AI 中文总结
研究曲面上三点涡旋坍缩现象，证明常曲率非负曲面存在自相似坍缩，双曲平面不存在特定自相似坍缩解，确立任意光滑曲面上三涡旋近自相似坍缩的存在性。
AI 中文摘要
点涡旋是描述二维理想流体动力学的重要简化模型。已知在欧几里得平面\(\mathbb{R}^2\)上存在三涡旋构型会出现有限时间奇点，即坍缩到一个点，且仅自相似坍缩。本文研究该现象在曲面上的持续程度。表明自相似坍缩是常曲率非负曲面（平面和球面）的普遍特征。相反，在双曲平面上，不存在关于由测地距离解析函数定义的任何距离变量的自相似坍缩解。最后，确立了嵌入\(\mathbb{R}^3\)的任意光滑曲面上三涡旋近自相似坍缩的存在性。
英文摘要
Point vortices represent an important reduced model describing two-dimensional ideal fluid dynamics. It is well known that there exist three-vortex configurations on the Euclidean plane $\mathbb{R}^2$ that exhibit finite-time singularities, i.e., collapse to a single point. Moreover, in $\mathbb{R}^2$, such collapses occur only self-similarly. Here, we investigate the extent to which this phenomenon persists on curved surfaces. We show that self-similar collapse is a universal feature of surfaces of nonnegative constant curvature, namely the plane and the sphere. In contrast, on the hyperbolic plane, it is shown that self-similar collapsing solutions do not exist with respect to any distance variable defined by an analytic function of the geodesic distance. Finally, we establish the existence of nearly self-similar collapse of three vortices on arbitrary smooth surfaces embedded in $\mathbb{R}^3$.
Comments17 pages
