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

Emeric Roulley

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arXiv · 2208.13109

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

Emeric Roulley

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Abstract

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.

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Emeric Roulley. 2022-08-28. Periodic and quasi-periodic Euler-$\alpha$ flows close to Rankine vortices. https://arxiv.org/abs/2208.13109

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