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August  2020, 13(4): 759-793. doi: 10.3934/krm.2020026

Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 South Korea

2. 

School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, South Korea

3. 

College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

* Corresponding author: Doheon Kim

Communicated by Huijiang Zhao

Received  August 2019 Revised  February 2020 Published  May 2020

Fund Project: The work of S.-Y. Ha was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korean Government(MSIP)(No.2017R1A5A1015626), and the work of D. Kim was supported by a KIAS Individual Grant (MG073901) at Korea Institute for Advanced Study, and the work of Weiyuan Zou is supported by the Fundamental Research Funds for the Central Universities ZY1937

We study slow flocking phenomenon arising from the dynamics of Cucker-Smale (CS) ensemble with chemotactic movements in a self-consistent temperature field. For constant temperature field, our situation reduces to the previous CS model with chemotactic movements. When a large CS ensemble with chemotactic movements is placed in a self-consistent temperature field, the dynamics of the CS ensemble can be effectively described by the kinetic thermodynamic CS (TCS) equation with chemotactic movements, which corresponds to the coupled collisional transport-reaction diffusion system. For the proposed coupled model, we provide a global solvability of strong solutions and their asymptotic flocking estimates which exhibit slow algebraic relaxation toward the flocking state. Our analytical results show that asymptotic flocking is robust with respect to a small perturbation of a constant temperature.

Citation: Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic & Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026
References:
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W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.  Google Scholar

[2]

F. A. C. C. Chalub and K. Kang, Global convergence of a kinetic model of chemotaxis to a perturbed Keller-Segel model., Nonlinear Anal., 64 (2006), 686-695.  doi: 10.1016/j.na.2005.04.048.  Google Scholar

[3]

F. A. C. C. ChalubP. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsch. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.  Google Scholar

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[5]

C.-C. ChenS.-Y. Ha and X. Zhang, The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements, Commun. Pure Appl. Anal., 17 (2018), 505-538.  doi: 10.3934/cpaa.2018028.  Google Scholar

[6]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

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[8]

Y.-P. ChoiS.-Y. HaJ. Jung and J. Kim, Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.  doi: 10.1088/1361-6544/aafaae.  Google Scholar

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[11]

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[13]

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[14]

S.-Y. HaJ. KimC. MinT. Ruggeri and X. Zhang, A global existence of classical solutions to the hydrodynamic Cucker-Smale model in presence of a temperature field, Anal. Appl. (Singap.), 16 (2018), 757-805.  doi: 10.1142/S0219530518500033.  Google Scholar

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S.-Y. HaJ. KimC. MinT. Ruggeri and X. Zhang, Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.  Google Scholar

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S.-Y. HaJ. 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

[17]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis., Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108.  doi: 10.3934/dcdsb.2009.12.77.  Google Scholar

[18]

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

[19]

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

[20]

S. C. Hille, Local well-posedness of kinetic chemotaxis models, J. Evol. Eqn., 8 (2008), 423-448.  doi: 10.1007/s00028-008-0358-7.  Google Scholar

[21]

T. HillenP. Hinow and Z.-A. Wang, Mathematical analysis of a kinetic model for cell movement in network tissues, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1055-1080.  doi: 10.3934/dcdsb.2010.14.1055.  Google Scholar

[22]

T. Hillen, Hyperbolic models for chemosensitive movement, Math. Models Methods Appl. Sci., 12 (2002), 1007-1034.  doi: 10.1142/S0218202502002008.  Google Scholar

[23]

H. HwangK. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosenstive movement, SIAM. J. Math. Anal., 36 (2005), 1177-1199.  doi: 10.1137/S0036141003431888.  Google Scholar

[24]

H. HwangK. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.  doi: 10.3934/dcdsb.2005.5.319.  Google Scholar

[25]

E. F. Keller and L. A. Segel, Initiation of slide mode aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[26]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[27]

N. E. LeonardD. A. PaleyF. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.  Google Scholar

[28]

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

[29]

R. Olfati-SaberJ. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE, 95 (2007), 215-233.  doi: 10.1109/JPROC.2006.887293.  Google Scholar

[30]

R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automat. Control, 49 (2004), 1520-1533.  doi: 10.1109/TAC.2004.834113.  Google Scholar

[31]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[32]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control. Syst. Mag., 27 (2007), 89-105.  doi: 10.1109/MCS.2007.384123.  Google Scholar

[33]

B. Perthame, Tranport equations in biology, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[34]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564.  doi: 10.1007/s10492-004-6431-9.  Google Scholar

