Relative decay conditions on Liouville type theorem for the steady Navier-Stokes system

Abstract

In this paper we prove Liouville type theorem for the stationary Navier-Stokes equations in ${\mathbb{R}}^3$ under the assumptions on the relative decays of velocity, pressure and the head pressure. More precisely, we show that any smooth solution $(u,p)$ of the stationary Navier-Stokes equations satisfying $u(x)\to 0$ as $|x|\to +\infty$ and the condition of finite Dirichlet integral ${\int }_{{\mathbb{R}}^3}|\nabla u{|}^{2}dx<+\infty$ is trivial, if either $|u|/|Q|=O(1)$ or $|p|/|Q|=O(1)$ as $|x|\to \infty$, where $|Q|=\frac{1}{2}|u{|}^2+p$ is the head pressure.

9 pages