Nonlocal cross-interaction systems on graphs: Nonquadratic Finslerian structure and nonlinear mobilities

TL;DR

This paper extends a Wasserstein-type quasi-metric to model the evolution of two interacting species with nonlinear mobilities on graphs, interpreting the system as a gradient flow in a Finslerian framework.

Contribution

It introduces a novel Finslerian structure for nonlocal cross-interaction systems on graphs, accommodating nonlinear mobilities and multiple species.

Findings

Extended quasi-metric to two-species systems with nonlinear mobilities

Provided a rigorous gradient flow interpretation in a Finslerian setting

Applicable to arbitrary positive measure graphs

Abstract

We study the evolution of a system of two species with nonlinear mobility and nonlocal interactions on a graph whose vertices are given by an arbitrary, positive measure. To this end, we extend a recently introduced $2$-Wasserstein-type quasi-metric on generalized graphs, which is based on an upwind-interpolation, to the case of two-species systems, concave, nonlinear mobilities, and $p\ne 2$. We provide a rigorous interpretation of the interaction system as a gradient flow in the Finslerian setting, arising from the new quasi-metric.