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 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.
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Jonas Bierler, Bogdan-Vasile Matioc, The multiphase Muskat problem with equal viscosities in two dimensions. Interfaces Free Bound. 24 (2022), no. 2, pp. 163–196DOI 10.4171/IFB/469