Analysis of a fractional cross-diffusion system for multi-species populations

TL;DR

This paper proves the global existence of weak solutions for a fractional cross-diffusion system modeling multi-species populations with long-range interactions, derived from particle systems with Lévy noise.

Contribution

It introduces a novel existence proof for fractional cross-diffusion systems using a three-level approximation, entropy estimates, and a new compactness lemma in unbounded domains.

Findings

Existence of weak solutions established for the system.

Methodology applicable to systems with fractional diffusion.

Framework derived from particle systems with Lévy noise.

Abstract

The global in time existence of weak solutions to a cross-diffusion system with fractional diffusion in the whole space is proved. The equations describe the evolution of multi-species populations in the regime of large-distance interactions; they have been derived in the many-particle limit from moderately interacting particle systems with L\'evy noise. The existence proof is based on a three-level approximation scheme, entropy and moment estimates, and a new Aubin-Lions compactness lemma in the whole space.