Well-posedness for the Brinkman-Cahn-Hilliard system with unmatched viscosities

This paper addresses the well-posedness of a diffuse interface model for the motion of binary fluids with different viscosities. The system consists of the Brinkman–Darcy law governing the fluid velocity, nonlinearly coupled with a convective Cahn–Hilliard equation for the difference of the fluid concentrations. In a three-dimensional bounded domain, for the Brinkman–Cahn–Hilliard system with logarithmic free energy density, we prove global existence and uniqueness of weak solutions and we establish global existence of (unique) 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.