  We prove the existence of a weak solution to the compressible Navier--Stokes
system with singular pressure that explodes when density achieves its
congestion level. This is a quantity whose initial value evolves according to
the transport equation. We then prove that the "stiff pressure" limit gives
rise to the two-phase compressible/incompressible system with congestion
constraint describing the free interface. We prescribe of the velocity at the
boundary and the value of density at the inflow part of the boundary of a
general bounded $C^2$ domain. There are no restrictions on the size of the
boundary conditions.

Analysis of PDEs