Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations

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 $\left(u,b\right)\in C\left(\left[0,T\right[;H^r\left({ℝ}^2\right)\right)$ with $r>2$ remains in $H^r\left({ℝ}^2\right)$ up to time $T$ under the assumption that

$${\int }_0^T\frac{\parallel \nabla u\left(t\right){\parallel }_{\infty }^{\frac{1}{2}}}{log\left(e+\parallel \nabla u\left(t\right){\parallel }_{\infty }\right)}dt<+\infty .$$

This regularity criterion may stand as a great improvement over the usual BKM regularity criterion, which states that if ${\int }_0^T\parallel \nabla \times u\left(t\right){\parallel }_{\infty }dt<+\infty$ then the solution $\left(u,b\right)\in C\left(\left[0,T\right[;H^r\left({ℝ}^2\right)\right)$ 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.