Mathematics

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.