  Considering the isentropic Euler equations of compressible fluid dynamics
with geometric effects included, we establish the existence of entropy
solutions for a large class of initial data. We cover fluid flows in a nozzle
or in spherical symmetry when the origin r=0 is included. These partial
differential equations are hyperbolic, but fail to be strictly hyperbolic when
the fluid mass density vanishes and vacuum is reached. Furthermore, when
geometric effects are taken into account, the sup-norm of solutions can not be
controlled since there exist no invariant regions. To overcome these
difficulties and to establish an existence theory for solutions with
arbitrarily large amplitude, we search for solutions with finite mass and total
energy. Our strategy of proof takes advantage of the particular structure of
the Euler equations, and leads to a versatile framework covering general
compressible fluid problems. We establish first higher-integrability estimates
for the mass density and the total energy. Next, we use arguments from the
theory of compensated compactness and Young measures, extended here to
sequences of solutions with finite mass and total energy. The third ingredient
of the proof is a characterization of the unbounded support of entropy
admissible Young measures. This requires the study of singular products
involving measures and principal values.
