Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids

We investigate a version of one velocity Baer-Nunziato model with dissipation for the mixture of two compressible fluids with the goal to prove for it the existence of weak solutions for arbitrary large initial data on a large time interval. We transform the one velocity Baer-Nunziato system to another "more academic" system which possesses the clear "Navier-Stokes structure". We solve the new system by adapting to its structure the Lions approach for solving the (mono-fluid) compressible Navier-Stokes equations. An extension of the theory of renormalized solutions to the transport equation to more continuity equations with renormalizing functions of several variables is essential in this process. We derive a criterion of almost uniqueness for the renormalized solutions to the pure transport equation without the classical assumption on the boundedness of the divergence of the transporting velocity. This result does not follow from the DiPerna-Lions transport theory and it is of independent interest. This criterion plays the crucial role in the identification of the weak solutions to the original one velocity Baer-Nunziato problem starting from the weak solutions of the academic problem. As far as we know, this is the first result on the existence of weak solutions for a version of the one velocity bi-fluid system of the Baer-Nunziato type in the mathematical literature.