Homogenization and localization in locally periodic transport
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 1-30.

In this paper, we study the homogenization and localization of a spectral transport equation posed in a locally periodic heterogeneous domain. This equation models the equilibrium of particles interacting with an underlying medium in the presence of a creation mechanism such as, for instance, neutrons in nuclear reactors. The physical coefficients of the domain are $\epsilon$-periodic functions modulated by a macroscopic variable, where $\epsilon$ is a small parameter. The mean free path of the particles is also of order $\epsilon$. We assume that the leading eigenvalue of the periodicity cell problem admits a unique minimum in the domain at a point ${x}_{0}$ where its hessian matrix is positive definite. This assumption yields a concentration phenomenon around ${x}_{0}$, as $\epsilon$ goes to $0$, at a new scale of the order of $\sqrt{\epsilon }$ which is superimposed with the usual $\epsilon$ oscillations of the homogenized limit. More precisely, we prove that the particle density is asymptotically the product of two terms. The first one is the leading eigenvector of a cell transport equation with periodic boundary conditions. The second term is the first eigenvector of a homogenized diffusion equation in the whole space with quadratic potential, rescaled by a factor $\sqrt{\epsilon }$, i.e., of the form $exp\left(-\frac{1}{2\epsilon }M\left(x-{x}_{0}\right)·\left(x-{x}_{0}\right)\right)$, where $M$ is a positive definite matrix. Furthermore, the eigenvalue corresponding to this second term gives a first-order correction to the eigenvalue of the heterogeneous spectral transport problem.

DOI : https://doi.org/10.1051/cocv:2002016
Classification : 35B27,  82D75
Mots clés : homogenization, localization, transport
@article{COCV_2002__8__1_0,
author = {Allaire, Gr\'egoire and Bal, Guillaume and Siess, Vincent},
title = {Homogenization and localization in locally periodic transport},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1--30},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
doi = {10.1051/cocv:2002016},
zbl = {1065.35042},
mrnumber = {1932943},
language = {en},
url = {http://www.numdam.org/item/COCV_2002__8__1_0/}
}
Allaire, Grégoire; Bal, Guillaume; Siess, Vincent. Homogenization and localization in locally periodic transport. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 1-30. doi : 10.1051/cocv:2002016. http://www.numdam.org/item/COCV_2002__8__1_0/

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

[2] G. Allaire and G. Bal, Homogenization of the critically spectral equation in neutron transport. ESAIM: M2AN 33 (1999) 721-746. | Numdam | MR 1726482 | Zbl 0931.35010

[3] G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion. Comput. Methods Appl. Mech. Engrg. 187 (2000) 91-117. | MR 1765549 | Zbl 1126.82346

[4] G. Allaire and A. Piatnitski, Uniform spectral asymptotics for singularly perturbed locally periodic operators. Com. Partial Differential Equations 27 (2002) 705-725. | MR 1900560 | Zbl 1026.35012

[5] P. Anselone, Collectively compact operator approximation theory. Prentice-Hall, Englewood Cliffs, NJ (1971). | MR 443383 | Zbl 0228.47001

[6] G. Bal, Couplage d'équations et homogénéisation en transport neutronique, Ph.D. Thesis. Paris 6 (1997).

[7] , Homogenization of a spectral equation with drift in linear transport. ESAIM: COCV 6 (2001) 613-627. | Numdam | MR 1872390 | Zbl 0988.35022

[8] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Boundary layer and homogenization of transport processes. Publ. RIMS Kyoto Univ. (1979) 53-157. | MR 533346 | Zbl 0408.60100

[9] Y. Capdeboscq, Homogénéisation des modèles de diffusion en neutronique, Ph.D. Thesis. Paris 6 (1999).

[10] F. Chatelin, Spectral approximation of linear operators. Academic Press (1983). | MR 716134 | Zbl 0517.65036

[11] R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Springer Verlag, Berlin (1993). | MR 1295030 | Zbl 0683.35001

[12] P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes. Indiana Univ. Math. J. 49 (2000) 1175-1198. | MR 1803225 | Zbl 0971.82035

[13] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View. Springer-Verlag, New York, Berlin (1981). | MR 628000 | Zbl 0461.46051

[14] 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

[15] F. Golse, B. Perthame and R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport. C. R. Acad. Sci. Paris (1985) 341-344. | Zbl 0591.45007

[16] T. Goudon and A. Mellet, Diffusion approximation in heterogeneous media. Asymptot. Anal. (to appear). | MR 1878799 | Zbl 1009.35010

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

[18] S. Kozlov, Reductibility of quasiperiodic differential operators and averaging. Transc. Moscow Math. Soc. 2 (1984) 101-126. | MR 737902 | Zbl 0566.35036

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

[20] , Neutron transport and diffusion in inhomogeneous media (2). Nuclear Sci. Engrg. (1976) 357-368.

[21] E. Larsen and J. Keller, Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. (1974) 75-81. | MR 339741

[22] M. Mokhtar-Kharoubi, Les équations de la neutronique, Thèse de Doctorat d'État. Paris XIII (1987).

[23] M. Mokhtar-Kharoubi, Mathematical topics in neutron transport theory. World Scientific Publishing Co. Inc., River Edge, NJ (1997). | Zbl 0997.82047

[24] A. Piatnitski, Asymptotic behaviour of the ground state of singularly perturbed elliptic equations. Commun. Math. Phys. 197 (1998) 527-551. | MR 1652771 | Zbl 0937.58023

[25] J.E. Potter, Matrix quadratic solutions, J. SIAM Appl. Math. 14 (1966) 496-501. | MR 201457 | Zbl 0144.02001

[26] D.L. Russel, Mathematics of finite-dimensional control systems, theory and design. Lecture Notes in Pure Appl. Math. 43 (1979). | Zbl 0408.93002

[27] R. Sentis, Study of the corrector of the eigenvalue of a transport operator. SIAM J. Math. Anal. (1985) 151-166. | MR 772875 | Zbl 0609.45002