From exclusion to the heat equation with non-linear Robin boundary conditions

Salvador Beatriz, Instito Superior Tecnico

We present an exclusion process defined on a one-dimensional lattice, which we call bulk, that is attached at each boundary point to a stochastic boundary reservoir. Each reservoir interacts with the bulk on a window of fixed size $l$ close to the respective boundary point at which it is attached. We obtain the hydrodynamic limit for this process whose hydrodynamic equation is described by the heat equation with non-linear Robin boundary conditions. The novelty here is that we do not put any restrictions on the choice of boundary rates apart from guaranteeing the irreducibility of the process, extending the results of \cite{Nahum2020}. Due to the freedom of choice of rates at the boundary, an interesting phenomenon is observed: we present an example for which we have more than one stationary profile but the hydrodynamic equation has a unique weak solution. This is joint work with Claudio Landim and João Mangi.

Area: CS43 - Hydrodynamic limits (Simone Floreani)

Keywords: Hydrodynamic Limit; Exclusion Process; Heat equation; Non-linear Robin boundary condition

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