Relaxation of the incompressible porous media equation

It was shown recently by Cordoba, Faraco and Gancedo that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework, uses ideas from the theory of laminates, in particular T4 configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding T4 configurations. We then use this to construct weak solutions to the unstable interface problem (the Muskat problem), as a byproduct shedding new light on the gradient flow approach introduced by Otto.Comment: 19 pages, no figures, some misprints correcte

The failure of spectral synthesis on some types of discrete Abelian groups

AbstractThe problem of spectral synthesis on arbitrary Abelian groups is solved in the negative