On Liouville type theorems for the stationary MHD and Hall-MHD systems

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics(MHD) system in $\Bbb R^3$. Let $(v, B, p)$ be a smooth solution to the stationary MHD equations in $\Bbb R^3$. We show that if there exist smooth matrix valued potential functions ${\bf \Phi}$, ${\bf \Psi}$ such that $\nabla \cdot {\bf \Phi} =v$ and $\nabla \cdot {\bf \Psi}= B$, whose $L^6$ mean oscillations have certain growth condition near infinity, namely $$-\!\!\!\!\!\int_{B(r)} |\mathbf{\Phi} - \mathbf{\Phi}_{ B(r)} |^6 dx + -\!\!\!\!\!\int_{B(r)} |\mathbf{\Psi}- \mathbf{\Psi}_{ B(r)} |^6 dx\le C r\quad \forall 1< r< +\infty,$$ then $v=B= 0$ and $p=$constant. With additional assumption of $$r^{-8}\int_{B(r)}|B-B_{B(r)}|^6dx\to 0\quad \mathrm{as}\quad r\to+\infty,$$ similar result holds also for the Hall-MHD system.