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 > d 2 + 1 , the existence and uniqueness of solution in C ( [ 0 , T ] ; H s ) ∩ 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.

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