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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

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