Quasi-periodic dynamics of desingularized vortex models

Abstract

Sufficient conditions for the existence of quasi-periodic solutions of two different desingularized vortex models for 2-dimensional Euler flows are derived. One of these models is the vortex blob model for the evolution of a periodic vortex sheet and the other is a second order elliptic moment model (DEMM) for the evolution of widely separated vortex regions. The method involves the identification of the well-known point vortex Hamiltonian term in both models. A transformation to new canonical variables (the JL-coordinates) and the definition of special open sets in phase space (the cone sets) puts the Hamiltonians considered into nearly integrable form. KAM-theory is used to prove the desired results for arbitrary degrees of freedom and almost arbitrary circulations in these models. A rigorous validification of the DEMM assumption is obtained. In view of the lack of a rigorous theory for vortex sheet roll-up past the critical time, the dynamical system approach presented here provides an alternative method for studying the macroscopic structures formed in the post-critical period.