The Beale-Kato-Majda criterion to the 3D Magneto-hydrodynamics equations

Abstract

We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, i. e. $\sup_{j\in\Z}\int_0^T\|Δ_j(\na\times u)\|_\infty dt$, where ${\Delta }_j$ is a frequency localization on $|\xi |\approx 2^j$.

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