Numerical study of singularity formation in two vortex sheets

The evolution of vortex sheets is described by a nonlinear singular integrodifferential equation called the Birkhoff-Rott equation. The initial-value problem for the Birkhoff-Rott equation seems to be ill-posed due to the Kelvin-Helmholtz instability. The preceding result by Krasny has numerically shown that a single vortex sheet forms a singularity at a finite time by using the point-vortex approximation. In this study, we investigate the formation of singularities in two vortex sheets with Krasny's method. We consider the initial condition perturbed by solutions of the linearized Birkhoff-Rott equations and see how the process of the singularity formation varies depending on the amplitude of the perturbation and the distance of two vortex sheets.