On long-time asymptotics for viscous hydrodynamic models of collective behavior with damping and nonlocal interactions

TL;DR

This paper studies the long-term behavior of viscous hydrodynamic models for collective behavior, proving existence, uniqueness, and convergence of solutions to stationary states under certain conditions.

Contribution

It establishes the global existence of weak solutions and their convergence to stationary states with fixed mass and center of mass in a viscous, damped hydrodynamic model.

Findings

Existence of global weak solutions under specified conditions

Uniqueness of stationary solutions with compact support

Convergence of solutions to stationary states over time

Abstract

Hydrodynamic systems arising in swarming modelling include nonlocal forces in the form of attractive-repulsive potentials as well as pressure terms modelling strong local repulsion. We focus on the case where there is a balance between nonlocal attraction and local pressure in presence of confinement in the whole space. Under suitable assumptions on the potentials and the pressure functions, we show the global existence of weak solutions for the hydrodynamic model with viscosity and linear damping. By introducing linear damping in the system, we ensure the existence and uniqueness of stationary solutions with compactly supported density, fixed mass and center of mass. The associated velocity field is zero in the support of the density. Moreover, we show that global weak solutions converge for large times to the set of these stationary solutions in a suitable sense. In particular cases, we can identify the limiting density uniquely as the global minimizer of the free energy with the right mass and center of mass.