Second proof of the global regularity of the two-dimensional MHD system with full diffusion and arbitrary weak dissipation

In regards to the mathematical issue of whether a system of equations admits a unique solution for all time or not, given an arbitrary initial data sufficiently smooth, the case of the magnetohydrodynamics system may be arguably more difficult than that of the Navier–Stokes equations. In the last several years, an explosive amount of work by many mathematicians was devoted to make progress toward the global well-posedness of the two-dimensional magnetohydrodynamics system with diffusion in terms of a full Laplacian but with zero dissipation; nevertheless, this problem remains open. The purpose of this manuscript is to provide a second proof of the global well-posedness in case the diffusion is in the form of a full Laplacian, and the dissipation is in the form of a fractional Laplacian with an arbitrary small power. In contrast to the first proof of this result in the literature that took advantage of the property of a heat kernel, the main tools in this manuscript consist of Besov space techniques, in particular fractional chain rule, which has been proven to possess potentials to lead to resolutions of difficult problems, in particular of fluid dynamics partial differential equations.

Keywords

Besov space, fractional Laplacians, global regularity, magnetohydrodynamics system, Navier–Stokes equations

2010 Mathematics Subject Classification

35B65, 35Q35, 35Q61

Full Text (PDF format)

Received 23 June 2017

Accepted 21 June 2018

Published 3 January 2019