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

In this paper, we consider a compressible one velocity Baer–Nunziato type system with dissipation describing the evolution of a mixture of two compressible heat conducting fluids. The complete existence proof for weak solutions to this system was addressed as an open problem in [12, 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 full system applying also the almost compactness property introduced by Vasseur et al. [19]. Finally, the existence of a weak solution to the original one velocity Baer–Nunziato system is shown using the almost uniqueness property of renormalized solutions to pure transport equations.