Entropy solutions to macroscopic IPM

We consider the incompressible porous media equation (IPM) describing the evolution of an incompressible fluid in a porous medium subject to gravity. The initial data of our interest consists of a (not necessarily flat) interface separating a heavier fluid with homogeneous density ${\rho }_{+}>0$ from a lighter fluid with homogeneous density ${\rho }_{-}\in (0,{\rho }_{+})$, with the heavier one being above the lighter one. Due to the gravity term this situation is in real world scenarios unstable and mathematically ill-posed as an initial value problem.

The talk addresses the question of recovering well-posedness on the level of averaged solutions (convex integration subsolutions) by means of a selection based on maximal potential energy dissipation. We will see that this criterion leads to a nonlocal hyperbolic conservation law - the "macroscopic IPM" system - which is consistent with the relaxation of Otto based on the gradient flow structure of IPM. In the second part of the talk we will discuss the construction of an entropy solution for macroscopic IPM emanating from a real analytic initial interface.

This is based on a joint work with Angel Castro and Daniel Faraco.