The multiphase Muskat problem with equal viscosities in two dimensions

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