Long time evolution of concentrated vortex rings with large radius
Paolo Buttà, Guido Cavallaro, Carlo Marchioro
TL;DR
We address the long-time evolution of concentrated vorticity in axisymmetric Euler flow without swirl by (i) placing the initial vorticity on $N$ annuli of radius $r_0$ and thickness $\\varepsilon$ with $r_0=|\\log\\varepsilon|^\\alpha$, $\\alpha>1$, and (ii) proving convergence to the planar point-vortex dynamics for times on the order of $\\log|\\log\\varepsilon|$. The approach combines a decomposition of the velocity into self-induced and external components, rigorous control of the vorticity mass and moment of inertia, and an iterative scheme that extends the convergence horizon across successive time windows, ultimately achieving long-time convergence under the given scaling. The centers $z_i(t)$ follow the point-vortex ODE $\\dot z_i(t)=\\sum_{j\\ne i} a_j K(z_i(t)-z_j(t))$ with $K(x)=-\\frac{1}{2\\pi}\\nabla^{\\perp}\\log|x|$, and the work shows that the vorticity remains localized near these moving centers up to times $\\zeta\\log|\\log\\varepsilon|$, improving prior results that required stronger radius growth. The results bridge 3D axisymmetric vortex rings and 2D point-vortex dynamics, providing a rigorous justification for long-time behavior and related leapfrogging phenomena in the high-radius regime.
Abstract
We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on $N$ annuli of radii of the order of $r_0$ and thickness $\varepsilon$. We prove that when $r_0= |\log \varepsilon|^α$, $α>1$, the vorticity field of the fluid converges for $\varepsilon \to 0$ to the point vortex model, in an interval of time which diverges as $\log|\log\varepsilon|$. This generalizes previous result by Cavallaro and Marchioro in [J. Math. Phys. 62, 053102, (2021)], that assumed $α>2$ and in which the convergence was proved for short times only.