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Global Regularity for a Class of Generalized Magnetohydrodynamic Equations

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Abstract

It remains unknown whether or not smooth solutions of the 3D incompressible MHD equations can develop finite-time singularities. One major difficulty is due to the fact that the dissipation given by the Laplacian operator is insufficient to control the nonlinearity and for this reason the 3D MHD equations are sometimes regarded as “supercritical”. This paper presents a global regularity result for the generalized MHD equations with a class of hyperdissipation. This result is inspired by a recent work of Terence Tao on a generalized Navier–Stokes equations (T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equations, arXiv: 0906.3070v3 [math.AP] 20 June 2009), but the result for the MHD equations is not completely parallel to that for the Navier–Stokes equations. Besov space techniques are employed to establish the result for the MHD equations.

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Correspondence to Jiahong Wu.

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Communicated by D. Chae

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Wu, J. Global Regularity for a Class of Generalized Magnetohydrodynamic Equations. J. Math. Fluid Mech. 13, 295–305 (2011). https://doi.org/10.1007/s00021-009-0017-y

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  • DOI: https://doi.org/10.1007/s00021-009-0017-y

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