On Liouville type theorem for the stationary Navier-Stokes equations

Dongho Chae, Joerg Wolf

Published 2018-11-22Version 1

In this paper we prove a Liouville type theorem in $\mathbb{R}^3$. Let $(u, p)$ be a smooth solution to the stationary Navier-Stokes equations in $\mathbb{R}^3$. We show that if there exists matrix valued potential function $\mathbf{V}$ such that $ \nabla \cdot \mathbf{V} =u$, whose $L^6$ mean oscillation has certain growth condition near infinity, namely $$\int_{B(r)} |\mathbf{V}- \mathbf{V}_{ B(r)} |^6 dx \le C r\quad \forall 1< r< +\infty,$$ then $u\equiv 0$.