@article {2024,
	title = {KAM for Vortex Patches},
	journal = {Regular and Chaotic Dynamics},
	volume = {29},
	year = {2024},
	month = {2024/08/01},
	pages = {654 - 676},
	abstract = {<p>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 {\textendash} Carath{\'e}odory 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.</p>},
	isbn = {1468-4845},
	url = {https://doi.org/10.1134/S1560354724540013},
	author = {Massimiliano Berti}
}
