A well-posedness theory in measures for some kinetic models of collective motion

By José A. Cañizo, José Antonio Carrillo, Jesús Rosado. 2009

320 citations recorded by the source corpus.

We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory, we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hy...

Subfields: Computer Networks and Communications, Condensed Matter Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/0002d2bba6998c726d897cf11cf22b68b10186fa

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Networks and Communications

In Condensed Matter Physics

In Modeling and Simulation

Citations in the corpus

Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming (2010, methodology): “We refer to the review [12] for the different connections between these models and for alarger set of references.”
On collision-avoiding initial configurations to Cucker-Smale type flocking models (2012, background): “Introduction Flocking phenomena [3, 11, 12, 13, 14, 32, 33, 34] such as flocking of birds, schooling of fishes, and swarming of bacteria are often observed in complex biological systems, and are some of the collective self-organizing phenomena in biological, mechanical, and social groups of self-pro”
Extension of the Cucker-Smale Control Law to Space Flight Formations (2009, unknown): “3 6:7 10 10 10 [0–300] 35.”
Measure Solutions for Some Models in Population Dynamics (2011, methodology): “Similar ideas were already used for studying the mean-field limit of kinetic equations such as the classical Vlasov equation [16] and more recently in swarming models [8].”
Cucker-Smale model with normalized communication weights and time delay (2016, methodology): “ can be expressed as a pushforward of the initial datum f 0 by the characteristic map Z(t;;), i.e., f t = Z(t;;)#f 0, and that this formulation is equivalent to the weak formulation (3.1), see, e.g., [7]. 3.2. Stability estimates for the kinetic equation (1.8). In this section we derive the stabilit”
Emergent dynamics of the Cucker-Smale flocking model and its variants (2016, unknown): “two measures ρ1,ρ2 ∈ M(R2d), i.e., dBL(ρ1,ρ2) := sup h∈S Z Rd×Rd hdρ1 − Z Rd×Rd hdρ2 , where S is given by S := ( h: R2d→ R : khkL∞ ≤ 1 and Lip(h) := sup x6= y |h(x) −h(y)| |x−y| ≤ 1 ) . Theorem 5.1. [8, 41] Given f0 ∈ M(R2d) compactly supported, take a sequence of µN 0 of measures of the form: µN 0”