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 H-s, s > d/2 + 1, the existence and uniqueness of solution in C([0, T]; H-s) boolean AND L-2(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. (C) 2012 Elsevier Masson SAS. All rights reserved.Mathematics, AppliedSCI(E)EI4ARTICLE3367-3843