  We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg
equations with a non-monotone pressure law. This system governs the
low-Reynolds number dynamics of a compressible viscous fluid that may take
either a liquid or a vapour state. For a porous domain that is perforated by
cavities with diameter proportional to their mutual distance the homogenization
limit is analyzed. We extend the results for compressible one-phase flow with
polytropic pressure laws and prove that the effective motion is governed by a
nonlocal version of the Cahn-Hilliard equation. Crucial for the analysis is the
convolution-like structure of the nonlocal capillarity term that allows to
equip the system with a generalized convex free energy. Moreover, the
capillarity term accounts not only for the energetic interaction within the
fluid but also for the interaction with a solid wall boundary.

Analysis of PDEs