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 \(R^3\) . Let ( u , p ) be a smooth stationary solution to the Navier–Stokes equations in \(R^3\) , and \(Q=1/2 |u|^2 +p\) is its head pressure, which vanishes near infinity. We assume \(∫ R^3 |∇ u|^2 dx<+∞ ,\) and there exists \(α >0 \) , \(C>0\) and \(R>0\) such that \( |Q(x)| ≥ C Q L^∞|x|-α\) for all \(|x|>R\) . Suppose, furthermore, there exists \(β \) such that either \(|u(x)|=O( |x|-β)\) with \(β ≥ α /2\) or \(|∇ Q(x)|=O( |x|-β)\) with \(β ≥ 2α \) respectively as \(|x|→ +∞ \) . Then, we show that u is zero or a constant respectively on \(R^3\) .