Beyond the $L^{9/5}$ Vorticity Criterion in the Stationary Navier--Stokes Liouville Problem

Abstract

Let $(v,p)$ be a smooth stationary Navier--Stokes solution in ${\mathbb{R}}^3$ with $v(x)\to 0$ as $|x|\to \infty$. We prove that $v\equiv 0$ if its vorticity $\omega :=\nabla \times v$ belongs to $L^{s,\infty }({\mathbb{R}}^3)$ for some $9/5\le s\le 1.87$. This gives a global weak-Lorentz vorticity criterion beyond $9/5$ requiring neither smallness nor an additional Fubini-type hypothesis. The finite Dirichlet integral is recovered from the vorticity hypothesis rather than assumed. In particular, $|\omega (x)|=O(|x{|}^{-\alpha })$ implies triviality for $\alpha \ge 300/187$, crossing the $|x{|}^{-5/3}$ vorticity-decay scale. The proof combines this automatic finite-energy upgrade with a new variable-denominator Bernoulli--vorticity quotient identity. Its completed-square form gives nonnegative defect terms, and localizing the concavity of the denominator creates a positive measure on a Bernoulli level surface. A pressure-weighted convex extension of this identity yields a quantitative normalized-vorticity trace and quotient-gradient control. These estimates are then coupled to the Bernoulli-gradient identity through a nonlinear feedback estimate.

arXiv commentv3: 32 pages. Extended the vorticity range to $9/5\le s\le 1.87$ and simplified the proof using quantitative weighted trace estimates. Our use of the Bernoulli-gradient identity was inspired by N. Lerner's arXiv:2601.13916. The author thanks Prof. Raphaël Danchin for bringing this reference to attention