March 2018, 17(2): 505-538. doi: 10.3934/cpaa.2018028

The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements

1. 

Department of Mathematics, National Taiwan University and National Center for Theoretical Sciences, No. 1 Sec. 4, Roosevelt Road, Taipei 10617, Taiwan

2. 

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

3. 

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

* Corresponding author: Xiongtao Zhang

Received  January 2017 Revised  September 2017 Published  March 2018

We present a coupled kinetic-macroscopic equation describing the dynamic behaviors of Cucker-Smale(in short C-S) ensemble undergoing velocity jumps and chemotactic movements. The proposed coupled model consists of a kinetic C-S equation supplemented with a turning operator for the kinetic density of C-S particles, and a reaction-diffusion equation for the chemotactic density. We study a global existence of strong solutions for the proposed model, when initial data is sufficiently regular, compactly supported in velocity and has finite mass and energy. The turning operator can screw up the velocity alignment, and result in a dispersed state. However, under suitable structural assumptions on the turning kernel and ansatz for the reaction term, the effects of the turning operator can vanish asymptotically due to the diffusion of chemical substances. In this situation, velocity alignment can emerge algebraically slow. We also present parabolic and hyperbolic Keller-Segel models with alignment dissipation in two scaling limits.

Citation: Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028
References:
[1]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.

[2]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Differ. Equ., 257 (2014), 2225-2255.

[3]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids, Discret. Contin. Dyn. Syst., 34 (2014), 4419-4458.

[4]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.

[5]

J. A. CarrilloM. FornasierJ. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.

[6]

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.

[7]

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.

[8]

F. A. C. C. Chalub and J. F. Rodrigues, A short description of kinetic models for chemotaxis. Hyperbolic Probl. and Regul. Quest. , (eds. M. Padula, L. Zanghirati), Birkhäuser Verlag, (2007), 59–68.

[9]

P.-H. Chavanis and C. Sire, Kinetic and hydrodynamic models of chemotactic aggregation, Physica A, 384 (2007), 199-222.

[10]

M. Copeland and D. Weibel, Bacterial swarming: a model system for studying dynamic self-assembly, Soft Matter, 5 (2009), 1174-1187.

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control., 52 (2007), 852-862.

[12]

Y. Dolak and T. Hillen, Cattaneo models for chemosensitive movement: Numerical solution and pattern formation, J. Math. Biol., 46 (2003), 153-170.

[13]

R. Erban and H. G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math., 65 (2004), 361-391.

[14]

F. FilbetP. Laurencot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.

[15]

M. FornasierJ. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31.

[16]

S.-Y. HaJ. JeongS. E. NohQ. Xiao and X. Zhang, Emergent dynamics of infinitely many Cucker-Smale particles in a random environment, J. Differ. Equ., 262 (2017), 2554-2591.

[17]

S.-Y. HaM.-J. Kang and B. Kwon, A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Models Methods Appl. Sci., 24 (2014), 2311-2359.

[18]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.

[19]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108.

[20]

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.

[21]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. and Relat. Model., 1 (2008), 415-435.

[22]

A. M. Hein, S. B. Rosenthal, G. I. Hagstrom, A. Berdahl, C. J. Torney and I. D. Couzin, The evolution of distributed sensing and collective computation in animal populations, eLIFE, 4 (2015), e10955.

[23]

S. C. Hille, Local well-posedness of kinetic chemotaxis models, J. Evol. Eqn., 8 (2008), 423-448.

[24]

T. HillenP. Hinow and Z.-A. Wang, Mathematical analysis of a kinetic model for cell movement in network tissues, DCDS-B, 14 (2010), 1055-1080.

[25]

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

[26]

H. HwangK. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, DCDS-B, 5 (2005), 319-334.

[27]

E. F. Keller and L. A. Segel, Initiation of slide mode aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.

[28]

E. F. Keller and L.A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.

[29]

A. Kolpas, M. Busch, H. Li, I. D. Couzin, L. Petzold and J. Moehlis, How the spatial position of individuals affects their influence on swarms: A mumerical comparison of two popular swarm dynamics models, PLOS One, 8 (2013), e58525.

[30]

N. E. LeonardD. PaleyA. F. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.

[31]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, In Kinet. Theor. and the Boltzmann Equ., Lect. Notes in Math. , 1048 (1984), Springer, Berlin, Heidelberg.

[32]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.

[33]

L. PereaG. Gómez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formation, J. Guid., Control. and Dyn., 32 (2009), 526-536.

[34]

J. ParkH. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Autom. Control., 55 (2010), 2617-2623.

[35]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion: Spatial patterns in the dynamics of engineered and biological networks, IEEE Control. Syst. Mag., 27 (2007), 89-105.

[36]

B. Perthame, Tranport equations in biology, Birkhäuser (2006).

[37]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. of Math., 49 (2004), 539-564.

