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

Abstract

In this paper we prove Liouville type theorem for the double Beltrami solutions to the stationary Hall-MHD equations in ${\mathbb{R}}^3$. Let $(u,B)$ be a smooth double Beltrami solution to the stationary Hall-MHD equations in ${\mathbb{R}}^3$, satisfying ${\int }_{{\mathbb{R}}^3}(|u{|}^q+|B{|}^q)dx<+\infty$ 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.

There is a calculation mistake, and the result is not yet valid