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

Abstract

We discover new cancellations upon \(H^{2}(\mathbb {R}^{n})\)-estimate of the Hall term, \(n \in \{2,3\}\). Consequently, first, we derive a regularity criterion for the 3-dimensional Hall-magnetohydrodynamics system in terms of horizontal components of velocity and magnetic fields. Second, we are able to 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 \(\alpha > \frac{1}{2}\) despite the fact that \((-\Delta )^{\frac{3}{2}}\) is the critical diffusive strength. 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 result requires is \(11+ \epsilon \) for any \(\epsilon > 0\), while the sum for the classical \(2\frac{1}{2}\)-dimensional Hall-magnetohydrodynamics system is 12 considering \(-\Delta u\) and \(-\Delta b\), of which its global regularity issue remains an outstanding open problem.