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Section: New Results

Microscopic derivation of moving interfaces problems

In [15], M. Simon and her coauthors derive the porous medium equation from an interacting particle system which belongs to the family of kinetically constrained lattice gases. It was already proved in the literature that the macroscopic density profile is governed by the porous medium equation for initial densities uniformly bounded away from 0 and 1. Here we consider the more general case where the density can take those extreme values. The solutions display a richer behavior, like moving interfaces, finite speed of propagation and breaking of regularity. Since standard techniques cannot be straightforwardly applied, we present a way to generalize the relative entropy method, by involving approximations of solutions to the hydrodynamic equation, instead of exact solutions.

In [16], M. Simon and her coauthors study the hydrodynamic limit for a similar one-dimensional exclusion process but with an even more restricting dynamical constraint: this process with degenerate jump rates admits transient states, which it eventually leaves to reach an ergodic component if the initial macroscopic density is larger than a critical value, or one of its absorbing states otherwise. They show that, for initial profiles smooth enough and uniformly larger than the critical density, the macroscopic density profile evolves under the diffusive time scaling according to a fast diffusion equation. The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.

These two macroscopic behaviors belong to the class of moving interfaces problems, which are particularly hard to derive from the microscopic point of view.