# Hydrodynamics of Porous Medium Model with slow reservoirs

## TL;DR

This paper studies the hydrodynamic limits of a porous medium model with reservoirs, using a modified entropy method to handle blocked configurations and derive boundary-dependent macroscopic behavior.

## Contribution

It introduces a perturbed dynamics approach that allows the entropy method to be applied to porous medium models with reservoirs.

## Key findings

- Hydrodynamic behavior follows the porous medium equation with boundary conditions.
- Modified dynamics effectively handle blocked configurations.
- Results extend understanding of porous medium models with reservoir interactions.

## Abstract

We analyze the hydrodynamic behavior of the porous medium model in a discrete space $\{0,\ldots, n\}$, where the sites $0$ and $n$ stand for reservoirs. Our strategy relies on the entropy method of Guo, Papanicolau and Varadhan. However, this method cannot be straightforwardly applied, since there are configurations that do not evolve according to the dynamics (blocked configurations). In order to avoid this problem, we slightly perturbed the dynamics in such a way that the macroscopic behavior of the system keeps following the porous medium equation, but with boundary conditions which depend on the reservoirs strength's.