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

Abstract

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 $\nabla \cdot \mathbf{\Phi }=v$ and $\nabla \cdot \mathbf{\Psi }=B$, whose $L^6$ mean oscillations have certain growth condition near infinity, namely $$-{\int }_{B(r)}|\mathbf{\Phi }-{\mathbf{\Phi }}_{B(r)}{|}^{6}dx+-{\int }_{B(r)}|\mathbf{\Psi }-{\mathbf{\Psi }}_{B(r)}{|}^{6}dx\le Cr\forall 1<r<+\infty ,$$ then $v=B=0$ and $p=$constant. With additional assumption of $$r^{-8}{\int }_{B(r)}|B-B_{B(r)}{|}^{6}dx\to 0\mathrm{as}r\to +\infty ,$$ similar result holds also for the Hall-MHD system.

16 pages. arXiv admin note: text overlap with arXiv:1811.09051