Time quasi-periodic vortex patches of Euler equation in the plane

TL;DR

This paper proves the existence of time quasi-periodic vortex patch solutions for the 2D Euler equations near elliptical vortices, employing a novel symplectic reduction technique to handle degeneracies in an infinite-dimensional Hamiltonian setting.

Contribution

It introduces a new symplectic reduction method for handling degeneracies in KAM theory applied to PDEs, specifically for vortex patches in fluid dynamics.

Findings

Existence of quasi-periodic vortex patch solutions near elliptical vortices.

Development of a symplectic reduction technique for infinite-dimensional Hamiltonian systems.

Overcoming degeneracies caused by angular momentum conservation in KAM proofs.

Abstract

We prove the existence of time quasi-periodic vortex patch solutions of the 2$d$-Euler equations 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-Carath\'eodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in a infinite dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself. This is the first time such an idea is implemented in KAM for PDEs. Other difficulties are the lack of rotational symmetry of the equation and the presence of hyperbolic/elliptic normal modes. The latter difficulties -- as well as the degeneracy of a normal frequency -- are absent in other vortex patches problems which have been recently studied using the formulation introduced in this paper.