Fluid relaxation approximation of the Busenberg--Travis cross-diffusion system

TL;DR

This paper rigorously derives a fluid relaxation approximation for the Busenberg--Travis cross-diffusion system using compressible Navier--Stokes--Korteweg equations, revealing its entropy structure and asymptotic behavior.

Contribution

It introduces a novel approximation of the cross-diffusion system via fluid dynamics equations with a detailed entropy analysis.

Findings

Rigorous proof of the asymptotic limit using compactness and entropy methods

Derivation of energy and entropy inequalities for the system

Identification of the double entropy structure in the limiting system

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.