Liouville type theorems for stationary Navier–Stokes equations

Abstract

We show that any smooth stationary solution of the 3D incompressible Navier–Stokes equations in the whole space, the half space, or a periodic slab must vanish under the condition that for some $0 \le \delta \le 1<L$ and $q=6(3-\delta )/(6-\delta )$,

$$\begin{aligned} \liminf _{R \rightarrow \infty } \frac{1}{R} \Vert u\Vert ^{3-\delta }_{L^{q}(R<|x|<LR)}=0. \end{aligned}$$

We also prove sufficient conditions allowing shrinking radii ratio $L= 1+R^{-\alpha }$. Similar results hold on a slab with zero boundary condition by assuming stronger decay rates. We do not assume global bound of the velocity. The key is to estimate the pressure locally in the annuli with radii ratio L arbitrarily close to 1.