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

TL;DR

This paper establishes a Liouville type theorem for stationary Navier-Stokes equations in three dimensions, showing solutions are trivial under certain decay conditions on velocity, pressure, and head pressure.

Contribution

It introduces new decay conditions involving the head pressure that guarantee trivial solutions, extending previous Liouville theorems for the Navier-Stokes system.

Findings

Solutions with finite energy and decay conditions on velocity and pressure are trivial.

The decay conditions involve the ratio of velocity or pressure to the head pressure being bounded.

The theorem applies to smooth solutions in inity with decay at infinity.

Abstract

In this paper we prove Liouville type theorem for the stationary Navier-Stokes equations in $\Bbb R^3$ 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_{\Bbb 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 $|Q|=\frac12 |u|^2 +p$ is the head pressure.