On existence of weak solutions to a Baer-Nunziato type system

TL;DR

This paper proves the global existence of weak solutions for a dissipative Baer-Nunziato type system modeling two compressible gases, solving an open problem by transforming and applying advanced PDE techniques.

Contribution

It establishes the first comprehensive proof of weak solutions for the system with large initial data, using a novel transformation and adaptation of Navier-Stokes solution methods.

Findings

Proved existence of weak solutions for the system.

Transformed the system into a Navier-Stokes-Fourier structure.

Applied Feireisl-Lions approach to the transformed system.

Abstract

In this paper, a dissipative version of a compressible one velocity Baer--Nunziato type system for a mixture of two compressible heat conducting gases is considered. The complete existence proof for weak solutions to this system was addressed as an open problem in [6, Section 5]. The purpose of this paper is to prove the global in time existence of weak solutions to the one velocity Baer--Nunziato type system for arbitrary large initial data. The goal is achieved in three steps. Firstly, the given system is transformed into a new one which possesses the "Navier-Stokes-Fourier" structure. Secondly, the new system is solved by an adaptation of the Feireisl--Lions approach for solving the compressible Navier--Stokes equations. Eventually, the existence of a weak solution to the original one velocity Baer--Nunziato system using the almost uniqueness property of renormalized solutions to pure transport equations.