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Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation
Stability and modeling error for the Boltzmann equation
1. | École Nationale de l'Industrie Minérale, Laboratoire de Mécanique, Thermique et Matériaux, Avenue Hadj Ahmed Cherkaoui - BP 753, Agdal, Rabat, Morocco |
2. | Département de mathématiques et de génie industriel, École Polytechnique de Montréal, C.P. 6079, succursale centre-ville, Montréal, Québec, Canada, H3C 3A7, Canada |
References:
[1] |
K. Assi and M. Laforest, Modeling error in $L^1$ for a hierarchy of 1-D discrete velocity models,, Transp. Th. and Stat. Phys., 38 (2009), 245.
doi: 10.1080/00411450903238665. |
[2] |
K. Assi and M. Laforest, Accuracy of modeling error estimates for hierarchies of discrete velocity models,, in Problems: Theory, 67 (2009), 369.
|
[3] |
J. T. Beale, Large time behavior of discrete velocity Boltzmann equations,, Commun. Math. Phys., 106 (1986), 659.
doi: 10.1007/BF01463401. |
[4] |
J.-M. Bony, Solutions globales bornées pour les modèles discrets de l'équation de Boltzmann, en dimension $1$ d'espace,, Journées équations aux dérivées partielles (Saint Jean de Monts, XVI (1987).
|
[5] |
A. Bressan, T. P. Liu and T. Yang, $L^1$ stability estimates for $n \times n$ conservation laws,, Arch. Rat. Mech. Anal., 149 (1999), 1.
doi: 10.1007/s002050050165. |
[6] |
M. Chae and S. Y. Ha, Stability estimates of the Boltzmann equation in the half Space,, Quart. Appl. Math., 65 (2007), 653.
|
[7] |
G. Chen, C. Christoforou and Y. Q. Zhang, Continuous dependence of entropy solutions to the Euler equations on the adiabatic exponent and Mach number,, Arch. Rat. Mech. Anal., 189 (2008), 97.
doi: 10.1007/s00205-007-0098-9. |
[8] |
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 697.
doi: 10.1002/cpa.3160180408. |
[9] |
S. Y. Ha and A. E. Tzavaras, Lyapunov functionals and $L^1$-stability for discrete velocity Boltzmann equations,, Comm. Math. Phy., 239 (2003), 65.
doi: 10.1007/s00220-003-0866-9. |
[10] |
S. Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates,, J. Differential Equations, 215 (2005), 178.
|
[11] |
R. Illner and S. Rjasanow, Random discrete velocity models: Possible bridges between the Boltzmann equation, discrete velocity models and particle simulation?,, in Nonlinear Kinetic Theory and Math. Asp. of Hyperbolic Systems (eds. V. C. Boffi, (1992), 152.
|
[12] |
M. Laforest, A posteriori error estimate for front-tracking: Systems of conservation laws,, SIAM J. Math. Anal., 35 (2004), 1347.
doi: 10.1137/S0036141002416870. |
[13] |
M. Laforest, An a posteriori error estimate for Glimm's scheme,, in Proceedings of the $11$th International Conference on Hyperbolic Problems: Theory, (2008), 643.
doi: 10.1007/978-3-540-75712-2_64. |
[14] |
T. P. Liu and T. Yang, $L^1$ stability for $2 \times 2$ systems of hyperbolic conservation laws,, J. of AMS, 12 (1999), 729.
doi: 10.1090/S0894-0347-99-00292-1. |
[15] |
D. Pelletier, S. Étienne, Q. Hay and J. Borggaard, The sensitivity equation in fluid mechanics,, Eur. J. Comp. Mech., 17 (2008), 31.
doi: 10.3166/remn.17.31-61. |
[16] |
J. Polewczak, Classical solution of the nonlinear Boltzmann equation in all $\mathbbR^3$ : Asymptotic behavior of solutions,, J. Stat. Phys., 50 (1988), 611.
doi: 10.1007/BF01026493. |
[17] |
C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Fluid Mechanics, I (2002), 71.
