# American Institute of Mathematical Sciences

September  2008, 1(3): 415-435. doi: 10.3934/krm.2008.1.415

## From particle to kinetic and hydrodynamic descriptions of flocking

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, South Korea 2 Department of Mathematics, Institute for Physical Science & Technology and Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD 20742, United States

Received  June 2008 Revised  June 2008 Published  August 2008

We discuss the Cucker-Smale's (C-S) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasov-type kinetic model for the C-S particle model and prove it exhibits time-asymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the C-S Vlasov-type kinetic model and prove flocking behavior without closure of higher moments.
Citation: Seung-Yeal Ha, Eitan Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic & Related Models, 2008, 1 (3) : 415-435. doi: 10.3934/krm.2008.1.415
 [1] Chjan C. Lim, Junping Shi. The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 717-740. doi: 10.3934/dcdsb.2009.11.717 [2] Mehdi Badsi, Martin Campos Pinto, Bruno Després. A minimization formulation of a bi-kinetic sheath. Kinetic & Related Models, 2016, 9 (4) : 621-656. doi: 10.3934/krm.2016010 [3] Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic & Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901 [4] Nicola Bellomo, Abdelghani Bellouquid, Juanjo Nieto, Juan Soler. Modeling chemotaxis from $L^2$--closure moments in kinetic theory of active particles. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 847-863. doi: 10.3934/dcdsb.2013.18.847 [5] François Gay-Balmaz, Tudor S. Ratiu. Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 2011, 3 (1) : 41-79. doi: 10.3934/jgm.2011.3.41 [6] Matthew M. Dunlop, Andrew M. Stuart. The Bayesian formulation of EIT: Analysis and algorithms. Inverse Problems & Imaging, 2016, 10 (4) : 1007-1036. doi: 10.3934/ipi.2016030 [7] Haiyang Wang, Jianfeng Zhang. Forward backward SDEs in weak formulation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1021-1049. doi: 10.3934/mcrf.2018044 [8] Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419 [9] Azmy S. Ackleh, Ben G. Fitzpatrick, Horst R. Thieme. Rate distributions and survival of the fittest: a formulation on the space of measures. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 917-928. doi: 10.3934/dcdsb.2005.5.917 [10] André Nachbin, Roberto Ribeiro-Junior. A boundary integral formulation for particle trajectories in Stokes waves. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3135-3153. doi: 10.3934/dcds.2014.34.3135 [11] Lorenzo Brasco, Filippo Santambrogio. An equivalent path functional formulation of branched transportation problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 845-871. doi: 10.3934/dcds.2011.29.845 [12] Xiaoying Han, Jinglai Li, Dongbin Xiu. Error analysis for numerical formulation of particle filter. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1337-1354. doi: 10.3934/dcdsb.2015.20.1337 [13] Andaluzia Matei, Mircea Sofonea. Dual formulation of a viscoplastic contact problem with unilateral constraint. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1587-1598. doi: 10.3934/dcdss.2013.6.1587 [14] Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343 [15] Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373 [16] Wenjun Xia, Jinzhi Lei. Formulation of the protein synthesis rate with sequence information. Mathematical Biosciences & Engineering, 2018, 15 (2) : 507-522. doi: 10.3934/mbe.2018023 [17] Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381 [18] Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 [19] Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35 [20] Xiaoliang Cheng, Rongfang Gong, Weimin Han. A new Kohn-Vogelius type formulation for inverse source problems. Inverse Problems & Imaging, 2015, 9 (4) : 1051-1067. doi: 10.3934/ipi.2015.9.1051

2018 Impact Factor: 1.38