Mean-field limits: from particle descriptions to macroscopic equations

TL;DR

This paper rigorously derives macroscopic equations from particle models of swarming behavior, using advanced mathematical tools to handle nonlocal interactions and dissipative effects.

Contribution

It introduces a novel rigorous derivation of pressureless Euler-type and aggregation equations from particle descriptions with nonlocal interactions.

Findings

Successful derivation of macroscopic equations from particle models

Use of discrete modulated kinetic energy and bounded Lipschitz distance for control

Establishment of mathematical framework for nonlocal swarming models

Abstract

We rigorously derive pressureless Euler-type equations with nonlocal dissipative terms in velocity and aggregation equations with nonlocal velocity fields from Newton-type particle descriptions of swarming models with alignment interactions. We crucially make use of a discrete version of a modulated kinetic energy together with the bounded Lipschitz distance for measures in order to control terms in its time derivative due to the nonlocal interactions.