[35]

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

[36]

T. VicsekA. CzirokE. Ben-JacobI. Cohen and O. Schochet, Novel type of phase transitions in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[37]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

show all references

References:
[1]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.  Google Scholar

[2]

F. A. C. C. Chalub and K. Kang, Global convergence of a kinetic model of chemotaxis to a perturbed Keller-Segel model., Nonlinear Anal., 64 (2006), 686-695.  doi: 10.1016/j.na.2005.04.048.  Google Scholar

[3]

F. A. C. C. ChalubP. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsch. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.  Google Scholar

[4]

F. A. C. C. Chalub and J. F. Rodrigues, A short description of kinetic models for chemotaxis, in Hyperbolic Problems and Regularity Questions, Birkhäuser Basel, 2007, 59–68. doi: 10.1007/978-3-7643-7451-8_7.  Google Scholar

[5]

C.-C. ChenS.-Y. Ha and X. Zhang, The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements, Commun. Pure Appl. Anal., 17 (2018), 505-538.  doi: 10.3934/cpaa.2018028.  Google Scholar

[6]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[7]

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

[8]

Y.-P. ChoiS.-Y. HaJ. Jung and J. Kim, Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.  doi: 10.1088/1361-6544/aafaae.  Google Scholar

[9]

M. Copeland and D. Weibel, Bacterial swarming: A model system for studying dynamic self-assembly, Soft Matter, 5 (2009), 1174-1187.  doi: 10.1039/B812146J.  Google Scholar

[10]

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

[11]

Y. Dolak and T. Hillen, Cattaneo models for chemosensitive movement: Numerical solution and pattern formation, J. Math. Biol., 46 (2003), 153-170.  doi: 10.1007/s00285-002-0173-7.  Google Scholar

[12]

R. Erban and H. G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math., 65 (2004), 361-391.  doi: 10.1137/S0036139903433232.  Google Scholar

[13]

F. FilbetP. Laurencot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.  doi: 10.1007/s00285-004-0286-2.  Google Scholar

[14]

S.-Y. HaJ. KimC. MinT. Ruggeri and X. Zhang, A global existence of classical solutions to the hydrodynamic Cucker-Smale model in presence of a temperature field, Anal. Appl. (Singap.), 16 (2018), 757-805.  doi: 10.1142/S0219530518500033.  Google Scholar

[15]

S.-Y. HaJ. KimC. MinT. Ruggeri and X. Zhang, Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.  Google Scholar

[16]

S.-Y. HaJ. 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

[17]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis., Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108.  doi: 10.3934/dcdsb.2009.12.77.  Google Scholar

[18]

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

[19]

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

[20]

S. C. Hille, Local well-posedness of kinetic chemotaxis models, J. Evol. Eqn., 8 (2008), 423-448.  doi: 10.1007/s00028-008-0358-7.  Google Scholar

[21]

T. HillenP. Hinow and Z.-A. Wang, Mathematical analysis of a kinetic model for cell movement in network tissues, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1055-1080.  doi: 10.3934/dcdsb.2010.14.1055.  Google Scholar

[22]

T. Hillen, Hyperbolic models for chemosensitive movement, Math. Models Methods Appl. Sci., 12 (2002), 1007-1034.  doi: 10.1142/S0218202502002008.  Google Scholar

[23]

H. HwangK. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosenstive movement, SIAM. J. Math. Anal., 36 (2005), 1177-1199.  doi: 10.1137/S0036141003431888.  Google Scholar

[24]

H. HwangK. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.  doi: 10.3934/dcdsb.2005.5.319.  Google Scholar

[25]

E. F. Keller and L. A. Segel, Initiation of slide mode aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[26]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[27]

N. E. LeonardD. A. PaleyF. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.  Google Scholar

[28]

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

[29]

R. Olfati-SaberJ. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE, 95 (2007), 215-233.  doi: 10.1109/JPROC.2006.887293.  Google Scholar

[30]

R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automat. Control, 49 (2004), 1520-1533.  doi: 10.1109/TAC.2004.834113.  Google Scholar

[31]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[32]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control. Syst. Mag., 27 (2007), 89-105.  doi: 10.1109/MCS.2007.384123.  Google Scholar

[33]

B. Perthame, Tranport equations in biology, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[34]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564.  doi: 10.1007/s10492-004-6431-9.  Google Scholar

[35]

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

[36]

T. VicsekA. CzirokE. Ben-JacobI. Cohen and O. Schochet, Novel type of phase transitions in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[37]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

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