American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On global solutions to the Vlasov-Poisson system with radiation damping
  • KRM Home
  • This Issue
  • Next Article
    Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling
October  2018, 11(5): 1157-1181. doi: 10.3934/krm.2018045

Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit

Seung-Yeal Ha 1,2,, , Jeongho Kim 3, and  Xiongtao Zhang 4,

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea
2. 

Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea
3. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea
4. 

Center for Mathematical Sciences, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China

* Corresponding author

Received  April 2017 Revised  July 2017 Published  May 2018

Fund Project: The work of S.-Y.- Ha is supported by the Samsung Science and Technology Foundation under project number SSTF-BA1401-03.

Keywords: Uniform stability, ℓp-stability, flocking, mean-field limit, the CuckerSmale model.
Mathematics Subject Classification: Primary: 34D05, 35Q70; Secondary: 34D06.
Citation: Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic & Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045
References:
[1]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Physics, 51 (2010), 103301, 17pp doi: 10.1063/1.3496895.  Google Scholar

[2]

H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang, Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids, Discrete and Continuous Dynamical System, 34 (2014), 4419-4458.  doi: 10.3934/dcds.2014.34.4419.  Google Scholar

[3]

H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[4]

A. Bressan, T.-P. Liu and T. Yang, L1 stability estimates for n × n conservation laws, Arch. Ration. Mech. Anal., 149 (1999), 1-22.  doi: 10.1007/s002050050165.  Google Scholar

[5]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[6]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12.  Google Scholar

[7]

J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models and Methods in Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[8]

Y.-P. Choi, S.-Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles, Vol. I -Advances in Theory, Models, Applications (tentative title), Series: Modeling and Simulation in Science and Technology, (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhäuser Basel, (2017), 299–331.  Google Scholar

[9]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[10]

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

[11]

P. Degond and S. Motsch, Large scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[12]

R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.  Google Scholar

[13]

S.-Y. Ha, $L_1$ stability of the Boltzmann equation for the hard-sphere model, Arch. Ration. Mech. Anal., 173 (2004), 279-296.  doi: 10.1007/s00205-004-0321-x.  Google Scholar

[14]

S.-Y. Ha, B. Kwon and M.-J. Kang, Emergent dynamics for the hydrodynamic Cucker-Smale system in a moving domain, SIAM. J. Math. Anal., 47 (2015), 3813-3831.  doi: 10.1137/140984403.  Google Scholar

[15]

S.-Y. Ha, B. Kwon and M.-J. Kang, A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Mod. Meth. Appl. Sci., 24 (2014), 2311-2359.  doi: 10.1142/S0218202514500225.  Google Scholar

[16]

S.-Y. Ha, C. Lattanzio, B. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.  doi: 10.1090/S0033-569X-2010-01200-7.  Google Scholar

[17]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[18]

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

[19]

S.-Y. Ha and M. Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dyn. Differential Equations, 22 (2010), 325-330.  doi: 10.1007/s10884-009-9142-9.  Google Scholar

[20]

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

[21]

S.-Y. Ha and A. E. Tzavaras, Lyapunov functionals and L1-stability for discrete velocity Boltzmann equations, Comm. Math. Phys., 239 (2003), 65-92.  doi: 10.1007/s00220-003-0866-9.  Google Scholar

[22]

J. Hale, Ordinary Differential Equations, Dover, 1997. Google Scholar

[23]

E. Justh and P. Krishnaprasad, A simple control law for UAV formation flying, Technical Report, 2002-38 (http://www.isr.umd.edu) Google Scholar

[24]

N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.  Google Scholar

[25]

T.-P. Liu and T. Yang, Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math., 52 (1999), 1553-1586.  doi: 10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S.  Google Scholar

[26]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[27]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[28]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic theories and the Boltzmann Equation, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1048 (1984), 60–110. doi: 10.1007/BFb0071878.  Google Scholar

[29]

D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105.   Google Scholar

[30]

L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. of Guidance, Control and Dynamics, 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[31]

J. Toner and Y. Tu, Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E., 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[32]

T. Vicsek, Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[33]

C. Villani, Optimal Transport, Old and New, Springer-Verlag, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

show all references

References:
[1]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Physics, 51 (2010), 103301, 17pp doi: 10.1063/1.3496895.  Google Scholar

