Nonlocal cross-diffusion systems for multi-species populations and networks

TL;DR

This paper analyzes nonlocal cross-diffusion systems on the torus, establishing global existence, uniqueness, and localization limits, with applications in population dynamics and neuroscience, using entropy-based methods.

Contribution

It provides new existence and uniqueness results for nonlocal cross-diffusion systems with nondifferentiable kernels, and explores the localization limit to derive local systems.

Findings

Proved global existence of weak solutions.

Established weak-strong uniqueness via relative entropy.

Analyzed the localization limit to connect nonlocal and local systems.

Abstract

Nonlocal cross-diffusion systems on the torus, arising in population dynamics and neuroscience, are analyzed. The global existence of weak solutions, the weak-strong uniqueness, and the localization limit are proved. The kernels are assumed to be positive definite and in detailed balance. The proofs are based on entropy estimates coming from Shannon-type and Rao-type entropies, while the weak-strong uniqueness result follows from the relative entropy method. The existence and uniqueness theorems hold for nondifferentiable kernels. The associated local cross-diffusion system, derived in the localization limit, is also discussed.