SIAM Review, Volume 65, Issue 1, Page 227-257, February 2023.
In this paper, we prove that in bounded planar domains with $C^{2,\alpha}$ boundary, for almost every initial condition in the sense of the Lebesgue measure, the point-vortex system has a global solution, meaning that there is no collision between two point-vortices or with the boundary. This extends the work previously done in [C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, Springer-Verlag, 1984] for the disk. The proof requires the construction of a regularized dynamics that approximates the real dynamics and some strong inequalities for the Green's function of the domain. The establishment of some useful estimates is discussed and the details of the proof are given in the original article [M. Donati, SIAM J. Math. Anal., 54 (2022), pp. 79--113].

中文翻译：

---

有界平面域中点涡碰撞的不可能性

SIAM Review，第 65 卷，第 1 期，第 227-257 页，2023 年 2 月。
在本文中，我们证明在具有 $C^{2,\alpha}$ 边界的有界平面域中，对于勒贝格测度意义上的几乎每个初始条件，点涡系统都有一个全局解，这意味着有两个点涡之间或与边界之间没有碰撞。这扩展了之前在 [C. Marchioro 和 M. Pulvirenti，二维流体动力学中的涡流方法，Springer-Verlag，1984] 用于磁盘。该证明需要构造一个正则化动力学，该动力学近似于域的格林函数的真实动力学和一些强不等式。讨论了一些有用估计的建立，并在原始文章 [M. Donati，SIAM J. Math。分析，54 (2022)，第 79--113 页]。