The Second Iterate of the Muskat Equation in Supercritical Spaces

The ill-posedness for the Muskat problem in spaces that are supercritical with respect to the scaling is studied. The main result of the paper establishes that for a sequence of approximations of the Muskat equation obtained via Taylor expansion, their corresponding second Picard’s iterate is discontinuous around the origin in a certain family of supercritical spaces approaching a critical space.