The Muskat equation models the interaction of two incompressible fluids with equal viscosity propagating in porous medium, governed by Darcy’s law. The Peskin problem describes the flow of a Stokes fluid through the heart valves.  In this talk, we investigate the small data critical regularity theory for these two models, and in particular, the desingularization of interfaces with small corners. The first part is joint work with E. Garcia-Juarez, J. Gomez-Serrano and B. Pausader. The second part is joint work with E. Garcia-Juarez.