# On the Vanishing of some D-solutions to the Stationary Magnetohydrodynamics System

## TL;DR

This paper proves that certain stationary magnetohydrodynamics solutions in a specific domain must be trivial if they are swirl-free or axially symmetric, extending Liouville type theorems.

## Contribution

It establishes new Liouville theorems for stationary MHD solutions under symmetry conditions, broadening understanding of solution triviality.

## Key findings

- D-solutions are trivial if swirl-free
- Liouville theorem holds for swirl-free velocity and axially symmetric magnetic field
- Results extend to boundary value problems in slab domains

## Abstract

In this paper, we study the stationary magnetohydrodynamics system in $\mathbb{R}^2\times\mathbb{T}$. We prove trivialness of D-solutions (the velocity field $u$ and the magnetic field $h$) when they are swirl-free. Meanwhile, this Liouville type theorem also holds provided $u$ is swirl-free and $h$ is axially symmetric, or both $u$ and $h$ are axially symmetric. Our method is also valid for certain related boundary value problems in the slab $\mathbb{R}^2\times[-\pi,\,\pi]$.