#### Vol. 11, No. 4, 2018

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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.

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