Hill--Norbury 族附近的光滑且旋转的稳态涡环
Smooth and swirling steady vortex rings near the Hill--Norbury family
- Imperial College London（帝国理工学院）
- ETH Zürich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究三维欧拉方程中 Hill--Norbury 涡环族的去奇异化，证明小通量成员是无限阶光滑行波解的极限，并首次验证该族在光滑涡环中不孤立。
AI中文摘要：
本文致力于研究三维欧拉方程轴对称行波解空间中，在 Hill 的零通量球形涡和 Norbury 的小通量涡环附近解的局部性质。上述族由理想化的、无旋转且非光滑的对象组成，其方位角相对涡度由紧致涡核的缩放特征函数给定。我们面对去奇异化问题，既在无旋转解子空间内，也允许引入旋转，并证明 Hill--Norbury 族中每个足够小通量的成员都是欧拉方程行波解的极限，这些行波解无限阶光滑且具有紧致的环面涡度支撑。然而，在后一种旋转情形下，揭示了一种奇特的灵活性：总涡度可能沿在自然去奇异化拓扑中收敛的序列发散。我们的构造基于 Stokes 流函数的五维重构，融合了精细的薄域分析、奇异参数牛顿型非线性反演和拓扑不动点论证。据作者所知，这是首次验证小通量 Hill--Norbury 族在光滑涡环集合中并非处处孤立。
英文摘要：
An investigation of local aspects of the space of axisymmetric traveling-wave solutions to the three-dimensional Euler equations near Hill's zero-flux spherical vortex and Norbury's small-flux vortex rings is the endeavor undertaken herein. The aforementioned family consists of idealized, swirl-free, and nonsmooth objects, their azimuthal relative vorticity being prescribed by a scaled characteristic function of the compact vortex core. We confront the desingularization question, both within the swirl-free solution subspace and upon admitting swirl, and prove that each member of the Hill--Norbury family of sufficiently small flux is the limit of traveling-wave solutions to the Euler equations that are smooth to infinite order and have compact toroidal vorticity support. In the latter swirling regime, however, a curious flexibility is revealed: total vorticity may diverge along sequences converging in a natural desingularization topology. Our construction, founded upon a five-dimensional reformulation of the Stokes stream function, is an amalgamation of delicate thin-domain analysis, singular-parameter Newton-type nonlinear inversion, and topological fixed-point arguments. To the best of the authors' knowledge, this constitutes the first verification that the small-flux Hill--Norbury family is nowhere isolated from the collection of smooth vortex rings.
