Liouville--type Theorems for Steady MHD and Hall--MHD Equations in $\R^2 \times \T$

TL;DR

This paper establishes Liouville--type theorems for stationary MHD and Hall--MHD equations in a slab, showing solutions must be constant under certain symmetry or boundedness conditions, extending previous results to broader cases.

Contribution

It extends Liouville--type theorems for stationary MHD and Hall--MHD equations to cases with unbounded local Dirichlet integrals, using Saint--Venant's estimates.

Findings

Solutions are constant under symmetry or boundedness assumptions.

Extends previous finite-energy results to unbounded cases.

Develops new estimates for handling magnetic field terms.

Abstract

In this paper, we study the Liouville--type theorems for three--dimensional stationary incompressible MHD and Hall--MHD systems in a slab with periodic boundary condition. We show that, under the assumptions that $(u^\theta,b^\theta)$ or $(u^r,b^r)$ is axisymmetric, or $(ru^r,rb^r)$ is bounded, any smooth bounded solution to the MHD or Hall--MHD system with local Dirichlet integral growing as an arbitrary power function must be constant. This hugely improves the result of \cite[Theorem 1.2]{pan2021Liouville}, where the Dirichlet integral of $\mathbf{u}$ is assumed to be finite. Motivated by \cite[Bang--Gui--Wang--Xie, 2022, {\it arXiv:2205.13259}]{bang2022Liouvilletype}, our proof relies on establishing Saint--Venant's estimates associated with our problem, and the result in the current paper extends that for stationary Navier--Stokes equations shown by \cite{bang2022Liouvilletype} to MHD and Hall--MHD equations. To achieve this, more intricate estimates are needed to handle the terms involving $\mathbf{b}$ properly.