Persisting entropy structure for nonlocal cross-diffusion systems

TL;DR

This paper establishes conditions under which nonlocal cross-diffusion systems retain the entropy structure of local models, aiding in model derivation and solution existence proofs.

Contribution

It provides a framework to ensure nonlocal systems inherit entropy, linking nonlocal and local models, and offers a regularisation scheme for PDE analysis.

Findings

Nonlocal systems can inherit entropy from local models.

The framework applies to population models and approximations.

It completes the derivation of the SKT model from nonlocal to local form.

Abstract

For cross-diffusion systems possessing an entropy (i.e. a Lyapunov functional)we study nonlocal versions and exhibit sufficient conditions to ensure that thenonlocal version inherits the entropy structure. These nonlocal systems can beunderstood as population models per se or as approximation of the classical ones.With the preserved entropy, we can rigorously link the approximating nonlocalversion to the classical local system. From a modelling perspective this gives away to prove a derivation of the model and from a PDE perspective this providesa regularisation scheme to prove the existence of solutions. A guiding example isthe SKT model [22]. In this context we answer positively the question raised byFontbona and M{\'e}l{\'e}ard [12] for the derivation and thus complete the derivation.