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Session S06 - Interacting Stochastic Systems

Wednesday, July 14, 14:00 ~ 14:35 UTC-3

Hydrodynamics of Porous Medium with Slow Boundary

We analyze the hydrodynamic behavior of the porous medium model (PMM) in a discrete uni-dimensional lattice. 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). 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 (PME), but with boundary conditions that depend on the boundary strength. These boundary conditions are Robin, Neumann, and Dirichlet. We also prove the convergence of the weak solution of the porous medium equation with a type of Robin boundary conditions, by tuning a parameter either to zero or to infinity. The convergence is in the strong sense, concerning the $L^2-$norm, and the limiting function solves the same equation with Neumann (resp. Dirichlet) boundary conditions when the parameter is taken to zero (resp. infinity).