Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-29T02:26:15.087Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation

Published online by Cambridge University Press:  15 September 2003

Thierry Goudon  and
Antoine Mellet
Thierry Goudon
Affiliation:
CNRS, Université des Sciences et Technologies Lille 1, UFR Mathématiques Pures et Appliquées, Cité Scientifique, 59655 Villeneuve-d'Ascq Cedex, France; [email protected]. INRIA-Sophia, Project Caiman.
Antoine Mellet
Affiliation:
Mathématiques pour l'Industrie et la Physique, UMR 5640, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France; [email protected].
Get access

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.

Keywords

Boltzmann equationdiffusion approximationhomogenizationdrift-diffusion equation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 371 - 398
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allaire, G., Homogenization and two scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. CrossRef
Allaire, G. and Bal, G., Homogenization of the criticality spectral equation in neutron transport. ESAIM: M2AN 33 (1999) 721-746. Announced in Homogénéisation d'une équation spectrale du transport neutronique. CRAS, Vol. 325 (1997) 1043-1048. CrossRef
Allaire, G., Bal, G. and Siess, V., Homogenization and localization in locally periodic transport. ESAIM: COCV 8 (2002) 1-30. CrossRef
Allaire, G. and Capdeboscq, Y., Homogeneization of a spectral problem for a multigroup neutronic diffusion model. Comput. Methods Appl. Mech. Engrg. 187 (2000) 91-117. CrossRef
G. Bal, Couplage d'équations et homogénéisation en transport neutronique. Thèse de doctorat de l'Université Paris 6 (1997).
Bal, G., Homogenization of a spectral equation with drift in linear transport. ESAIM: COCV 6 (2001) 613-627. CrossRef
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.
Bardos, C., Golse, F., Perthame, B. and Sentis, R., The nonaccretive radiative transfer equations: Existence of solutions ans Rosseland approximations. J. Funct. Anal. 77 (1988) 434-460. CrossRef
Bensoussan, A., Lions, J.-L. and Papanicolaou, G., Boundary layers and homogenization of transport processes. Publ. Res. Inst. Math. Sci. 15 (1979) 53-157. CrossRef
H. Brézis, Analyse fonctionnelle, Théorie et applications. Masson (1993).
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.
Y. Capdeboscq, Homogénéisation des modèles de diffusion en neutronique. Thèse Université Paris 6 (1999).
C. Cercignani, The Boltzmann equation and its applications. Springer-Verlag, Appl. Math. Sci. 67 (1988).
F. Chalub, P. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits. Preprint.
J.-F. Collet, Work in preparation. Personal communication.
R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 3. Masson (1985).
Degond, P., Goudon, T. and Poupaud, F., Diffusion limit for non homogeneous and non reversible processes. Indiana Univ. Math. J. 49 (2000) 1175-1198.
Di Perna, R., Lions, P.-L. and Meyer, Y., $L\sp p$ regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 271-287. CrossRef
Dumas, L. and Golse, F., Homogenization of transport equations. SIAM J. Appl. Math. 60 (2000) 1447-1470. CrossRef
R. Edwards, Functional analysis, Theory and applications. Dover (1994).
Evans, L.C., The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359-375. CrossRef
Evans, L.C., Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992) 245-265. CrossRef
Gérard, P. and Golse, F., Averaging regularity results for pdes under transversality assumptions. Comm. Pure Appl. Math. 45 (1992) 1-26. CrossRef
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.
Golse, F., Lions, P.-L., Perthame, B. and Sentis, R., Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76 (1988) 110-125. CrossRef
Golse, F. and Poupaud, F., Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac. Asymptot. Anal. 6 (1992) 135-160.
Goudon, T. and Mellet, A., Diffusion approximation in heterogeneous media. Asymptot. Anal. 28 (2001) 331-358.
T. Goudon and A. Mellet, On fluid limit for the semiconductors Boltzmann equation. J. Differential Equations (to appear).
Goudon, T. and Poupaud, F., Approximation by homogeneization and diffusion of kinetic equations. Comm. Partial Differential Equations 26 (2001) 537-569. CrossRef
T. Goudon and F. Poupaud, Homogenization of transport equations; weak mean field approximation. Preprint.
Krein, M. and Rutman, M., Linear operator leaving invariant a cone in a Banach space. AMS Transl. 10 (1962) 199-325.
R. Kubo, H-Theorems for Markoffian Processes, in Perspectives in Statistical Physics, edited by H. Raveché. North Holland (1981).
E. Larsen, Neutron transport and diffusion in heterogeneous media (1). J. Math. Phys. (1975) 1421-1427.
E. Larsen, Neutron transport and diffusion in heterogeneous media (2). Nuclear Sci. Engrg. (1976) 357-368.
Larsen, E. and Keller, J., Asymptotic solution of neutron transport processes for small free paths. J. Math. Phys. 15 (1974) 75-81. CrossRef
Larsen, E. and Williams, M., Neutron drift in heterogeneous media. Nuclear Sci. Engrg. 65 (1978) 290-302. CrossRef
Lions, P.-L. and Toscani, G., Diffuse limit for finite velocity Boltzmann kinetic models. Rev. Mat. Ib. 13 (1997) 473-513. CrossRef
Nguetseng, G., A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. CrossRef
Petterson, R., Existence theorems for the linear, space-inhomogeneous transport equation. IMA J. Appl. Math. 30 (1983) 81-105. CrossRef
Poupaud, F., Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers. Asymptot. Anal. 4 (1991) 293-317.
Ringeisen, E. and Sentis, R., On the diffusion approximation of a transport process without time scaling. Asymptot. Anal. 5 (1991) 145-159.
L. Tartar, Remarks on homogenization, in Homogenization and effective moduli of material and media. Springer, IMA Vol. in Math. and Appl. (1986) 228-246.
E. Wigner, Nuclear reactor theory. AMS (1961).