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Hausdorff Measure of the Singular Set in the Incompressible Magnetohydrodynamic Equations

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Abstract

We derive a local energy inequality for weak solutions of the three dimensional magnetohydrodynamic equations. Combining Biot–Savart law, interpolation inequalities and the local energy inequality, we prove a partial regularity theorem for suitable weak solutions. Furthermore, we obtain an improved estimate for the logarithmic Hausdorff dimension of the singular set of suitable weak solutions.

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Authors and Affiliations

  1. Department of Mathematics, Yonsei University, Seoul, Republic of Korea

    Hi Jun Choe

  2. School of Mathematics, Korea Institute for Advanced Study, Seoul, Republic of Korea

    Minsuk Yang

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  1. Hi Jun Choe
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  2. Minsuk Yang
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Correspondence to Minsuk Yang.

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Communicated by L. Caffarelli

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Choe, H.J., Yang, M. Hausdorff Measure of the Singular Set in the Incompressible Magnetohydrodynamic Equations. Commun. Math. Phys. 336, 171–198 (2015). https://doi.org/10.1007/s00220-015-2307-y

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  • DOI: https://doi.org/10.1007/s00220-015-2307-y

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