December  2021, 20(12): 4209-4237. doi: 10.3934/cpaa.2021156

Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit

1. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

2. 

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

3. 

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea

* Corresponding author

Received  March 2021 Revised  August 2021 Published  December 2021 Early access  September 2021

Fund Project: The work of S.-Y. Ha was supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881), and the work of J. Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science and ICT (NRF-2020R1A4A3079066)

We present a uniform(-in-time) stability of the relativistic Cucker-Smale (RCS) model in a suitable framework and study its application to a uniform mean-field limit which lifts earlier classical results for the CS model in a relativistic setting. For this, we first provide a sufficient framework for an exponential flocking for the RCS model in terms of the diameters of state observables, coupling strength and communication weight function, and then we use the obtained exponential flocking estimate to derive a uniform $ \ell_{q,p} $-stability of the RCS model under appropriate conditions on initial data and system parameters. As an application of the derived uniform $ \ell_{q,p} $-stability estimate, we show that a uniform mean-field limit of the RCS model can be made for some admissible class of solutions uniformly in time. This justifies a formal derivation of the kinetic RCS equation [18] in a rigorous setting.

Citation: 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
References:
[1]

J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137–185.

[2]

H. Ahn, S.-Y. Ha, M. Kang and W. Shim, Emergent behaviors of relativistic flocks on Riemannian manifolds, preprint.

[3]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds, and swarms. On the kinetic theory approach towards research perspective, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[4]

B. AylajN. BellomoL. Gibell and A. Reali, On a unified multiscale vision of behavioral crowds, Math. Models Methods Appl. Sci., 30 (2020), 1-22.  doi: 10.1142/s0218202520500013.

[5]

N. Bellomo, S.-Y. Ha and N. Outada, Towards a mathematical theory of behavioral swarms, ESAIM Control Optim. Calc. Var., 26 (2020), 19 pp. doi: 10.1051/cocv/2020071.

[6]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuation in the $1/n$ limit of interacting classical particles, Commun. Math. Phys., 56 (1977), 101-113. 

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562–564.

[8]

J. A. CarrilloM. FornasierJ. 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.

[9]

Y.-P. ChoiD. KaliseJ. Peszek and A. A. Peters, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981.  doi: 10.1137/19M1241799.

[10]

F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math, 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[11]

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

[12]

R. L. Dobrushin, Vlasov equations, Func. Anal. Appl., 13 (1979), 115-123. 

[13]

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

[14]

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

[15]

D. FangS.-Y. Ha and S. Jin, Emergent behaviors of the Cucker-Smale ensemble under attractive-repulsive couplings and Rayleigh frictions, Math. Models Methods Appl. Sci., 29 (2019), 1349-1385.  doi: 10.1142/S0218202519500234.

[16]

S.-Y. HaJ. KimC. MinT. Ruggeri and X. Zhang, Uniform stability and mean-field limit of a thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.

[17]

S.-Y. HaJ. KimJ. Park and X. Zhang, Uniform stability and mean-field limit for the augmented Kuramoto model, Netw. Heterog. Media, 13 (2018), 297-322.  doi: 10.3934/nhm.2018013.

[18]

S.-Y. Ha, J. Kim and T. Ruggeri, Kinetic and hydrodynamic models for the relativistic Cucker-Smale ensemble and emergent dynamics, preprint.

[19]

S.-Y. HaJ. Kim and T. Ruggeri, From the Relativistic Mixture of Gases to the Relativistic Cucker-Smale Flocking, Arch. Rational Mech. Anal, 235 (2019), 1661-1706.  doi: 10.1007/s00205-019-01452-y.

[20]

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.

[21]

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. 

[22]

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.

[23]

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

[24]

H. Neunzert, An Introduction to the Nonlinear Boltzmann-Vlasov Equation, Kinetic Theories and the Boltzmann Equation, Springer, Berlin, Heidelberg, 1984. doi: 10.1007/BFb0071878.

