Seminario EDP UAM-ICMAT
Leapfrogging vortex rings for 3D Euler equations
Ponente: Zineb Hassaina (UGr)Fecha: viernes 22 de mayo de 2026 - 12:30Lugar: Aula Gris 2, ICMAT
Resumen:
In this talk, I will present a rigorous construction of time-periodic leapfrogging vortex rings for the three-dimensional incompressible Euler equations. More precisely, we prove the existence of solutions in which two coaxial vortex rings periodically exchange positions, as observed in experiments and numerical simulations. The construction relies on a desingularization of two interacting vortex filaments within the contour dynamics formulation, leading to a Hamiltonian description of nearly concentric vortex rings. A central difficulty arises from a singular small-divisor problem in the linearized dynamics, where the effective time scale degenerates with the ring thickness parameter. We overcome this issue by combining a degenerate KAM-type analysis with pseudo-differential techniques and a Nash-Moser iteration scheme. This approach yields families of nontrivial time-periodic solutions in an almost uniformly translating frame and thereby provides a rigorous mathematical construction of classical leapfrogging motion for axisymmetric Euler flows without swirl, with no restriction on the time interval of existence. The talk is based on a joint work with Claudia García and Taoufik Hmidi.
