Liouville-type theorems for the stationary fractional Navier-Stokes equations in $\mathbb{R}^n$

Abstract

We establish Liouville-type theorems for the stationary fractional Navier-Stokes equations in $\mathbb{R}^n$ under suitable integrability conditions on the velocity field $u$ and a large-scale Morrey-type bound on the fractional energy. As a corollary, these assumptions are automatically satisfied if $u \in \dot{H}^{\frac{\alpha}{2}}(\mathbb{R}^n)$, yielding Liouville-type results under the finite fractional energy condition for $\frac{n}{3} \le \alpha < \frac{n+2}{3}$, where $\alpha$ denotes the order of the fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$. This range reflects a scaling-critical correspondence between Liouville-type theorems in the finite-energy setting and the threshold arising in partial regularity theory. The proof relies on direct kernel estimates for the commutator of the fractional Laplacian, based on a dyadic decomposition of the tail term, which remain valid in the hyper-dissipative case. The argument also uses a bootstrap argument that propagates integrability from near the scaling-invariant exponent down to lower exponents, including the Sobolev embedding exponent.