  We prove a priori estimates for the compressible Euler equations modeling the
motion of a liquid with moving physical vacuum boundary in an unbounded initial
domain. The liquid is under influence of gravity but without surface tension.
Our fluid is not assumed to be irrotational. But the physical sign condition
needs to be assumed on the free boundary. We generalize the method used in
\cite{LL} to prove the energy estimates in an unbounded domain up to arbitrary
order. To our knowledge, this result appears to be the first that concerns a
priori energy bounds for the compressible water wave. In addition to that, the
a priori energy estimates are in fact uniform in the sound speed $\kappa$. As a
consequence, we obtain the convergence of solutions of compressible Euler
equations with a free boundary to solutions of the incompressible equations,
generalizing the result of \cite{LL} to when you have an unbounded domain. On
the other hand, we prove that there are initial data satisfying the
compatibility condition in some weighted Sobolev spaces, and this will
propagate within a short time interval, which is essential for proving long
time existence for slightly compressible irrotational water waves.

Analysis of PDEs