Homogenization and diffusion asymptotics of the linear Boltzmann equation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 371-398.

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.

DOI : https://doi.org/10.1051/cocv:2003018
Classification : 35Q35,  82C70,  76P05,  74Q99,  35B27
Mots clés : Boltzmann equation, diffusion approximation, homogenization, drift-diffusion equation
@article{COCV_2003__9__371_0,
     author = {Goudon, Thierry and Mellet, Antoine},
     title = {Homogenization and diffusion asymptotics of the linear {Boltzmann} equation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {371--398},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     doi = {10.1051/cocv:2003018},
     zbl = {1070.35032},
     mrnumber = {1988668},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2003018/}
}
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
PY  - 2003
DA  - 2003///
SP  - 371
EP  - 398
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2003018/
UR  - https://zbmath.org/?q=an%3A1070.35032
UR  - https://www.ams.org/mathscinet-getitem?mr=1988668
UR  - https://doi.org/10.1051/cocv:2003018
DO  - 10.1051/cocv:2003018
LA  - en
ID  - COCV_2003__9__371_0
ER  - 
Goudon, Thierry; Mellet, Antoine. Homogenization and diffusion asymptotics of the linear Boltzmann equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 371-398. doi : 10.1051/cocv:2003018. http://www.numdam.org/articles/10.1051/cocv:2003018/

[1] G. Allaire, Homogenization and two scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | MR 1185639 | Zbl 0770.35005

[2] G. Allaire and G. Bal, 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. | Zbl 0888.45002

[3] G. Allaire, G. Bal and V. Siess, Homogenization and localization in locally periodic transport. ESAIM: COCV 8 (2002) 1-30. | Numdam | MR 1932943 | Zbl 1065.35042

[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. | MR 1765549 | Zbl 1126.82346

[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: COCV 6 (2001) 613-627. | Numdam | MR 1872390 | Zbl 0988.35022

[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. | MR 910950 | Zbl 0654.65095

[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. | MR 933978 | Zbl 0655.35075

[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. | MR 533346 | Zbl 0408.60100

[10] H. Brézis, Analyse fonctionnelle, Théorie et applications. Masson (1993). | MR 697382 | Zbl 0511.46001

[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. | MR 1912416 | Zbl 1066.82530

[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). | MR 1313028 | Zbl 0646.76001

[14] F. Chalub, P. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits. Preprint. | MR 2065025 | Zbl 1052.92005

[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). | Zbl 0642.35001

[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. | MR 1803225 | Zbl 0971.82035

[18] R. Di Perna, P.-L. Lions and Y. Meyer, L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 271-287. | EuDML 78254 | Numdam | MR 1127927 | Zbl 0763.35014

[19] L. Dumas and F. Golse, Homogenization of transport equations. SIAM J. Appl. Math. 60 (2000) 1447-1470. | MR 1760042 | Zbl 0964.35016

[20] R. Edwards, Functional analysis, Theory and applications. Dover (1994). | MR 1320261 | Zbl 0189.12103

[21] L.C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359-375. | MR 1007533 | Zbl 0679.35001

[22] L.C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992) 245-265. | MR 1159184 | Zbl 0796.35011

[23] P. Gérard and F. Golse, Averaging regularity results for pdes under transversality assumptions. Comm. Pure Appl. Math. 45 (1992) 1-26. | MR 1135922 | Zbl 0832.35020

[24] F. Golse, From kinetic to macroscopic models, in Kinetic equations and asymptotic theory, edited by B. Perthame and L. Desvillettes. Gauthier-Villars, Appl. Math. 4 (2000) 41-121. | MR 2065070 | Zbl 0979.82048

[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. | MR 923047 | Zbl 0652.47031

[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. | MR 1193108 | Zbl 0784.35084

[27] T. Goudon and A. Mellet, Diffusion approximation in heterogeneous media. Asymptot. Anal. 28 (2001) 331-358. | MR 1878799 | Zbl 1009.35010

[28] T. Goudon and A. Mellet, On fluid limit for the semiconductors Boltzmann equation. J. Differential Equations (to appear). | MR 1968313 | Zbl 1013.82024

[29] T. Goudon and F. Poupaud, Approximation by homogeneization and diffusion of kinetic equations. Comm. Partial Differential Equations 26 (2001) 537-569. | MR 1842041 | Zbl 0988.35023

[30] T. Goudon and F. Poupaud, Homogenization of transport equations; weak mean field approximation. Preprint. | MR 2111918 | Zbl 1077.35019

[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). | MR 626365

[33] E. Larsen, Neutron transport and diffusion in heterogeneous media (1). J. Math. Phys. (1975) 1421-1427. | MR 391839

[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. | MR 339741

[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. | EuDML 39532 | MR 1617393 | Zbl 0896.35109

[38] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | MR 990867 | Zbl 0688.35007

[39] R. Petterson, Existence theorems for the linear, space-inhomogeneous transport equation. IMA J. Appl. Math. 30 (1983) 81-105. | MR 711104 | Zbl 0528.76083

[40] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers. Asymptot. Anal. 4 (1991) 293-317. | MR 1127004 | Zbl 0762.35092

[41] E. Ringeisen and R. Sentis, On the diffusion approximation of a transport process without time scaling. Asymptot. Anal. 5 (1991) 145-159. | MR 1136360 | Zbl 0757.35001

[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. | MR 859418 | Zbl 0652.35012

[43] E. Wigner, Nuclear reactor theory. AMS (1961).

Cité par Sources :