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 ${\mathbb{R}}^3$. Let $(v,B,p)$ be a smooth solution to the stationary MHD equations in ${\mathbb{R}}^3$. We show that if there exist smooth matrix valued potential functions $\mathbf{\Phi }$, $\mathbf{\Psi }$ such that $\mathrm{\nabla }\cdot \mathbf{\Phi }=v$ and $\mathrm{\nabla }\cdot \mathbf{\Psi }=B$, whose $L^6$ mean oscillations have certain growth condition near infinity, namely $$-\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{B(r)}\mathrm{\mid }\mathbf{\Phi }-{\mathbf{\Phi }}_{B(r)}{\mathrm{\mid }}^{6}dx+-\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{B(r)}\mathrm{\mid }\mathbf{\Psi }-{\mathbf{\Psi }}_{B(r)}{\mathrm{\mid }}^{6}dx\le Cr\mathrm{\forall }1<r<+\mathrm{\infty },$$ then $v=B=0$ and $p=$constant. With additional assumption of $$r^{-8}{\int }_{B(r)}\mathrm{\mid }B-B_{B(r)}{\mathrm{\mid }}^{6}dx\to 0\mathrm{a}\mathrm{s}r\to +\mathrm{\infty },$$ similar result holds also for the Hall-MHD system.