Vol. 11, No. 4, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations

Léo Agélas

Vol. 11 (2018), No. 4, 899–918
Abstract

The question of whether the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only magnetic diffusion can develop a finite-time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. In this paper, we derive a regularity criterion less restrictive than the Beale–Kato–Majda (BKM) regularity criterion type, namely any solution $\left(u,b\right)\in C\left(\left[0,T\left[;{H}^{r}\left({ℝ}^{2}\right)\right)$ with $r>2$ remains in ${H}^{r}\left({ℝ}^{2}\right)$ up to time $T$ under the assumption that

${\int }_{0}^{T}\frac{\parallel \nabla u\left(t\right){\parallel }_{\infty }^{\frac{1}{2}}}{log\left(e+\parallel \nabla u\left(t\right){\parallel }_{\infty }\right)}\phantom{\rule{0.3em}{0ex}}dt<+\infty .$

This regularity criterion may stand as a great improvement over the usual BKM regularity criterion, which states that if ${\int }_{0}^{T}\phantom{\rule{0.3em}{0ex}}\parallel \nabla ×u\left(t\right){\parallel }_{\infty }\phantom{\rule{0.3em}{0ex}}dt<+\infty$ then the solution $\left(u,b\right)\in C\left(\left[0,T\left[;{H}^{r}\left({ℝ}^{2}\right)\right)$ with $r>2$ remains in ${H}^{r}\left({ℝ}^{2}\right)$ up to time $T$. Furthermore, our result applies also to a class of equations arising in hydrodynamics and studied by Elgindi and Masmoudi (2014) for their ${L}^{\infty }$ ill-posedness.

Keywords
MHD, Navier–Stokes, Euler, BKM criterion
Mathematical Subject Classification 2010
Primary: 35Q31, 35Q61