A well-posedness theory in measures for some kinetic models of collective motion

TL;DR

This paper establishes a well-posedness framework for kinetic models of collective motion, incorporating various interaction effects, and demonstrates convergence from particle systems to kinetic equations and hydrodynamic limits.

Contribution

It introduces a measure-based well-posedness theory for complex kinetic models of collective behavior, including interaction, velocity adaptation, and self-propulsion effects.

Findings

Proves existence and uniqueness of solutions in measure spaces.

Shows convergence of particle systems to kinetic equations.

Establishes local-in-time hydrodynamic limit.

Abstract

We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models.