We study the time evolution of an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside $N$ small disjoint rings of thickness $\varepsilon$ and vorticity mass of the order of $|\log\varepsilon|^{ -1}$. When $\varepsilon \to 0$ we show that the motion of each vortex ring converges to a simple translation with constant speed (depending on the single ring) along the symmetry axis. We obtain a sharp localization of the vorticity support at time $t$ in the radial direction, whereas we state only a concentration property in the axial direction. This is obtained for arbitrary (but fixed) intervals of time. This study is the completion of a previous paper arXiv:1904.04785 [math-ph], where a sharp localization of the vorticity support was obtained both along the radial and axial directions, but the convergence for $\varepsilon\to 0$ worked only for short times.
Categories: math.AP math-ph math.MP