# American Institute of Mathematical Sciences

June  2014, 9(2): 269-297. doi: 10.3934/nhm.2014.9.269

## Dynamics and kinetic limit for a system of noiseless $d$-dimensional Vicsek-type particles

 1 Dipartimento di Matematica, Università della Calabria, Campus di Arcavacata, Ponte P. Bucci - cubo 30B, 87036 Arcavacata di Rende (CS), Italy 2 Dipartimento di Matematica, Università di Roma Tre, L.go S.Murialdo 1, 00146 Roma, Italy

Received  November 2012 Revised  February 2014 Published  July 2014

We analyze the continuous time evolution of a $d$-dimensional system of $N$ self propelled particles with a kinematic constraint on the velocities inspired by the original Vicsek's one [29]. Interactions among particles are specified by a pairwise potential in such a way that the velocity of any given particle is updated to the weighted average velocity of all those particles interacting with it. The weights are given in terms of the interaction rate function. The interaction is not of mean field type and the system is non-Hamiltonian. When the size of the system is fixed, we show the existence of an invariant manifold in the phase space and prove its exponential asymptotic stability. In the kinetic limit we show that the particle density satisfies a nonlinear kinetic equation of Vlasov type, under suitable conditions on the interaction. We study the qualitative behaviour of the solution and we show that the Boltzmann-Vlasov entropy is strictly decreasing in time.
Citation: Michele Gianfelice, Enza Orlandi. Dynamics and kinetic limit for a system of noiseless $d$-dimensional Vicsek-type particles. Networks and Heterogeneous Media, 2014, 9 (2) : 269-297. doi: 10.3934/nhm.2014.9.269
##### References:

show all references

##### References:
 [1] Alethea B. T. Barbaro, Pierre Degond. Phase transition and diffusion among socially interacting self-propelled agents. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1249-1278. doi: 10.3934/dcdsb.2014.19.1249 [2] José A. Carrillo, M. R. D’Orsogna, V. Panferov. Double milling in self-propelled swarms from kinetic theory. Kinetic and Related Models, 2009, 2 (2) : 363-378. doi: 10.3934/krm.2009.2.363 [3] Paolo Buttà, Franco Flandoli, Michela Ottobre, Boguslaw Zegarlinski. A non-linear kinetic model of self-propelled particles with multiple equilibria. Kinetic and Related Models, 2019, 12 (4) : 791-827. doi: 10.3934/krm.2019031 [4] Pierre Degond, Angelika Manhart, Hui Yu. A continuum model for nematic alignment of self-propelled particles. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1295-1327. doi: 10.3934/dcdsb.2017063 [5] Seung-Yeal Ha, Eitan Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic and Related Models, 2008, 1 (3) : 415-435. doi: 10.3934/krm.2008.1.415 [6] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations and Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83 [7] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations and Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59 [8] Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic and Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381 [9] Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic and Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 [10] Seung-Yeal Ha, Doron Levy. Particle, kinetic and fluid models for phototaxis. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 77-108. doi: 10.3934/dcdsb.2009.12.77 [11] María del Mar González, Regis Monneau. Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1255-1286. doi: 10.3934/dcds.2012.32.1255 [12] Pierre Degond, Simone Goettlich, Axel Klar, Mohammed Seaid, Andreas Unterreiter. Derivation of a kinetic model from a stochastic particle system. Kinetic and Related Models, 2008, 1 (4) : 557-572. doi: 10.3934/krm.2008.1.557 [13] Dirk Helbing, Jan Siegmeier, Stefan Lämmer. Self-organized network flows. Networks and Heterogeneous Media, 2007, 2 (2) : 193-210. doi: 10.3934/nhm.2007.2.193 [14] Piotr B. Mucha. Limit of kinetic term for a Stefan problem. Conference Publications, 2007, 2007 (Special) : 741-750. doi: 10.3934/proc.2007.2007.741 [15] Arnaud Debussche, Julien Vovelle. Diffusion limit for a stochastic kinetic problem. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2305-2326. doi: 10.3934/cpaa.2012.11.2305 [16] Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic and Related Models, 2021, 14 (3) : 541-570. doi: 10.3934/krm.2021015 [17] Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic and Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011 [18] David Cowan. Rigid particle systems and their billiard models. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 111-130. doi: 10.3934/dcds.2008.22.111 [19] Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao, Xiongtao Zhang. A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data. Communications on Pure and Applied Analysis, 2020, 19 (2) : 835-882. doi: 10.3934/cpaa.2020039 [20] Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure and Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028

2021 Impact Factor: 1.41