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September  2020, 15(3): 389-404. doi: 10.3934/nhm.2020024

A BGK kinetic model with local velocity alignment forces

Young-Pil Choi 1,, and  Seok-Bae Yun 2,

1. 

Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea
2. 

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

* Corresponding author: Young-Pil Choi

Received  January 2020 Revised  May 2020 Published  September 2020 Early access  September 2020

Fund Project: Young-Pil Choi is supported by National Research Foundation of Korea(NRF) grants funded by the Korea government(MSIP) (No. 2017R1C1B2012918), POSCO Science Fellowship of POSCO TJ Park Foundation, and Yonsei University Research Fund of 2019-22-02. Seok-Bae Yun is supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-02

The global Cauchy problem for a local alignment model with a relaxational inter-particle interaction operator is considered. More precisely, we consider the global-in-time existence of weak solutions of BGK model with local velocity-alignment term when the initial data have finite mass, momentum, energy, and entropy. The analysis involves weak/strong compactness based on the velocity moments estimates and several velocity-weighted $ L^\infty $ estimates.

Keywords: Kinetic flocking model, BGK model of Boltzmann equation, weak solution.
Mathematics Subject Classification: 35Q20, 35Q92, 35Q83, 82B40.
Citation: Young-Pil Choi, Seok-Bae Yun. A BGK kinetic model with local velocity alignment forces. Networks & Heterogeneous Media, 2020, 15 (3) : 389-404. doi: 10.3934/nhm.2020024
References:
[1]

J. Bang and S.-B. Yun, Stationary solutions for the ellipsoidal BGK model in a slab, J. Differential Equations, 261 (2016), 5803-5828.  doi: 10.1016/j.jde.2016.08.022.  Google Scholar

[2]

A. Bellouquid, Global existence and large-time behavior for BGK model for a gas with non-constant cross section, Transport Theory Statist. Phys., 32 (2003), 157-184.  doi: 10.1081/TT-120019041.  Google Scholar

[3]

P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude process in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.   Google Scholar

[4]

J. A. Carrillo, Y.-P. Choi and S. P. Pérez, A review on attractive-repulsive hydrodynamics for consensus in collective behavior, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 259-298.   Google Scholar

[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.  Google Scholar

[6]

Y.-P. Choi, Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916.  doi: 10.1088/0951-7715/29/7/1887.  Google Scholar

[7]

Y.-P. Choi, Uniform-in-time bound for kinetic flocking models, Appl. Math. Lett., 103 (2020), 106164, 9 pp. doi: 10.1016/j.aml.2019.106164.  Google Scholar

[8]

Y.-P. Choi, J. Lee and S.-B. Yun, Strong solutions to the inhomogeneous Navier-Stokes-BGK system, Nonlinear Anal. Real World Appl., 57 (2021), 103196. doi: 10.1016/j.nonrwa.2020.103196.  Google Scholar

[9]

Y.-P. Choi and S.-B. Yun, Existence and hydrodynamic limit for a Paveri-Fontana type kinetic traffic model, preprint, arXiv: 1911.05572. Google Scholar

[10]

Y.-P. Choi and S.-B. Yun, Global existence of weak solutions for Navier-Stokes-BGK system, Nonlinearity, 33 (2020), 1925-1955.  doi: 10.1088/1361-6544/ab6c38.  Google Scholar

[11]

Y.-P. Choi, S.-Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 299-331.   Google Scholar

[12]

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

[13]

F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1998), 110-125.  doi: 10.1016/0022-1236(88)90051-1.  Google Scholar

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S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Comm. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[15]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[16]

T. K. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, Hyperbolic Conservation Laws and Related Analysis with Applications, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 227-242.  doi: 10.1007/978-3-642-39007-4_11.  Google Scholar

[17]

T. K. Karper, A. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale flocking model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.  Google Scholar

[18]

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.  Google Scholar

[19]

S. J. Park and S.-B. Yun, Cauchy problem for ellipsoidal BGK model for polyatomic particles, J. Differential Equations, 266 (2019), 7678-7708.  doi: 10.1016/j.jde.2018.12.013.  Google Scholar

[20]

S.-L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis, Transport. Res., 9 (1975), 225-235.  doi: 10.1016/0041-1647(75)90063-5.  Google Scholar

[21]

B. Perthame, Global existence to the BGK model of Boltzmann equation, J. Differential Equations, 82 (1989), 191-205.  doi: 10.1016/0022-0396(89)90173-3.  Google Scholar

[22]

B. Perthame and M. Pulvirenti, Weighted $L^\infty$ bounds and uniqueness for the Boltzmann BGK model, Arch. Rational Mech. Anal., 125 (1993), 289-295.  doi: 10.1007/BF00383223.  Google Scholar

[23]

S. Ukai, Stationary solutions of the BGK model equation on a finite interval with large boundary data, Transport theory Statist. Phys., 21 (1992), 487-500.  doi: 10.1080/00411459208203795.  Google Scholar

[24]

S.-B. Yun, Cauchy problem for the Boltzmann-BGK model near a global Maxwellian, J. Math. Phys., 51 (2010), 123514, 24 pp. doi: 10.1063/1.3516479.  Google Scholar

[25]

S.-B. Yun, Classical solutions for the ellipsoidal BGK model with fixed collision frequency, J. Differential Equations, 259 (2015), 6009-6037.  doi: 10.1016/j.jde.2015.07.016.  Google Scholar

[26]

S.-B. Yun, Ellipsoidal BGK model for polyatomic molecules near Maxwellians: A dichotomy in the dissipation estimate, J. Differential Equations, 266 (2019), 5566-5614.  doi: 10.1016/j.jde.2018.10.036.  Google Scholar

