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Relative entropy method for the relaxation limit of hydrodynamic models
A BGK kinetic model with local velocity alignment forces
1. | Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea |
2. | Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea |
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.
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. |
[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. |
[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.
|
[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.
|
[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] |
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. |
[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. |
[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. |
[9] |
Y.-P. Choi and S.-B. Yun, Existence and hydrodynamic limit for a Paveri-Fontana type kinetic traffic model, preprint, arXiv: 1911.05572. |
[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. |
[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.
|
[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] |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[27] |
S.-B. Yun,
Ellipsoidal BGK model near a global Maxwellian, SIAM J. Math. Anal., 47 (2015), 2324-2354.
doi: 10.1137/130932399. |
[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. |
[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. |
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. |
[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. |
[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.
|
[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.
|
[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] |
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. |
[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. |
[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. |
[9] |
Y.-P. Choi and S.-B. Yun, Existence and hydrodynamic limit for a Paveri-Fontana type kinetic traffic model, preprint, arXiv: 1911.05572. |
[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. |
[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.
|
[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] |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[27] |
S.-B. Yun,
Ellipsoidal BGK model near a global Maxwellian, SIAM J. Math. Anal., 47 (2015), 2324-2354.
doi: 10.1137/130932399. |
[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. |
[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. |
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