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 ε and vorticity mass of the order of | log ε| - 1. When ε→ 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 [5], where a sharp localization of the vorticity support was obtained both along the radial and axial directions, but the convergence for ε→ 0 worked only for short times.
Global time evolution of concentrated vortex rings / Butta', P.; Cavallaro, G.; Marchioro, C.. - In: ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK. - ISSN 0044-2275. - 73:2(2022). [10.1007/s00033-022-01708-w]
Global time evolution of concentrated vortex rings
Butta' P.;Cavallaro G.
;Marchioro C.
2022
Abstract
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 ε and vorticity mass of the order of | log ε| - 1. When ε→ 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 [5], where a sharp localization of the vorticity support was obtained both along the radial and axial directions, but the convergence for ε→ 0 worked only for short times.File | Dimensione | Formato | |
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