Special Seminar

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Desingularization of time-periodic and leapfrogging vortex motion via KAM tools

Speaker: Zineb Hassainia, NYU Abu Dhabi

Location: Online

Videoconference link: https://nyu.zoom.us/j/92420699179

Date: Friday, November 21, 2025, 10 a.m.

Synopsis:

In this talk, I will present recent advances in the analytical description of complex, long-lived coherent vortex structures governed by the planar Euler equations. We focus on the construction of time-periodic and leapfrogging vortex-patch solutions obtained by desingularizing classical point-vortex configurations. This is achieved within the natural Hamiltonian framework of contour dynamics and by applying tools from Kolmogorov–Arnold–Moser (KAM) theory. In particular, a degenerate KAM theory is employed to control the linearized dynamics around concentric configurations, which gives rise to quasi-linear transport equations with time–space periodic coefficients. Using a Nash–Moser iterative scheme, we construct families of these leapfrogging vortex-patch solutions, providing a rigorous resolution to a long-standing open problem in the study of inviscid flows and vortex dynamics.  In the second part, I will discuss a new application of KAM theory which, for the first time, desingularizes periodic orbits of the Euler equations in bounded domains.

 

Zoom link: https://nyu.zoom.us/j/92420699179