New Liouville type theorems for the stationary MHD equations in $\mathbb{R}^3$

TL;DR

This paper proves new Liouville theorems for stationary magnetohydrodynamics equations in three-dimensional space, emphasizing the role of low-frequency velocity components and extending previous results.

Contribution

It introduces two novel Liouville theorems for solutions with finite energy, focusing on the significance of low-frequency velocity parts in MHD equations.

Findings

Low-frequency velocity dominates the Liouville behavior.

Established conditions under which solutions must be trivial.

Improved upon previous Liouville results by Chae-Weng.

Abstract

We research the Liouville type problem for the 3D stationary MHD equations in the frequency space. We establish two new Liouville type theorems for solutions with finite Dirichlet energy. Specifically, we show that the low-frequency part of the velocity field plays the leading role in a Liouville theory for MHD equations and then improve the results of Chae-Weng \cite{Chae-W}.