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On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases
Local sensitivity analysis for the Cucker-Smale model with random inputs
1. | Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea |
2. | Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea |
3. | Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA |
We present pathwise flocking dynamics and local sensitivity analysis for the Cucker-Smale(C-S) model with random communications and initial data. For the deterministic communications, it is well known that the C-S model can model emergent local and global flocking dynamics depending on initial data and integrability of communication function. However, the communication mechanism between agents is not a priori clear and needs to be figured out from observed phenomena and data. Thus, uncertainty in communication is an intrinsic component in the flocking modeling of the C-S model. In this paper, we provide a class of admissible random uncertainties which allows us to perform the local sensitivity analysis for flocking and establish stability to the random C-S model with uncertain communication.
References:
[1] |
S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee,
On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.
doi: 10.4310/CMS.2012.v10.n2.a10. |
[2] |
S. M. Ahn and S. -Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp.
doi: 10.1063/1.3496895. |
[3] |
G. Albi, L. Pareschi and M. Zanella, Uncertain quantification in control problems for flocking models, Math. Probl. Eng., 2015 (2015), Art. ID 850124, 14pp.
doi: 10.1155/2015/850124. |
[4] |
J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani,
Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.
doi: 10.1137/090757290. |
[5] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming. Mathematical modeling of collective behavior in socio-economic and life sciences, in Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Inc., Boston, MA, (2010), 297-336. Google Scholar |
[6] |
J. A. Carrillo, L. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Preprint. Google Scholar |
[7] |
J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko,
Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models and Methods in Appl. Sci., 26 (2016), 1191-1218.
doi: 10.1142/S0218202516500287. |
[8] |
Y. -P. Choi, S. -Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol. Ⅰ -Theory, Models, Applications, Series: Modeling and Simulation in Science and Technology (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhauser-Springer, 1 (2017), 299-331. |
[9] |
F. Cucker and J.-G Dong,
On flocks influenced by closest neighbors, Math. Models Methods Appl. Sci., 26 (2016), 2685-2708.
doi: 10.1142/S0218202516500639. |
[10] |
F. Cucker and J.-G Dong,
A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.
doi: 10.1109/TAC.2011.2107113. |
[11] |
F. Cucker and E. Mordecki,
Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.
doi: 10.1016/j.matpur.2007.12.002. |
[12] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[13] |
P. Degond and S. Motsch,
Large-scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1022.
doi: 10.1007/s10955-008-9529-8. |
[14] |
R. Duan, M. Fornasier and G. Toscani,
A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145.
doi: 10.1007/s00220-010-1110-z. |
[15] |
S. -Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, To appear in Kinetic Relat. Models. Google Scholar |
[16] |
S.-Y. Ha, B. Kwon and M.-J. Kang,
A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Mod. Meth. Appl. Sci., 24 (2014), 2311-2359.
doi: 10.1142/S0218202514500225. |
[17] |
S.-Y. Ha, K. Lee and D. Levy,
Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.
doi: 10.4310/CMS.2009.v7.n2.a9. |
[18] |
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.
doi: 10.4310/CMS.2009.v7.n2.a2. |
[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] |
J. Hu and S. Jin,
A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), 150-168.
doi: 10.1016/j.jcp.2016.03.047. |
[21] |
J. Hu and S. Jin, Uncertainty quantification for kinetic equations, in Uncertainty Quantification for Kinetic and Hyperbolic Equations, (eds S. Jin and L. Pareschi), 14 (2018), 193-229.
doi: 10.1007/978-3-319-67110-9_6. |
[22] |
J. Hu, S. Jin and D. Xiu,
A stochastic Galerkin method for Hamiltonian-Jacobi equations with uncertainty, SIAM. J. Sci. Comput., 37 (2015), 2246-2269.
doi: 10.1137/140990930. |
[23] |
H. -N. Huang, S. A. M. Marcantognini and N. J. Young, Chain rules for higher derivatives, The Mathematical Intelligencer, 28 (2006), 61-69, http://ambio1.leeds.ac.uk/~nicholas/abstracts/FaadiBruno3.pdf.
doi: 10.1007/BF02987158. |
[24] |
S. Jin, J. -G. Liu and Z. Ma, Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro-macro decomposition based asymptotic preserving method, Research in Math. Sci., 4 (2017), Paper No. 15, 25 pp.
doi: 10.1186/s40687-017-0105-1. |
[25] |
S. Jin and L. Liu,
An asymptotic-preserving stochastic Galerkin method for the semicondutor Boltzmann equation with random inputs and diffusive scalings, Multiscale Model. Simu., 15 (2017), 157-183.
doi: 10.1137/15M1053463. |
[26] |
S. Jin, M. -B. Tran and E. Zuazua, A local sensivity analysis for a damped wave equation with random initial input, Preprint. Google Scholar |
[27] |
S. Jin, D. Xiu and X. Zhu,
Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, J. Comput. Phys., 289 (2015), 35-52.
