Kinetic and Related Models (KRM)

Regularity criteria for the 3D MHD equations via partial derivatives. II

Pages: 291 - 304, Volume 7, Issue 2, June 2014      doi:10.3934/krm.2014.7.291

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Xuanji Jia - Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China (email)
Yong Zhou - Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China (email)

Abstract: In this paper we continue studying regularity criteria for the 3D MHD equations via partial derivatives of the velocity or the pressure. We obtain some new regularity criteria which improve the related results in [1,3,9,11,17]. Precisely, we first prove that if for any $i,\,j,\,k\in \{1,2,3\}$ there holds $(\frac{\partial u_1}{\partial x_i},\,\frac{\partial u_2}{\partial x_j},\,\frac{\partial u_3}{\partial x_k}) \in L_T^{\alpha,\gamma}$ with $\frac{2}{\alpha}+\frac{3}{\gamma}\leq 1+\frac{1}{\gamma},~2\leq \gamma\leq \infty$, then the solution $(u,b)$ is smooth on $\mathbb{R}^3\times(0,T]$. Secondly, we show that any component (resp. components) of $(\frac{\partial u_1}{\partial x_i},\,\frac{\partial u_2}{\partial x_j},\,\frac{\partial u_3}{\partial x_k})$ in the criterion above can be replaced by the corresponding velocity component (resp. components) which is (resp. are) in the space $L_T^{\alpha',\gamma'}$with $\frac{2}{\alpha'}+\frac{3}{\gamma'}\leq 1$, $3< \gamma'\leq \infty$. Fianlly, we obtain a Ladyzhenskaya-Prodi-Serrin type regularity condition involving two components of the gradient of pressure, which in fact partially answers an open question proposed in [9] and improves Theorem 3.3 in Berselli and Galdi's article [1].

Keywords:  MHD equations, regularity criteria, partial derivatives.
Mathematics Subject Classification:  Primary: 35Q35, 35B65; Secondary: 76W05.

Received: May 2013;      Revised: November 2013;      Available Online: March 2014.

References