Well-posedness of the Hele–Shaw–Cahn–Hilliard system

We study the well-posedness of the Hele–Shaw–Cahn–Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in Hs, , the existence and uniqueness of solution in C([0,T];Hs)∩L2(0,T;Hs+2) that is global in time in the two dimensional case (d=2) and local in time in the three dimensional case (d=3) are established. Several blow-up criterions in the three dimensional case are provided as well. One of the tools that we utilized is the Littlewood–Paley theory in order to establish certain key commutator estimates.