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

Abstract

We study the two-dimensional stationary incompressible Navier--Stokes equations on $\mathbb R^2$ with fractional dissipation $(-\Delta)^s$. 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 $u\equiv0$ 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 $0<s<\frac13$, we combine an $\mathrm L^2$-estimate derived from the equation with a stream-function truncation argument. For $\frac13\leq s\leq\frac23$, we use a localized energy estimate whose boundary terms are supported on expanding annuli. For $\frac23<s<1$, 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 $1\le r\le2$.