Communications in Mathematical Sciences

Volume 17 (2019)

Number 4

Conditional regularity for the 3D incompressible MHD equations via partial components

Pages: 1025 – 1043



Huifang Wang (School of Law, Politics and Public Management, Huaiyin Normal University, Huai’an, Jiangsu, China)

Yafei Li (Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang, China)

Zheng Guang Guo (Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang, China; and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China)

Zdenĕk Skalák (Institute of Hydrodynamics, Czech Academy of Sciences, Prague, Czech Republic)


In this paper we establish some new regularity criteria for the three dimensional incompressible magnetohydrodynamic (MHD) equations. Particularly, we prove that if $\nabla u_3$ and the horizontal magnetic field $b_h = (b_1, b_2)$ satisfy certain integrable conditions with respect to space and time variables in Lebesgue spaces, then a weak solution $(u, b)$ is actually regular. Moreover, we obtain a regularity criterion in the framework of scaling invariance.


MHD equations, regularity criteria, partial components

2010 Mathematics Subject Classification

35B65, 35Q35, 76D05

The third-named author is partially supported by the National Natural Science Foundation of China (11301394), and China Postdoctoral Science Foundation (2017M620149 and 2018T110387).

The fourth-named author is supported by the Grant Agency of the Czech Republic through grant 18-09628S, and by the Czech Academy of Sciences through RVO:67985874.

Received 15 August 2018

Accepted 24 March 2019

Published 25 October 2019