A system of continuity equations with nonlocal interactions of Morse type

TL;DR

This paper analyzes a system of two coupled continuity equations with nonlocal Morse-type interactions, establishing existence, uniqueness, stability, and a particle approximation scheme for solutions in the context of multi-population modeling.

Contribution

It introduces a novel particle scheme and proves convergence of gradient flow solutions for a nonlocal Morse interaction system, extending the mathematical understanding of such models.

Findings

Proved existence and uniqueness of solutions in Wasserstein space.

Developed a particle scheme that converges to the gradient flow solutions.

Addressed the Lipschitz singularity in the kernel with a non-standard ODE formulation.

Abstract

We study a system of two continuity equations with nonlocal velocity fields using interaction potentials of both attractive and repulsive Morse type. Such a system is of interest in many contexts in multi-population modelling. We prove existence, uniqueness and stability in the 2-Wasserstein spaces of probability measures via Jordan-Kinderlehrer-Otto scheme and gradient flow solutions in the spirit of the Ambrosio-Gigli-Savar\'e theory. We then formulate a deterministic particle scheme for this model and prove that gradient flow solutions are obtained in the many particle limit by discrete densities constructed out of moving particles satisfying a suitable system of ODEs. The ODE system is formulated in a non standard way in order to bypass the Lipschitz singularity of the kernel, with difference quotients of the kernel replacing its derivative.