# American Institute of Mathematical Sciences

September  2020, 19(9): 4455-4478. doi: 10.3934/cpaa.2020203

## Improved blow up criterion for the three dimensional incompressible magnetohydrodynamics system

 1 Department of Mathematics, Hangzhou Dianzi University, Hangzhou, 310018, China 2 Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

* Corresponding author

Received  November 2019 Revised  March 2020 Published  June 2020

Fund Project: The first author is supported by NSFC grant 11701131. The second author is supported by China Postdoctoral Science Foundation grant 2019TQ0042

In this work, we study the regularity criterion for the 3D incompressible MHD equations. By making use of the structure of the system, we obtain a criterion that is imposed on the magnetic vector field and only one component of the velocity vector field, both in scaling invariant spaces. Moreover, the norms imposed on the magnetic vector field are the Lebesgue and anisotropic Lebesgue norms. This improved the result of our previous blow up criterion in [15], in which the magnetic vector field is bounded in critical Sobolev spaces.

Citation: Bin Han, Na Zhao. Improved blow up criterion for the three dimensional incompressible magnetohydrodynamics system. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4455-4478. doi: 10.3934/cpaa.2020203
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