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2018, 13(3): 379-407.
doi: 10.3934/nhm.2018017
Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication
| 1. | Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, Korea |
| 2. | Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea |
| 3. | Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea |
| 4. | Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea |
We study dynamical behaviors of the ensemble of thermomechanical Cucker-Smale (in short TCS) particles with singular power-law communication weights in velocity and temperatures. For the particle TCS model, we present several sufficient frameworks for the global regularity of solution and a finite-time breakdown depending on the blow-up exponents in the power-law communication weights at the origin where the relative spatial distances become zero. More precisely, when the blow-up exponent in velocity communication weight is greater than unity and the blow-up exponent in temperature communication weights is more than twice of blow-up exponent in velocity communication, we show that there will be no finite time collision between particles, unless there are collisions initially. In contrast, when the blow-up exponent of velocity communication weight is smaller than unity, we show that there can be a collision in finite time. For the kinetic TCS equation, we present a local-in-time existence of a unique weak solution using the suitable regularization and compactness arguments.
Keywords:
Collision,
flocking,
the thermomechanical Cucker-Smale model,
flocking model,
thermodynamics.
Mathematics Subject Classification:
70F99, 92B25.
Citation:
Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks and Heterogeneous Media,
2018, 13
(3)
: 379-407.
doi: 10.3934/nhm.2018017
![]()
References:
| [1] |
S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee,
On collision-avoiding initial configurations to Cucker-Smale type flocking models, Comm. Math. Sci., 10 (2012), 625-643.
doi: 10.4310/CMS.2012.v10.n2.a10. |
| [2] |
J. A. Carrillo, Y. -P. Choi and M. Hauray, Local well-posedness of the generalized CuckerSmale model with singular kernels, Mathematical Modeling of Complex Systems, 17-35,
ESAIM Proc. Surveys, 47, EDP Sci., Les Ulis, 2014.
doi: 10.1051/proc/201447002. |
| [3] |
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., 37 (2017), 317-328.
doi: 10.1016/j.nonrwa.2017.02.017. |
| [4] |
J. A. Carrillo, Y. -P. Choi and S. Pérez, A review on attractive-repulsive hydrodynamics for
consensus in collective behavior, 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), 259-298. |
| [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] |
Y.-P. Choi,
Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916.
doi: 10.1088/0951-7715/29/7/1887. |
| [7] |
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. |
| [8] |
F. Cucker and J.-G. Dong,
Avoiding collisions in flocks, IEEE Trans. Automatic Control, 55 (2010), 1238-1243.
doi: 10.1109/TAC.2010.2042355. |
| [9] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
| [10] |
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. |
| [11] |
M. Fornasier, J. Haskovec and G. Toscani,
Fluid dynamic description of flocking via Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31.
doi: 10.1016/j.physd.2010.08.003. |
| [12] |
S. -Y. Ha, J. Kim, C. Min, T. Ruggeri and X. Zhang, Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Submitted. |
| [13] |
S.-Y. Ha, J. Kim and T. Ruggeri,
Emergent behaviors of Thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.
doi: 10.1137/17M111064X. |
| [14] |
S.-Y. Ha, J. 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. |
| [15] |
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. |
| [16] |
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. |
| [17] |
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. |
| [18] |
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. |
| [19] |
S.-Y. Ha and T. Ruggeri,
Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. An., 223 (2017), 1397-1425.
doi: 10.1007/s00205-016-1062-3. |
| [20] |
Z. Li and S.-Y. Ha,
On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709.
doi: 10.1090/qam/1401. |
| [21] |
Z. Li and X. Xue,
Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.
doi: 10.1137/100791774. |
| [22] |
S. Motsch and E. Tadmor,
Heterophilious dynamics: Enhanced Consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
| [23] |
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. |
| [24] |
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. |
| [25] |
J. Peszek,
Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), 3671-3686.
doi: 10.1137/15M1009299. |
| [26] |
J. Peszek,
Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differ. Equat., 257 (2014), 2900-2925.
doi: 10.1016/j.jde.2014.06.003. |
| [27] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719.
doi: 10.1137/060673254. |
| [28] |
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. |
| [29] |
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. |
| [30] |
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. |
| [31] |
C. Villani,
Topics in Optimal Transportation, American Mathematical Society, 2003.
doi: 10.1007/b12016. |
| [32] |
C. Villani,
Optimal Transport, Old and New, Springer-Verlag, 2009.
doi: 10.1007/978-3-540-71050-9. |
show all references
References:
| [1] |
S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee,
On collision-avoiding initial configurations to Cucker-Smale type flocking models, Comm. Math. Sci., 10 (2012), 625-643.
doi: 10.4310/CMS.2012.v10.n2.a10. |
| [2] |
J. A. Carrillo, Y. -P. Choi and M. Hauray, Local well-posedness of the generalized CuckerSmale model with singular kernels, Mathematical Modeling of Complex Systems, 17-35,
ESAIM Proc. Surveys, 47, EDP Sci., Les Ulis, 2014.
doi: 10.1051/proc/201447002. |
| [3] |
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., 37 (2017), 317-328.
doi: 10.1016/j.nonrwa.2017.02.017. |
| [4] |
J. A. Carrillo, Y. -P. Choi and S. Pérez, A review on attractive-repulsive hydrodynamics for
consensus in collective behavior, 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), 259-298. |
| [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] |
Y.-P. Choi,
Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916.
doi: 10.1088/0951-7715/29/7/1887. |
| [7] |
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. |
| [8] |
F. Cucker and J.-G. Dong,
Avoiding collisions in flocks, IEEE Trans. Automatic Control, 55 (2010), 1238-1243.
doi: 10.1109/TAC.2010.2042355. |
| [9] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
| [10] |
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. |
| [11] |
M. Fornasier, J. Haskovec and G. Toscani,
Fluid dynamic description of flocking via Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31.
doi: 10.1016/j.physd.2010.08.003. |
| [12] |
S. -Y. Ha, J. Kim, C. Min, T. Ruggeri and X. Zhang, Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Submitted. |
| [13] |
S.-Y. Ha, J. Kim and T. Ruggeri,
Emergent behaviors of Thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.
doi: 10.1137/17M111064X. |
| [14] |
S.-Y. Ha, J. 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. |
| [15] |
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. |
| [16] |
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. |
| [17] |
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. |
| [18] |
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. |
| [19] |
S.-Y. Ha and T. Ruggeri,
Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. An., 223 (2017), 1397-1425.
doi: 10.1007/s00205-016-1062-3. |
| [20] |
Z. Li and S.-Y. Ha,
On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709.
doi: 10.1090/qam/1401. |
| [21] |
Z. Li and X. Xue,
Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.
doi: 10.1137/100791774. |
| [22] |
S. Motsch and E. Tadmor,
Heterophilious dynamics: Enhanced Consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
| [23] |
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. |
| [24] |
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. |
| [25] |
J. Peszek,
Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), 3671-3686.
doi: 10.1137/15M1009299. |
| [26] |
J. Peszek,
Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differ. Equat., 257 (2014), 2900-2925.
doi: 10.1016/j.jde.2014.06.003. |
| [27] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719.
doi: 10.1137/060673254. |
| [28] |
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. |
| [29] |
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. |
| [30] |
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. |
| [31] |
C. Villani,
Topics in Optimal Transportation, American Mathematical Society, 2003.
doi: 10.1007/b12016. |
| [32] |
C. Villani,
Optimal Transport, Old and New, Springer-Verlag, 2009.
doi: 10.1007/978-3-540-71050-9. |
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