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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 8

### The regularity criteria on the magnetic field to the $3D$ incompressible MHD equations

Pages: 2257 – 2280

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n8.a8

#### Authors

#### Abstract

This note is devoted to studying the regularity conditions of the mild solution $(u,B)$ to the $3D$ incompressible MHD equations. More precisely, for the $3D$ incompressible MHD equations, [He and Xin, *J. Diff. Eqs.*, 213(2):235–254, 2005] (see also [Zhou, *Discrete Contin. Dyn. Syst.*, 12: 881–886, 2005]) proved that the velocity field is dominant in the MHD fluids; meanwhile, the effect of the magnetic field $B$ is vague. In this note, we shall establish the regularity criteria for the MHD equations in terms of

$\int^{T\ast}_0 {\lVert u_3 \rVert}^p_{\dot{H}^{\frac{1}{2} + \frac{2}{p}} (\mathbb{R}^3)} +{\lVert {\lvert \partial_3 \rvert}^{-\frac{1}{2}-\delta} B_h \rVert}^p_{\dot{H}^ {1+ \frac{2}{p}+\delta} (\mathbb{R}^3)} \mathbb{d}s \lt \infty$, with $p\in (2,\infty), \delta=3(\frac{1}{r} - \frac{1}{2} ) \gt 0$, here $r$ sufficiently close to $2$.

This result follows along the lines of [Chemin and Zhang, *Ann. Sci. Éc. Norm. Supér.*, 49:131–167, 2016], [Chemin et al., *Arch. Ration Mech. Anal.*, 224(3):871–905, 2017] and [Han et al., *Arch. Ration. Mech. Anal.*, 231:939–970, 2019], which partially improved the works of [Yamazaki, *Bull. Sci. Math.*, 140:575–614, 2016] and [Liu, *J. Diff. Eqs.*, 260:6989–7019, 2016].

#### Keywords

incompressible MHD equations in $3D$, regularity criteria

#### 2010 Mathematics Subject Classification

60F10, 60J75, 62P10, 92C37

The first and second authors were supported by NSFC grants with No. 11471126 and No. 11971199.

Received 1 June 2018

Accepted 3 September 2019

Published 3 February 2020