Invariant KAM tori around annular vortex patches for 2D Euler equations

TL;DR

This paper constructs quasi-periodic vortex patch solutions with one hole for the 2D Euler equations, using advanced mathematical techniques to handle complex interactions and small divisor problems.

Contribution

It introduces a novel application of Nash-Moser and KAM theory to vortex patches with holes, addressing technical challenges from interface interactions.

Findings

Existence of invariant KAM tori around annular vortex patches.

Application of Nash-Moser scheme to fluid interface problems.

Resolution of a new small divisor problem in vortex dynamics.

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, 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.