Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 3 (2019), 843-866.

A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment

Alessio Figalli and Moon-Jin Kang

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider the kinetic Cucker–Smale model with local alignment as a mesoscopic description for the flocking dynamics. The local alignment was first proposed by Karper, Mellet and Trivisa (2014), as a singular limit of a normalized nonsymmetric alignment introduced by Motsch and Tadmor (2011). The existence of weak solutions to this model was obtained by Karper, Mellet and Trivisa (2014), and in the same paper they showed the time-asymptotic flocking behavior. Our main contribution is to provide a rigorous derivation from a mesoscopic to a macroscopic description for the Cucker–Smale flocking models. More precisely, we prove the hydrodynamic limit of the kinetic Cucker–Smale model with local alignment towards the pressureless Euler system with nonlocal alignment, under a regime of strong local alignment. Based on the relative entropy method, a main difficulty in our analysis comes from the fact that the entropy of the limit system has no strict convexity in terms of density variable. To overcome this, we combine relative entropy quantities with the 2-Wasserstein distance.

Article information

Anal. PDE, Volume 12, Number 3 (2019), 843-866.

Received: 22 January 2018
Revised: 23 April 2018
Accepted: 29 June 2018
First available in Project Euclid: 25 October 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q70: PDEs in connection with mechanics of particles and systems
Secondary: 35B25: Singular perturbations

hydrodynamic limit kinetic Cucker–Smale model local alignment pressureless Euler system relative entropy Wasserstein distance


Figalli, Alessio; Kang, Moon-Jin. A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment. Anal. PDE 12 (2019), no. 3, 843--866. doi:10.2140/apde.2019.12.843.

Export citation


  • L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Birkhäuser, Basel, 2005.
  • F. Berthelin and A. Vasseur, “From kinetic equations to multidimensional isentropic gas dynamics before shocks”, SIAM J. Math. Anal. 36:6 (2005), 1807–1835.
  • F. Bouchut, “On zero pressure gas dynamics”, pp. 171–190 in Advances in kinetic theory and computing, edited by B. Perthame, Ser. Adv. Math. Appl. Sci. 22, World Sci., River Edge, NJ, 1994.
  • F. Bouchut and F. James, “Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness”, Comm. Partial Differential Equations 24:11-12 (1999), 2173–2189.
  • L. Boudin, “A solution with bounded expansion rate to the model of viscous pressureless gases”, SIAM J. Math. Anal. 32:1 (2000), 172–193.
  • Y. Brenier and E. Grenier, “Sticky particles and scalar conservation laws”, SIAM J. Numer. Anal. 35:6 (1998), 2317–2328.
  • J. A. Cañizo, J. A. Carrillo, and J. Rosado, “A well-posedness theory in measures for some kinetic models of collective motion”, Math. Models Methods Appl. Sci. 21:3 (2011), 515–539.
  • E. Carlen, M. C. Carvalho, P. Degond, and B. Wennberg, “A Boltzmann model for rod alignment and schooling fish”, Nonlinearity 28:6 (2015), 1783–1803.
  • J. A. Carrillo, M. Fornasier, J. Rosado, and G. Toscani, “Asymptotic flocking dynamics for the kinetic Cucker–Smale model”, SIAM J. Math. Anal. 42:1 (2010), 218–236.
  • J. A. Carrillo, Y.-P. Choi, and T. K. Karper, “On the analysis of a coupled kinetic-fluid model with local alignment forces”, Ann. Inst. H. Poincaré Anal. Non Linéaire 33:2 (2016), 273–307.
  • F. Cucker and S. Smale, “Emergent behavior in flocks”, IEEE Trans. Automat. Control 52:5 (2007), 852–862.
  • T. Do, A. Kiselev, L. Ryzhik, and C. Tan, “Global regularity for the fractional Euler alignment system”, Arch. Ration. Mech. Anal. 228:1 (2018), 1–37.
  • R. Duan, M. Fornasier, and G. Toscani, “A kinetic flocking model with diffusion”, Comm. Math. Phys. 300:1 (2010), 95–145.
  • M. Fornasier, J. Haskovec, and G. Toscani, “Fluid dynamic description of flocking via the Povzner–Boltzmann equation”, Phys. D 240:1 (2011), 21–31.
  • T. Goudon, P.-E. Jabin, and A. Vasseur, “Hydrodynamic limit for the Vlasov–Navier–Stokes equations, II: Fine particles regime”, Indiana Univ. Math. J. 53:6 (2004), 1517–1536.
  • S.-Y. Ha and J.-G. Liu, “A simple proof of the Cucker–Smale flocking dynamics and mean-field limit”, Commun. Math. Sci. 7:2 (2009), 297–325.
  • S.-Y. Ha and E. Tadmor, “From particle to kinetic and hydrodynamic descriptions of flocking”, Kinet. Relat. Models 1:3 (2008), 415–435.
  • S.-Y. Ha, F. Huang, and Y. Wang, “A global unique solvability of entropic weak solution to the one-dimensional pressureless Euler system with a flocking dissipation”, J. Differential Equations 257:5 (2014), 1333–1371.
  • S.-Y. Ha, M.-J. Kang, and B. Kwon, “A hydrodynamic model for the interaction of Cucker–Smale particles and incompressible fluid”, Math. Models Methods Appl. Sci. 24:11 (2014), 2311–2359.
  • S.-Y. Ha, Z. Li, M. Slemrod, and X. Xue, “Flocking behavior of the Cucker–Smale model under rooted leadership in a large coupling limit”, Quart. Appl. Math. 72:4 (2014), 689–701.
  • S.-Y. Ha, M.-J. Kang, and B. Kwon, “Emergent dynamics for the hydrodynamic Cucker–Smale system in a moving domain”, SIAM J. Math. Anal. 47:5 (2015), 3813–3831.
  • S.-Y. Ha, J. Jeong, S. E. Noh, Q. Xiao, and X. Zhang, “Emergent dynamics of Cucker–Smale flocking particles in a random environment”, J. Differential Equations 262:3 (2017), 2554–2591.
  • F. Huang and Z. Wang, “Well posedness for pressureless flow”, Comm. Math. Phys. 222:1 (2001), 117–146.
  • P.-E. Jabin, Équations de transport modélisant des particules en interaction dans un fluide et comportement asymptotiques, Ph.D. thesis, Université Paris VI, 2000.
  • P.-E. Jabin and T. Rey, “Hydrodynamic limit of granular gases to pressureless Euler in dimension 1”, Quart. Appl. Math. 75:1 (2017), 155–179.
  • M.-J. Kang, “From the Vlasov–Poisson equation with strong local alignment to the pressureless Euler–Poisson system”, Appl. Math. Lett. 79 (2018), 85–91.
  • M.-J. Kang and A. F. Vasseur, “Asymptotic analysis of Vlasov-type equations under strong local alignment regime”, Math. Models Methods Appl. Sci. 25:11 (2015), 2153–2173.
  • T. K. Karper, A. Mellet, and K. Trivisa, “Existence of weak solutions to kinetic flocking models”, SIAM J. Math. Anal. 45:1 (2013), 215–243.
  • T. K. Karper, A. Mellet, and K. Trivisa, “On strong local alignment in the kinetic Cucker–Smale model”, pp. 227–242 in Hyperbolic conservation laws and related analysis with applications, edited by G.-Q. G. Chen et al., Springer Proc. Math. Stat. 49, Springer, 2014.
  • T. K. Karper, A. Mellet, and K. Trivisa, “Hydrodynamic limit of the kinetic Cucker–Smale flocking model”, Math. Models Methods Appl. Sci. 25:1 (2015), 131–163.
  • A. Mellet and A. Vasseur, “Asymptotic analysis for a Vlasov–Fokker–Planck/compressible Navier–Stokes system of equations”, Comm. Math. Phys. 281:3 (2008), 573–596.
  • S. Motsch and E. Tadmor, “A new model for self-organized dynamics and its flocking behavior”, J. Stat. Phys. 144:5 (2011), 923–947.
  • F. Poupaud and M. Rascle, “Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients”, Comm. Partial Differential Equations 22:1-2 (1997), 337–358.
  • D. Poyato and J. Soler, “Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker–Smale models”, Math. Models Methods Appl. Sci. 27:6 (2017), 1089–1152.
  • J. Silk, A. Szalay, and Y. B. Zeldovich, “Large-scale structure of the universe”, Sci. Amer. 249:4 (1983), 72–80.
  • E. Tadmor and C. Tan, “Critical thresholds in flocking hydrodynamics with non-local alignment”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372:2028 (2014), art. id. 20130401.
  • A. F. Vasseur, “Recent results on hydrodynamic limits”, pp. 323–376 in Handbook of differential equations: evolutionary equations, IV, edited by C. M. Dafermos and M. Pokorný, Elsevier/North-Holland, Amsterdam, 2008.
  • T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, “Novel type of phase transition in a system of self-driven particles”, Phys. Rev. Lett. 75:6 (1995), 1226–1229.
  • C. Villani, Optimal transport: old and new, Grundlehren der Mathematischen Wissenschaften 338, Springer, 2009.
  • E. Weinan, Y. G. Rykov, and Y. G. Sinai, “Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics”, Comm. Math. Phys. 177:2 (1996), 349–380.
  • M. Zavlanos, M. Egerstedt, and G. J. Pappas, “Graph-theoretic connectivity control of mobile robot networks”, Proc. IEEE 99:9 (2011), 1525–1540.
  • Y. B. Zeldovich, Astron. Astrophys. 5:1 (1970), 84–89.