Cahn-Hilliard-Darcy systems

Maurizio Grasselli (Politecnico di Milano)

La conferenza è dedicata ai 75 anni del prof. Gianni Gilardi.

Abstract. In a two-dimensional Hele-Shaw cell, provided that the viscous forces dominate the inertial ones, the well-known Navier-Stokes-Cahn-Hilliard system (or model H) for an incompressible binary flow can be approximated by the so-called Cahn-Hilliard-Hele-Shaw system. This consists of a Darcy’s law coupled with an advective Cahn-Hilliard equation through a capillary force known as Korteweg force. Similar systems, called in general Cahn-Hilliard-Darcy (CHD) systems, are also used to describe fluid flow in a porous medium as well as to model solid tumor growth through diffuse interfaces. I intend to review and discuss a number of recent results on CHD systems, focusing on uniqueness and regularity issues in the case of a singular (i.e. Flory-Huggins) potential. Some open issues will also be mentioned.