Regularity of the Kelvin-Helmholtz problem for the Euler 2D equation

We prove that for any solution u locally defined in time of the Kelvin{Helmholtz problem for the Euler 2d equation in the plane, then the curve of discontinuity of u and the density of the vortex sheet are analytic (under holder a priori estimates for the curve of discontinuity). We also give a partial result for a solution u defined in a half interval [O; T[.