Leapfrogging motion for the 3D Euler equations (Parts I and II)
数学专题报告
报告题目(Title)：Leapfrogging motion for the 3D Euler equations (Parts I and II)
报告人(Speaker)：Prof. Claudia García (Part I, University of Granada, Spain) and Prof. Zineb Hassainia (Part II, University of Granada, Spain)
地点(Place)：Zoom Meeting ID: 831 854 09607 | Password: 123456
时间(Time)：2026年10月19日（周一）15:00-17:00
邀请人(Inviter)：薛留堂
报告摘要
In this talk, we focus on the existence of time-periodic leapfrogging vortex rings for the three-dimensional incompressible Euler equations, thereby providing a rigorous realization of a phenomenon first conjectured by Helmholtz (1858). In the leapfrogging motion, two coaxial vortex rings periodically exchange positions, a striking behavior repeatedly observed in experiments and numerical simulations, yet lacking complete mathematical justification. Our construction relies on a desingularization of two interacting vortex filaments within the contour dynamics formulation, which yields a Hamiltonian description of nearly concentric vortex rings. The main difficulty stems from a singular small-divisor problem arising in the linearized transport dynamics, where the effective time scale degenerates with the ring thickness parameter. To overcome this obstruction, we develop a degenerate KAM-type analysis combined with pseudo-differential operator techniques to control the linearized dynamics around symmetric configurations. Combining these tools with a Nash-Moser iteration scheme, we construct families of nontrivial time-periodic solutions in an almost uniformly translating frame. This establishes the first rigorous construction of classical leapfrogging motion for axisymmetric Euler flows without swirl, with no restriction on the time interval of existence. This is a joint work with Taoufik Hmidi.
