Analysis & PDE

• Anal. PDE
• Volume 11, Number 4 (2018), 899-918.

Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations

Léo Agélas

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

$∫ 0 T ∥ ∇ u ( t ) ∥ ∞ 1 2 log ( e + ∥ ∇ u ( t ) ∥ ∞ ) d t < + ∞ .$

This regularity criterion may stand as a great improvement over the usual BKM regularity criterion, which states that if $∫ 0 T ∥ ∇ × u ( t ) ∥ ∞ d t < + ∞$ then the solution $( u , b ) ∈ C ( [ 0 , T [ ; H r ( ℝ 2 ) )$ 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 $L ∞$ ill-posedness.

Article information

Source
Anal. PDE, Volume 11, Number 4 (2018), 899-918.

Dates
Revised: 12 September 2017
Accepted: 14 November 2017
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.apde/1517454158

Digital Object Identifier
doi:10.2140/apde.2018.11.899

Mathematical Reviews number (MathSciNet)
MR3749371

Zentralblatt MATH identifier
1383.35144

Citation

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

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