  This paper studies the dynamics of an incompressible fluid driven by gravity
and capillarity forces in a porous medium. The main interest is the
stabilization of the fluid in Rayleigh-Taylor unstable situations where the
fluid lays on top of a dry region. An important feature considered here is that
the layer of fluid is under an impervious wall. This physical situation have
been widely study by mean of thin film approximations in the case of small
characteristic high of the fluid considering its strong interaction with the
fixed boundary. Here, instead of considering any simplification leading to
asymptotic models, we deal with the complete free boundary problem. We prove
that, if the fluid interface is smaller than an explicit constant, the solution
is global in time and it becomes instantly analytic. In particular, the fluid
does not form drops in finite time. Our results are stated in terms of Wiener
spaces for the interface together with some non-standard Wiener-Sobolev
anisotropic spaces required to describe the regularity of the fluid pressure
and velocity. These Wiener-Sobolev spaces are of independent interest as they
can be useful in other problems. Finally, let us remark that our techniques do
not rely on the irrotational character of the fluid in the bulk and they can be
applied to other free boundary problems.
