We show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff-Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $\omega^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities $\omega^0_\varepsilon\in C^\infty_c(\mathbb{R}^2)$ converging to~$\omega^0_{\mathrm{sing}}$ distributionally as $\eps\to 0^+$, and show that the corresponding solutions $\omega_\eps(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff--Rott system with initial datum $\omega^0_{\mathrm{sing}}$. Based on joint work with Alberto Enciso and Antonio Fernandez.

Xmas PDE Colloquium - coffee break after (14:45 - 15:15, SRZ Lounge)