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.

23 pages, no figures