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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

J. T. Beale, Large time behavior of discrete velocity Boltzmann equations, Commun. Math. Phys., 106 (1986), 659-678. doi: 10.1007/BF01463401.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[6]

M. Chae and S. Y. Ha, Stability estimates of the Boltzmann equation in the half Space, Quart. Appl. Math., 65 (2007), 653-682.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations, 215 (2005), 178-205.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Polewczak, Classical solution of the nonlinear Boltzmann equation in all $\mathbbR^3$ : Asymptotic behavior of solutions, J. Stat. Phys., 50 (1988), 611-632. doi: 10.1007/BF01026493.  Google Scholar

[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.  Google Scholar

[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), 245-272. doi: 10.1080/00411450903238665.  Google Scholar

[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.  Google Scholar

[3]

J. T. Beale, Large time behavior of discrete velocity Boltzmann equations, Commun. Math. Phys., 106 (1986), 659-678. doi: 10.1007/BF01463401.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[6]

M. Chae and S. Y. Ha, Stability estimates of the Boltzmann equation in the half Space, Quart. Appl. Math., 65 (2007), 653-682.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations, 215 (2005), 178-205.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Polewczak, Classical solution of the nonlinear Boltzmann equation in all $\mathbbR^3$ : Asymptotic behavior of solutions, J. Stat. Phys., 50 (1988), 611-632. doi: 10.1007/BF01026493.  Google Scholar

[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.  Google Scholar

[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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