[2]

H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang, Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids, Discrete and Continuous Dynamical System, 34 (2014), 4419-4458.  doi: 10.3934/dcds.2014.34.4419.  Google Scholar

[3]

H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[4]

A. Bressan, T.-P. Liu and T. Yang, L1 stability estimates for n × n conservation laws, Arch. Ration. Mech. Anal., 149 (1999), 1-22.  doi: 10.1007/s002050050165.  Google Scholar

[5]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[6]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12.  Google Scholar

[7]

J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models and Methods in Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[8]

Y.-P. Choi, S.-Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles, Vol. I -Advances in Theory, Models, Applications (tentative title), Series: Modeling and Simulation in Science and Technology, (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhäuser Basel, (2017), 299–331.  Google Scholar

[9]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[10]

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

[11]

P. Degond and S. Motsch, Large scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[12]

R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.  Google Scholar

[13]

S.-Y. Ha, $L_1$ stability of the Boltzmann equation for the hard-sphere model, Arch. Ration. Mech. Anal., 173 (2004), 279-296.  doi: 10.1007/s00205-004-0321-x.  Google Scholar

[14]

S.-Y. Ha, B. Kwon and M.-J. Kang, Emergent dynamics for the hydrodynamic Cucker-Smale system in a moving domain, SIAM. J. Math. Anal., 47 (2015), 3813-3831.  doi: 10.1137/140984403.  Google Scholar

[15]

S.-Y. Ha, B. Kwon and M.-J. Kang, A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Mod. Meth. Appl. Sci., 24 (2014), 2311-2359.  doi: 10.1142/S0218202514500225.  Google Scholar

[16]

S.-Y. Ha, C. Lattanzio, B. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.  doi: 10.1090/S0033-569X-2010-01200-7.  Google Scholar

[17]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[18]

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

[19]

S.-Y. Ha and M. Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dyn. Differential Equations, 22 (2010), 325-330.  doi: 10.1007/s10884-009-9142-9.  Google Scholar

[20]

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

[21]

S.-Y. Ha and A. E. Tzavaras, Lyapunov functionals and L1-stability for discrete velocity Boltzmann equations, Comm. Math. Phys., 239 (2003), 65-92.  doi: 10.1007/s00220-003-0866-9.  Google Scholar

[22]

J. Hale, Ordinary Differential Equations, Dover, 1997. Google Scholar

[23]

E. Justh and P. Krishnaprasad, A simple control law for UAV formation flying, Technical Report, 2002-38 (http://www.isr.umd.edu) Google Scholar

[24]

N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.  Google Scholar

[25]

T.-P. Liu and T. Yang, Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math., 52 (1999), 1553-1586.  doi: 10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S.  Google Scholar

[26]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[27]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[28]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic theories and the Boltzmann Equation, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1048 (1984), 60–110. doi: 10.1007/BFb0071878.  Google Scholar

[29]

D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105.   Google Scholar

[30]

L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. of Guidance, Control and Dynamics, 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[31]

J. Toner and Y. Tu, Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E., 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[32]

T. Vicsek, Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[33]

C. Villani, Optimal Transport, Old and New, Springer-Verlag, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[1]

Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013
[2]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381
[3]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023
[4]

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 & Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039
[5]

Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299
[6]

Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic & Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385
[7]

Michael Herty, Mattia Zanella. Performance bounds for the mean-field limit of constrained dynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2023-2043. doi: 10.3934/dcds.2017086
[8]

Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126
[9]

Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic & Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052
[10]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157
[11]

Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086
[12]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080
[13]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[14]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303
[15]

Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019111
[16]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929
[17]

Seung-Yeal Ha, Ho Lee, Seok Bae Yun. Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 115-143. doi: 10.3934/dcds.2009.24.115
[18]

Seung-Yeal Ha, Mitsuru Yamazaki. $L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 353-364. doi: 10.3934/dcdsb.2009.11.353
[19]

Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099
[20]

Max-Olivier Hongler. Mean-field games and swarms dynamics in Gaussian and non-Gaussian environments. Journal of Dynamics & Games, 2020, 7 (1) : 1-20. doi: 10.3934/jdg.2020001

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (146)
  • HTML views (211)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences