On Liouville type theorems for the stationary MHD and the Hall-MHD systems in $\mathbb{R}^3$

TL;DR

This paper establishes Liouville type theorems for stationary MHD and Hall-MHD systems in three-dimensional space, showing solutions are trivial under certain growth conditions, thus extending previous results with a refined iterative approach.

Contribution

It generalizes earlier Liouville theorems for MHD systems by incorporating Hall effects and using a refined iteration method under growth conditions.

Findings

Solutions are trivial under specified growth conditions.

The results extend previous Liouville theorems to Hall-MHD systems.

A refined iteration argument is employed in the proofs.

Abstract

In this paper we prove a Liouville type theorem for the stationary MHD and the stationary Hall-MHD systems. Assuming suitable growth condition at infinity for the mean oscillations for the potential functions, we show that the solutions are trivial. These results generalize the previous results obtained by two of the current authors in [6]. To prove our main theorems we use a refined iteration argument.