Liouville-type theorem for steady helically symmetric MHD system in $\mathbb{R}^{3}$

TL;DR

This paper proves that bounded, smooth, helically symmetric solutions to the steady MHD system in three-dimensional space must be constant, extending previous Liouville-type results from Navier-Stokes to MHD systems.

Contribution

It extends Liouville-type theorems from Navier-Stokes to magnetohydrodynamics, showing bounded solutions are necessarily constant under helically symmetric conditions.

Findings

Bounded smooth solutions are constant vectors.

Extension of previous Navier-Stokes results to MHD system.

Use of Saint-Venant estimate to analyze solution growth.

Abstract

We show that any bounded smooth helically symmetric solution $(\Bu, \Bh)$ in $\mathbb{R}^3$ must be constant vectors. This is an extension of previous result \cite[Theorem 1.1]{HWXAHE} from Navier-Stokes system to MHD system. The proof relies on establishing a Saint-Venant type estimate to characterize the growth of Dirichlet integral of nontrivial solutions.