Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Claudia García, Zineb Hassainia, Taoufik Hmidi

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.

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

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.
Paper Structure (58 sections, 68 theorems, 1416 equations, 4 figures)

This paper contains 58 sections, 68 theorems, 1416 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main contribution
  3. Previous results on vortex rings
  4. Sketch of the proof
  5. Differences and challenges compared to previous works
  6. Axisymmetric 3D Euler equations
  7. Axisymmetric formulation
  8. Hamiltonian reformulation
  9. Derivation of the contour dynamics equation
  10. Filament dynamics
  11. Symmetry reduction for suitable 4D Hamiltonian systems
  12. Filament dynamics and scaling
  13. Periodic motion for the unperturbed system
  14. Periodic motion for the perturbed system
  15. Symmetry reduction and time scaling in the patch pairs motion
  16. ...and 43 more sections

Key Result

Theorem 1.1

Let $0<a<b$ and $\kappa>0$ be fixed. Then there there exists $\varepsilon_0>0$ such that, every $\varepsilon\in(0,\varepsilon_0)$, there is a Borel set $\mathcal{C}_\varepsilon\subset (a,b)$ of asymptotically full Lebesgue measure, with the following property: for every $\lambda \in \mathcal{C}_\varepsilon$, there exists a global solution of the axisymmetric Euler equation intro:eq:tilde-q of the

Figures (4)

  • Figure 1: Evolution of the points $(P_1,P_2)$ for the parameters $\kappa=0.4$ and $\varepsilon=0.05$. The green diamonds and blue points represent the vortex ring positions $P_1$, $P_2$, respectively. The right-hand panel displays the evolution of $(P_1, P_2)$ in the translating frame of reference, where a periodic motion is observed.
  • Figure 2: Parameters as in Figure \ref{['fig-VR']}, namely $\kappa=0.4$ and $\varepsilon=0.05$. he left-hand panel shows the periodic evolution of $\tau\mapsto P_{1,1}(\tau)$ and $\tau\mapsto P_{2,1}(\tau)$while the right-hand panel displays the periodic function $\tau\mapsto (P_{2,2}-P_{1,2})(\tau)$.
  • Figure 3: Periodic evolution of the two vortex rings described in Theorem \ref{['th-main1']}, with parameters $\kappa = 0.4$ and $\varepsilon = 0.05$. The left‑hand panel shows the initial configuration, while the right‑hand panel depicts the state at time $t = T$.
  • Figure 4: Illustration of our solution ansatz in the coordinates $(\varrho,z)$. The separation between the centers of the two rings in both the $\varrho$- and $z$‑directions is of order $|\ln\varepsilon|^{-1/2}$, while each vortex‑ring cross‑section has area of order $\varepsilon^{2}$. The corresponding potential vorticity $\mathbf{q}$ attains amplitude of order $\varepsilon^{-2}$ on each ring.

Theorems & Definitions (121)

  • Theorem 1.1
  • Remark 1.1: Non-rigid periodicity
  • Remark 1.2: Elliptical perturbation
  • Remark 1.3: Speed modulation
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 111 more