Abstract:
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.
Keywords:KAM for PDEs, Euler equations, vortex patches, quasi-periodic solutions
References