[27]

S.-B. Yun, Ellipsoidal BGK model near a global Maxwellian, SIAM J. Math. Anal., 47 (2015), 2324-2354.  doi: 10.1137/130932399.  Google Scholar

[28]

X. Zhang, On the Cauchy problem of the Vlasov-Posson-BGK system: Global existence of weak solutions, J. Stat. Phys., 141 (2010), 566-588.  doi: 10.1007/s10955-010-0064-z.  Google Scholar

[29]

X. Zhang and S. Hu, $L^p$ solutions to the Cauchy problem of the BGK equation, J. Math. Phys., 48 (2007), 113304, 17 pp. doi: 10.1063/1.2816261.  Google Scholar

show all references

References:
[1]

J. Bang and S.-B. Yun, Stationary solutions for the ellipsoidal BGK model in a slab, J. Differential Equations, 261 (2016), 5803-5828.  doi: 10.1016/j.jde.2016.08.022.  Google Scholar

[2]

A. Bellouquid, Global existence and large-time behavior for BGK model for a gas with non-constant cross section, Transport Theory Statist. Phys., 32 (2003), 157-184.  doi: 10.1081/TT-120019041.  Google Scholar

[3]

P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude process in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.   Google Scholar

[4]

J. A. Carrillo, Y.-P. Choi and S. P. Pérez, A review on attractive-repulsive hydrodynamics for consensus in collective behavior, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 259-298.   Google Scholar

[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.  Google Scholar

[6]

Y.-P. Choi, Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916.  doi: 10.1088/0951-7715/29/7/1887.  Google Scholar

[7]

Y.-P. Choi, Uniform-in-time bound for kinetic flocking models, Appl. Math. Lett., 103 (2020), 106164, 9 pp. doi: 10.1016/j.aml.2019.106164.  Google Scholar

[8]

Y.-P. Choi, J. Lee and S.-B. Yun, Strong solutions to the inhomogeneous Navier-Stokes-BGK system, Nonlinear Anal. Real World Appl., 57 (2021), 103196. doi: 10.1016/j.nonrwa.2020.103196.  Google Scholar

[9]

Y.-P. Choi and S.-B. Yun, Existence and hydrodynamic limit for a Paveri-Fontana type kinetic traffic model, preprint, arXiv: 1911.05572. Google Scholar

[10]

Y.-P. Choi and S.-B. Yun, Global existence of weak solutions for Navier-Stokes-BGK system, Nonlinearity, 33 (2020), 1925-1955.  doi: 10.1088/1361-6544/ab6c38.  Google Scholar

[11]

Y.-P. Choi, S.-Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 299-331.   Google Scholar

[12]

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

[13]

F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1998), 110-125.  doi: 10.1016/0022-1236(88)90051-1.  Google Scholar

[14]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Comm. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[15]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[16]

T. K. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, Hyperbolic Conservation Laws and Related Analysis with Applications, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 227-242.  doi: 10.1007/978-3-642-39007-4_11.  Google Scholar

[17]

T. K. Karper, A. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale flocking model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.  Google Scholar

[18]

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.  Google Scholar

[19]

S. J. Park and S.-B. Yun, Cauchy problem for ellipsoidal BGK model for polyatomic particles, J. Differential Equations, 266 (2019), 7678-7708.  doi: 10.1016/j.jde.2018.12.013.  Google Scholar

[20]

S.-L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis, Transport. Res., 9 (1975), 225-235.  doi: 10.1016/0041-1647(75)90063-5.  Google Scholar

[21]

B. Perthame, Global existence to the BGK model of Boltzmann equation, J. Differential Equations, 82 (1989), 191-205.  doi: 10.1016/0022-0396(89)90173-3.  Google Scholar

[22]

B. Perthame and M. Pulvirenti, Weighted $L^\infty$ bounds and uniqueness for the Boltzmann BGK model, Arch. Rational Mech. Anal., 125 (1993), 289-295.  doi: 10.1007/BF00383223.  Google Scholar

[23]

S. Ukai, Stationary solutions of the BGK model equation on a finite interval with large boundary data, Transport theory Statist. Phys., 21 (1992), 487-500.  doi: 10.1080/00411459208203795.  Google Scholar

[24]

S.-B. Yun, Cauchy problem for the Boltzmann-BGK model near a global Maxwellian, J. Math. Phys., 51 (2010), 123514, 24 pp. doi: 10.1063/1.3516479.  Google Scholar

[25]

S.-B. Yun, Classical solutions for the ellipsoidal BGK model with fixed collision frequency, J. Differential Equations, 259 (2015), 6009-6037.  doi: 10.1016/j.jde.2015.07.016.  Google Scholar

[26]

S.-B. Yun, Ellipsoidal BGK model for polyatomic molecules near Maxwellians: A dichotomy in the dissipation estimate, J. Differential Equations, 266 (2019), 5566-5614.  doi: 10.1016/j.jde.2018.10.036.  Google Scholar

[27]

S.-B. Yun, Ellipsoidal BGK model near a global Maxwellian, SIAM J. Math. Anal., 47 (2015), 2324-2354.  doi: 10.1137/130932399.  Google Scholar

[28]

X. Zhang, On the Cauchy problem of the Vlasov-Posson-BGK system: Global existence of weak solutions, J. Stat. Phys., 141 (2010), 566-588.  doi: 10.1007/s10955-010-0064-z.  Google Scholar

[29]

X. Zhang and S. Hu, $L^p$ solutions to the Cauchy problem of the BGK equation, J. Math. Phys., 48 (2007), 113304, 17 pp. doi: 10.1063/1.2816261.  Google Scholar

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