Non-Linear Singularity Formation for Circular Vortex Sheets

TL;DR

This paper investigates how non-linear effects cause singularity formation in circular vortex sheets governed by the Birkhoff-Rott equation, revealing mechanisms behind their breakdown and stability properties.

Contribution

It extends previous linear stability analysis by exploring non-linear dynamics and singularity formation using a complex Burgers equation approach.

Findings

Singularity formation is driven by a complex Burgers equation in high-frequency regimes.

Circular vortex sheets exhibit marginal linear stability unlike flat sheets.

The study enhances understanding of vortex sheet breakdown mechanisms.

Abstract

We study the evolution of vortex sheets according to the Birkhoff-Rott equation, which describe the motion of sharp shear interfaces governed by the incompressible Euler equation in two dimensions. In a recent work, the authors demonstrated within this context a marginal linear stability of circular vortex sheets, standing in sharp contrast with classical instability of the flat vortex sheet, which is known as the Kelvin-Helmholtz instability. This article continues that analysis by investigating how non-linear effects induce singularity formation near the circular vortex sheet. In high-frequency regimes, the singularity formation is primarily driven by a complex-valued, conjugated Burgers equation, which we study by modifying a classical argument from hyperbolic conservation laws. This provides a deeper understanding of the mechanisms driving the breakdown of circular vortex sheets, which are observed both numerically and experimentally.