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

TL;DR

This paper establishes the existence of weak solutions for a compressible Navier--Stokes system with singular pressure and explores the transition to a two-phase incompressible system with congestion constraints, accommodating general boundary conditions.

Contribution

It proves the existence of solutions for a complex fluid system with singular pressure and derives the two-phase limit with congestion constraints under general boundary conditions.

Findings

Existence of weak solutions for the compressible Navier--Stokes system with singular pressure.

Derivation of the two-phase compressible/incompressible system as a limit.

No restrictions on boundary condition sizes.

Abstract

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.