Analysis & PDE
- Anal. PDE
- Volume 11, Number 4 (2018), 899-918.
Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations
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 with remains in up to time under the assumption that
This regularity criterion may stand as a great improvement over the usual BKM regularity criterion, which states that if then the solution with remains in up to time . Furthermore, our result applies also to a class of equations arising in hydrodynamics and studied by Elgindi and Masmoudi (2014) for their ill-posedness.
Anal. PDE, Volume 11, Number 4 (2018), 899-918.
Received: 16 January 2017
Revised: 12 September 2017
Accepted: 14 November 2017
First available in Project Euclid: 1 February 2018
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Agélas, Léo. Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations. Anal. PDE 11 (2018), no. 4, 899--918. doi:10.2140/apde.2018.11.899. https://projecteuclid.org/euclid.apde/1517454158