  In this paper, we consider a class of models for multiphase fluids, in the
framework of mixture theory. The considered system, in its more general form,
contains both the gradient of a hydrostatic pressure, generated by an
incompressibility constraint, and the gradient of a compressible pressure
depending on the volume fractions of some of the different phases. To approach
these systems, we define an approximation based on the \emph{Leray} projection,
which involves the use of the \emph{Lax} symbolic symmetrizer for hyperbolic
systems and paradifferential techniques. In two space dimensions, we prove its
well-posedness and convergence to the unique classical solution to the original
system. In the last part, we shortly discuss the difficulties in the three
dimensional case.
