The leapfrogging conjecture for concentrated vortex rings

Let $N>1$ and $0<\varepsilon\ll 1$. Suppose that the initial vorticity $\omega_0^\varepsilon$ is decomposed into $N$ blobs, each with circulation $\gamma>0$, sharply concentrated around points $X_{i,0}^\varepsilon$, and that these points have mutual distance of order $1/\sqrt{|\ln \varepsilon|}$. Let $\omega^\varepsilon(t,\cdot)$ be the unique solution of the axisymmetric Euler system

$$\partial_t\omega^\varepsilon+\frac{1}{|\ln\varepsilon|}u^\varepsilon\cdot\nabla\omega^\varepsilon=0,$$

with the corresponding incompressibility and vorticity relations. Leapfrogging conjecture. The solution $\omega^\varepsilon(t,\cdot)$ remains concentrated around $N$ points $X_i^\varepsilon(t)$, whose trajectories are governed by

$$\frac{\mathrm d}{\mathrm d t}X_i^\varepsilon(t)=\frac{\gamma}{4\pi X_{i,r}^\varepsilon(t)}e_z+\frac{\gamma}{2\pi|\ln\varepsilon|}\sum_{j\ne i}\frac{\bigl(X_i^\varepsilon(t)-X_j^\varepsilon(t)\bigr)^\perp}{\bigl|X_i^\varepsilon(t)-X_j^\varepsilon(t)\bigr|^2},$$

with $X_i^\varepsilon(0)=X_{i,0}^\varepsilon$. This conjecture formalizes the expected leapfrogging motion of several concentrated vortex rings: their common self-induced vertical motion is combined with the mutual point-vortex rotation, which is expected to produce complete rotations around one another while the rings travel vertically. The source provides experimental and numerical support for finite times, but does not state a resolution of the conjecture.