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

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 $\omega =\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}v\in L^q({\mathbb{R}}^3)$ for some $\frac{3}{2}\le q<3$, and $\mathrm{\mid }v(x)\mathrm{\mid }\to 0$ as $\mathrm{\mid }x\mathrm{\mid }\to +\mathrm{\infty }$, then either $v=0$ on ${\mathbb{R}}^3$, or $\int_{\Bbb R^6} \Phi_+ dxdy=\int_{\Bbb R^6} \Phi_- dxdy=+\infty$,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.