The three-dimensional Cahn-Hilliard-Brinkman system with unmatched viscosities

This paper is focused on a diffuse interface model for the motion of binary fluids with different viscosities. The system consists of the Brinkman-Darcy equations governing the fluid velocity, nonlinearly coupled with a convective Cahn-Hilliard equation for the difference of the fluid concentrations. For the three-dimensional Cahn-Hilliard-Brinkman system with free energy density of logarithmic type we prove the well-posedness of weak solutions and we establish the global-in-time existence of strong solutions. Furthermore, we discuss the validity of the separation property from the pure states, which occurs instantaneously in dimension two and asymptotically in dimension three