Stability and modeling error for the Boltzmann equation

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June 2014, 7(2): 401-414. doi: 10.3934/krm.2014.7.401

Stability and modeling error for the Boltzmann equation

El Miloud Zaoui ^1, and Marc Laforest ^2,

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

Received August 2012 Revised November 2013 Published March 2014

Abstract
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We show that the residual measures the difference in $L^1$ between the solutions to two different Boltzmann models of rarefied gases. This work is an extension of earlier work by Ha on the stability of Boltzmann's model, and more specifically on a nonlinear interaction functional that controls the growth of waves. The two kinetic models that are compared in this research are given by (possibly different) inverse power laws, such as the hard spheres and pseudo-Maxwell models. The main point of the estimate is that the modeling error is measured a posteriori, that is to say, the difference between the solutions to the first and second model can be bounded by a term that depends on only one of the two solutions. This work allows the stability estimate to be used to assess uncertainty, modeling or numerical, present in the solution of the first model without solving the second model.

Keywords: modeling error, a posteriori, Error estimate, stability., Boltzmann equation.

Mathematics Subject Classification: Primary: 82B40, 35B35; Secondary: 65M1.

Citation: El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401

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), 245-272. doi: 10.1080/00411450903238665.
[2]	K. Assi and M. Laforest, Accuracy of modeling error estimates for hierarchies of discrete velocity models, in Problems: Theory, Numerics and Applications, Proceedings of Symposia in Applied Mathematics (eds. E. Tadmor, J. G. Liu and A. Tzavaras), AMS, 67 (2009), 369-378.
[3]	J. T. Beale, Large time behavior of discrete velocity Boltzmann equations, Commun. Math. Phys., 106 (1986), 659-678. 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, 1987), Palaiseau, École Polytechnique, 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), 1-22. 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), 653-682.
[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), 97-130. 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), 697-715. 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), 65-92. 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), 178-205.
[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, F. Bampi and G. Toscani), World Sci. Publ., Singapore, 1992, 152-158.
[12]	M. Laforest, A posteriori error estimate for front-tracking: Systems of conservation laws, SIAM J. Math. Anal., 35 (2004), 1347-1370. 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, Numerics and Applications, (eds. S. Benzoni-Gavage and D. Serre), Springer-Verlag, (2008), 643-651. 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), 729-774. 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), 31-62. doi: 10.3166/remn.17.31-61.
[16]	J. Polewczak, Classical solution of the nonlinear Boltzmann equation in all $\mathbb{R}^3$ : Asymptotic behavior of solutions, J. Stat. Phys., 50 (1988), 611-632. doi: 10.1007/BF01026493.
[17]	C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Fluid Mechanics, (eds. by S. Friedlander and D. Serre), I (2002), 71-74. 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).

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), 245-272. doi: 10.1080/00411450903238665.
[2]	K. Assi and M. Laforest, Accuracy of modeling error estimates for hierarchies of discrete velocity models, in Problems: Theory, Numerics and Applications, Proceedings of Symposia in Applied Mathematics (eds. E. Tadmor, J. G. Liu and A. Tzavaras), AMS, 67 (2009), 369-378.
[3]	J. T. Beale, Large time behavior of discrete velocity Boltzmann equations, Commun. Math. Phys., 106 (1986), 659-678. 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, 1987), Palaiseau, École Polytechnique, 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), 1-22. 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), 653-682.
[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), 97-130. 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), 697-715. 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), 65-92. 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), 178-205.
[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, F. Bampi and G. Toscani), World Sci. Publ., Singapore, 1992, 152-158.
[12]	M. Laforest, A posteriori error estimate for front-tracking: Systems of conservation laws, SIAM J. Math. Anal., 35 (2004), 1347-1370. 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, Numerics and Applications, (eds. S. Benzoni-Gavage and D. Serre), Springer-Verlag, (2008), 643-651. 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), 729-774. 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), 31-62. doi: 10.3166/remn.17.31-61.
[16]	J. Polewczak, Classical solution of the nonlinear Boltzmann equation in all $\mathbb{R}^3$ : Asymptotic behavior of solutions, J. Stat. Phys., 50 (1988), 611-632. doi: 10.1007/BF01026493.
[17]	C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Fluid Mechanics, (eds. by S. Friedlander and D. Serre), I (2002), 71-74. 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).

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