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 & 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. Google Scholar

[2]

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

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

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. Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

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