  We study the free boundary problem for the equations of compressible Euler
equations with a vacuum boundary condition. Our main goal is to recover in
Eulerian coordinates the earlier well-posedness result obtained by Lindblad
[Lindblad H., Commun. Math. Phys. 260 (2005), 319-392] for the isentropic Euler
equations and extend it to the case of full gas dynamics. For technical
simplicity we consider the case of an unbounded domain whose boundary has the
form of a graph and make short comments about the case of a bounded domain. We
prove the local-in-time existence in Sobolev spaces by the technique applied
earlier to weakly stable shock waves and characteristic discontinuities. It
contains, in particular, the reduction to a fixed domain, using the "good
unknown" of Alinhac, and a suitable Nash-Moser-type iteration scheme. A certain
modification of such an approach is caused by the fact that the symbol
associated to the free surface is not elliptic. This approach is still directly
applicable to the relativistic version of our problem in the setting of special
relativity and we briefly discuss its extension to general relativity.
