On a diffuse interface model for a two-phase flow of compressible viscous fluids

We consider a model of a binary mixture of compressible, viscous, and macroscopically immiscible fluids based on the diffuse interface approximation, where the difference in concentrations of the two fluids plays the role of the order parameter. The resulting system consists of the compressible Navier-Stokes equations governing the motion of the mixture coupled with the Cahn-Hilliard equation for the order parameter. We prove existence of global-in-time weak (distributional) solutions of the problem without any restriction on the size of initial data.