  We propose a model of a density-dependent compressible-incompressible fluid,
which is intended as a simplified version of models based on mixture theory as,
for instance, those arising in the study of biofilms, tumor growth and
vasculogenesis. Though our model is, in some sense, close to the
density-dependent incompressible Euler equations, it presents some differences
that require a different approach from an analytical point of view. In this
paper, we establish a result of local existence and uniqueness of solutions in
Sobolev spaces to our model, using paradifferential techniques. Besides, we
show the convergence of both a continuous version of the Chorin-Temam
projection method, viewed as a singular perturbation type approximation, and
the 'artificial compressibility method'.
