Liouville type theorems for stationary Navier-Stokes equations

TL;DR

This paper establishes Liouville type theorems for stationary solutions of the 3D incompressible Navier-Stokes equations, showing solutions must vanish under certain decay conditions without requiring global bounds.

Contribution

It introduces new decay conditions involving local pressure estimates that guarantee triviality of stationary solutions in various domains.

Findings

Solutions vanish under specified decay conditions.

Local pressure estimates are crucial for the proofs.

Results extend to slabs with boundary conditions.

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 $q=6(3-\delta)/(6-\delta)$, $$\liminf_{R \to \infty} \frac 1R \|u\|^{3-\delta}_{L^{q}(R<|x|<LR)}=0.$$ 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.