Liouville Type Theorem for the Steady Fractional Compressible MHD Equations in $\mathbb{R}^{3}$

Abstract

This paper is concerned with the Liouville-type problem for the stationary fractional compressible magnetohydrodynamics (MHD) equations. The main difficulty comes from the nonlocal fractional Laplace operator $(-Δ)^s$. To overcome it, we combine the Caffarelli-Silvestre extension technique with truncation arguments. Under suitable regularity and decay conditions, we prove that the only solution is trivial, establishing a Liouville-type theorem.