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

top
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

top

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

top
  1. G. Allaire, Homogenization and two scale convergence. SIAM J. Math. Anal.23 (1992) 1482-1518.  
  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.  
  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.  
  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.  
  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.  
  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.  
  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.  
  10. H. Brézis, Analyse fonctionnelle, Théorie et applications. Masson (1993).  
  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.  
  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).  
  14. F. Chalub, P. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits. Preprint.  
  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).  
  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.  
  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.  
  19. L. Dumas and F. Golse, Homogenization of transport equations. SIAM J. Appl. Math.60 (2000) 1447-1470.  
  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.  
  22. L.C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A120 (1992) 245-265.  
  23. P. Gérard and F. Golse, Averaging regularity results for pdes under transversality assumptions. Comm. Pure Appl. Math.45 (1992) 1-26.  
  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.  
  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.  
  27. T. Goudon and A. Mellet, Diffusion approximation in heterogeneous media. Asymptot. Anal.28 (2001) 331-358.  
  28. T. Goudon and A. Mellet, On fluid limit for the semiconductors Boltzmann equation. J. Differential Equations (to appear).  
  29. T. Goudon and F. Poupaud, Approximation by homogeneization and diffusion of kinetic equations. Comm. Partial Differential Equations26 (2001) 537-569.  
  30. T. Goudon and F. Poupaud, Homogenization of transport equations; weak mean field approximation. Preprint.  
  31. M. Krein and M. Rutman, Linear operator leaving invariant a cone in a Banach space. AMS Transl.10 (1962) 199-325.  
  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.  
  38. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20 (1989) 608-623.  
  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.  
  41. E. Ringeisen and R. Sentis, On the diffusion approximation of a transport process without time scaling. Asymptot. Anal.5 (1991) 145-159.  
  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).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.