On Compressible and Incompressible Vortex Sheets

Abstract

We introduce the main known results of the theory of incompressible and compressible vortex sheets. Moreover, we present recent results obtained by the author with J.F. Coulombel about compressible vortex sheets in two space dimensions, under a supersonic condition that precludes violent instabilities. The problem is a nonlinear free boundary hyperbolic problem with two difficulties: the free boundary is characteristic and the Lopatinski condition holds only in a weak sense, yielding losses of derivatives. In [18, 20] we prove the existence of such piecewise smooth solutions to the Euler equations close enough to stationary vortex sheets. Since the a priori estimates for the linearized equations exhibit a loss of regularity, our existence result is proved by using a suitable modification of the Nash-Moser iteration scheme.