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

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

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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