On the Liouville type theorems for the stationary Navier-Stokes equations in R3

Abstract

In this paper we prove Liouville type theorems for the stationary solution to the Navier-Stokes equations in $R^3$. Let $(u,p)$ a smooth stationary solution to the Navier-Stokes equations in $R^3$, satisfying ${\int }_{R^3}|\nabla u{|}^{2}dx<+\infty$, and $Q=\frac{1}{2}|u{|}^2+p$ is its head pressure, which vanishes at infinity. Then, we prove various Liouville type theorems, using the function ξ satisfying the Osgood criterion. These theorems improve some of previously known results. The proofs use the level set methods and the coarea formula.