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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
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
Received: 04.02.2024 Accepted: 30.04.2024
Citation:
Massimiliano Berti, “KAM for Vortex Patches”, Regul. Chaotic Dyn., 29:4 (2024), 654–676
Linking options:
https://www.mathnet.ru/eng/rcd1274 https://www.mathnet.ru/eng/rcd/v29/i4/p654
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Abstract page: | 24 | References: | 12 |
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