doi: 10.1016/j.jcp.2015.02.023. |
[28] |
S. Jin, D. Xiu and X. Zhu,
A well-balanced stochastic Galerkin method for scalar hyperbolic balance laws with random inputs, J. Sci. Comput., 67 (2016), 1198-1218.
doi: 10.1007/s10915-015-0124-2. |
[29] |
S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple Scales, To appear in SIAM J. Math. Anal. Google Scholar |
[30] |
M.-J. Kang and A. Vasseur,
Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.
doi: 10.1142/S0218202515500542. |
[31] |
T. Karper, A. Mellet and K. Trivisa,
Hydrodynamic limit of the kinetic Cucker-Smale model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.
doi: 10.1142/S0218202515500050. |
[32] |
T. Karper, A. Mellet and K. Trivisa,
Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2013), 215-243.
doi: 10.1137/120866828. |
[33] |
T. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 227-242.
doi: 10.1007/978-3-642-39007-4_11. |
[34] |
N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. 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. |
[35] |
Q. Li and L. Wang,
Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertainty Quantification, 5 (2017), 1193-1219.
doi: 10.1137/16M1106675. |
[36] |
L. Liu and S. Jin, Hypocoercivity based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs, Preprint. Google Scholar |
[37] |
S. Motsch and E. Tadmor,
Heterophilious dynamics: Enhanced Consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[38] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
[39] |
D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105. Google Scholar |
[40] |
L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. of Guidance, Control and Dynamics, 32 (2009), 527-537. Google Scholar |
[41] |
A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Introduction to sensivity analysis, Global sensivity analysis, The Primer, (2008), 1-51. |
[42] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719.
doi: 10.1137/060673254. |
[43] |
R. Shu and S. Jin, Uniform regularity in the random space and spectral accuracy of the stochastic Galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime, Preprint. Google Scholar |
[44] |
E. Tadmor, Mathematical aspects of self-organized dynamics: consensus, emergence of leaders, and social hydrodynamics, SIAM News, 48 (2015). Google Scholar |
[45] |
J. Toner and Y. Tu,
Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E, 58 (1998), 4828-4858.
doi: 10.1103/PhysRevE.58.4828. |
[46] |
T. Vicsek, E. Ben-Jacob Czirók, I. Cohen and O. Schochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
[47] |
D. Xiu, Numerical Methods fo Stochastic Computations, Princeton University Presss, 2010. |
[48] |
D. Xiu and G. E. Karniadakis,
The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM. J. Scientific Computiong, 24 (2002), 619-644.
doi: 10.1137/S1064827501387826. |
show all references
References:
[1] |
S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee,
On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.
doi: 10.4310/CMS.2012.v10.n2.a10. |
[2] |
S. M. Ahn and S. -Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp.
doi: 10.1063/1.3496895. |
[3] |
G. Albi, L. Pareschi and M. Zanella, Uncertain quantification in control problems for flocking models, Math. Probl. Eng., 2015 (2015), Art. ID 850124, 14pp.
doi: 10.1155/2015/850124. |
[4] |
J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani,
Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.
doi: 10.1137/090757290. |
[5] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming. Mathematical modeling of collective behavior in socio-economic and life sciences, in Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Inc., Boston, MA, (2010), 297-336. Google Scholar |
[6] |
J. A. Carrillo, L. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Preprint. Google Scholar |
[7] |
J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko,
Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models and Methods in Appl. Sci., 26 (2016), 1191-1218.
doi: 10.1142/S0218202516500287. |
[8] |
Y. -P. Choi, S. -Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol. Ⅰ -Theory, Models, Applications, Series: Modeling and Simulation in Science and Technology (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhauser-Springer, 1 (2017), 299-331. |
[9] |
F. Cucker and J.-G Dong,
On flocks influenced by closest neighbors, Math. Models Methods Appl. Sci., 26 (2016), 2685-2708.
doi: 10.1142/S0218202516500639. |
[10] |
F. Cucker and J.-G Dong,
A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.
doi: 10.1109/TAC.2011.2107113. |
[11] |
F. Cucker and E. Mordecki,
Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.
doi: 10.1016/j.matpur.2007.12.002. |
[12] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[13] |
P. Degond and S. Motsch,
Large-scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1022.
doi: 10.1007/s10955-008-9529-8. |
[14] |
R. Duan, M. Fornasier and G. Toscani,
A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145.
doi: 10.1007/s00220-010-1110-z. |
[15] |
S. -Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, To appear in Kinetic Relat. Models. Google Scholar |
[16] |
S.-Y. Ha, B. Kwon and M.-J. Kang,
A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Mod. Meth. Appl. Sci., 24 (2014), 2311-2359.
