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

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 $-\Delta b$.