KAM for vortex patches (Q6612705)

The survey focuses on recent advances in studying quasi-periodic vortex patch solutions of the 2d-Euler equation in \(\mathbb{R}^{2},\) close to uniformly rotating Kirchhoff elliptical vortices, whose aspect ratios belong to a set of asymptotically full Lebesgue measure. Main results in tis direction were obtained by the author the survey jointly with \textit{M. Berti} et al. [Invent. Math. 233, No. 3, 1279--1391 (2023; Zbl 1530.76015)]. \N\NThe 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.\N\NIn this survey author presents the main result and accompanied ideas of the cited above work, in particular, they prove that the bifurcation of an abundance of time quasi-periodic vortex patch solutions of the 2d-Euler equations in \(\mathbb{R}^{2}\) close to uniformly rotating Kirchhoff elliptical vortices, and that these solutions compiles a family of exact vortex patch solutions to the 2d-Euler equation discovered by Kirchhoff in 1874, which are steady in a rotating moving frame, and whose linearized equations possess infinitely many purely imaginary eigenvalues (elliptic normal frequencies) and finitely many real eigenvalues (hyperbolic directions), whose number tends to infinity as the aspect ratio of the ellipse increases to infinity. The time quasi-periodic vortex patch solutions of 2d-Euler constructed in the cited above work are not steady in any moving frame and exist for aspect ratios belonging to a set of asymptotically full Lebesgue measure. The author's key novelty in overcoming the related degeneracy problem consists in performing a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux-Caratheodory theorem of symplectic rectification, valid in finite dimension. All these results obtained are formulated as main Theorems 1 and 2. The author also states that the result mentioned above is the first one concerning the existence of quasi-periodic vortex patches for Euler equations.