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

Two-phase compressible/incompressible Navier--Stokes system with inflow-outflow boundary conditions