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Global regularity for a class of generalized magnetohydrodynamic equations. (English) Zbl 1270.35371

Summary: 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 T. Tao on a generalized Navier-Stokes equations [“Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equations”, arXiv: 0906.3070v3 [math.AP] (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.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
76W05 Magnetohydrodynamics and electrohydrodynamics
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