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Global regularity for logarithmically critical 2D MHD equations with zero viscosity

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Abstract

In this article, the two-dimensional magneto-hydrodynamic (MHD) equations are considered with only magnetic diffusion. Here the magnetic diffusion is given by \({\mathfrak D}\) a Fourier multiplier whose symbol m is given by \(m(\xi )=|\xi |^2\log (e+|\xi |^2)^\beta \). We prove that there exists an unique global solution in \(H^s(\mathbb {R}^2)\) with \(s>2\) for these equations when \(\beta >1\). This result improves the previous works which require that \(m(\xi )=|\xi |^{2\beta }\) with \(\beta >1\) and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely \(m(\xi )=|\xi |^2\).

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Acknowledgments

The author would like to thank the referee for his precious suggestions and corrections.

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  1. Department of Mathematics, IFP Energies nouvelles, 1 & 4 avenue Bois Préau, 92852, Rueil-Malmaison Cedex, France

    Léo Agélas

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  1. Léo Agélas
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Correspondence to Léo Agélas.

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Communicated by A. Constantin.

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Agélas, L. Global regularity for logarithmically critical 2D MHD equations with zero viscosity. Monatsh Math 181, 245–266 (2016). https://doi.org/10.1007/s00605-016-0958-1

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