On the regularity of the solutions of the Hall-magnetohydrodynamics system

Abstract

Due to the lack of sufficient dissipation and diffusion in controlling all nonlinear terms including the Hall term, the global regularity of the 3-D Hall-magnetohydrodynamics (Hall-MHD) system remains an outstanding open problem. Establishing the global regularity criterion in terms of $\nabla b_h$ is also challenging due to the inherent complexities introduced by the highly nonlinear Hall term, as described in [29, Remark 2.3(1)]. However, we obtain several regularity results, including for $\nabla b_h$, in both the 3-D Hall equation and 3-D Hall-MHD system, along with key cancellations that might play an important role in gaining a better understanding of nonlinear phenomena for the Hall-MHD system.