Liouville-type theorems for steady MHD and Hall-MHD equations in [formula omitted

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 (uθ,bθ) or (ur,br) is axisymmetric, or (rur,rbr) 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 improves the result of Theorem 1.2 in Pan (2021) [33], where the Dirichlet integral of u is assumed to be finite. Inspired by Bang et al. (2022) [3], 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 [3] to MHD and Hall-MHD equations. To achieve this, more intricate estimates are needed to handle the terms involving b properly.