A one-phase Stefan problem with non-linear diffusion from highly competing two-species particle systems

TL;DR

This paper derives a one-phase Stefan problem with non-linear diffusion from a highly competitive two-species particle system using hydrodynamic limits, revealing new insights into free boundary problems in interacting particle models.

Contribution

It introduces a novel derivation of a Stefan-type free boundary problem with non-linear diffusion from microscopic particle interactions under strong competition.

Findings

Derivation of a one-phase Stefan problem with non-linear diffusion

Connection between microscopic particle dynamics and macroscopic free boundary problems

Use of hydrodynamic limit to link particle systems with PDE models

Abstract

We consider an interacting particle system with two species under strong competition dynamics between the two species. Then, through the hydrodynamic limit procedure for the microscopic model, we derive a one-phase Stefan type free boundary problem with non-linear diffusion, letting the competition rate divergent. Non-linearity of diffusion comes from a zero-range dynamics for one species while we impose the other species to weakly diffuse according to the Kawasaki dynamics for technical reasons, which macroscopically corresponds to the vanishing viscosity method.