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

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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, namely

$$\sup_{j\in\mathbb{Z}}\int_0^T\|\Delta_j(\nabla\times u)\|_\infty dt,$$

where Δ j is the frequency localization operator in the Littlewood-Paley decomposition.

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088, P. R. China

    Qionglei Chen & Changxing Miao

  2. School of Mathematical Science, Peking University, Beijing, 100871, P. R. China

    Zhifei Zhang

Authors
  1. Qionglei Chen
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  2. Changxing Miao
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  3. Zhifei Zhang
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Correspondence to Changxing Miao.

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Communicated by P. Constantin

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Chen, Q., Miao, C. & Zhang, Z. The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations. Commun. Math. Phys. 275, 861–872 (2007). https://doi.org/10.1007/s00220-007-0319-y

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  • DOI: https://doi.org/10.1007/s00220-007-0319-y

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