Liouville Theorems for the Stationary Fractional Magnetohydrodynamic Equations

We establish Liouville-type theorems for the three-dimensional stationary fractional magnetohydrodynamic (MHD) system with different orders of dissipation. Our first result shows that every smooth solution satisfying the natural homogeneous Sobolev energy conditions is trivial whenever $1\le \alpha \le 5\mathrm{/}3$ and $3\le \alpha +2\beta \le 5$, without any additional integrability assumptions on the velocity or magnetic fields. This extends previous results to new parameter regimes, including cases with $\beta <1$. Beyond this region, we prove a complementary Liouville theorem under suitable integrability conditions only on the radial components of the velocity and magnetic fields. The proofs rely on the Caffarelli–Silvestre extension and localized energy estimates.