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

On the mean field limit for Cucker-Smale models

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.

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. BolleyJ. 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. CañizoJ. 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 CostanzoM. MenciE. MessinaR. 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. GolseC. 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. GolseC. 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. GolseT. 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. HaJ. 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. PiccoliF. 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. BolleyJ. 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. CañizoJ. 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 CostanzoM. MenciE. MessinaR. 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. GolseC. 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. GolseC. 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. GolseT. 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. HaJ. 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. PiccoliF. 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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