On Liouville type theorems for the steady Navier-Stokes equations in ${\mathbb{R}}^3$

Abstract

In this paper we prove three different Liouville type theorems for the steady Navier-Stokes equations in ${\mathbb{R}}^3$. In the first theorem we improve logarithmically the well-known $L^{\frac{9}{2}}({\mathbb{R}}^3)$ result. In the second theorem we present a sufficient condition for the trivially of the solution($v=0$) in terms of the head pressure, $Q=\frac{1}{2}|v{|}^2+p$. The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee $v=0$.

17 pages