Remarks on the Liouville type problem in the stationary 3D Navier-Stokes equations

Abstract

We study the Liouville type problem for the stationary 3D Navier-Stokes equations on ${\mathbb{R}}^3$. Specifically, we prove that if $v$ is a smooth solution to (NS) satisfying $\omega =\mathrm{curl}v\in L^q({\mathbb{R}}^3)$ for some $\frac{3}{2}\le q<3$, and $|v(x)|\to 0$ as $|x|\to +\infty$, then either $v=0$ on ${\mathbb{R}}^3$, or ${\int }_{{\mathbb{R}}^6}{\Phi }_{+}dxdy={\int }_{{\mathbb{R}}^6}{\Phi }_{-}dxdy=+\infty$, where $\Phi (x,y):=\frac{1}{4\pi }\frac{\omega (x)\cdot (x-y)\times (v(y)\times \omega (y))}{|x-y{|}^3}$, and ${\Phi }_{\pm }:=\max \{0,\pm \Phi \}$. The proof uses crucially the structure of nonlinear term of the equations.

12 pages