doi: 10.3934/krm.2021011

A mean-field limit of the particle swarmalator model

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, and Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

2. 

Research Institute of Basic Sciences, Seoul National University, Seoul, 08826, Republic of Korea

3. 

Institute of New Media and Communications, Seoul National University, Seoul, 08826, Republic of Korea

4. 

Department of Mathematics and Research Institute of Natural Sciences, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Republic of Korea

5. 

Center for Mathematical Sciences, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan, 430074, China

* Corresponding author: Jinwook Jung

Received  June 2020 Revised  November 2020 Published  March 2021

We present a mean-field limit of the particle swarmalator model introduced in [46] with singular communication weights. For a mean-field limit, we employ a probabilistic approach for the propagation of molecular chaos and suitable cut-offs in singular terms, which results in the validation of the mean-field limit. We also provide a local-in-time well-posedness of strong and weak solutions to the derived kinetic swarmalator equation.

Citation: Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, doi: 10.3934/krm.2021011
References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137.  Google Scholar

[2]

G. Ajmone MarsanN. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Math. Models Methods Appl. Sci., 26 (2016), 1051-1093.  doi: 10.1142/S0218202516500251.  Google Scholar

[3]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.  Google Scholar

[4]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci., 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105.  Google Scholar

[5]

N. Bellomo and S.-Y. Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770.  doi: 10.1142/S0218202517500154.  Google Scholar

[6]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006, 29 pp. doi: 10.1142/S0218202511400069.  Google Scholar

[7]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.  doi: 10.1007/s002110050002.  Google Scholar

[8]

D. BenedettoE. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3.  Google Scholar

[9]

D. BenedettoE. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6.  Google Scholar

[10]

A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65.  doi: 10.4310/CMS.2010.v8.n1.a4.  Google Scholar

[11]

A. L. BertozziT. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., 64 (2011), 45-83.  doi: 10.1002/cpa.20334.  Google Scholar

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N. Boers and P. Pickl, On mean field limits for dynamical systems, J. Stat. Phys., 164 (2016), 1-16.  doi: 10.1007/s10955-015-1351-5.  Google Scholar

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L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier- Stokes equations, Differ. Integral Equ., 22 (2009), 1247-1271.   Google Scholar

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J. C. Bronski, L. Deville and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-$N$ Kuramoto model, Chaos, 22 (2012), 033133, 17 pp. doi: 10.1063/1.4745197.  Google Scholar

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J. A. Carrillo, M. Chipot and Y. Huang, On global minimizers of repulsive–attractive power–law interaction energies, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130399, 13 pp. doi: 10.1098/rsta.2013.0399.  Google Scholar

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J. A. CarrilloY.-P. ChoiP. B. Mucha and Jan 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

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A. CavagnaL. D. CastelloI. GiardinaT. GrigeraA. JelicS. MelilloT. MoraL. ParisiE. SilvestriM. Viale and A. M. Walczak, Flocking and turning: A new model for self-organized collective motion, J. Stat. Phys., 158 (2015), 601-627.  doi: 10.1007/s10955-014-1119-3.  Google Scholar

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Y.-P. ChoiS.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.  Google Scholar

[20]

Y.-P. ChoiS.-Y. HaS. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011.  Google Scholar

[21]

T. ChampionL. D. Pascale and P. Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. Math. Anal., 40 (2008), 1-20.  doi: 10.1137/07069938X.  Google Scholar

[22]

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

[23]

P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1022.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[24]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[25]

R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.  doi: 10.1007/BF01077243.  Google Scholar

[26]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.  Google Scholar

[27]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.  Google Scholar

[28]

F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X.  Google Scholar

[29]

R. DuanM. 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

[30]

S.-Y. HaJ. JungJ. KimJ. Park and X. Zhang, Emergent behaviors of the swarmalator model for position-phase aggregation, Math. Models Methods Appl. Sci., 29 (2019), 2225-2269.  doi: 10.1142/S0218202519500453.  Google Scholar

[31]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.  Google Scholar

[32]

S.-Y. HaJ. KimP. Pickl and X. Zhang, A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication, Kinet. Relat. Models, 12 (2019), 1045-1067.  doi: 10.3934/krm.2019039.  Google Scholar

[33]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.  Google Scholar

[34]

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

[35]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal., 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.  Google Scholar

[36]

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

[37]

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

[38]

P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with $W^{-1, \infty}$ kernels, Invent. Math., 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y.  Google Scholar

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Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Theoretical Physics, 30 (1975), 420–422. doi: 10.1007/BFb0013365.  Google Scholar

[40]

D. Lazarovici and P. Pickl, A mean field limit for the Vlasov-Poisson system, Arch. Ration. Mech. Anal., 225 (2017), 1201-1231.  doi: 10.1007/s00205-017-1125-0.  Google Scholar

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G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.  doi: 10.1016/j.matpur.2006.01.005.  Google Scholar

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

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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

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H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation (ed. C. Cercignani), Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1048 (1984), 60–110. doi: 10.1007/BFb0071878.  Google Scholar

