发表机构
University of Macau; Soochow University; Shanghai Jiao Tong University（澳门大学; 苏州大学; 上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在五维及以上空间中，满足横向界$|u(x)|\leq C |x'|^{-1}$的稳态Navier-Stokes方程光滑解必为零，并利用局部正则性定理消除线奇异性，推广了外部区域的渐近展开结果。
AI 中文摘要
对于$n\geq5$，设$(u,p)$是$\R^n\setminus\{x'=0\}$中稳态不可压缩Navier-Stokes方程的光滑解，其中$\{x'=0\}$为$x_n$轴。我们证明尺度不变的横向界$|u(x)|\leq C |x'|^{-1}$迫使$u\equiv0$。该结果依赖于一个定量的局部正则性定理，该定理表明横向界条件在更小的球内给出$u$和$\nabla u$的一致控制，并消除了沿轴的可能奇异性。证明结合了跨轴的弱延拓、伴随漂移算子的近似格林函数、总水头压力的一侧界、局部化Frehse-Růžička加权估计以及有限Stokes-Morrey自举。在任意大尺度上应用局部定理可得到全空间刚性和线奇异性的可去性。这与[1]中在$|u(x)|\leq C |x|^{-1}$情形下外部区域上的渐近展开相结合，在条件$|u(x)|\leq C |x'|^{-1}$下得到外部区域中的相同展开。
英文摘要
For $n\geq5$, let $(u,p)$ be a smooth solution of the stationary incompressible Navier-Stokes equations in $\R^n\setminus\{x'=0\}$, where $\{x'=0\}$ is the $x_n$-axis. We prove that the scale-invariant transverse bound $|u(x)|\leq C |x'|^{-1}$ forces $u\equiv0$. This result relies on a quantitative local regularity theorem, which shows that the transverse bound condition yields uniform control of $u$ and $\nabla u$ in a smaller ball and removes the possible singularity along the axis. The proof combines weak extension across the axis, approximate Green functions for an adjoint drift operator, a one-sided bound for the total head pressure, a localized Frehse-Růžička weighted estimate, and a finite Stokes-Morrey bootstrap. Applying the local theorem at arbitrarily large scales yields whole-space rigidity and removability of line singularities. This together with the asymptotic expansions on the exterior domains in the case $|u(x)|\leq C |x|^{-1}$ in [1], yields the same expansion in exterior domains under the condition $|u(x)|\leq C |x'|^{-1}$.
