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

Xuanji Jia; Yong Zhou

doi:10.3934/krm.2014.7.291

Article Contents

2014, Volume 7, Issue 2: 291-304. Doi: 10.3934/krm.2014.7.291

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

Xuanji Jia^1, and
Yong Zhou^2,

1.
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
2.
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang

Received: April 30, 2013

Revised: October 31, 2013

Published: May 2014

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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].

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Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76W05.

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References

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