  In this paper we study the two-dimensional multiphase Muskat problem
describing the motion of three immiscible fluids with general viscosities in a
vertical homogeneous porous medium under the influence of gravity. Employing
Rellich type identities in the regime where the fluids are ordered according to
their viscosities, respectively a Neumann series argument when the fluids are
not ordered by viscosity, we may recast the governing equations as a strongly
coupled nonlinear and nonlocal evolution problem for the functions that
parameterize the sharp interfaces that separate the fluids. This problem is of
parabolic type if the Rayleigh-Taylor condition is satisfied at each interface.
Based on this property, we then show that the multiphase Muskat problem is
well-posed in all $L_2$-subcritical Sobolev spaces and that it features some
parabolic smoothing properties.
