  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.
