Geometric refinements of Liouville-type theorems for the stationary Navier--Stokes equations in $\mathbb{R}^3$

Abstract

We prove Liouville-type theorems for smooth solutions $(u,p)$ of the stationary Navier--Stokes equations in $\mathbb R^3$ satisfying the finite Dirichlet energy condition and the uniform decay condition. Our results give geometric refinements of two recent Osgood-type criteria, namely the relative decay criterion and the weighted integrability criterion formulated in terms of the head pressure $Q=\frac{1}{2}|u|^2+p$. The proof exploits several properties of the head pressure and the scalar triple-product structure of $(u\timesω)\cdot\nabla Q$. Combined with an Osgood-type representation of the total vorticity energy, this geometric structure yields Liouville-type criteria involving only the tangential interaction of $u$ and $ω$ over the superlevel sets of $|Q|$. In the relative decay setting, they lead to Osgood-scale smallness conditions that allow subcritical growth beyond the corresponding uniform bounds. In the weighted integrability setting, the previous conditions involving the full velocity and velocity gradient are refined to involve only the tangential components of $u$ and $ω$ along the level surfaces of $Q$, together with an angular factor measuring their relative orientation.