{L^{\alpha,\gamma}}$和$\|\|u\|\|_{L^{\alpha,\gamma}}$分别为$2/\alpha+3/\gamma=2$和$2/\alpha+3/\gamma=1$的缩放维度为零。
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.