2025/05/24 by Enciso, Alberto, Fernández, Antonio J., Meyer, David
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.18655
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 ω0sing, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities ω0ε ∈ C^∞c(ℝ2) converging to ω0sing distributionally as ε → 0+, and show that the corresponding solutions ωε(x,t) to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff-Rott system with initial datum ω0sing. The regularization relies on a layer construction designed to exploit the key observation that the Kelvin-Helmholtz instability has a strongly anisotropic effect: while vorticities must be analytic in the "tangential" direction, the way layers can be arranged in the "normal" direction is essentially arbitrary.