  In this paper, we establish a priori estimates for the three-dimensional
compressible Euler equations with moving physical vacuum boundary, the
$\gamma$-gas law equation of state for $\gamma=2$ and the general initial
density $\ri \in H^5$. Because of the degeneracy of the initial density, we
investigate the estimates of the horizontal spatial and time derivatives and
then obtain the estimates of the normal or full derivatives through the
elliptic-type estimates. We derive a mixed space-time interpolation inequality
which play a vital role in our energy estimates and obtain some extra estimates
for the space-time derivatives of the velocity in $L^3$.
