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October  2018, 38(10): 5245-5260. doi: 10.3934/dcds.2018232

Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings

Ioannis Markou

Institute of Applied and Computational Mathematics, Foundation for Research and Technology - Hellas, N. Plastira 100, Vassilika Vouton, GR - 700 13, Heraklion, Crete, Greece

Received  March 2018 Revised  June 2018 Published  July 2018

Collision avoidance is an interesting feature of the Cucker-Smale (CS) model of flocking that has been studied in many works, e.g. [2,1,4,6,7,20,21,22]. In particular, in the case of singular interactions between agents, as is the case of the CS model with communication weights of the type $ψ(s) = s^{-α}$ for $α ≥ 1$, it is important for showing global well-posedness of the underlying particle dynamics. In [4], a proof of the non-collision property for singular interactions is given in the case of the linear CS model, i.e. when the velocity coupling between agents $i,j$ is $v_{j}-v_{i}$. This paper can be seen as an extension of the analysis in [4]. We show that particles avoid collisions even when the linear coupling in the CS system has been substituted with the nonlinear term $Γ(·)$ introduced in [12] (typical examples being $Γ(v) = v|v|^{2(γ -1)}$ for $γ ∈ (\frac{1}{2},\frac{3}{2})$), and prove that no collisions can happen in finite time when $α ≥ 1$. We also show uniform estimates for the minimum inter-particle distance, for a communication weight with expanded singularity $ψ_{δ}(s) = (s-δ)^{-α}$, when $α ≥ 2γ$, $δ ≥ 0$.

Keywords: Nonlinear Cucker-Smale system, emergent behavior, collision avoidance, singular communication weight.
Mathematics Subject Classification: 82C22, 92D50.
Citation: Ioannis Markou. Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5245-5260. doi: 10.3934/dcds.2018232
References:
[1]

M. Agueh, R. Illner and A. Richardson, Analysis and simulation of a refined flocking and swarming model of Cucker-Smale type, Kinet. Relat. Models, 4 (2011), 1-16. doi: 10.3934/krm.2011.4.1. Google Scholar

[2]

S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643. doi: 10.4310/CMS.2012.v10.n2.a10. Google Scholar

[3]

J. A. Carrillo, Y.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model, ESAIM:Proc., 47 (2014), 17-35. doi: 10.1051/proc/201447002. Google Scholar

[4]

J. A. Carrillo, Y.-P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328. doi: 10.1016/j.nonrwa.2017.02.017. Google Scholar

[5]

Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033. doi: 10.3934/krm.2017040. Google Scholar

[6]

F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129. doi: 10.1109/TAC.2011.2107113. Google Scholar

[7]

F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete Contin. Dyn. Syst., 34 (2014), 1009-1020. doi: 10.3934/dcds.2014.34.1009. Google Scholar

[8]

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

[9]

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

[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]

R. Erban, J. Haskovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557. doi: 10.1137/15M1030467. Google Scholar

[12]

S.-Y. Ha, T. Ha and J.-H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113. Google Scholar

[13]

S.-Y. Ha, T. Ha and J.-H. Kim, Asymptotic dynamics for the Cucker-Smale-type model with the Rayleigh friction, J. Phys. A, 43 (2010), 315201, 19pp. doi: 10.1088/1751-8113/43/31/315201. Google Scholar

[14]

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

[15]

S.-Y. Ha and J.-G. Liu, A simple proof of the 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

[16]

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

[17]

M. Hauray and P.-E. Jabin, Particles approximations of Vlasov equations with singular forces: Propagation of chaos, Ann. Sci. Éc. Norm. Supér., 48 (2015), 891-940. doi: 10.24033/asens.2261. Google Scholar

[18]

P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinet. Relat. Models, 7 (2014), 661-711. doi: 10.3934/krm.2014.7.661. Google Scholar

[19]

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

[20]

P. B. Mucha and J. Peszek, The Cucker-Smale equation: Singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), 273-308. doi: 10.1007/s00205-017-1160-x. Google Scholar

[21]

J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differential Equations, 257 (2014), 2900-2925. doi: 10.1016/j.jde.2014.06.003. Google Scholar

[22]

J. Peszek, Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), 3671-3686. doi: 10.1137/15M1009299. Google Scholar

[23]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719. doi: 10.1137/060673254. Google Scholar

[24]

T. V. Ton, N. T. H. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73. doi: 10.1142/S0219530513500255. Google Scholar

[25]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, 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

show all references

References:
[1]

M. Agueh, R. Illner and A. Richardson, Analysis and simulation of a refined flocking and swarming model of Cucker-Smale type, Kinet. Relat. Models, 4 (2011), 1-16. doi: 10.3934/krm.2011.4.1. Google Scholar

[2]

S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643. doi: 10.4310/CMS.2012.v10.n2.a10. Google Scholar

[3]

J. A. Carrillo, Y.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model, ESAIM:Proc., 47 (2014), 17-35. doi: 10.1051/proc/201447002. Google Scholar

[4]

J. A. Carrillo, Y.-P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328. doi: 10.1016/j.nonrwa.2017.02.017. Google Scholar

[5]

Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033. doi: 10.3934/krm.2017040. Google Scholar

[6]

F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129. doi: 10.1109/TAC.2011.2107113. Google Scholar

[7]

F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete Contin. Dyn. Syst., 34 (2014), 1009-1020. doi: 10.3934/dcds.2014.34.1009. Google Scholar

[8]

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

[9]

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

[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]

R. Erban, J. Haskovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557. doi: 10.1137/15M1030467. Google Scholar

[12]

S.-Y. Ha, T. Ha and J.-H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113. Google Scholar

[13]

S.-Y. Ha, T. Ha and J.-H. Kim, Asymptotic dynamics for the Cucker-Smale-type model with the Rayleigh friction, J. Phys. A, 43 (2010), 315201, 19pp. doi: 10.1088/1751-8113/43/31/315201. Google Scholar

[14]

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

[15]

S.-Y. Ha and J.-G. Liu, A simple proof of the 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

[16]

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

[17]

M. Hauray and P.-E. Jabin, Particles approximations of Vlasov equations with singular forces: Propagation of chaos, Ann. Sci. Éc. Norm. Supér., 48 (2015), 891-940. doi: 10.24033/asens.2261. Google Scholar

[18]

P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinet. Relat. Models, 7 (2014), 661-711. doi: 10.3934/krm.2014.7.661. Google Scholar

[19]

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

[20]

P. B. Mucha and J. Peszek, The Cucker-Smale equation: Singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), 273-308. doi: 10.1007/s00205-017-1160-x. Google Scholar

[21]

J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differential Equations, 257 (2014), 2900-2925. doi: 10.1016/j.jde.2014.06.003. Google Scholar

[22]

J. Peszek, Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), 3671-3686. doi: 10.1137/15M1009299. Google Scholar

[23]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719. doi: 10.1137/060673254. Google Scholar

[24]

T. V. Ton, N. T. H. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73. doi: 10.1142/S0219530513500255. Google Scholar

[25]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, 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

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