Periodic and quasi-periodic Euler-$\alpha$ flows close to Rankine vortices

In the present contribution, we first prove the existence of $\mathbf{m}$-fold simply-connected V‑states close to the unit disc for Euler-$\alpha$ equations. These solutions are implicitly obtained as bifurcation curves from the circular patches. We also prove the existence of quasi-periodic in time vortex patches close to the Rankine vortices provided that the scale parameter $\alpha$ belongs to a suitable Cantor-like set of almost full Lebesgue measure. The techniques used to prove this result are borrowed from the Berti–Bolle theory in the context of KAM for PDEs.

Keywords

KAM theory, Nash–Moser scheme, bifurcation theory, vortex patches

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Received 10 September 2022

Published 1 November 2023