[25]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guid. Control Dyn., 32 (2009), 527-537. 

[26] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[27]

C. W. Reynolds, Flocks, Herds and Schools: A Distributed Behavioral Model, Proceeding SIGGRAPH 87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques, 1987.

[28]

S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[29]

E. Tadmor and C. Tan, Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130401.  doi: 10.1098/rsta.2013.0401.

[30]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

[31]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.

[32]

T. VicsekA. CzirókE. Ben-JacobI. 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]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. 

[34]

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

[35]

A. T. Winfree, The Geometry of Biological Time, Springer, New York, 1980.

[36]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. 

show all references

References:
[1]

J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137–185.

[2]

H. Ahn, S.-Y. Ha, M. Kang and W. Shim, Emergent behaviors of relativistic flocks on Riemannian manifolds, preprint.

[3]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds, and swarms. On the kinetic theory approach towards research perspective, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[4]

B. AylajN. BellomoL. Gibell and A. Reali, On a unified multiscale vision of behavioral crowds, Math. Models Methods Appl. Sci., 30 (2020), 1-22.  doi: 10.1142/s0218202520500013.

[5]

N. Bellomo, S.-Y. Ha and N. Outada, Towards a mathematical theory of behavioral swarms, ESAIM Control Optim. Calc. Var., 26 (2020), 19 pp. doi: 10.1051/cocv/2020071.

[6]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuation in the $1/n$ limit of interacting classical particles, Commun. Math. Phys., 56 (1977), 101-113. 

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562–564.

[8]

J. A. CarrilloM. FornasierJ. 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.

[9]

Y.-P. ChoiD. KaliseJ. Peszek and A. A. Peters, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981.  doi: 10.1137/19M1241799.

[10]

F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math, 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[11]

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

[12]

R. L. Dobrushin, Vlasov equations, Func. Anal. Appl., 13 (1979), 115-123. 

[13]

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

[14]

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

[15]

D. FangS.-Y. Ha and S. Jin, Emergent behaviors of the Cucker-Smale ensemble under attractive-repulsive couplings and Rayleigh frictions, Math. Models Methods Appl. Sci., 29 (2019), 1349-1385.  doi: 10.1142/S0218202519500234.

[16]

S.-Y. HaJ. KimC. MinT. Ruggeri and X. Zhang, Uniform stability and mean-field limit of a thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.

[17]

S.-Y. HaJ. KimJ. Park and X. Zhang, Uniform stability and mean-field limit for the augmented Kuramoto model, Netw. Heterog. Media, 13 (2018), 297-322.  doi: 10.3934/nhm.2018013.

[18]

S.-Y. Ha, J. Kim and T. Ruggeri, Kinetic and hydrodynamic models for the relativistic Cucker-Smale ensemble and emergent dynamics, preprint.

[19]

S.-Y. HaJ. Kim and T. Ruggeri, From the Relativistic Mixture of Gases to the Relativistic Cucker-Smale Flocking, Arch. Rational Mech. Anal, 235 (2019), 1661-1706.  doi: 10.1007/s00205-019-01452-y.

[20]

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.

[21]

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. 

[22]

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.

[23]

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

[24]

H. Neunzert, An Introduction to the Nonlinear Boltzmann-Vlasov Equation, Kinetic Theories and the Boltzmann Equation, Springer, Berlin, Heidelberg, 1984. doi: 10.1007/BFb0071878.

[25]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guid. Control Dyn., 32 (2009), 527-537. 

[26] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[27]

C. W. Reynolds, Flocks, Herds and Schools: A Distributed Behavioral Model, Proceeding SIGGRAPH 87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques, 1987.

[28]

S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[29]

E. Tadmor and C. Tan, Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130401.  doi: 10.1098/rsta.2013.0401.

[30]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

[31]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.

[32]

T. VicsekA. CzirókE. Ben-JacobI. 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]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. 

[34]

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

[35]

A. T. Winfree, The Geometry of Biological Time, Springer, New York, 1980.

[36]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. 

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