The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations

By Qionglei Chen, Changxing Miao, Zhifei Zhang. 2007

159 citations recorded by the source corpus.

AbstractWe 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, namely $$\sup_{j\in\mathbb{Z}}\int_0^T\|\Delta_j(\nabla\times u)\|_\infty dt,$$where Δj is the frequency localization operator in the Littlewood-Paley decomposition.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bb783b03a123178028bb2f86d8d29f3ece67554d

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Regularity criteria for the solutions to the 3D MHD equations in the multiplier space (2010, background): “Part of them are listed here: Chen et al. [ 1 ] proved regularity by adding condition on ∆j( ∇× u); He and Wang [3] extended it to the weak-L p space L p,∞ ;W ang [9] proved regularity for u ∈ L 2 (0 ,T , BM O); Wu [10] considered the velocity field being in the homogeneous Besov space; regularity w”
On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space (2010, background): “Later, Zhou [36] improved these regularity criteria imposed only on u or its gradient ∇ u , and He and Wang [11] improved them to weak L p spaces, imposed only on u or its gradient ∇ u ; Chen, Miao and Zhang [6] also improved the criteria to a more general Besov-type space on Littlewood-Paley decomp”
BKM's Criterion and Global Weak Solutions for Magnetohydrodynamics with Zero Viscosity (2009, methodology): “Especially, for fully viscous magnetohydrodynamic equations, the analogy of Beale-Kato-Majda’s criterion is studied by Chen-Miao-Zhang [6] under the assumption only on the vorticity of velocity field, where the authors made clever use of Littlewood-Paley theory to avoid using the logarithmic Sobolev”