In a two-dimensional Hele-Shaw cell, provided that the viscous forces dominate the iner- tial ones, the well-known Navier-Stokes-Cahn-Hilliard system for an incompressible binary flow can be approximated by the so-called Cahn-Hilliard-Hele-Shaw (CHHS) system. In three dimensions, the CHHS system is used to describe fluid flow in a porous medium as well as it is a cornerstone of solid tumor growth modeling through diffuse interfaces. I intend to present some recent results on a nonlocal CHHS system characterized by degenerate mobility, singular potential, and nonconstant kinematic viscosity. "Nonlocal" means that the demixing effects in the free energy are represented by a (spatially) nonlocal term. Well-posedness and regularity issues will be discussed. Also, a comparison with the results obtained for similar systems will be made. This is a joint project with C. Cavaterra and S. Frigeri (Universita degli Studi di Milano).
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