  A mathematical model describing the flow of two-phase fluids in a bounded
container $\Omega$ is considered under the assumption that the phase transition
process is influenced by inertial effects. The model couples a variant of the
Navier-Stokes system for the velocity $u$ with an Allen-Cahn-type equation for
the order parameter $\varphi$ relaxed in time in order to introduce inertia.
The resulting model is characterized by second-order material derivatives which
constitute the main difficulty in the mathematical analysis. Actually, in order
to obtain a tractable problem, a viscous relaxation term is included in the
phase equation. The mathematical results consist in existence of weak solutions
in 3D and, under additional assumptions, existence and uniqueness of strong
solutions in 2D. A partial characterization of the long-time behavior of
solutions is also given and in particular some issues related to dissipation of
energy are discussed.
