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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control., 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[37]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control., 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[37]

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

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