  We study the two-dimensional multiphase Muskat problem describing the motion
of three immiscible fluids with equal viscosities in a vertical homogeneous
porous medium identified with $\mathbb{R}^2$ under the effect of gravity. We
first formulate the governing equations as a strongly coupled evolution problem
for the functions that parameterize the sharp interfaces between the fluids.
Afterwards we prove that the problem is of parabolic type and establish its
well-posedness together with two parabolic smoothing properties. For solutions
that are not global we exclude, in a certain regime, that the interfaces come
into contact along a curve segment.
