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.
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