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In this paper we consider the Muskat problem describing the motion of twounbounded immiscible fluid layers with equal viscosities in vertical orhorizontal two-dimensional geometries. We first prove that the mathematicalmodel can be formulated as an evolution problem for the sharp interfaceseparating the two fluids, which turns out to be, in a suitable functionalanalytic setting, quasilinear and of parabolic type. Based upon theseproperties, we then establish the local well-posedness of the problem forarbitrary large initial data and show that the solutions become instantlyreal-analytic in time and space. Our method allows us to choose the initialdata in the class $H^s,$ $s\in(3/2,2)$, when neglecting surface tension,respectively in $H^s,$ $s\in(2,3),$ when surface tension effects are included.Besides, we provide new criteria for the global existence of solutions.

The Muskat problem in 2D: equivalence of formulations, well-posedness, and regularity results