The Peskin problem describes the flow of a Stokes fluid through the heart valves. We begin by presenting the simpler 2D model and investigate its small data critical regularity theory, with initial data possibly containing small corners. We then present the 3D problem and describe the challenges that arise to proving global well-posedness.

The first part is joint work with Eduardo Garcia-Juarez, and the second is on-going work with Eduardo Garcia-Juarez and Yoichiro Mori