Liouville-type theorems for the stationary ideal compressible MHD equations

Abstract

We study Liouville-type theorems for the stationary ideal compressible magnetohydrodynamics (MHD) equations in $R^n$ for $n\ge 1$. In particular, we improve the theorems of Cai et al. (2024) (specifically Theorems 1.1 and 1.2). We remove symmetry assumptions such as axial symmetry without swirl and establish Liouville-type theorems under significantly weaker integrability conditions. We derive mean value identities and corresponding monotonicity properties to prove that smooth solutions satisfying a vanishing energy-type condition at infinity must be trivial. The results extend to lower-dimensional reduced models derived from the MHD system.