Remarks on the global regularity issue of the two and a half dimensional Hall-magnetohydrodynamics system

TL;DR

This paper investigates the regularity of solutions to the 2.5D Hall-magnetohydrodynamics system, discovering cancellations in the Hall term that lead to new regularity criteria and global well-posedness results.

Contribution

It introduces novel regularity criteria based on partial components of the magnetic and velocity fields and proves global well-posedness with hyper-diffusion in the magnetic field.

Findings

New regularity criteria involving only specific components of the magnetic and velocity fields.

Discovery of cancellations in the Hall term enabling improved regularity results.

Proof of global well-posedness with hyper-diffusion in the magnetic field for the 2.5D system.

Abstract

Whether or not the solution to the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system starting from smooth initial data preserves its regularity for all time remains a challenging open problem. Although the research direction on component reduction of regularity criterion for Navier-Stokes equations and magnetohydrodynamics system have caught much attention recently, the Hall term has presented much difficulty. In this manuscript we discover a certain cancellation within the Hall term and obtain various new regularity criterion: first, in terms of a gradient of only the third component of the magnetic field; second, in terms of only the third component of the current density; third, in terms of only the third component of the velocity field; fourth, in terms of only the first and second components of the velocity field. As another consequence of the cancellation that we discovered, we are able to prove the global well-posedness of the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system with hyper-diffusion only for the magnetic field in the horizontal direction; we also obtained an analogous result in the 3-dimensional case via discovery of additional cancellations. These results extend and improve various previous works.