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- W2793901589 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-Madja (BKM) regularity criterion type, namely any solution $(u,b) in C([0,T[;H^r(R^2))$ with $r>2$ remains in $H^r(R^2)$ up to time $T$ under the assumption that $dsp int_0^T frac{|nabla u(t)|_infty^{frac{1}{2}}}{log (e+|nabla u(t)|_infty)} dt 2$ remains in $H^r(R^2)$ up to time $T$. Indeed, in virtue of $nabla u = {rm R} ((nabla times u), {rm Id})$ with ${rm R}$ Riesz transform on matrix-valued functions, we expect that the blow-up rate at a time $T$ of $|nabla u(t)|_infty$ be the same as the one of $|nabla times u(t)|_infty$. Furthermore, our result applies also to a class of equations arising in hydrodynamics and studied in cite{EM} for their $L^infty$ ill-posedness." @default.
- W2793901589 created "2018-03-29" @default.
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- W2793901589 date "2018-01-12" @default.
- W2793901589 modified "2023-10-17" @default.
- W2793901589 title "Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations" @default.
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- W2793901589 doi "https://doi.org/10.2140/apde.2018.11.899" @default.
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