Leapfrogging vortex rings for the three‐dimensional incompressible Euler equations

Dávila, Juan; Pino, Manuel del; Musso, Monica; Wei, Juncheng

doi:10.1002/cpa.22199

Cited by 2 publications

(1 citation statement)

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“…Moreover, beside the convergence is proved for short times only, it is possible to chose initial data for which the convergence to a periodic solution cover several periods. This provide a mathematical justification of the leapfrogging, which was first proved in [13] (but with different techniques and scaling). For α " 0 (see [2] for a vortex ring alone and [8,4] for several rings) the dynamics converges to simple translations parallel to the symmetry axis with constant speed, while the case 0 ă α ă 1 has not been studied explicitly, but we believe that the behavior is analogous to the case α " 0.…”

Section: Introductionmentioning

confidence: 92%

Global time evolution of concentrated vortex rings

Buttà

Cavallaro

Marchioro

2022

Z. Angew. Math. Phys.

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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 $$\varepsilon $$ ε and vorticity mass of the order of $$|\log \varepsilon |^{ -1}$$ | log ε | - 1 . When $$\varepsilon \rightarrow 0$$ 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 $$\varepsilon \rightarrow 0$$ ε → 0 worked only for short times.

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