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October  2020, 13(5): 889-931. doi: 10.3934/krm.2020031

Quantitative local sensitivity estimates for the random kinetic Cucker-Smale model with chemotactic movement

1. 

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

2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

3. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Bora Moon

Received  November 2019 Revised  March 2020 Published  August 2020

Fund Project: The work of S.-Y. Ha was supported by the NRF grant (2017R1A2B2001864) and the work of B. Moon was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585)

In this paper, we present quantitative local sensitivity estimates for the kinetic chemotaxis Cucker-Smale(CCS) equation with random inputs. In the absence of random inputs, the kinetic CCS model exhibits velocity alignment under suitable structural assumptions on the turning kernel and reaction term despite of the random effect due to a turning operator. We provide a global existence of a regular solution with slow velocity alignment for the random kinetic CCS model within the proposed framework. Moreover, we investigate the propagation of regularity and stability of infinitesimal variations in random space.

Citation: Seung-Yeal Ha, Bora Moon. Quantitative local sensitivity estimates for the random kinetic Cucker-Smale model with chemotactic movement. Kinetic and Related Models, 2020, 13 (5) : 889-931. doi: 10.3934/krm.2020031
References:
[1]

G. Albi, L. Pareschi and M. Zanella, Uncertainty quantification in control problems for flocking models, Math. Probl. Eng., (2015), Art. ID 850124, 14 pp. doi: 10.1155/2015/850124.

[2]

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

[3]

V. CalvezB. Perthame and S. Yasuda, Traveling wave and aggregation in a flux-limited Keller-Segel model, Kinet. Relat. Models, 11 (2018), 891-909.  doi: 10.3934/krm.2018035.

[4]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531.  doi: 10.4208/cicp.oa-2017-0244.

[5]

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.

[6]

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.

[7]

P.-H. Chavanis and C. Sire, Kinetic and hydrodynamic models of chemotactic aggregation, Physica A, 384 (2007), 59-68.  doi: 10.1016/j.physa.2007.05.069.

[8]

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.

[9]

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

[10]

Z. Ding, S.-Y. Ha and S. Jin, A local sensitivity analysis in the Landau damping for the kinetic Kuramoto equation, Submitted.

[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. B. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math., 65 (2004), 361-391.  doi: 10.1137/S0036139903433232.

[13]

S.-Y. Ha and S. Jin, Local sensitivity analysis for the Cucker-Smale model with random inputs, Kinet. Relat. Models, 11 (2018), 859-889.  doi: 10.3934/krm.2018034.

[14]

S.-Y. HaS. Jin and J. Jung, A local sensitivity analysis for the kinetic Cucker-Smale equation with random inputs, J. Differential Equations, 265 (2018), 3618-3649.  doi: 10.1016/j.jde.2018.05.013.

[15]

S.-Y. HaS. Jin and J. Jung, A local sensitivity analysis for the kinetic Kuramoto model with random inputs, Networks and Heterogeneous Media, 14 (2019), 317-340.  doi: 10.3934/nhm.2019013.

[16]

S.-Y. Ha, S. Jin and J. Jung, Local sensitivity analysis for the Kuramoto-Daido model with random inputs in a large coupling regime, to appear in SIAM J. Math. Anal. doi: 10.1137/18M1173435.

[17]

S.-Y. HaS. JinJ. Jung and W. Shim, A local sensitivity analysis for the hydrodynamics Cucker-Smale equation with random inputs, J. Differ. Equations, 268 (2020), 636-679.  doi: 10.1016/j.jde.2019.08.031.

[18]

S.-Y. Ha, D. Kim and B. Moon, Interplay of random inputs and adaptive couplings in the Winfree model, Submitted.

[19]

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.

[20]

S.-Y. Ha, W. J. Shim, Q. Xiao and Y. Zhang, Pathwise robustness on the instability of the incoherent state to the random Kuramoto-Sakaguchi-Fokker-Planck equation, Submitted.

[21]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. and Relat. Model., 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[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. doi: 10.7554/eLife.10955.

[23]

J. Hu and S. Jin, Uncertainty quantification for kinetic equations, SEMA SIMAI Springer Ser., 14, Springer, Cham, 2017,193–229. doi: 10.1007/978-3-319-67110-9_6.

[24]

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

[25]

H. J. HwangK. Kang and Z.-A. Wang, 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.

[26]

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.

[27]

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.

[28]

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 numerical comparison of two popular swarm dynamics models, PLOS One, 8 (2013), e58525. doi: 10.1371/journal.pone.0058525.

