Localized Mixing Zone for Muskat Bubbles and Turned Interfaces

We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh¿Taylor and smoothness breakdown exhibited in (Castro et al. in Arch Ration Mech Anal 208(3):805¿909, 2013, Castro et al. in Ann Math. (2) 175(2):909¿948, 2012). At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method.