Beyond the $L^{9/5}$ Vorticity Criterion in the Stationary Navier--Stokes Liouville Problem
Year: 2026
Type: preprint
Abstract: Let $(v,p)$ be a smooth stationary Navier--Stokes solution in $\mathbb R^3$ with $v(x)\to0$ as $|x|\to\infty$. We prove that $v\equiv0$ if its vorticity $ω:=\nabla \times v$ belongs to $L^{s,\infty}(\mathbb R^3)$ for some $9/5\le s\le1.87$. 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, $|ω(x)|=O(|x|^{-α})$ implies triviality for $α\ge300/187$, 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.
Source: arXiv (Cornell University)
Author: Wu, Wangzhe
Institution: Ningbo University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204158803