Global existence for a two-phase flow model with cross diffusion

In this work we study a degenerate pseudo-parabolic system with cross diffusion describing the evolution of the densities of an unsaturated two-phase flow mixture with dynamic capillary pressure in porous medium with saturation-dependent relaxation parameter and hypocoercive diffusion operator modeling cross diffusion. The equations are derived in a thermodynamically correct way from mass conservation laws. Global-in-time existence of weak solutions to the system in a bounded domain with equilibrium boundary conditions is shown. The main tools of the analysis are an entropy inequality and a crucial apriori bound which allows for controlling the degeneracy.