Relative Decay Conditions on Liouville Type Theorem for the Steady Navier–Stokes System

In this paper we prove Liouville type theorem for the stationary Navier–Stokes equations in \(\mathbb {R}^3\) under assumptions on relative decays of velocity, pressure and head pressure. More precisely, show that any smooth solution (u, p) satisfying \(u(x) \rightarrow 0\) as \(|x|\rightarrow +\infty \) condition finite Dirichlet integral \(\int _{\mathbb {R}^3} | \nabla u|^2 dx <+\infty is trivial, if either \(|u(x)|/|Q(x)|=O(1)\) or \(|p(x)|/|Q(x)| =O(1) \infty \), where \(|Q|=\frac{1}{2} |u|^2 +p\)