On the Finite-Time Singularities of Point-Vortex Systems I: The Half-Plane
Abstract
In this paper, the first in a series devoted to finite-time singularities of point-vortex systems in domains with boundaries, we establish the existence of finite-time interior collapse for three vortices in the upper half-plane ℍ+. Our approach treats the boundary-induced interactions as a perturbation of the classical self-similar Gröbli collapse in the full plane. Under suitable conditions on the vortex circulations and collapse profile, we prove the existence of a one-dimensional local strong stable manifold of the augmented rescaled system, which generates exact solutions in which three distinct vortices collapse simultaneously at any prescribed interior point z*∈ℍ+. Moreover, these solutions are asymptotically self-similar, with the characteristic square-root collapse rate O(√(T−t)).
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BibTeX
@misc{yang2026finitetime,
title = {{On the Finite-Time Singularities of Point-Vortex Systems I: The Half-Plane}},
author = {Cheng Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01014},
eprint = {2611.01014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01014},
}
RIS
TY - UNPB
TI - On the Finite-Time Singularities of Point-Vortex Systems I: The Half-Plane
AU - Cheng Yang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01014
UR - https://arxiv.org/abs/2611.01014
AB - In this paper, the first in a series devoted to finite-time singularities of point-vortex systems in domains with boundaries, we establish the existence of finite-time interior collapse for three vortices in the upper half-plane ℍ^+. Our approach treats the boundary-induced interactions as a perturbation of the classical self-similar Gröbli collapse in the full plane. Under suitable conditions on the vortex circulations and collapse profile, we prove the existence of a one-dimensional local strong stable manifold of the augmented rescaled system, which generates exact solutions in which three distinct vortices collapse simultaneously at any prescribed interior point z^*∈ℍ^+. Moreover, these solutions are asymptotically self-similar, with the characteristic square-root collapse rate O(√(T−t)).
ER -