Another remark on the global regularity issue of the Hall-magnetohydrodynamics system

TL;DR

This paper establishes new regularity results for the Hall-magnetohydrodynamics system by discovering cancellations in the $H^2$ estimate of the Hall term, leading to improved criteria and global regularity proofs in specific dimensions.

Contribution

It introduces novel cancellations in the Hall term estimates, enabling new regularity criteria and global regularity results for 2.5D Hall-MHD and electron MHD systems with fractional diffusion.

Findings

Derived a regularity criterion based on horizontal components in 3D Hall-MHD.

Proved global regularity for 2.5D electron MHD with fractional magnetic diffusion.

Extended global regularity results to 2.5D Hall-MHD with fractional Laplacian diffusion.

Abstract

We discover cancellations upon $H^{2}(\mathbb{R}^{n})$-estimate of the Hall term for $n \in \{2,3\}$. As its consequence, first, we derive a regularity criterion for the 3-dimensional Hall-magnetohydrodynamics system in terms of only horizontal components of velocity and magnetic fields. Second, we prove the global regularity of the $2\frac{1}{2}$-dimensional electron magnetohydrodynamics system with magnetic diffusion $(-\Delta)^{\frac{3}{2}} (b_{1}, b_{2}, 0) + (-\Delta)^{\alpha} (0, 0, b_{3})$ for $\alpha > \frac{1}{2}$. Lastly, we extend this result to the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system with $-\Delta u$ replaced by $(-\Delta)^{\alpha} (u_{1}, u_{2}, 0) -\Delta (0, 0, u_{3})$ for $\alpha > \frac{1}{2}$. The sum of the derivatives in diffusion that our global regularity result requires is $11+ \epsilon$ for any $\epsilon > 0$ while the analogous sum for the classical $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system is 12 considering $-\Delta u$ and $-\Delta b$.