  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.
