The Global Weak-Lorentz Vorticity Endpoint in the Stationary Navier--Stokes Liouville Problem

Abstract

Let $(v,p)$ be a smooth stationary Navier--Stokes flow in $\mathbb R^3$ that vanishes at infinity, and set $\omega :=\operatorname{curl}v$. We prove the endpoint implication \[ \omega\in L^{9/5,\infty}(\mathbb R^3) \quad\Longrightarrow\quad v\equiv0. \] This weak-Lorentz condition is invariant under the far-field rescaling associated with the borderline vorticity decay $|\omega(x)|=O(|x|^{-5/3})$ and strictly extends the $L^{9/5}$ vorticity criterion of Chae--Wolf. It also removes the relative smallness condition from the critical pointwise criterion of Kozono--Terasawa--Wakasugi: the decay $|\omega(x)|=O(|x|^{-5/3})$ alone implies $v\equiv0$. No finite Dirichlet energy is assumed. In our proof, we firstly use the endpoint Biot--Savart mapping and the critical annular Lorentz estimate of Seregin--Wang to yield finite Dirichlet energy. A logarithmic bound for the cumulative $L^{9/5}$ mass then selects blow-down scales whose stationary Euler limit satisfies Bernoulli companion laws on the exterior region. Together with the inherited weak endpoint bounds, these laws force the limiting energy flux to vanish. A harmonic-cutoff identity transfers this vanishing to the original scale and yields zero Dirichlet energy.