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

TL;DR

This paper proves the existence of special vortex patch solutions near classical vortices for Euler-$ alpha$ equations, including quasi-periodic solutions, using bifurcation theory and KAM techniques, expanding understanding of vortex dynamics.

Contribution

It introduces new existence results for vortex patches near Rankine vortices and circular patches in Euler-$\alpha$ equations, employing bifurcation and KAM methods.

Findings

Existence of $\mathbf{m}$-fold V-states close to the unit disc.

Existence of quasi-periodic vortex patches near Rankine vortices.

Solutions are obtained via bifurcation and KAM theory.

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.