Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids with general boundary data

TL;DR

This paper establishes the global existence of weak solutions for a bi-fluid model describing two non-interacting compressible viscous fluids with general boundary conditions, extending classical methods to more complex geometries.

Contribution

It adapts the Lions-Feireisl approach and generalizes the theory of renormalized solutions for transport equations to a bi-fluid system with complex boundary conditions.

Findings

Proves global existence of weak solutions for the bi-fluid model.

Extends the theory of renormalized solutions for transport equations.

Applicable to complex geometries like curved pipes with variable cross sections.

Abstract

We prove global existence of weak solutions for a version of one velocity Baer-Nunziato system with dissipation describing a mixture of two non interacting viscous compressible fluids in a piecewise regular Lipschitz domain with general inflow/outfow boundary conditions. The geometrical setting is general enough to comply with most current domains important for applications as, for example, (curved) pipes of picewise regular and axis-dependent cross sections. As far as the existence proof is concerned, we adapt to the system the nowaday's classical Lions-Feireisl approach to the compressible Navier-Stokes equations which is combined with a generalization of the theory of renormalized solutions to the transport equations in the spirit of Vasseur-Wen-Yu. The results related to the families of transport equations presented in this paper extend/improve some of statements of the theory of renormalized solutions, and they are therefore of independent interest.