Liouville type theorem for double Beltrami solutions of the Hall-MHD system in $\Bbb R^3$

Abstract

In this paper we prove Liouville type theorem for the double Beltrami solutions to the stationary Hall-MHD equations in $\Bbb R^3$. Let $(u, B)$ be a smooth double Beltrami solution to the stationary Hall-MHD equations in $\Bbb R^3$, satisfying $\int_{\Bbb R^3} (|u|^q + |B|^q )dx <+\infty$ for some $q\in [2, 3)$, then $u=B=0$. In the case of $B=0$ the theorem reduces the previously known Liouville type result for the Beltrami solutions to the Euler equations.