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

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 $(r{u}^r,r{b}^r)$ 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.