Long time evolution of concentrated vortex rings with large radius

TL;DR

This paper investigates the long-term evolution of concentrated vortex rings with large radii in an incompressible fluid, showing convergence to a point vortex model over diverging time intervals as the vortex thickness shrinks.

Contribution

It extends previous results by proving convergence to the point vortex model for larger radii and over longer, diverging time intervals, under less restrictive conditions.

Findings

Vorticity converges to the point vortex model as vortex thickness approaches zero.

Convergence holds over time intervals that diverge as | log\u00b5a0 ext{a0}| log\u0000a0"],

conclusion that extends prior short-time results.

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 $r_0= |\log \varepsilon|^\alpha$, $\alpha>1$, 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 $\alpha>2$ and in which the convergence was proved for short times only.