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 (u,b) C([0,T[;Hr(2)) with r > 2 remains in Hr(2) up to time T under the assumption that

0T u(t)1 2 log(e + u(t))dt < +.

This regularity criterion may stand as a great improvement over the usual BKM regularity criterion, which states that if 0T× u(t)dt < + then the solution (u,b) C([0,T[;Hr(2)) with r > 2 remains in Hr(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 ill-posedness.

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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