发表机构
School of Mathematics and Statistics, Ningbo University（宁波大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对三维定常Navier--Stokes流的刘维尔问题，证明了涡度属于弱Lorentz空间$L^{9/5,∞}(\boldsymbol{R}^3)$即可推出流恒为零，推广了已有准则且去除了额外小性条件。
AI 中文摘要
设$(v,p)$是$\boldsymbol{R}^3$中在无穷远趋于零的光滑定常Navier--Stokes流，令$ω:=\text{curl}\thinspace v$。我们证明如下端点蕴含关系：$ω∈L^{9/5,∞}(\boldsymbol{R}^3) \boldsymbol{\text{蕴含}} v≡0$。该弱Lorentz条件在与边界涡度衰减$|ω(x)|=O(|x|^{-5/3})$相关的远场重标度下保持不变，严格推广了Chae--Wolf的$L^{9/5}$涡度准则；它还去除了Kozono--Terasawa--Wakasugi逐点临界准则中的相对小性条件：仅衰减$|ω(x)|=O(|x|^{-5/3})$即可推出$v≡0$，且无需假设有限Dirichlet能量。证明中，我们首先利用端点Biot--Savart映射与Seregin--Wang的临界环形Lorentz估计得到有限Dirichlet能量；随后通过累积$L^{9/5}$质量的对数界选取 blow-down 尺度，其定常Euler极限在外区域满足Bernoulli伴随律；结合继承的弱端点界，这些规律迫使极限能通量为零；最后通过调和截断恒等式将该零性传递到原尺度，得到零Dirichlet能量。
英文摘要
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.
Commentsv3: 32 pages. Extended the vorticity range to $9/5\le s\le1.87$ 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
