A naive parametrization for the vortex-sheet problem

TL;DR

This paper investigates the vortex sheet problem using a simple parametrization, establishing well-posedness for analytic initial data and demonstrating ill-posedness for non-analytic data within the Birkhoff-Rott framework.

Contribution

It introduces a naive parametrization approach that ensures well-posedness for analytic initial conditions and highlights ill-posedness issues for non-analytic cases.

Findings

Well-posedness for analytic initial data.

Ill-posedness for non-analytic initial data.

Analysis within the Birkhoff-Rott equations framework.

Abstract

We consider the dynamics of a vortex sheet that evolves by the Birkhoff-Rott equations. The fluid evolution is understood as a weak solution of the incompressible Euler equations where the vorticity is given by a delta function on a curve multiplied by an amplitude. The solutions we study are with finite energy, which implies zero mean amplitude. In this context we choose a parametrization for the motion of the vortex sheet for which the equation is well-posed for analytic initial data. For the equation of the amplitude we show ill-posedness for non-analytic initial data.