On Singular Vortex Patches, II: Long-time dynamics

TL;DR

This paper explores the long-term behavior of singular vortex patches, revealing phenomena such as periodicity, cusp formation, and spirals, building on prior work establishing their well-posedness.

Contribution

It provides new insights into the infinite-time dynamics of vortex patches, including explicit examples of complex behaviors.

Findings

Existence of time-periodic vortex patches

Cusp formation at exponential rate over infinite time

Spiral formation in the long-term evolution

Abstract

In a companion paper, we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the companion work. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time.