Global regularity of logarithmically supercritical MHD system with improved logarithmic powers

The magnetohydrodynamics system consists of a coupling of the Navier–Stokes equations and Maxwell’s equation from electromagnetism. We extend the work of [2] on the Navier-Stokes equations to the magnetohydrodynamics system to prove its global well-posedness with logarithmically supercritical dissipation and diffusion with the logarithmic power that is improved in contrast to the previous work of [14]. The main difficulty is that the method in [2] relies heavily on the symmetry within the Navier–Stokes equation, which is lacking in the magnetohydrodynamics system due to the non-linear terms that are mixed with both velocity and magnetic fields; this difficulty may be overcome by somehow taking advantage of the symmetry within the energy formulation of the magnetohydrodynamics system appropriately.

Keywords

fractional Laplacians, global regularity, magnetohydrodynamics system, Navier–Stokes equations, supercritical

2010 Mathematics Subject Classification

Primary 35B65. Secondary 35Q35.

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The author sincerely expresses his deep gratitude to Prof. Jiahong Wu for valuable discussion on the work of [1] as well as other related topics. The author also expresses deep gratitude to Prof. Jingna Li for her kindness and Jinan University for its hospitality throughout the author’s visit in May 2017, during which much of this work was completed. Finally, the author also expresses deep gratitude to the Editor and the anonymous referees for their valuable comments that greatly improved this manuscript.

Received 18 June 2017

Published 29 May 2018