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Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling
Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit
1. | Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea |
2. | Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea |
3. | Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea |
4. | Center for Mathematical Sciences, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China |
References:
[1] |
S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Physics, 51 (2010), 103301, 17pp
doi: 10.1063/1.3496895. |
[2] |
H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang,
Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids, Discrete and Continuous Dynamical System, 34 (2014), 4419-4458.
doi: 10.3934/dcds.2014.34.4419. |
[3] |
H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang,
Time-asymptotic interaction of flocking particles and incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.
doi: 10.1088/0951-7715/25/4/1155. |
[4] |
A. Bressan, T.-P. Liu and T. Yang,
L1 stability estimates for n × n conservation laws, Arch. Ration. Mech. Anal., 149 (1999), 1-22.
doi: 10.1007/s002050050165. |
[5] |
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. |
[6] |
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, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, (2010), 297–336.
doi: 10.1007/978-0-8176-4946-3_12. |
[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 -Advances in Theory, Models, Applications (tentative title), Series: Modeling and Simulation in Science and Technology, (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhäuser Basel, (2017), 299–331. |
[9] |
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. |
[10] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[11] |
P. Degond and S. Motsch,
Large scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.
doi: 10.1007/s10955-008-9529-8. |
[12] |
R. Duan, M. Fornasier and G. Toscani,
A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.
doi: 10.1007/s00220-010-1110-z. |
[13] |
S.-Y. Ha,
$L_1$ stability of the Boltzmann equation for the hard-sphere model, Arch. Ration. Mech. Anal., 173 (2004), 279-296.
doi: 10.1007/s00205-004-0321-x. |
[14] |
S.-Y. Ha, B. Kwon and M.-J. Kang,
Emergent dynamics for the hydrodynamic Cucker-Smale system in a moving domain, SIAM. J. Math. Anal., 47 (2015), 3813-3831.
doi: 10.1137/140984403. |
[15] |
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. |
[16] |
S.-Y. Ha, C. Lattanzio, B. Rubino and M. Slemrod,
Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.
doi: 10.1090/S0033-569X-2010-01200-7. |
[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 M. Slemrod,
Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dyn. Differential Equations, 22 (2010), 325-330.
doi: 10.1007/s10884-009-9142-9. |
[20] |
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. |
[21] |
S.-Y. Ha and A. E. Tzavaras,
Lyapunov functionals and L1-stability for discrete velocity Boltzmann equations, Comm. Math. Phys., 239 (2003), 65-92.
doi: 10.1007/s00220-003-0866-9. |
[22] |
J. Hale, Ordinary Differential Equations, Dover, 1997. Google Scholar |
[23] |
E. Justh and P. Krishnaprasad, A simple control law for UAV formation flying, Technical Report, 2002-38 (http://www.isr.umd.edu) Google Scholar |
[24] |
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. |
[25] |
T.-P. Liu and T. Yang,
Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math., 52 (1999), 1553-1586.
doi: 10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S. |
[26] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[27] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys., 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
[28] |
H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic theories and the Boltzmann Equation, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1048 (1984), 60–110.
doi: 10.1007/BFb0071878. |
[29] |
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 |
[30] |
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.
doi: 10.2514/1.36269. |
[31] |
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. |
[32] |
T. Vicsek, Czirók, E. Ben-Jacob, 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. |
[33] |
C. Villani, Optimal Transport, Old and New, Springer-Verlag, 2009.
doi: 10.1007/978-3-540-71050-9. |
show all references
References:
[1] |
S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Physics, 51 (2010), 103301, 17pp
doi: 10.1063/1.3496895. |
[2] |
H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang,
Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids, Discrete and Continuous Dynamical System, 34 (2014), 4419-4458.
doi: 10.3934/dcds.2014.34.4419. |
[3] |
H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang,
Time-asymptotic interaction of flocking particles and incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.
doi: 10.1088/0951-7715/25/4/1155. |
[4] |
A. Bressan, T.-P. Liu and T. Yang,
L1 stability estimates for n × n conservation laws, Arch. Ration. Mech. Anal., 149 (1999), 1-22.
doi: 10.1007/s002050050165. |
[5] |
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. |
[6] |
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, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, (2010), 297–336.
doi: 10.1007/978-0-8176-4946-3_12. |
[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 -Advances in Theory, Models, Applications (tentative title), Series: Modeling and Simulation in Science and Technology, (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhäuser Basel, (2017), 299–331. |
[9] |
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. |
[10] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[11] |
P. Degond and S. Motsch,
Large scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.
doi: 10.1007/s10955-008-9529-8. |
[12] |
R. Duan, M. Fornasier and G. Toscani,
A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.
doi: 10.1007/s00220-010-1110-z. |
[13] |
S.-Y. Ha,
$L_1$ stability of the Boltzmann equation for the hard-sphere model, Arch. Ration. Mech. Anal., 173 (2004), 279-296.
doi: 10.1007/s00205-004-0321-x. |
[14] |
S.-Y. Ha, B. Kwon and M.-J. Kang,
Emergent dynamics for the hydrodynamic Cucker-Smale system in a moving domain, SIAM. J. Math. Anal., 47 (2015), 3813-3831.
doi: 10.1137/140984403. |
[15] |
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. |
[16] |
S.-Y. Ha, C. Lattanzio, B. Rubino and M. Slemrod,
Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.
doi: 10.1090/S0033-569X-2010-01200-7. |
[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 M. Slemrod,
Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dyn. Differential Equations, 22 (2010), 325-330.
doi: 10.1007/s10884-009-9142-9. |
[20] |
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. |
[21] |
S.-Y. Ha and A. E. Tzavaras,
Lyapunov functionals and L1-stability for discrete velocity Boltzmann equations, Comm. Math. Phys., 239 (2003), 65-92.
doi: 10.1007/s00220-003-0866-9. |
[22] |
J. Hale, Ordinary Differential Equations, Dover, 1997. Google Scholar |
[23] |
E. Justh and P. Krishnaprasad, A simple control law for UAV formation flying, Technical Report, 2002-38 (http://www.isr.umd.edu) Google Scholar |
[24] |
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. |
[25] |
T.-P. Liu and T. Yang,
Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math., 52 (1999), 1553-1586.
doi: 10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S. |
[26] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[27] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys., 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
[28] |
H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic theories and the Boltzmann Equation, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1048 (1984), 60–110.
doi: 10.1007/BFb0071878. |
[29] |
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 |
[30] |
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.
doi: 10.2514/1.36269. |
[31] |
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. |
[32] |
T. Vicsek, Czirók, E. Ben-Jacob, 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. |
[33] |
C. Villani, Optimal Transport, Old and New, Springer-Verlag, 2009.
doi: 10.1007/978-3-540-71050-9. |
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