Boundary effects on the emergence of quasi-periodic solutions for Euler equations

TL;DR

This paper investigates how boundary effects influence the formation of quasi-periodic vortex patches near Rankine vortices in bounded domains, revealing their existence through advanced iterative methods.

Contribution

It demonstrates the existence of quasi-periodic vortex patches close to Rankine vortices in a bounded domain, utilizing a Nash-Moser scheme and analyzing boundary effects.

Findings

Existence of quasi-periodic vortex patches near Rankine vortices.

Boundary effects are crucial for vortex solution construction.

Solutions exist for a large measure set of vortex radii.

Abstract

In this paper, we highlight the importance of the boundary effects on the construction of quasi-periodic vortex patches solutions close to Rankine vortices and whose existence is not known in the whole space due to the resonances of the linear frequencies. Availing of the lack of invariance by radial dilation of Euler equations in the unit disc and using a Nash-Moser implicit function iterative scheme we show the existence of such structures when the radius of the Rankine vortex belongs to a suitable massive Cantor-like set with almost full Lebesgue measure.