Liouville Type Theorems for the Stationary Navier–Stokes Equations in $$\mathbb {R}^3$$

In this paper we prove Liouville type theorems for the stationary solution to the Navier–Stokes equations in $$\mathbb {R}^3$$ . Let (u, p) be a smooth stationary solution to the Navier–Stokes equations in $$\mathbb {R}^3$$ , and $$Q=\frac{1}{2} |u|^2 +p$$ is its head pressure, which vanishes near infinity. We assume $$\int _{\mathbb {R}^3} |\nabla u|^2 dx<+\infty ,$$ and there exists $$\alpha >0 $$ , $$C>0$$ and $$R>0$$ such that $$ |Q(x)| \ge C \Vert Q\Vert _{L^\infty }|x|^{-\alpha }$$ for all $$|x|>R$$ . Suppose, furthermore, there exists $$\beta $$ such that either $$|u(x)|=O( |x|^{-\beta })$$ with $$\beta \ge \frac{\alpha }{2}$$ or $$|\nabla Q(x)|=O( |x|^{-\beta })$$ with $$\beta \ge 2\alpha $$ respectively as $$|x|\rightarrow +\infty $$ . Then, we show that u is zero or a constant respectively on $$\mathbb {R}^3$$ .