  The phase separation of an isothermal incompressible binary fluid in a porous
medium can be described by the so-called Brinkman equation coupled with a
convective Cahn-Hilliard (CH) equation. The former governs the average fluid
velocity $\mathbf{u}$, while the latter rules evolution of $\varphi$, the
difference of the (relative) concentrations of the two phases. The two
equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular,
the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg
force) which is proportional to $\mu\nabla\varphi$, where $\mu$ is the chemical
potential. When the viscosity vanishes, then the system becomes the
Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the
theoretical and the numerical viewpoints. However, theoretical results on the
CHHS system are still rather incomplete. For instance, uniqueness of weak
solutions is unknown even in 2D. Here we replace the usual CH equation with its
physically more relevant nonlocal version. This choice allows us to prove more
about the corresponding nonlocal CHHS system. More precisely, we first study
well-posedness for the CHB system, endowed with no-slip and no-flux boundary
conditions. Then, existence of a weak solution to the CHHS system is obtained
as a limit of solutions to the CHB system. Stronger assumptions on the initial
datum allow us to prove uniqueness for the CHHS system. Further regularity
properties are obtained by assuming additional, though reasonable, assumptions
on the interaction kernel. By exploiting these properties, we provide an
estimate for the difference between the solution to the CHB system and the one
to the CHHS system with respect to viscosity.
