Analysis & PDE

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

Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations

Léo Agélas

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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 ( 2 ) ) with r > 2 remains in H 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 ( 2 ) ) with r > 2 remains in H 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

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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q31: Euler equations [See also 76D05, 76D07, 76N10] 35Q61: Maxwell equations

MHD Navier–Stokes Euler BKM criterion


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.

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