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, \(\rho^{-1 / 3} g(\rho) \rightarrow 0,\) and \(\left.\int^{\infty} \frac{d \rho}{\rho g(\rho)}=\infty\right)\), any solution with velocity and magnetic field growing at most like \(\rho^{\frac{2}{p}-\frac{1}{3}} g(\rho)^{\frac{3}{p}-1}\) 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.