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

Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities