A model of a non-isothermal two phase flow of compressible fluids

We introduce a simple model of the time evolution of a binary mixture of compressible fluids including the thermal effects. Despite its apparent simplicity, the model is thermodynamically consistent admitting an entropy balance equation. We introduce a suitable weak formulation of the problem based on a combination of the entropy inequality with the total energy conservation principle. Finally, we show compactness of any bounded family of weak solutions and establish a global existence result.