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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[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.Google Scholar

[23]

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

[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. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[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. Google Scholar

[34]

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

[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. Google Scholar

[36]

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

[37]

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

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[42]

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

[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. Google Scholar

[44]

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

[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. Google Scholar

[46]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[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.Google Scholar

[23]

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

[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. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[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. Google Scholar

[34]

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

[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. Google Scholar

[36]

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

[37]

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

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[42]

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

[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. Google Scholar

[44]

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

[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. Google Scholar

[46]

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

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