In this talk, 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 [2, Section 5].
The purpose of the talk is the presentation of most essential elements of the proof of the global in time existence of weak solutions to the one velocity Baer--Nunziato type system for arbitrary large initial data. Namely, the attention will be focused on the following three steps:
(i) Transformation of the given system into a new one which possesses the Navier-Stokes-Fourier structure;
(ii) Showing the existence of weak solutions of the new system by an adaptation of the approach used in the existence theory for the compressible Navier--Stokes--Fourier equations which is presented in [1];
(iii) Showing 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.

[1] E. Feireisl, A. Novotn\`y: Singular limits in thermodynamics of viscous fluids. Advances in Mathematical Fluid Mechanics. Birkh\auser Verlag, Basel, (2009).

[2] Y.-S. Kwon, A. Novotn\`y, C.H. Arthur Cheng: On weak solutions to a dissipative Baer--Nunziato--type system for a mixture of two compressible heat conducting gases,
\newblock Math. Models Methods Appl. Sci. 30 (2020) no. 8, 1517--1553.