-
Previous Article
A true three-scroll chaotic attractor coined
- DCDS-B Home
- This Issue
-
Next Article
A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems
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 |
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 [
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. |
[1] |
Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic and Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023 |
[2] |
Seung-Yeal Ha, Jeongho Kim, Peter Pickl, Xiongtao Zhang. A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication. Kinetic and Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039 |
[3] |
Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic and Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045 |
[4] |
Hyunjin Ahn, Seung-Yeal Ha, Jeongho Kim. Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4209-4237. doi: 10.3934/cpaa.2021156 |
[5] |
Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic and Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381 |
[6] |
Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168 |
[7] |
Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic and Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026 |
[8] |
Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic and Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 |
[9] |
Martin Friesen, Oleksandr Kutoviy. Stochastic Cucker-Smale flocking dynamics of jump-type. Kinetic and Related Models, 2020, 13 (2) : 211-247. doi: 10.3934/krm.2020008 |
[10] |
Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419 |
[11] |
Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223 |
[12] |
Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062 |
[13] |
Laure Pédèches. Asymptotic properties of various stochastic Cucker-Smale dynamics. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2731-2762. doi: 10.3934/dcds.2018115 |
[14] |
Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155 |
[15] |
Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic and Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027 |
[16] |
Zhisu Liu, Yicheng Liu, Xiang Li. Flocking and line-shaped spatial configuration to delayed Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3693-3716. doi: 10.3934/dcdsb.2020253 |
[17] |
Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure and Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028 |
[18] |
Franco Flandoli, Marta Leocata, Cristiano Ricci. The Vlasov-Navier-Stokes equations as a mean field limit. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3741-3753. doi: 10.3934/dcdsb.2018313 |
[19] |
Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic and Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011 |
[20] |
Moon-Jin Kang, Seung-Yeal Ha, Jeongho Kim, Woojoo Shim. Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1233-1256. doi: 10.3934/cpaa.2020057 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]