Liouville-Type Theorems for the Stationary Ideal Magnetohydrodynamics Equations in \(\textbf{R}^n\)

Abstract

In this paper, we establish Liouville-type theorems for the stationary ideal compressible magnetohydrodynamics system in \(\textbf{R}^n\) with \(n\in \{1, 2, 3\}\). We address various cases when the finite energy condition is in force and the stationary density function \(\rho \) satisfies \(\displaystyle \lim _{|x|\rightarrow \infty }\rho (x)=\rho _\infty \ge 0\). Our proof relies heavily on the good structure of the nonlinear magnetic force term and the usage of well-chosen smooth cut-off test functions.