A note on the uniqueness of weak solutions to a class of cross-diffusion systems

TL;DR

This paper proves the uniqueness of bounded weak solutions for a class of strongly coupled cross-diffusion systems, including models like gas mixtures and population dynamics, using advanced mathematical techniques.

Contribution

It establishes a new uniqueness result for a broad class of coupled parabolic equations with cross-diffusion and drift terms.

Findings

Uniqueness of solutions is proven for the considered class of systems.

The proof employs the $H^{-1}$ technique and the entropy method.

Results apply to models like Maxwell-Stefan and Shigesada-Kawasaki-Teramoto.

Abstract

The uniqueness of bounded weak solutions to strongly coupled parabolic equations in a bounded domain with no-flux boundary conditions is shown. The equations include cross-diffusion and drift terms and are coupled selfconsistently to the Poisson equation. The model class contains special cases of the Maxwell-Stefan equations for gas mixtures, generalized Shigesada-Kawasaki-Teramoto equations for population dynamics, and volume-filling models for ion transport. The uniqueness proof is based on a combination of the $H^{-1}$ technique and the entropy method of Gajewski.