# Fluid Relaxation Approximation of the Busenberg–Travis Cross-Diffusion System

## TL;DR

This paper shows how a system modeling population segregation can be approximated using fluid dynamics equations with specific entropy properties.

## Contribution

The paper introduces new energy and entropy inequalities that connect fluid dynamics to population segregation models.

## Key findings

- The Busenberg–Travis system is approximated using compressible Navier–Stokes–Korteweg equations.
- Energy and entropy inequalities are derived and shown to reduce to Boltzmann–Shannon and Rao entropy in the asymptotic limit.

## Abstract

The Busenberg–Travis cross-diffusion system for segregating populations is approximated by the compressible Navier–Stokes–Korteweg equations on the torus, including a density-dependent viscosity and drag forces. The Korteweg term can be associated to the quantum Bohm potential. The singular asymptotic limit is proved rigorously using compactness and relative entropy methods. The novelty is the derivation of energy and entropy inequalities, which reduce in the asymptotic limit to the Boltzmann–Shannon and Rao entropy inequalities, thus revealing the double entropy structure of the limiting Busenberg–Travis system.