Invariant KAM Tori Around Annular Vortex Patches for 2D Euler Equations

Abstract

We construct time quasi-periodic vortex patch solutions with one hole for the planar Euler equations. These structures are captured close to any annulus provided that its modulus belongs to a massive Borel set. The proof is based on Nash–Moser scheme and KAM theory applied with a Hamiltonian system governing the radial deformations of the patch. Compared to the scalar case discussed recently in Hassainia et al. (KAM theory for active scalar equations, arXiv:2110.08615), Hassainia and Roulley (Boundary effects on the existence of quasi-periodic solutions for Euler equations, arXiv:2202.10053), Hmidi and Roulley (Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations, arXiv:2110.13751) and Roulley (Dyn Partial Differ Equ 20(4):311–366, 2023), some technical issues emerge due to the interaction between the interfaces. One of them is related to a new small divisor problem in the second order Melnikov non-resonances condition coming from the transport equations advected with different velocities.