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-Madja (BKM) regularity criterion type, namely any solution $(u,b) \in C([0,T[;H^r(\R^2))$ with $r>2$ remains in $H^r(\R^2)$ up to time $T$ under the assumption that $\dsp \int_0^T \frac{\|\nabla u(t)\|_\infty^{\frac{1}{2}}}{\log (e+\|\nabla u(t)\|_\infty)} dt 2$ remains in $H^r(\R^2)$ up to time $T$. Indeed, in virtue of $\nabla u = {\rm R} ((\nabla \times u)\, {\rm Id})$ with ${\rm R}$ Riesz transform on matrix-valued functions, we expect that the blow-up rate at a time $T$ of $\|\nabla u(t)\|_\infty$ be the same as the one of $\|\nabla \times u(t)\|_\infty$. Furthermore, our result applies also to a class of equations arising in hydrodynamics and studied in \cite{EM} for their $L^\infty$ ill-posedness.