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K. P. O'Keeffe, J. H. Evers and T. Kolokolnikov, Ring states in swarmalator systems, Phys. Rev. E, 98 (2018), 022203. doi: 10.1103/PhysRevE.98.022203.  Google Scholar

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K. P. O'Keeffe, H. Hong and S. H. Strogatz, Oscillators that sync and swarm, Nature Communications, 8 (2017), 1504. doi: 10.1038/s41467-017-01190-3.  Google Scholar

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show all references

References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137.  Google Scholar

[2]

G. Ajmone MarsanN. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Math. Models Methods Appl. Sci., 26 (2016), 1051-1093.  doi: 10.1142/S0218202516500251.  Google Scholar

[3]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.  Google Scholar

[4]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci., 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105.  Google Scholar

[5]

N. Bellomo and S.-Y. Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770.  doi: 10.1142/S0218202517500154.  Google Scholar

[6]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006, 29 pp. doi: 10.1142/S0218202511400069.  Google Scholar

[7]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.  doi: 10.1007/s002110050002.  Google Scholar

[8]

D. BenedettoE. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3.  Google Scholar

[9]

D. BenedettoE. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6.  Google Scholar

[10]

A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65.  doi: 10.4310/CMS.2010.v8.n1.a4.  Google Scholar

[11]

A. L. BertozziT. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., 64 (2011), 45-83.  doi: 10.1002/cpa.20334.  Google Scholar

[12]

N. Boers and P. Pickl, On mean field limits for dynamical systems, J. Stat. Phys., 164 (2016), 1-16.  doi: 10.1007/s10955-015-1351-5.  Google Scholar

[13]

L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier- Stokes equations, Differ. Integral Equ., 22 (2009), 1247-1271.   Google Scholar

[14]

J. C. Bronski, L. Deville and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-$N$ Kuramoto model, Chaos, 22 (2012), 033133, 17 pp. doi: 10.1063/1.4745197.  Google Scholar

[15]

J. Buck and E. Buck, Biology of sychronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.  Google Scholar

[16]

J. A. Carrillo, M. Chipot and Y. Huang, On global minimizers of repulsive–attractive power–law interaction energies, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130399, 13 pp. doi: 10.1098/rsta.2013.0399.  Google Scholar

[17]

J. A. CarrilloY.-P. ChoiP. B. Mucha and Jan 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

[18]

A. CavagnaL. D. CastelloI. GiardinaT. GrigeraA. JelicS. MelilloT. MoraL. ParisiE. SilvestriM. Viale and A. M. Walczak, Flocking and turning: A new model for self-organized collective motion, J. Stat. Phys., 158 (2015), 601-627.  doi: 10.1007/s10955-014-1119-3.  Google Scholar

[19]

Y.-P. ChoiS.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.  Google Scholar

[20]

Y.-P. ChoiS.-Y. HaS. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011.  Google Scholar

[21]

T. ChampionL. D. Pascale and P. Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. Math. Anal., 40 (2008), 1-20.  doi: 10.1137/07069938X.  Google Scholar

[22]

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

[23]

P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1022.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[24]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[25]

R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.  doi: 10.1007/BF01077243.  Google Scholar

[26]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.  Google Scholar

[27]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.  Google Scholar

[28]

F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X.  Google Scholar

[29]

R. DuanM. 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

[30]

S.-Y. HaJ. JungJ. KimJ. Park and X. Zhang, Emergent behaviors of the swarmalator model for position-phase aggregation, Math. Models Methods Appl. Sci., 29 (2019), 2225-2269.  doi: 10.1142/S0218202519500453.  Google Scholar

[31]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.  Google Scholar

[32]

S.-Y. HaJ. KimP. Pickl and X. Zhang, A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication, Kinet. Relat. Models, 12 (2019), 1045-1067.  doi: 10.3934/krm.2019039.  Google Scholar

[33]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.  Google Scholar

[34]

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

[35]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal., 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.  Google Scholar

[36]

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

[37]

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

[38]

P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with $W^{-1, \infty}$ kernels, Invent. Math., 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y.  Google Scholar

[39]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Theoretical Physics, 30 (1975), 420–422. doi: 10.1007/BFb0013365.  Google Scholar

[40]

D. Lazarovici and P. Pickl, A mean field limit for the Vlasov-Poisson system, Arch. Ration. Mech. Anal., 225 (2017), 1201-1231.  doi: 10.1007/s00205-017-1125-0.  Google Scholar

[41]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.  doi: 10.1016/j.matpur.2006.01.005.  Google Scholar

[42]

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

[43]

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

[44]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation (ed. C. Cercignani), Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1048 (1984), 60–110. doi: 10.1007/BFb0071878.  Google Scholar

[45]

K. P. O'Keeffe, J. H. Evers and T. Kolokolnikov, Ring states in swarmalator systems, Phys. Rev. E, 98 (2018), 022203. doi: 10.1103/PhysRevE.98.022203.  Google Scholar

[46]

K. P. O'Keeffe, H. Hong and S. H. Strogatz, Oscillators that sync and swarm, Nature Communications, 8 (2017), 1504. doi: 10.1038/s41467-017-01190-3.  Google Scholar

[47]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Syst. Mag., 27 (2007), 89-105.   Google Scholar

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