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 $1\leqα\leq 5/3$ and $3\leqα+2β\leq5$, without any additional integrability assumptions on the velocity or magnetic fields. This extends previous results to new parameter regimes, including cases with $β<1$. 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.