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

Abstract

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 $H^s$, $s>\frac{d}{2}+1$, the existence and uniqueness of solution in $C([0,T];H^s)\cap L^2(0,T;H^{s+2})$ 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.