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 ℝ³ with v(x)→0 as |x|→∞. We prove that v≡0 if its vorticity ω:=∇ × v belongs to L^s,∞(ℝ³) for some 9/5≤ s≤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, |ω(x)|=O(|x|^−α) implies triviality for α≥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.

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BibTeX RIS

BibTeX

@misc{wu2026beyond,
  title = {{Beyond the $L^9/5$ Vorticity Criterion in the Stationary Navier--Stokes Liouville Problem}},
  author = {Wangzhe Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.22471},
  eprint = {2608.22471},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.22471},
}

RIS

TY  - UNPB
TI  - Beyond the L^9/5 Vorticity Criterion in the Stationary Navier--Stokes Liouville Problem
AU  - Wangzhe Wu
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2608.22471
UR  - https://arxiv.org/abs/2608.22471
AB  - Let (v,p) be a smooth stationary Navier--Stokes solution in ℝ^3 with v(x)→0 as |x|→∞. We prove that v≡0 if its vorticity ω:=∇ × v belongs to L^s,∞(ℝ^3) for some 9/5≤ s≤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, |ω(x)|=O(|x|^−α) implies triviality for α≥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.
ER  -

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