  In this paper we prove full local well-posedness for the Cauchy problem for
the compressible 3D Euler equation, i.e. local existence, uniqueness, and
continuous dependence on initial data, with initial velocity, density and
vorticity $(\mathbf{v}_0, \rho_0, \mathbf{w}_0) \in H^{2+} \times H^{2+} \times
H^{2}$, improving on the regularity conditions of \cite{WQEuler}. The
continuous dependence on initial data for rough solutions of the compressible
Euler system is new, even with the same regularity conditions as in
\cite{WQEuler}. In addition, we prove new local well-posedness results for the
3D compressible Euler system with entropy.
