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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.18655 |
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Table of Contents:
- 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 $ω^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities $ω^0_\varepsilon \in C^\infty_c(\mathbb{R}^2)$ converging to $ω^0_{\mathrm{sing}}$ distributionally as $\varepsilon \to 0^+$, and show that the corresponding solutions $ω_\varepsilon(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff-Rott system with initial datum $ω^0_{\mathrm{sing}}$. 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.