[38]

W. Ren and R. W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control., 50 (2005), 655-661.

[39]

G. Rosen, On the propagation theory for bands of chemotactic bacteria, Math. Biosci., 20 (1974), 185-189.

[40]

R. O. SaberJ. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. of the IEEE, 95 (2007), 215-233.

[41]

R. O. Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control., 49 (2004), 1520-1533.

[42]

E. Steager, C. Kim and M. Kim, Dynamics of pattern formation in bacterial swarms, Phys. of Fluids, 20 (2008), 073601.

[43]

M. J. TindallaP. K. MainiaS. L. Porterb and J. L. Armitageb, Overview of mathematical approaches used to model bacterial chemotaxis Ⅱ: Bacterial populations, Bull. of Math. Biol., 70 (2008), 1570-1607.

[44]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.

[45]

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.

[46]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.

show all references

References:
[1]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.

[2]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Differ. Equ., 257 (2014), 2225-2255.

[3]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids, Discret. Contin. Dyn. Syst., 34 (2014), 4419-4458.

[4]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.

[5]

J. A. CarrilloM. FornasierJ. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.

[6]

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.

[7]

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.

[8]

F. A. C. C. Chalub and J. F. Rodrigues, A short description of kinetic models for chemotaxis. Hyperbolic Probl. and Regul. Quest. , (eds. M. Padula, L. Zanghirati), Birkhäuser Verlag, (2007), 59–68.

[9]

P.-H. Chavanis and C. Sire, Kinetic and hydrodynamic models of chemotactic aggregation, Physica A, 384 (2007), 199-222.

[10]

M. Copeland and D. Weibel, Bacterial swarming: a model system for studying dynamic self-assembly, Soft Matter, 5 (2009), 1174-1187.

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control., 52 (2007), 852-862.

[12]

Y. Dolak and T. Hillen, Cattaneo models for chemosensitive movement: Numerical solution and pattern formation, J. Math. Biol., 46 (2003), 153-170.

[13]

R. Erban and H. G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math., 65 (2004), 361-391.

[14]

F. FilbetP. Laurencot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.

[15]

M. FornasierJ. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31.

[16]

S.-Y. HaJ. JeongS. E. NohQ. Xiao and X. Zhang, Emergent dynamics of infinitely many Cucker-Smale particles in a random environment, J. Differ. Equ., 262 (2017), 2554-2591.

[17]

S.-Y. HaM.-J. Kang and B. Kwon, A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Models Methods Appl. Sci., 24 (2014), 2311-2359.

[18]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.

[19]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108.

[20]

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.

[21]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. and Relat. Model., 1 (2008), 415-435.

[22]

A. M. Hein, S. B. Rosenthal, G. I. Hagstrom, A. Berdahl, C. J. Torney and I. D. Couzin, The evolution of distributed sensing and collective computation in animal populations, eLIFE, 4 (2015), e10955.

[23]

S. C. Hille, Local well-posedness of kinetic chemotaxis models, J. Evol. Eqn., 8 (2008), 423-448.

[24]

T. HillenP. Hinow and Z.-A. Wang, Mathematical analysis of a kinetic model for cell movement in network tissues, DCDS-B, 14 (2010), 1055-1080.

[25]

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

[26]

H. HwangK. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, DCDS-B, 5 (2005), 319-334.

[27]

E. F. Keller and L. A. Segel, Initiation of slide mode aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.

[28]

E. F. Keller and L.A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.

[29]

A. Kolpas, M. Busch, H. Li, I. D. Couzin, L. Petzold and J. Moehlis, How the spatial position of individuals affects their influence on swarms: A mumerical comparison of two popular swarm dynamics models, PLOS One, 8 (2013), e58525.

[30]

N. E. LeonardD. PaleyA. F. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.

[31]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, In Kinet. Theor. and the Boltzmann Equ., Lect. Notes in Math. , 1048 (1984), Springer, Berlin, Heidelberg.

[32]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.

[33]

L. PereaG. Gómez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formation, J. Guid., Control. and Dyn., 32 (2009), 526-536.

[34]

J. ParkH. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Autom. Control., 55 (2010), 2617-2623.

[35]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion: Spatial patterns in the dynamics of engineered and biological networks, IEEE Control. Syst. Mag., 27 (2007), 89-105.

[36]

B. Perthame, Tranport equations in biology, Birkhäuser (2006).

[37]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. of Math., 49 (2004), 539-564.

[38]

W. Ren and R. W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control., 50 (2005), 655-661.

[39]

G. Rosen, On the propagation theory for bands of chemotactic bacteria, Math. Biosci., 20 (1974), 185-189.

[40]

R. O. SaberJ. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. of the IEEE, 95 (2007), 215-233.

[41]

R. O. Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control., 49 (2004), 1520-1533.

[42]

E. Steager, C. Kim and M. Kim, Dynamics of pattern formation in bacterial swarms, Phys. of Fluids, 20 (2008), 073601.

[43]

M. J. TindallaP. K. MainiaS. L. Porterb and J. L. Armitageb, Overview of mathematical approaches used to model bacterial chemotaxis Ⅱ: Bacterial populations, Bull. of Math. Biol., 70 (2008), 1570-1607.

[44]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.

[45]

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.

[46]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.

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