A Liouville theorem for the two-dimensional stationary hypodissipative Navier--Stokes system

We study the two-dimensional stationary incompressible Navier–Stokes equations on ${\mathbb{R}}^2$ with fractional dissipation $(-\mathrm{\Delta }{)}^s$. In the full range $s\in (0,1)$, we prove that every smooth solution satisfying the natural energy condition $u\in {\dot{\mathrm{H}}}^s({\mathbb{R}}^2;{\mathbb{R}}^2)$ has $u\equiv 0$ and constant pressure. This is a fractional counterpart of the planar finite-Dirichlet theorem of Gilbarg and Weinberger at $s=1$. The proof uses different arguments in three ranges. For $0<s<\frac{1}{3}$, we combine an ${\mathrm{L}}^2$-estimate derived from the equation with a stream-function truncation argument. For $\frac{1}{3}\le s\le \frac{2}{3}$, we use a localized energy estimate whose boundary terms are supported on expanding annuli. For $\frac{2}{3}<s<1$, we establish regularity and decay via a Lorentz-space bootstrap and then apply the maximum principle to the vorticity. We also treat the stationary damped Euler system at $s=0$ by combining the Bernoulli identity with a cut-off argument under an annular growth condition that includes $u\in {\mathrm{L}}^r({\mathbb{R}}^2)$ for every $1\le r\le 2$.