Global regularity for generalized Hall magneto-hydrodynamics systems

Abstract

In this article, we consider the tridimensional generalized Hall magneto-hydrodynamics (Hall-MHD) system, with $(-\Delta)^\alpha u$ and $(-\Delta)^\beta b$. For $\alpha\ge 5/4$, $\beta\ge 7/4$, we obtain the global regularity of classical solutions. For $0<\alpha<5/4$ and $1/2<\beta<7/4$, with small data, the system also possesses global classical solutions. In addition, for the standard Hall-MHD system, $\alpha=\beta=1$, by adding a suitable condition, we give a positive answer to the open question in [3]. At last, we study the regularity criterions of generalized Hall-MHD system. In particular, we prove the regularity criterion in terms of horizontal gradient $\nabla_{h}u,\nabla_{h}b$ for $1<\alpha<5/4$, $5/4\le\beta<7/4$.

Keywords

• Generalized Hall-mhd system
• global regularity
• classical solutions