Stability of a cross-diffusion system and approximation by repulsive random walks: a duality approach

TL;DR

This paper develops duality estimates to analyze the stability of cross-diffusion systems for two species and demonstrates convergence of repulsive random walks to these systems, providing both qualitative and quantitative results.

Contribution

It introduces a duality approach to establish stability, uniqueness, and approximation results for cross-diffusion systems and their stochastic counterparts.

Findings

Controlled the evolution of solution differences using duality estimates.

Proved convergence of one-dimensional repulsive random walks to cross-diffusion systems.

Provided both rough and refined quantitative estimates for the approximation.

Abstract

We consider conservative cross-diffusion systems for two species where individual motion rates depend linearly on the local density of the other species. We develop duality estimates and obtain stability and approximation results. We first control the time evolution of the gap between two bounded solutions by means of its initial value. As a by product, we obtain a uniqueness result for bounded solutions valid for any space dimension, under a non-perturbative smallness assumption. Using a discrete counterpart of our duality estimates, we prove the convergence of random walks with local repulsion in one dimensional discrete space to cross-diffusion systems. More precisely, we prove quantitative estimates for the gap between the stochastic process and the cross-diffusion system. We give first rough but general estimates; then we use the duality approach to obtain fine estimates under less general conditions.