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

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.

BD Entropy and Bernis-Friedman Entropy

“…Then using the energy estimate and the BD entropy, we can pass to the limit and get the global existence of non-negative solutions of the system studied in [11]. Note that compressible Navier-Stokes system with constant viscosities and non-local term has been studied recently in [8] together with the long-time behavior of its solutions. The details will be given in the forthcoming paper [1].…”

“…On peut par exemple étudier ces systèmes et montrer l'existence globale de solutions faibles du modèle de lubrification en partant de solutions faibles positives du modèle de Saint-Venant associé. On peut également considérer des systèmes avec des termes non-locaux en s'inspirant de résultats récents, voir par exemple [11], [8].…”

High-Order Well-Balanced Finite-Volume Schemes for Hydrodynamic Equations With Nonlocal Free Energy

“…For the overdamped Keller-Segel model there are basically three possible regimes [9,10], which result from adequately tuning the parameters α in the kernel W (x) and m in the pressure P (ρ): diffusion-dominated regime (m > 1 − α), balanced regime (m = 1 − α) where a critical mass separates self-similar and blowup behavior, and aggregationdominated regime (m < 1 − α). Results with the momentum equation, and thus inertia included, are still quite limited in the literature, with only some specific scenarios studied [14,21]. In our previous work [20] we investigated the role of inertia for a choice of parameters of α = 0.5, m = 1.5 and α = −0.5, m = 1.3, which led to diffusion-and aggregation-dominated regimes, respectively.…”

“…is formally dissipated (see [15,21,46]) as The paper is structured as follows. First, in section 2 we explain the construction of our well-balanced high-order finite-volume scheme.…”

Continued Gravitational Collapse for Newtonian Stars

“…Without the free boundary, in the context of finite-time break up of C 1 -solutions for the gravitational Euler-Poisson system with a fixed background we refer to [12] and references therein. There are various models in the literature where the stabilizing effects of the pressure are contrasted to the attractive effects of a nonlocal interaction; we refer the reader to [7][8][9] for a review and many references for different choices of repulsive/attractive potentials.…”