  Isothermal compressible two-phase flows in a capillary are modeled with and
without phase transition in the presence of gravity, employing Darcy's law for
the velocity field. It is shown that the resulting systems are
thermodynamically consistent in the sense that the available energy is a strict
Lyapunov functional. In both cases, the equilibria with flat interface are
identified. It is shown that the problems are well-posed in an $L_p$-setting
and generate local semiflows in the proper state manifolds. The main result
concerns the stability of equilibria with flat interface, i.e. the
Rayleigh-Taylor instability.
