# First-order aggregation models with alignment

@article{Fetecau2016FirstorderAM,
title={First-order aggregation models with alignment},
author={Razvan C. Fetecau and Weiran Sun and Changhui Tan},
journal={Physica D: Nonlinear Phenomena},
year={2016},
volume={325},
pages={146-163}
}
• Published 2016
• Mathematics
• Physica D: Nonlinear Phenomena
Abstract We include alignment interactions in a well-studied first-order attractive–repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the population density, becomes implicit, and can have non-unique solutions. We investigate the well-posedness of the model and show rigorously how it can be obtained as a macroscopic limit of a second-order kinetic equation. We work within the space of… Expand
12 Citations

#### Figures from this paper

Small inertia regularization of an anisotropic aggregation model
• Mathematics, Physics
• 2016
We consider an anisotropic first-order ODE aggregation model and its approximation by a second-order relaxation system. The relaxation model contains a small parameter $\varepsilon$, which can beExpand
Global Regularity for 1D Eulerian Dynamics with Singular Interaction Forces
• Computer Science, Mathematics
• SIAM J. Math. Anal.
• 2018
It is proved that global regularity persists for more general EPA models with class of singular alignment terms as well as natural attraction/repulsion terms, and to incorporate the attractive/repulsive potential. Expand
Some aspects of the inertial spin model for flocks and related kinetic equations
• Physics, Mathematics
• 2019
In this paper we study the macroscopic behavior of the inertial spin (IS) model. This model has been recently proposed to describe the collective dynamics of flocks of birds, and its main feature isExpand
Hydrodynamic limit of a coupled Cucker–Smale system with strong and weak internal variable relaxation
• Mathematics
• 2021
In this paper, we present the hydrodynamic limit of a multiscale system describing the dynamics of two populations of agents with alignment interactions and the effect of an internal variable. ItExpand
Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces
• Physics, Mathematics
• 2019
We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from theExpand
Filippov flows and mean-field limits in the kinetic singular Kuramoto model
The agent-based singular Kuramoto model was proposed in [60] as a singular version of the Kuramoto model of coupled oscillators that is consistent with Hebb's rule of neuroscience. In such paper, theExpand
An asymptotic preserving scheme for kinetic models with singular limit
• Mathematics
• 2017
We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations byExpand
Agent-based modelling for epidemiological applications
An agent-based model of short-range mosquito host-seeking behaviour in the presence of Long-Lasting Insecticidal Nets (LLINs) is introduced and the model is extended to quantify the overall impact of intervention strategies in community-level situations. Expand
Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives
This paper presents a review and critical analysis on the modeling of the dynamics of vehicular traffic, human crowds and swarms seen as living and, hence, complex systems. It contains a survey of ...
Large friction limit of presureless Euler equations with nonlocal forces
We rigorously show a large friction limit of hydrodynamic models with alignment, attractive, and repulsive effects. More precisely, we consider pressureless Euler equations with nonlocal forces andExpand

#### References

SHOWING 1-10 OF 69 REFERENCES
First-order aggregation models and zero inertia limits
• Mathematics
• 2015
Abstract We consider a first-order aggregation model in both discrete and continuum formulations and show rigorously how it can be obtained as zero inertia limits of second-order models. In theExpand
Swarming Patterns in a Two-Dimensional Kinematic Model for Biological Groups
• Mathematics, Computer Science
• SIAM J. Appl. Math.
• 2004
A continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior, which takes the form of a conservation law in two spatial dimensions. Expand
Asymptotic Dynamics of Attractive-Repulsive Swarms
• Mathematics, Computer Science
• SIAM J. Appl. Dyn. Syst.
• 2009
An analytical upper bound is derived for the finite blow-up time after which the solution forms one or more $\delta$-functions of the conservation equation. Expand
A Continuum Three-Zone Model for Swarms
• Mathematics, Medicine
• Bulletin of mathematical biology
• 2012
We present a progression of three distinct three-zone, continuum models for swarm behavior based on social interactions with neighbors in order to explain simple coherent structures in popularExpand
The derivation of swarming models: Mean-field limit and Wasserstein distances
• Mathematics
• 2014
These notes are devoted to a summary on the mean-field limit of large ensembles of interacting particles with applications in swarming models. We first make a summary of the kinetic models derived asExpand
A non-local model for a swarm
• Mathematics
• 1999
Abstract. This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists ofExpand
Swarm dynamics and equilibria for a nonlocal aggregation model
• Mathematics
• 2011
We consider the aggregation equation ρt −∇ ·(ρ∇K ∗ ρ) = 0i nR n , where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentialsExpand
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations
• Mathematics
• 2011
In this paper we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interactingExpand
Large time behavior of nonlocal aggregation models with nonlinear diffusion
• Mathematics, Computer Science
• Networks Heterog. Media
• 2008
The aim of this paper is to establish rigorous results on the large time behavior of nonlocal models for aggregation, including the possible presence of nonlinear diffusion terms modeling local Expand
State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System
• Mathematics, Physics
• 2007
We study a class of swarming problems wherein particles evolve dynamically via pairwise interaction potentials and a velocity selection mechanism. We find that the swarming system undergoes variousExpand