Desingularization of time-periodic vortex motion in bounded domains via KAM tools

TL;DR

This paper demonstrates the desingularization of time-periodic vortex trajectories into vortex patches in bounded domains using KAM and Nash-Moser techniques, addressing an open problem in Euler dynamics.

Contribution

It introduces a novel method to desingularize vortex trajectories into patches and constructs synchronized multi-vortex periodic motions in generic domains.

Findings

Existence of non-rigid time-periodic vortex motion in bounded domains

Application of KAM and Nash-Moser schemes to vortex desingularization

Construction of symmetric multi-vortex periodic solutions

Abstract

We examine the Euler equations within a simply-connected bounded domain. The dynamics of a single point vortex are governed by a Hamiltonian system, with most of its energy levels corresponding to time-periodic motion. We show that for the single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches. We provide concrete examples of these non-degeneracy conditions, which are satisfied by a broad class of domains, including convex ones. The proof uses Nash-Moser scheme and KAM techniques, in the spirit of the recent work of Hassainia-Hmidi-Masmoudi on the leapfrogging motion, combined with complex geometry tools. Additionally, we employ a vortex duplication mechanism to generate synchronized time-periodic motion of multiple vortices. This approach can be, for instance, applied to desingularize the motion of two symmetric dipoles (with four vortices) in a disc or a rectangle. To our knowledge, this is the first result showing the existence of non-rigid time-periodic motion for Euler equations in generic simply-connected bounded domain. This answers an open problem that has been pointed in the literature, for example by Bartsch-Sacchet.