Liouville Type Theorems for the Stationary Navier–Stokes Equations in R³

In this paper we prove Liouville type theorems for the stationary solution to the Navier–Stokes equations in \ (R³\). Let (u, p) be a smooth stationary solution to the Navier–Stokes equations in \ (R³\), and \ (Q=12 |u|² +p\) is its head pressure, which vanishes near infinity. We assume \ (ₑ³ | u|² dx0 \), \ (C>0\) and \ (R>0\) such that \ (|Q (x) | C Q ₋^ |x|^- \) for all \ (|x|>R\). Suppose, furthermore, there exists \ (\) such that either \ (|u (x) |=O (|x|^-) \) with \ (2\) or \ (| Q (x) |=O (|x|^-) \) with \ (2 \) respectively as \ (|x| + \). Then, we show that u is zero or a constant respectively on \ (R³\).