Splitting Schemes & Segregation In Reaction-(Cross-)Diffusion Systems

TL;DR

This paper introduces a variational splitting scheme combining ODEs and optimal transport to analyze reaction-cross diffusion systems, proving existence of weak solutions and conservation of segregation even with vacuum.

Contribution

It presents a novel splitting scheme that handles initial data without support restrictions and proves segregation conservation in reaction-cross diffusion systems.

Findings

Existence of weak solutions for broad initial data

Segregation is conserved even with vacuum

The scheme effectively combines ODEs and optimal transport methods

Abstract

One of the most fascinating phenomena observed in reaction-diffusion systems is the emergence of segregated solutions, i.e. population densities with disjoint supports. We analyse such a reaction cross-diffusion system. In order to prove existence of weak solutions for a wide class of initial data without restriction about their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum.