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May  2022, 27(5): 2873-2889. doi: 10.3934/dcdsb.2021164

On the mean field limit for Cucker-Smale models

Roberto Natalini 1,, and  Thierry Paul 2,

1. 

Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini, 19, 00185, Rome, Italy
2. 

CNRS & LJLL Sorbonne Université, 4, place Jussieu, 75005 Paris, France

* Corresponding author: Roberto Natalini

Received  March 2021 Published  May 2022 Early access  June 2021

In this note, we consider generalizations of the Cucker-Smale dynamical system and we derive rigorously in Wasserstein's type topologies the mean-field limit (and propagation of chaos) to the Vlasov-type equation introduced in [13]. Unlike previous results on the Cucker-Smale model, our approach is not based on the empirical measures, but, using an Eulerian point of view introduced in [9] in the Hamiltonian setting, we show the limit providing explicit constants. Moreover, for non strictly Cucker-Smale particles dynamics, we also give an insight on what induces a flocking behavior of the solution to the Vlasov equation to the - unknown a priori - flocking properties of the original particle system.

Keywords: Cucker-Smale system, mean-field limit, Vlasov equations, Wasserstein topology, flocking properties.
Mathematics Subject Classification: Primary: 92D50, 35Q83; Secondary: 35Q92, 82C05.
Citation: Roberto Natalini, Thierry Paul. On the mean field limit for Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2873-2889. doi: 10.3934/dcdsb.2021164
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second Edition, Lectures in Mathematics ETH Zürich. Birlhäuser Verlag, Berlin, 2008.

[2]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-lipschitz forces & swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210.  doi: 10.1142/S0218202511005702.

[3]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod. Meth. Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.

[4]

F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[5]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[6]

E. Di Costanzo, M. Menci, E. Messina, R. Natalini and A. Vecchio, A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 443-472.  doi: 10.3934/dcdsb.2019189.

[7]

R. L. Dobrušin, Vlasov equations, (Russian), Funktsional. Anal. i Prilozhen, 13 (1979), 48-58. 

[8]

F. Golse, C. Mouhot and V. Ricci, Empirical measures and Vlasov hierarchies, Kinet. Relat. Models, 6 (2013), 919-943.  doi: 10.3934/krm.2013.6.919.

[9]

F. Golse, C. Mouhot and T. Paul, On the mean field and classical limits of quantum mechanics, Comm. Math. Phys., 343 (2016), 165-205.  doi: 10.1007/s00220-015-2485-7.

[10]

F. Golse, T. Paul and M. Pulvirenti, On the derivation of the Hartree equation from the $N$-body Schrödinger equation: Uniformity in the Planck constant, J. Funct. Anal., 275 (2018), 1603-1649.  doi: 10.1016/j.jfa.2018.06.008.

[11]

S.-Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.

[12]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[13]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[14]

P.-E. Jabin and Z. Wang, Mean field limit and propagation of chaos for Vlasov systems with bounded forces, J. Funct. Anal., 271 (2016), 3588-3627.  doi: 10.1016/j.jfa.2016.09.014.

[15]

P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with $W{-1, \infty}$ kernels, Invent. Math., 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y.

[16]

B. Piccoli, F. Rossi and E. Trélat, Control to flocking of the kinetic Cucker-Smale model, SIAM J. Math. Anal., 47 (2015), 4685-4719.  doi: 10.1137/140996501.

[17]

A.-S. Sznitman, Topics in propagation of chaos, École d'Été de Probabilités de Saint-Flour XIX–-1989, Lecture Notes in Math. 1464, Springer, Berlin, 1991,165–251. doi: 10.1007/BFb0085169.

[18]

C. Villani, Topics in Optimal Transportation, Amer. Math. Soc., Providence (RI), 2003. doi: 10.1090/gsm/058.

[19]

C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second Edition, Lectures in Mathematics ETH Zürich. Birlhäuser Verlag, Berlin, 2008.

[2]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-lipschitz forces & swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210.  doi: 10.1142/S0218202511005702.

[3]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod. Meth. Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.

[4]

F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[5]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[6]

E. Di Costanzo, M. Menci, E. Messina, R. Natalini and A. Vecchio, A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 443-472.  doi: 10.3934/dcdsb.2019189.

[7]

R. L. Dobrušin, Vlasov equations, (Russian), Funktsional. Anal. i Prilozhen, 13 (1979), 48-58. 

[8]

F. Golse, C. Mouhot and V. Ricci, Empirical measures and Vlasov hierarchies, Kinet. Relat. Models, 6 (2013), 919-943.  doi: 10.3934/krm.2013.6.919.

[9]

F. Golse, C. Mouhot and T. Paul, On the mean field and classical limits of quantum mechanics, Comm. Math. Phys., 343 (2016), 165-205.  doi: 10.1007/s00220-015-2485-7.

[10]

F. Golse, T. Paul and M. Pulvirenti, On the derivation of the Hartree equation from the $N$-body Schrödinger equation: Uniformity in the Planck constant, J. Funct. Anal., 275 (2018), 1603-1649.  doi: 10.1016/j.jfa.2018.06.008.

[11]

S.-Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.

[12]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[13]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[14]

P.-E. Jabin and Z. Wang, Mean field limit and propagation of chaos for Vlasov systems with bounded forces, J. Funct. Anal., 271 (2016), 3588-3627.  doi: 10.1016/j.jfa.2016.09.014.

[15]

P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with $W{-1, \infty}$ kernels, Invent. Math., 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y.

[16]

B. Piccoli, F. Rossi and E. Trélat, Control to flocking of the kinetic Cucker-Smale model, SIAM J. Math. Anal., 47 (2015), 4685-4719.  doi: 10.1137/140996501.

[17]

A.-S. Sznitman, Topics in propagation of chaos, École d'Été de Probabilités de Saint-Flour XIX–-1989, Lecture Notes in Math. 1464, Springer, Berlin, 1991,165–251. doi: 10.1007/BFb0085169.

[18]

C. Villani, Topics in Optimal Transportation, Amer. Math. Soc., Providence (RI), 2003. doi: 10.1090/gsm/058.

[19]

C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

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