Convergence of a degenerate microscopic dynamics to the porous medium equation

TL;DR

This paper proves that a degenerate exclusion process with boundary states converges to the porous medium equation, extending previous results to include extreme densities and employing a generalized relative entropy method.

Contribution

It generalizes the hydrodynamic limit from bounded away from zero and one densities to include boundary cases with degenerate dynamics, using an innovative approximation approach.

Findings

Hydrodynamic limit is governed by the porous medium equation with boundary densities.

The standard techniques are insufficient for degenerate cases, requiring a generalized entropy method.

The approach handles moving interfaces and finite propagation speed in solutions.

Abstract

We derive the porous medium equation from an interacting particle system which belongs to the family of exclusion processes, with nearest neighbor exchanges. The particles follow a degenerate dynamics, in the sense that the jump rates can vanish for certain configurations, and there exist blocked configurations that cannot evolve. In [Gon\c{c}alves-Landim-Toninelli '09] it was proved that the macroscopic density profile in the hydrodynamic limit is governed by the porous medium equation (PME), for initial densities uniformly bounded away from $0$ and $1$. In this paper we consider the more general case where the density can take those extreme values. In this context, the PME solutions display a richer behavior, like moving interfaces, finite speed of propagation and breaking of regularity. As a consequence, the standard techniques that are commonly used to prove this hydrodynamic limits cannot be straightforwardly applied to our case. We present here a way to generalize the \emph{relative entropy method}, by involving approximations of solutions to the hydrodynamic equation, instead of exact solutions.