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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[33]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[33]

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

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