Coupled Self-Organized Hydrodynamics and Navier-Stokes models: local well-posedness and the limit from the Self-Organized Kinetic-fluid models

TL;DR

This paper establishes the local well-posedness of coupled self-organized hydrodynamics and Navier-Stokes equations and rigorously justifies the hydrodynamic limit from kinetic-fluid models, advancing the mathematical understanding of particle-fluid interactions.

Contribution

It proves local existence and uniqueness of solutions for the coupled model and provides the first rigorous derivation of the hydrodynamic limit from kinetic-fluid systems.

Findings

Proved local well-posedness of the coupled SOH-NS system.

Established the hydrodynamic limit from kinetic-fluid models.

Provided a rigorous mathematical foundation for the model's derivation.

Abstract

A coupled system of self-organized hydrodynamics and Navier-Stokes equations (SOH-NS), which models self-propelled particles in a viscous fluid, was recently derived by Degond et al. \cite{DMVY-2017-arXiv}, starting from a micro-macro particle system of Vicsek-Navier-Stokes model, through an intermediate step of a self-organized kinetic-kinetic model by multiple coarse-graining processes. We first transfer SOH-NS into a non-singular system by stereographic projection, then prove the local in time well-posedness of classical solutions by energy method. Furthermore, employing the Hilbert expansion approach, we justify the hydrodynamic limit from the self-organized kinetic-fluid model to macroscopic dynamics. This provides the first analytically rigorous justification of the modeling and asymptotic analysis in \cite{DMVY-2017-arXiv}.