Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation

Thierry Goudon; Antoine Mellet

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 9, page 371-398
  • ISSN: 1292-8119

Abstract

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We investigate the diffusion limit for general conservative Boltzmann equations with oscillating coefficients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coefficients may depend on both slow and fast variables. Passing to the limit, we are led to an effective drift-diffusion equation. We also describe the diffusive behaviour when the equilibrium function has a non-vanishing flux.

How to cite

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Goudon, Thierry, and Mellet, Antoine. "Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 371-398. <http://eudml.org/doc/90701>.

@article{Goudon2010,
abstract = { We investigate the diffusion limit for general conservative Boltzmann equations with oscillating coefficients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coefficients may depend on both slow and fast variables. Passing to the limit, we are led to an effective drift-diffusion equation. We also describe the diffusive behaviour when the equilibrium function has a non-vanishing flux. },
author = {Goudon, Thierry, Mellet, Antoine},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = { Boltzmann equation; diffusion approximation; homogenization; drift-diffusion equation.; Boltzmann equation; homogenization; drift-diffusion equation; semiconductors},
language = {eng},
month = {3},
pages = {371-398},
publisher = {EDP Sciences},
title = {Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation},
url = {http://eudml.org/doc/90701},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Goudon, Thierry
AU - Mellet, Antoine
TI - Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 371
EP - 398
AB - We investigate the diffusion limit for general conservative Boltzmann equations with oscillating coefficients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coefficients may depend on both slow and fast variables. Passing to the limit, we are led to an effective drift-diffusion equation. We also describe the diffusive behaviour when the equilibrium function has a non-vanishing flux.
LA - eng
KW - Boltzmann equation; diffusion approximation; homogenization; drift-diffusion equation.; Boltzmann equation; homogenization; drift-diffusion equation; semiconductors
UR - http://eudml.org/doc/90701
ER -

