Entropy solutions to the macroscopic incompressible porous media equation

We investigate maximal potential energy dissipation as a selection criterion for subsolutions (coarsegrained solutions) in the setting of the unstable Muskat problem.We show that both (a) imposing this criterion on the level of convex integration subsolutions and (b) the strategy of Otto based on a relaxation via minimizing movements lead to the same nonlocal conservation law.Our main result shows that this equation admits an entropy solution for unstable initial data with an analytic interface.1. Introduction 2241 2. Ill-posedness and relaxation 2247 3. The main result 2258 4. Level set formulation 2261 5. Existence of a solution for analytic graphs 2266 6. Justification of ansatzes 2278 Appendix A. The abstract Cauchy-Kovalevskaya theorem 2280 Appendix B. More on Otto's relaxation 2281 Appendix C. Rigorous energy dissipation 2286