Refined Liouville-Type Results for the Three-Dimensional Stationary MHD Equations

We prove refined Liouville-type theorems for smooth solutions to the three-dimensional stationary MHD equations. Under a mild growth condition involving a function g(ρ) (monotone, \(ρ-1 / 3 g(ρ) → 0,\) and \(.∫∞ d ρ/ρ g(ρ)=∞)\), any solution with velocity and magnetic field growing at most like \(ρ2/p-1/3 g(ρ)3/p-1\) for some 3/2 < p < 3 must be identically zero. This extends recent sharp Liouville theorems for the Navier-Stokes equations to the MHD case and allows for logarithmic or even weaker subcritical growth.