Title | KAM for Vortex Patches |

Publication Type | Journal Article |

Year of Publication | 2024 |

Authors | Berti, M |

Journal | Regular and Chaotic Dynamics |

Volume | 29 |

Issue | 4 |

Pagination | 654 - 676 |

Date Published | 2024/08/01 |

ISBN Number | 1468-4845 |

Abstract | In the last years substantial mathematical progress has been made in KAM theoryfor quasi-linear/fully nonlinearHamiltonian partial differential equations, notably forwater waves and Euler equations.In this survey we focus on recent advances in quasi-periodic vortex patchsolutions 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 symplecticchange of variables is a nonlinear modification of the transport flow generated by the angularmomentum itself. |

URL | https://doi.org/10.1134/S1560354724540013 |

## KAM for Vortex Patches

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