The Doubly Nonlocal Hele-Shaw–Cahn–Hilliard System with Singular Potential and Nonconstant Mobility

Abstract We present the rigorous asymptotic analysis in thin domains of a diffuse interface model of two-component Hele-Shaw flow based on an advective nonlocal Cahn–Hilliard equation with singular potential and nonconstant nondegenerate mobility for the relative concentration. The velocity is determined by a Stokes system in which the inhomogeneous viscosity is highly oscillating and dependent on the relative concentration. Using the notion of sigma-convergence for thin heterogeneous media, we obtain in the homogenization limit a new doubly nonlocal Hele-Shaw–Cahn–Hilliard-type model system containing an additional term arising from the dependence of the viscosity on the relative concentration. In the case when both the viscosity and the mobility coefficients do not depend on the relative concentration, we additionally prove that the new model is well posed and we establish the existence of global strong solutions.