Boundary Value Problems Related to the Muskat Problem

The Muskat problem is a two-phase moving boundary problem describing the motion of the flow of two immiscible fluids separated by a sharp interface. Since Darcy’s law is a widely used simplification for the relevant hydrodynamics under consideration, the corresponding velocity potentials are determined as solutions of elliptic boundary value problems. Of particular interest in applications are cases allowing delta distributions as boundary data. Applying tools from classical analysis, several such scenarios are discussed.