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doi: 10.3934/dcdsb.2021164
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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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

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