doi: 10.1016/S1874-5792(02)80004-0. |
[18] |
E. M. Zaoui and M. Laforest, Error estimation and adaptivity for the random discrete velocity model of the Boltzmann equation,, In preparation, (2014). Google Scholar |
show all references
References:
[1] |
K. Assi and M. Laforest, Modeling error in $L^1$ for a hierarchy of 1-D discrete velocity models,, Transp. Th. and Stat. Phys., 38 (2009), 245.
doi: 10.1080/00411450903238665. |
[2] |
K. Assi and M. Laforest, Accuracy of modeling error estimates for hierarchies of discrete velocity models,, in Problems: Theory, 67 (2009), 369.
|
[3] |
J. T. Beale, Large time behavior of discrete velocity Boltzmann equations,, Commun. Math. Phys., 106 (1986), 659.
doi: 10.1007/BF01463401. |
[4] |
J.-M. Bony, Solutions globales bornées pour les modèles discrets de l'équation de Boltzmann, en dimension $1$ d'espace,, Journées équations aux dérivées partielles (Saint Jean de Monts, XVI (1987).
|
[5] |
A. Bressan, T. P. Liu and T. Yang, $L^1$ stability estimates for $n \times n$ conservation laws,, Arch. Rat. Mech. Anal., 149 (1999), 1.
doi: 10.1007/s002050050165. |
[6] |
M. Chae and S. Y. Ha, Stability estimates of the Boltzmann equation in the half Space,, Quart. Appl. Math., 65 (2007), 653.
|
[7] |
G. Chen, C. Christoforou and Y. Q. Zhang, Continuous dependence of entropy solutions to the Euler equations on the adiabatic exponent and Mach number,, Arch. Rat. Mech. Anal., 189 (2008), 97.
doi: 10.1007/s00205-007-0098-9. |
[8] |
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 697.
doi: 10.1002/cpa.3160180408. |
[9] |
S. Y. Ha and A. E. Tzavaras, Lyapunov functionals and $L^1$-stability for discrete velocity Boltzmann equations,, Comm. Math. Phy., 239 (2003), 65.
doi: 10.1007/s00220-003-0866-9. |
[10] |
S. Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates,, J. Differential Equations, 215 (2005), 178.
|
[11] |
R. Illner and S. Rjasanow, Random discrete velocity models: Possible bridges between the Boltzmann equation, discrete velocity models and particle simulation?,, in Nonlinear Kinetic Theory and Math. Asp. of Hyperbolic Systems (eds. V. C. Boffi, (1992), 152.
|
[12] |
M. Laforest, A posteriori error estimate for front-tracking: Systems of conservation laws,, SIAM J. Math. Anal., 35 (2004), 1347.
doi: 10.1137/S0036141002416870. |
[13] |
M. Laforest, An a posteriori error estimate for Glimm's scheme,, in Proceedings of the $11$th International Conference on Hyperbolic Problems: Theory, (2008), 643.
doi: 10.1007/978-3-540-75712-2_64. |
[14] |
T. P. Liu and T. Yang, $L^1$ stability for $2 \times 2$ systems of hyperbolic conservation laws,, J. of AMS, 12 (1999), 729.
doi: 10.1090/S0894-0347-99-00292-1. |
[15] |
D. Pelletier, S. Étienne, Q. Hay and J. Borggaard, The sensitivity equation in fluid mechanics,, Eur. J. Comp. Mech., 17 (2008), 31.
doi: 10.3166/remn.17.31-61. |
[16] |
J. Polewczak, Classical solution of the nonlinear Boltzmann equation in all $\mathbbR^3$ : Asymptotic behavior of solutions,, J. Stat. Phys., 50 (1988), 611.
doi: 10.1007/BF01026493. |
[17] |
C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Fluid Mechanics, I (2002), 71.
doi: 10.1016/S1874-5792(02)80004-0. |
[18] |
E. M. Zaoui and M. Laforest, Error estimation and adaptivity for the random discrete velocity model of the Boltzmann equation,, In preparation, (2014). Google Scholar |
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