Uniqueness of 2-D compressible vortex sheets

We consider compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. Under a supersonic condition that precludes violent instabilities, in previous papers [3, 4] we have studied the linearized stability and proved the local existence of piecewise smooth solutions to the nonlinear problem. This is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. In the present paper we prove that sufficiently smooth solutions are unique