Massimiliano Berti

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.

We present the almost global in time existence result in [13] of small amplitude space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity and we describe the ideas of proof. This is based on a novel Hamiltonian paradifferential Birkhoff normal form approach for quasi-linear PDEs.