Nonlocal approximation of an anisotropic cross-diffusion system

TL;DR

This paper establishes the nonlocal approximation of an anisotropic cross-diffusion system relevant in population dynamics, using entropy methods to handle phase-separation phenomena and internal layers.

Contribution

It extends nonlocal approximation results to anisotropic systems with phase separation, a previously underexplored area.

Findings

Proves nonlocal-to-limit convergence for the anisotropic system

Develops an entropy dissipation identity for weak solutions

Addresses internal layers in population dynamics models

Abstract

Localisation limits and nonlocal approximations of degenerate parabolic systems have experienced a renaissance in recent years. However, only few results cover anisotropic systems. This work addresses this gap by establishing the nonlocal-to-limit for a specific anisotropic cross-diffusion system encountered in population dynamics featuring phase-separation phenomena, i.e., internal layers between different species. A critical element of the proof is an entropy dissipation identity, which we show to hold for any weak solution.