Liouville Theorems for the Stationary Fractional Hall-Magnetohydrodynamic Equations

In this paper we prove Liouville-type theorems for the stationary fractional Hall-magnetohydrodynamic equations in $\mathbb R^3$. Smooth solutions vanishing at infinity, with $u\in\dot H^{α/2}$ and $b\in\dot H^{β/2}$, are trivial for $1\leα\le\frac53$, $1\leβ<2$ and $α+2β\le5$, with no additional integrability assumption. The key step is to estimate the Hall boundary term through the Caffarelli--Silvestre extension of $b$.