Nonlinear degenerate cross-diffusion systems with nonlocal interaction

TL;DR

This paper establishes the global existence of weak solutions for a broad class of nonlinear, degenerate cross-diffusion systems with nonlocal interactions, applicable in social, biological, and financial contexts, using a semi-implicit scheme.

Contribution

It introduces a novel approach that relaxes previous assumptions, allowing for nonlocal interactions without requiring a gradient flow structure, and handles degenerate cross-diffusion systems.

Findings

Proves global-in-time existence of weak solutions.

Handles nonlocal interactions beyond gradient flow frameworks.

Extends previous results to more general degenerate systems.

Abstract

We investigate a class of systems of partial differential equations with nonlinear cross-diffusion and nonlocal interactions, which are of interest in several contexts in social sciences, finance, biology, and real world applications. Assuming a uniform "coerciveness" assumption on the diffusion part, which allows to consider a large class of systems with degenerate cross-diffusion (i.e. of porous medium type) and relaxes sets of assumptions previously considered in the literature, we prove global-in-time existence of weak solutions by means of a semi-implicit version of the Jordan-Kinderlehrer-Otto scheme. Our approach allows to consider nonlocal interaction terms not necessarily yielding a formal gradient flow structure.