Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations

Agélas, Léo

doi:10.2140/apde.2018.11.899

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Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations

Léo Agélas

Vol. 11 (2018), No. 4, 899–918

DOI: 10.2140/apde.2018.11.899

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

\int_{0}^{T} \frac{∥ \nabla u (t) ∥_{\infty}^{\frac{1}{2}}}{log (e + ∥ \nabla u (t) ∥_{\infty})} d t < + \infty .

This regularity criterion may stand as a great improvement over the usual BKM regularity criterion, which states that if $\int_{0}^{T} ∥ \nabla \times u (t) ∥_{\infty} d t < + \infty$ then the solution $(u, b) \in C ([0, T [; H^{r} (ℝ^{2}))$ with $r > 2$ remains in $H^{r} (ℝ^{2})$ 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

Milestones

Received: 16 January 2017

Revised: 12 September 2017

Accepted: 14 November 2017

Published: 12 January 2018

Authors

Léo Agélas
Department of Mathematics IFP Energies nouvelles Rueil-Malmaison France

Vol. 11, No. 4, 2018