Long time evolution of concentrated vortex rings with large radius

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 $\epsilon$. We prove that when $r_0=\mathrm{\mid }\log \epsilon {\mathrm{\mid }}^{\alpha }$, $\alpha >1$, the vorticity field of the fluid converges for $\epsilon \to 0$ to the point vortex model, in an interval of time which diverges as $\log \mathrm{\mid }\log \epsilon \mathrm{\mid }$. 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.