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

TL;DR

This paper establishes a Liouville type theorem for smooth double Beltrami solutions of the stationary Hall-MHD equations in three-dimensional space, showing that under certain integrability conditions, the solutions must be trivial.

Contribution

It proves a Liouville theorem for double Beltrami solutions to the stationary Hall-MHD equations, extending known results for Beltrami solutions of Euler equations.

Findings

Double Beltrami solutions with finite q-integral are trivial (zero)

Reduces to known Euler Beltrami results when magnetic field is zero

Provides conditions under which solutions must vanish

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.