Rigorous derivation of the leapfrogging motion for planar Euler equations

• 成果类型: Article
• 署名作者: Hassainia, Zineb; Hmidi, Taoufik; Masmoudi, Nader
• 署名单位: New York University; New York University Abu Dhabi; New York University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01368-3
• 发表日期: 2025
• 页码: 725-825
• 关键词: quasi-periodic solutions; vortex pairs; bifurcation; patches; FLOWS

摘要

This paper investigates the leapfrogging phenomenon in inviscid planar flows, specifically for the 2D Euler equations, showing that under suitable constraints, four concentrated vortex patches leapfrog indefinitely. Observing this system from a translating reference frame reveals a non-rigid time-periodic motion of the vortex patches. These solutions are shown to evolve near singular time-periodic relative equilibria of the point vortex system. Our proof hinges upon two main components. First, we desingularize a symmetric four point vortex configuration, which leapfrogs in accordance with Love's result (Proc. Lond. Math. Soc. 1:185-194, 1893), by concentrated vortex patches. Second, we extend KAM theory to tackle the small divisor problem complicated by the degeneracy in the time direction. This introduces new significant challenges to the KAM framework, differentiating our approach from previous work on quasi-linear systems such as (Berti et al. in Invent. Math. 233:1279-1391, 2023; Hassainia et al. in Mem. Am. Math. Soc., arXiv:2110.08615; Hmidi and Roulley in M & eacute;m. Soc. Math. Fr., 2021, arXiv:2110.13751). Notably, as the small parameter describing the patch thickness tends to zero, the system does not converge to a smooth equilibrium. Instead, it weakly approaches the Love singular configuration. Another serious difficulty stems from the degeneracy of the space modes +/- 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\pm 1$\end{document}, associated with translation invariance, at leading order. We tackle this issue by carefully examining the asymptotic behavior of the linearized operator and conducting a detailed analysis of the corresponding monodromy matrix. Our approach is both robust and flexible, providing a solution to a long-standing open problem in the study of inviscid flows and vortex dynamics.