References

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  1. G. Allaire, Homogenization and two scale convergence. SIAM J. Math. Anal.23 (1992) 1482-1518.  Zbl0770.35005
  2. G. Allaire and G. Bal, Homogenization of the criticality spectral equation in neutron transport. ESAIM: M2AN33 (1999) 721-746. Announced in Homogénéisation d'une équation spectrale du transport neutronique. CRAS, Vol. 325 (1997) 1043-1048.  
  3. G. Allaire, G. Bal and V. Siess, Homogenization and localization in locally periodic transport. ESAIM: COCV8 (2002) 1-30.  Zbl1065.35042
  4. G. Allaire and Y. Capdeboscq, Homogeneization of a spectral problem for a multigroup neutronic diffusion model. Comput. Methods Appl. Mech. Engrg.187 (2000) 91-117.  Zbl1126.82346
  5. G. Bal, Couplage d'équations et homogénéisation en transport neutronique. Thèse de doctorat de l'Université Paris 6 (1997).  
  6. G. Bal, Homogenization of a spectral equation with drift in linear transport. ESAIM: COCV6 (2001) 613-627.  Zbl0988.35022
  7. C. Bardos, F. Golse and B. Perthame, The Rosseland approximation for the radiative transfer equations. CPAM 40 (1987) 691-721; and CPAM 42 (1989) 891-894.  Zbl0654.65095
  8. C. Bardos, F. Golse, B. Perthame and R. Sentis, The nonaccretive radiative transfer equations: Existence of solutions ans Rosseland approximations. J. Funct. Anal.77 (1988) 434-460.  Zbl0655.35075
  9. A. Bensoussan, J.-L. Lions and G. Papanicolaou, Boundary layers and homogenization of transport processes. Publ. Res. Inst. Math. Sci.15 (1979) 53-157.  Zbl0408.60100
  10. H. Brézis, Analyse fonctionnelle, Théorie et applications. Masson (1993).  Zbl0511.46001
  11. Y. Capdeboscq, Homogenization of a spectral problem with drift. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 567-594; Announced in Homogenization of a diffusion equation with drift. CRAS, Vol. 327 (2000) 807-812.  Zbl1066.82530
  12. Y. Capdeboscq, Homogénéisation des modèles de diffusion en neutronique. Thèse Université Paris 6 (1999).  
  13. C. Cercignani, The Boltzmann equation and its applications. Springer-Verlag, Appl. Math. Sci. 67 (1988).  Zbl0646.76001
  14. F. Chalub, P. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits. Preprint.  Zbl1052.92005
  15. J.-F. Collet, Work in preparation. Personal communication.  
  16. R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 3. Masson (1985).  Zbl0642.35001
  17. P. Degond, T. Goudon and F. Poupaud, Diffusion limit for non homogeneous and non reversible processes. Indiana Univ. Math. J.49 (2000) 1175-1198.  Zbl0971.82035
  18. R. Di Perna, P.-L. Lions and Y. Meyer, L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire8 (1991) 271-287.  Zbl0763.35014
  19. L. Dumas and F. Golse, Homogenization of transport equations. SIAM J. Appl. Math.60 (2000) 1447-1470.  Zbl0964.35016
  20. R. Edwards, Functional analysis, Theory and applications. Dover (1994).  
  21. L.C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A111 (1989) 359-375.  Zbl0679.35001
  22. L.C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A120 (1992) 245-265.  Zbl0796.35011
  23. P. Gérard and F. Golse, Averaging regularity results for pdes under transversality assumptions. Comm. Pure Appl. Math.45 (1992) 1-26.  Zbl0832.35020
  24. F. Golse, From kinetic to macroscopic models, in Kinetic equations and asymptotic theory, edited by B. Perthame andL. Desvillettes. Gauthier-Villars, Appl. Math. 4 (2000) 41-121.  
  25. 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 (1988) 110-125.  Zbl0652.47031
  26. F. Golse and F. Poupaud, Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac. Asymptot. Anal.6 (1992) 135-160.  Zbl0784.35084
  27. T. Goudon and A. Mellet, Diffusion approximation in heterogeneous media. Asymptot. Anal.28 (2001) 331-358.  Zbl1009.35010
  28. T. Goudon and A. Mellet, On fluid limit for the semiconductors Boltzmann equation. J. Differential Equations (to appear).  Zbl1013.82024
  29. T. Goudon and F. Poupaud, Approximation by homogeneization and diffusion of kinetic equations. Comm. Partial Differential Equations26 (2001) 537-569.  Zbl0988.35023
  30. T. Goudon and F. Poupaud, Homogenization of transport equations; weak mean field approximation. Preprint.  Zbl1077.35019
  31. M. Krein and M. Rutman, Linear operator leaving invariant a cone in a Banach space. AMS Transl.10 (1962) 199-325.  Zbl0030.12902
  32. R. Kubo, H-Theorems for Markoffian Processes, in Perspectives in Statistical Physics, edited by H. Raveché. North Holland (1981).  
  33. E. Larsen, Neutron transport and diffusion in heterogeneous media (1). J. Math. Phys. (1975) 1421-1427.  
  34. E. Larsen, Neutron transport and diffusion in heterogeneous media (2). Nuclear Sci. Engrg. (1976) 357-368.  
  35. E. Larsen and J. Keller, Asymptotic solution of neutron transport processes for small free paths. J. Math. Phys.15 (1974) 75-81.  
  36. E. Larsen and M. Williams, Neutron drift in heterogeneous media. Nuclear Sci. Engrg.65 (1978) 290-302.  
  37. P.-L. Lions and G. Toscani, Diffuse limit for finite velocity Boltzmann kinetic models. Rev. Mat. Ib.13 (1997) 473-513.  Zbl0896.35109
  38. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20 (1989) 608-623.  Zbl0688.35007
  39. R. Petterson, Existence theorems for the linear, space-inhomogeneous transport equation. IMA J. Appl. Math.30 (1983) 81-105.  
  40. F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers. Asymptot. Anal.4 (1991) 293-317.  Zbl0762.35092
  41. E. Ringeisen and R. Sentis, On the diffusion approximation of a transport process without time scaling. Asymptot. Anal.5 (1991) 145-159.  Zbl0757.35001
  42. L. Tartar, Remarks on homogenization, in Homogenization and effective moduli of material and media. Springer, IMA Vol. in Math. and Appl. (1986) 228-246.  
  43. E. Wigner, Nuclear reactor theory. AMS (1961).  

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