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

Let $(v,p)$ be a smooth stationary Navier–Stokes flow in ${\mathbb{R}}^3$ that vanishes at infinity, and set $\omega :=\mathrm{curl}v$. We prove the endpoint implication $$\omega \in L^{9\mathrm{/}5,\mathrm{\infty }}({\mathbb{R}}^3)⟹v\equiv 0.$$ This weak-Lorentz condition is invariant under the far-field rescaling associated with the borderline vorticity decay $\mathrm{\mid }\omega (x)\mathrm{\mid }=O(\mathrm{\mid }x{\mathrm{\mid }}^{-5\mathrm{/}3})$ and strictly extends the $L^{9\mathrm{/}5}$ vorticity criterion of Chae–Wolf. It also removes the relative smallness condition from the critical pointwise criterion of Kozono–Terasawa–Wakasugi: the decay $\mathrm{\mid }\omega (x)\mathrm{\mid }=O(\mathrm{\mid }x{\mathrm{\mid }}^{-5\mathrm{/}3})$ alone implies $v\equiv 0$. 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 $L^{9\mathrm{/}5}$ 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.