Asymptotic Flocking Dynamics for the Kinetic Cucker-Smale Model

By José Antonio Carrillo, Massimo Fornasier, Jesús Rosado, Giuseppe Toscani. SIAM Journal on Mathematical Analysis, 2010

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In this paper, we analyze the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [IEEE Trans. Automat. Control, 52 (2007), pp. 852–862], which describes the collective behavior of an ensemble of organisms, animals, or devices. This kinetic version introduced in [S.-Y. Ha and E. Tadmor, Kinet. Relat. Models, 1 (2008), pp. 415–435] is here obtained starting from a Boltzmann-type equation. The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [F. Cucker and S. Smale, IEEE Trans. Automat. Control, 52 (2007), pp. 852–862] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast in velocity to the mean velocity of the initial condition, while in space they will converge towards a translational flocking solution.

Subfields: Computer Networks and Communications, Condensed Matter Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/93a025e260cb1726d78d51514a1ed3670b1e5bfc

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Citations in the corpus

Emergent Behavior of a Cucker-Smale Type Particle Model With Nonlinear Velocity Couplings (2010, background): “Later Cucker and Smale’s flocking estimate was further simplified and improved to cover general communication weights and hierarchy structures in [9]–[12].”
Fokker–Planck equations in the modeling of socio-economic phenomena (2017, background): “[28, 82, 84] In the biological context, a consistent part of the recent research in population and behavioral biology and ecology has been looking for the emergent behavior of bird flocks, fish schools or bacteria aggregations.[26, 27, 38, 39, 40, 60, 61, 62] Other important examples of emergent beh”
The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, methodology): “, We refer to [34 46, 45, 26] for further discussion about this model and qualitative properties.”
Flocking estimates for the Cucker–Smale model with time lag and hierarchical leadership (2017, background): “[15, 17, 27, 37, 40, 46, 50, 52]), stochastic terms ([18, 23, 25]), pedestrian crowds (see [16, 31]), infinite-dimensional kinetic models (see [1, 4, 9, 12, 22, 28, 49]), topological models ([5, 29]), control models (see [6, 10, 11, 43, 51]).”
A New Model for Self-organized Dynamics and Its Flocking Behavior (2011, methodology): “Other approaches, based on spectral analysis, 1- and ∞-based estimates were used in [6, 8, 14] to derive C-S flocking with a (refined version of) slowly decaying influence function (1.”
A Kinetic Flocking Model with Diffusion (2010, methodology): “As in [2], the evolution of the bird density can be described at a kinetic level by the following integro-differential equation of Boltzmann type: ∂tf + ξ · ∇xf = Q(f, f), (1.”
Cucker-Smale model with normalized communication weights and time delay (2016, background): “Another proof has been provided in [17], based on a bound by a system of dissipative differential inequalities, and, finally, the proof of [8] is based on bounding the maximal velocity.”
Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming (2010, methodology): “We refer to [16, 22, 11, 12] and references therein for further discussion about this model and qualitative properties.”
Extension of the Cucker-Smale Control Law to Space Flight Formations (2009, unknown): “3 6:7 10 10 10 [0–300] 35.”