  We are concerned with supersonic vortex sheets for the Euler equations of
compressible inviscid fluids in two space dimensions. For the problem with
constant coefficients we derive an evolution equation for the discontinuity
front of the vortex sheet. This is a pseudo-differential equation of order two.
In agreement with the classical stability analysis, if the jump of the
tangential component of the velocity satisfies $|[v\cdot\tau]|<2\sqrt{2}\,c$
(here $c$ denotes the sound speed) the symbol is elliptic and the problem is
ill-posed. On the contrary, if $|[v\cdot\tau]|>2\sqrt{2}\,c$, then the problem
is weakly stable, and we are able to derive a wave-type a priori energy
estimate for the solution, with no loss of regularity with respect to the data.
Then we prove the well-posedness of the problem, by showing the existence of
the solution in weighted Sobolev spaces.
