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Regular and Chaotic Dynamics, 2024, Volume 29, Issue 4, Pages 654–676
DOI: https://doi.org/10.1134/S1560354724540013
(Mi rcd1274)
 

This article is cited in 1 scientific paper (total in 1 paper)

Special Issue: 70 Years of KAM Theory (Issue Editors: Alessandra Celletti, Luigi Chierchia, and Dmitry Treschev)

KAM for Vortex Patches

Massimiliano Berti

SISSA, Via Bonomea 265, 34136 Trieste, Italy
Citations (1)
References:
Abstract: In the last years substantial mathematical progress has been made in KAM theory for quasi-linear/fully nonlinear Hamiltonian partial differential equations, notably for water waves and Euler equations. In this survey we focus on recent advances in quasi-periodic vortex patch solutions of the $2d$-Euler equation in $\mathbb R^2 $ close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux – Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in an infinite-dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself.
Keywords: KAM for PDEs, Euler equations, vortex patches, quasi-periodic solutions
Funding agency Grant number
PRIN 2020XBFL
This research was supported by the project PRIN 2020XBFL “Hamiltonian and Dispersive PDEs”.
Received: 04.02.2024
Accepted: 30.04.2024
Document Type: Article
Language: English
Citation: Massimiliano Berti, “KAM for Vortex Patches”, Regul. Chaotic Dyn., 29:4 (2024), 654–676
Citation in format AMSBIB
\Bibitem{Ber24}
\by Massimiliano Berti
\paper KAM for Vortex Patches
\jour Regul. Chaotic Dyn.
\yr 2024
\vol 29
\issue 4
\pages 654--676
\mathnet{http://mi.mathnet.ru/rcd1274}
\crossref{https://doi.org/10.1134/S1560354724540013}
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  • This publication is cited in the following 1 articles:
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    References:12
     
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