Hydrodynamic models of self-organized dynamics: derivation and existence theory

TL;DR

This paper derives and analyzes hydrodynamic models for self-propelled particles with alignment and attraction-repulsion, revealing their hyperbolic structure and proving local existence results in 2D and 3D.

Contribution

It introduces new scalings that incorporate non-local effects into hydrodynamic limits, leading to models with pressure, viscosity, and capillary forces, and establishes existence theorems.

Findings

Models are symmetrizable hyperbolic with viscosity.

Local existence proved for 2D viscous models.

Local existence proved for 3D inviscid models.

Abstract

This paper is concerned with the derivation and analysis of hydrodynamic models for systems of self-propelled particles subject to alignment interaction and attraction-repulsion. The starting point is the kinetic model considered in earlier work of Degond & Motsch with the addition of an attraction-repulsion interaction potential. Introducing different scalings than in Degond & Motsch, the non-local effects of the alignment and attraction-repulsion interactions can be kept in the hydrodynamic limit and result in extra pressure, viscosity terms and capillary force. The systems are shown to be symmetrizable hyperbolic systems with viscosity terms. A local-in-time existence result is proved in the 2D case for the viscous model and in the 3D case for the inviscid model. The proof relies on the energy method.