Saint-Venant Estimates and Liouville-Type Theorems for the Stationary Navier-Stokes Equation in $\mathbb{R}^3$

Abstract

We prove two Liouville type theorems for the stationary Navier-Stokes equations in $\mathbb{R}^3$ under some assumptions on 1) the growth of the $L^s$ mean oscillation of a potential function of the velocity field, or 2) the relative decay of the head pressure and the square of the velocity field at infinity. The main idea is to use Saint-Venant type estimates to characterize the growth of Dirichlet energy of nontrivial solutions. These assumptions are weaker than those previously known of a similar nature.