  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.
