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The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations. (English) Zbl 1138.76066
Summary: We study the blow-up criterion of smooth solutions to 3D MHD equations. By means of Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, namely
\[ \sup_{j\in\mathbb{Z}} \int_0^T\|\Delta_j(\nabla\times u)\|_\infty \,dt, \] where \(\Delta_{j}\) is the frequency localization operator in Littlewood-Paley decomposition.

MSC:
76W05 Magnetohydrodynamics and electrohydrodynamics
35Q35 PDEs in connection with fluid mechanics
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