# American Institue of Mathematical Sciences

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. Bae, Y.-P. Choi, S.-Y. Ha, 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. Bae, Y.-P. Choi, S.-Y. Ha, 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. Bae, Y.-P. Choi, S.-Y. Ha, M.-J. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177. [5] J. A. Carrillo, M. Fornasier, J. Rosado, 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, 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. Chalub, P. Markowich, B. Perthame, 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, C. Sire, Kinetic and hydrodynamic models of chemotactic aggregation, Physica A, 384 (2007), 199-222. [10] M. Copeland, D. Weibel, Bacterial swarming: a model system for studying dynamic self-assembly, Soft Matter, 5 (2009), 1174-1187. [11] F. Cucker, S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control., 52 (2007), 852-862. [12] Y. Dolak, T. Hillen, Cattaneo models for chemosensitive movement: Numerical solution and pattern formation, J. Math. Biol., 46 (2003), 153-170. [13] R. Erban, H. G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math., 65 (2004), 361-391. [14] F. Filbet, P. Laurencot, B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207. [15] M. Fornasier, J. Haskovec, G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31. [16] S.-Y. Ha, J. Jeong, S. E. Noh, Q. Xiao, X. Zhang, Emergent dynamics of infinitely many Cucker-Smale particles in a random environment, J. Differ. Equ., 262 (2017), 2554-2591. [17] S.-Y. Ha, M.-J. Kang, 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. Ha, K. Lee, D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. [19] S.-Y. Ha, D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108. [20] S.-Y. Ha, 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, 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. Hillen, P. Hinow, Z.-A. Wang, Mathematical analysis of a kinetic model for cell movement in network tissues, DCDS-B, 14 (2010), 1055-1080. [25] H. Hwang, K. Kang, A. Stevens, Global solutions of nonlinear transport equations for chemosenstive movement, SIAM. J. Math. Anal., 36 (2005), 1177-1199. [26] H. Hwang, K. Kang, A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, DCDS-B, 5 (2005), 319-334. [27] E. F. Keller, L. A. Segel, Initiation of slide mode aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. [28] E. F. Keller, 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. Leonard, D. Paley, A. F. Lekien, R. Sepulchre, D. M. Fratantoni, 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. Othmer, S. R. Dunbar, W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. [33] L. Perea, G. Gómez, P. Elosegui, Extension of the Cucker-Smale control law to space flight formation, J. Guid., Control. and Dyn., 32 (2009), 526-536. [34] J. Park, H. Kim, S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Autom. Control., 55 (2010), 2617-2623. [35] D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum, 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, 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. Saber, J. A. Fax, R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. of the IEEE, 95 (2007), 215-233. [41] R. O. Saber, 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. Tindalla, P. K. Mainia, S. L. Porterb, 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, Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858. [45] T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, O. Schochet, Novel type of phase transitions in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. [46] T. Vicsek, 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. Bae, Y.-P. Choi, S.-Y. Ha, 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. Bae, Y.-P. Choi, S.-Y. Ha, 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. Bae, Y.-P. Choi, S.-Y. Ha, M.-J. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177. [5] J. A. Carrillo, M. Fornasier, J. Rosado, 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, 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. Chalub, P. Markowich, B. Perthame, 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, C. Sire, Kinetic and hydrodynamic models of chemotactic aggregation, Physica A, 384 (2007), 199-222. [10] M. Copeland, D. Weibel, Bacterial swarming: a model system for studying dynamic self-assembly, Soft Matter, 5 (2009), 1174-1187. [11] F. Cucker, S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control., 52 (2007), 852-862. [12] Y. Dolak, T. Hillen, Cattaneo models for chemosensitive movement: Numerical solution and pattern formation, J. Math. Biol., 46 (2003), 153-170. [13] R. Erban, H. G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math., 65 (2004), 361-391. [14] F. Filbet, P. Laurencot, B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207. [15] M. Fornasier, J. Haskovec, G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31. [16] S.-Y. Ha, J. Jeong, S. E. Noh, Q. Xiao, X. Zhang, Emergent dynamics of infinitely many Cucker-Smale particles in a random environment, J. Differ. Equ., 262 (2017), 2554-2591. [17] S.-Y. Ha, M.-J. Kang, 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. Ha, K. Lee, D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. [19] S.-Y. Ha, D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108. [20] S.-Y. Ha, 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, 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. Hillen, P. Hinow, Z.-A. Wang, Mathematical analysis of a kinetic model for cell movement in network tissues, DCDS-B, 14 (2010), 1055-1080. [25] H. Hwang, K. Kang, A. Stevens, Global solutions of nonlinear transport equations for chemosenstive movement, SIAM. J. Math. Anal., 36 (2005), 1177-1199. [26] H. Hwang, K. Kang, A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, DCDS-B, 5 (2005), 319-334. [27] E. F. Keller, L. A. Segel, Initiation of slide mode aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. [28] E. F. Keller, 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. Leonard, D. Paley, A. F. Lekien, R. Sepulchre, D. M. Fratantoni, 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. Othmer, S. R. Dunbar, W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. [33] L. Perea, G. Gómez, P. Elosegui, Extension of the Cucker-Smale control law to space flight formation, J. Guid., Control. and Dyn., 32 (2009), 526-536. [34] J. Park, H. Kim, S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Autom. Control., 55 (2010), 2617-2623. [35] D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum, 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, 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. Saber, J. A. Fax, R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. of the IEEE, 95 (2007), 215-233. [41] R. O. Saber, 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. Tindalla, P. K. Mainia, S. L. Porterb, 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, Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858. [45] T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, O. Schochet, Novel type of phase transitions in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. [46] T. Vicsek, A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.
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