Communications in Mathematical Sciences
Volume 17 (2019)
Conditional regularity for the 3D incompressible MHD equations via partial components
Pages: 1025 – 1043
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