Time-periodic leapfrogging vortex rings in the 3D Euler equations

TL;DR

This paper rigorously proves the existence of time-periodic leapfrogging vortex rings in 3D Euler equations, confirming a long-standing conjecture and providing a mathematical foundation for observed vortex behaviors.

Contribution

It introduces a novel mathematical construction of leapfrogging vortex rings using desingularization, Hamiltonian dynamics, and advanced analytical techniques like KAM theory and Nash-Moser iteration.

Findings

First rigorous proof of classical leapfrogging vortex rings in Euler flows.

Construction of families of time-periodic vortex solutions.

No restrictions on the time interval of vortex leapfrogging.

Abstract

We prove 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.