doi: 10.1142/S0218202514500225. |
[17] |
S.-Y. Ha, K. Lee and D. Levy,
Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.
doi: 10.4310/CMS.2009.v7.n2.a9. |
[18] |
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.
doi: 10.4310/CMS.2009.v7.n2.a2. |
[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] |
J. Hu and S. Jin,
A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), 150-168.
doi: 10.1016/j.jcp.2016.03.047. |
[21] |
J. Hu and S. Jin, Uncertainty quantification for kinetic equations, in Uncertainty Quantification for Kinetic and Hyperbolic Equations, (eds S. Jin and L. Pareschi), 14 (2018), 193-229.
doi: 10.1007/978-3-319-67110-9_6. |
[22] |
J. Hu, S. Jin and D. Xiu,
A stochastic Galerkin method for Hamiltonian-Jacobi equations with uncertainty, SIAM. J. Sci. Comput., 37 (2015), 2246-2269.
doi: 10.1137/140990930. |
[23] |
H. -N. Huang, S. A. M. Marcantognini and N. J. Young, Chain rules for higher derivatives, The Mathematical Intelligencer, 28 (2006), 61-69, http://ambio1.leeds.ac.uk/~nicholas/abstracts/FaadiBruno3.pdf.
doi: 10.1007/BF02987158. |
[24] |
S. Jin, J. -G. Liu and Z. Ma, Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro-macro decomposition based asymptotic preserving method, Research in Math. Sci., 4 (2017), Paper No. 15, 25 pp.
doi: 10.1186/s40687-017-0105-1. |
[25] |
S. Jin and L. Liu,
An asymptotic-preserving stochastic Galerkin method for the semicondutor Boltzmann equation with random inputs and diffusive scalings, Multiscale Model. Simu., 15 (2017), 157-183.
doi: 10.1137/15M1053463. |
[26] |
S. Jin, M. -B. Tran and E. Zuazua, A local sensivity analysis for a damped wave equation with random initial input, Preprint. Google Scholar |
[27] |
S. Jin, D. Xiu and X. Zhu,
Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, J. Comput. Phys., 289 (2015), 35-52.
doi: 10.1016/j.jcp.2015.02.023. |
[28] |
S. Jin, D. Xiu and X. Zhu,
A well-balanced stochastic Galerkin method for scalar hyperbolic balance laws with random inputs, J. Sci. Comput., 67 (2016), 1198-1218.
doi: 10.1007/s10915-015-0124-2. |
[29] |
S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple Scales, To appear in SIAM J. Math. Anal. Google Scholar |
[30] |
M.-J. Kang and A. Vasseur,
Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.
doi: 10.1142/S0218202515500542. |
[31] |
T. Karper, A. Mellet and K. Trivisa,
Hydrodynamic limit of the kinetic Cucker-Smale model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.
doi: 10.1142/S0218202515500050. |
[32] |
T. Karper, A. Mellet and K. Trivisa,
Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2013), 215-243.
doi: 10.1137/120866828. |
[33] |
T. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 227-242.
doi: 10.1007/978-3-642-39007-4_11. |
[34] |
N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. 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. |
[35] |
Q. Li and L. Wang,
Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertainty Quantification, 5 (2017), 1193-1219.
doi: 10.1137/16M1106675. |
[36] |
L. Liu and S. Jin, Hypocoercivity based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs, Preprint. Google Scholar |
[37] |
S. Motsch and E. Tadmor,
Heterophilious dynamics: Enhanced Consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[38] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
[39] |
D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105. Google Scholar |
[40] |
L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. of Guidance, Control and Dynamics, 32 (2009), 527-537. Google Scholar |
[41] |
A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Introduction to sensivity analysis, Global sensivity analysis, The Primer, (2008), 1-51. |
[42] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719.
doi: 10.1137/060673254. |
[43] |
R. Shu and S. Jin, Uniform regularity in the random space and spectral accuracy of the stochastic Galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime, Preprint. Google Scholar |
[44] |
E. Tadmor, Mathematical aspects of self-organized dynamics: consensus, emergence of leaders, and social hydrodynamics, SIAM News, 48 (2015). Google Scholar |
[45] |
J. Toner and Y. Tu,
Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E, 58 (1998), 4828-4858.
doi: 10.1103/PhysRevE.58.4828. |
[46] |
T. Vicsek, E. Ben-Jacob Czirók, I. Cohen and O. Schochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
[47] |
D. Xiu, Numerical Methods fo Stochastic Computations, Princeton University Presss, 2010. |
[48] |
D. Xiu and G. E. Karniadakis,
The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM. J. Scientific Computiong, 24 (2002), 619-644.
doi: 10.1137/S1064827501387826. |
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