  This paper concerns the well-posedness theory of the motion of physical
vacuum for the compressible Euler equations with or without self-gravitation.
First, a general uniqueness theorem of classical solutions is proved for the
three dimensional general motion. Second, for the spherically symmetric
motions, without imposing the compatibility condition of the first derivative
being zero at the center of symmetry, a new local-in-time existence theory is
established in a functional space involving less derivatives than those
constructed for three-dimensional motions in \cite{10',7,16'} by constructing
suitable weights and cutoff functions featuring the behavior of solutions near
both the center of the symmetry and the moving vacuum boundary.
