American Institute of Mathematical Sciences

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September  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

Young-Pil Choi 1, , Seung-Yeal Ha 2,3, and  Jeongho Kim 4,,

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

* Corresponding author

Received  September 2017 Revised  October 2017 Published  July 2018

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

C. Villani, Topics in Optimal Transportation, American Mathematical Society, 2003. doi: 10.1007/b12016.  Google Scholar

[32]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

C. Villani, Topics in Optimal Transportation, American Mathematical Society, 2003. doi: 10.1007/b12016.  Google Scholar

[32]

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

Figure 1.  Initial configuration
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