This paper concerns the well-posedness theory of the motion of a 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
(Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller,
Arch Ration Mech Anal 206:515–616, 2012; Jang andMasmoudi,Well-posedness
of compressible Euler equations in a physical vacuum, 2008) by constructing suitable
weights and cutoff functions featuring the behavior of solutions near both the
center of the symmetry and the moving vacuum boundary.