[29]

B. Perthame and S. Yasuda, Stiff-response-induced instability for chemotactic bacteria and flux-limited Keller-Segel equation, Nonlinearity, 31 (2018), 4065-4089.  doi: 10.1088/1361-6544/aac760.

[30]

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Introduction to sensitivity analysis, Global Sensitivity Analysis. The Primer, (2008), 1–51.

[31]

R. C. Smith, Uncertainty quantification: Theory, implementation and applications, SIAM, 12 (2013).

[32]

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

[33]

B. Perthame, Transport Equations in Biology, Birkhäuser, 2006.

[34]

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.

[35]

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]

G. Albi, L. Pareschi and M. Zanella, Uncertainty quantification in control problems for flocking models, Math. Probl. Eng., (2015), Art. ID 850124, 14 pp. doi: 10.1155/2015/850124.

[2]

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

[3]

V. CalvezB. Perthame and S. Yasuda, Traveling wave and aggregation in a flux-limited Keller-Segel model, Kinet. Relat. Models, 11 (2018), 891-909.  doi: 10.3934/krm.2018035.

[4]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531.  doi: 10.4208/cicp.oa-2017-0244.

[5]

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.

[6]

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.

[7]

P.-H. Chavanis and C. Sire, Kinetic and hydrodynamic models of chemotactic aggregation, Physica A, 384 (2007), 59-68.  doi: 10.1016/j.physa.2007.05.069.

[8]

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.

[9]

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

[10]

Z. Ding, S.-Y. Ha and S. Jin, A local sensitivity analysis in the Landau damping for the kinetic Kuramoto equation, Submitted.

[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. B. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math., 65 (2004), 361-391.  doi: 10.1137/S0036139903433232.

[13]

S.-Y. Ha and S. Jin, Local sensitivity analysis for the Cucker-Smale model with random inputs, Kinet. Relat. Models, 11 (2018), 859-889.  doi: 10.3934/krm.2018034.

[14]

S.-Y. HaS. Jin and J. Jung, A local sensitivity analysis for the kinetic Cucker-Smale equation with random inputs, J. Differential Equations, 265 (2018), 3618-3649.  doi: 10.1016/j.jde.2018.05.013.

[15]

S.-Y. HaS. Jin and J. Jung, A local sensitivity analysis for the kinetic Kuramoto model with random inputs, Networks and Heterogeneous Media, 14 (2019), 317-340.  doi: 10.3934/nhm.2019013.

[16]

S.-Y. Ha, S. Jin and J. Jung, Local sensitivity analysis for the Kuramoto-Daido model with random inputs in a large coupling regime, to appear in SIAM J. Math. Anal. doi: 10.1137/18M1173435.

[17]

S.-Y. HaS. JinJ. Jung and W. Shim, A local sensitivity analysis for the hydrodynamics Cucker-Smale equation with random inputs, J. Differ. Equations, 268 (2020), 636-679.  doi: 10.1016/j.jde.2019.08.031.

[18]

S.-Y. Ha, D. Kim and B. Moon, Interplay of random inputs and adaptive couplings in the Winfree model, Submitted.

[19]

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.

[20]

S.-Y. Ha, W. J. Shim, Q. Xiao and Y. Zhang, Pathwise robustness on the instability of the incoherent state to the random Kuramoto-Sakaguchi-Fokker-Planck equation, Submitted.

[21]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. and Relat. Model., 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[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. doi: 10.7554/eLife.10955.

[23]

J. Hu and S. Jin, Uncertainty quantification for kinetic equations, SEMA SIMAI Springer Ser., 14, Springer, Cham, 2017,193–229. doi: 10.1007/978-3-319-67110-9_6.

[24]

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

[25]

H. J. HwangK. Kang and Z.-A. Wang, 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.

[26]

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.

[27]

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.

[28]

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 numerical comparison of two popular swarm dynamics models, PLOS One, 8 (2013), e58525. doi: 10.1371/journal.pone.0058525.

[29]

B. Perthame and S. Yasuda, Stiff-response-induced instability for chemotactic bacteria and flux-limited Keller-Segel equation, Nonlinearity, 31 (2018), 4065-4089.  doi: 10.1088/1361-6544/aac760.

[30]

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Introduction to sensitivity analysis, Global Sensitivity Analysis. The Primer, (2008), 1–51.

[31]

R. C. Smith, Uncertainty quantification: Theory, implementation and applications, SIAM, 12 (2013).

[32]

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

[33]

B. Perthame, Transport Equations in Biology, Birkhäuser, 2006.

[34]

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.

[35]

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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