# Homogenization and localization in locally periodic transport

Grégoire Allaire; Guillaume Bal; Vincent Siess

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

- Volume: 8, page 1-30
- ISSN: 1292-8119

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topAllaire, Grégoire, Bal, Guillaume, and Siess, Vincent. "Homogenization and localization in locally periodic transport." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 1-30. <http://eudml.org/doc/90646>.

@article{Allaire2010,

abstract = {
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
ε-periodic functions modulated by a macroscopic variable, where
ε is a small parameter. The mean free path of the particles is
also of order ε. We assume that the leading eigenvalue of the
periodicity cell problem admits a unique minimum in the domain at a
point x0 where its Hessian matrix is positive definite. This
assumption yields a concentration phenomenon around x0, as ε
goes to 0, at a new scale of the order of $\sqrt\{\varepsilon\}$ which is
superimposed with the usual ε 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 $\sqrt\{\varepsilon\}$, i.e., of the form
$\exp \left (- \frac \{1\} \{2 \varepsilon\} M (x-x_0)\cdot (x-x_0) \right )$,
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.
},

author = {Allaire, Grégoire, Bal, Guillaume, Siess, Vincent},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Homogenization; localization; transport.; transport; spectral transport equation; concentration phenomenon},

language = {eng},

month = {3},

pages = {1-30},

publisher = {EDP Sciences},

title = {Homogenization and localization in locally periodic transport},

url = {http://eudml.org/doc/90646},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Allaire, Grégoire

AU - Bal, Guillaume

AU - Siess, Vincent

TI - Homogenization and localization in locally periodic transport

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 8

SP - 1

EP - 30

AB -
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
ε-periodic functions modulated by a macroscopic variable, where
ε is a small parameter. The mean free path of the particles is
also of order ε. We assume that the leading eigenvalue of the
periodicity cell problem admits a unique minimum in the domain at a
point x0 where its Hessian matrix is positive definite. This
assumption yields a concentration phenomenon around x0, as ε
goes to 0, at a new scale of the order of $\sqrt{\varepsilon}$ which is
superimposed with the usual ε 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 $\sqrt{\varepsilon}$, i.e., of the form
$\exp \left (- \frac {1} {2 \varepsilon} M (x-x_0)\cdot (x-x_0) \right )$,
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.

LA - eng

KW - Homogenization; localization; transport.; transport; spectral transport equation; concentration phenomenon

UR - http://eudml.org/doc/90646

ER -

## References

top- G. Allaire, Homogenization and two scale convergence. SIAM23 (1992) 1482-1518. Zbl0770.35005
- G. Allaire and G. Bal, Homogenization of the critically spectral equation in neutron transport. ESAIM: M2AN33 (1999) 721-746. Zbl0931.35010
- G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion. Comput. Methods Appl. Mech. Engrg.187 (2000) 91-117. Zbl1126.82346
- G. Allaire and A. Piatnitski, Uniform spectral asymptotics for singularly perturbed locally periodic operators. Com. Partial Differential Equations27 (2002) 705-725. Zbl1026.35012
- P. Anselone, Collectively compact operator approximation theory. Prentice-Hall, Englewood Cliffs, NJ (1971). Zbl0228.47001
- G. Bal, Couplage d'équations et homogénéisation en transport neutronique, Ph.D. Thesis. Paris 6 (1997).
- height 2pt depth -1.6pt width 23pt, Homogenization of a spectral equation with drift in linear transport. ESAIM: COCV6 (2001) 613-627.
- A. Bensoussan, J.-L. Lions and G. Papanicolaou, Boundary layer and homogenization of transport processes. Publ. RIMS Kyoto Univ. (1979) 53-157. Zbl0408.60100
- Y. Capdeboscq, Homogénéisation des modèles de diffusion en neutronique, Ph.D. Thesis. Paris 6 (1999).
- F. Chatelin, Spectral approximation of linear operators. Academic Press (1983). Zbl0517.65036
- R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Springer Verlag, Berlin (1993).
- P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes. Indiana Univ. Math. J.49 (2000) 1175-1198. Zbl0971.82035
- J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View. Springer-Verlag, New York, Berlin (1981). Zbl0461.46051
- 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.
- 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. Zbl0591.45007
- T. Goudon and A. Mellet, Diffusion approximation in heterogeneous media. Asymptot. Anal. (to appear). Zbl1009.35010
- T. Goudon and F. Poupaud, Approximation by homogenization and diffusion of kinetic equations. Comm. Partial Differential Equations26 (2001) 537-569. Zbl0988.35023
- S. Kozlov, Reductibility of quasiperiodic differential operators and averaging. Transc. Moscow Math. Soc.2 (1984) 101-126. Zbl0566.35036
- E. Larsen, Neutron transport and diffusion in inhomogeneous media (1). J. Math. Phys. (1975) 1421-1427.
- height 2pt depth -1.6pt width 23pt, Neutron transport and diffusion in inhomogeneous media (2). Nuclear Sci. Engrg. (1976) 357-368.
- E. Larsen and J. Keller, Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. (1974) 75-81.
- M. Mokhtar-Kharoubi, Les équations de la neutronique, Thèse de Doctorat d'État. Paris XIII (1987).
- M. Mokhtar-Kharoubi, Mathematical topics in neutron transport theory. World Scientific Publishing Co. Inc., River Edge, NJ (1997).
- A. Piatnitski, Asymptotic behaviour of the ground state of singularly perturbed elliptic equations. Commun. Math. Phys.197 (1998) 527-551. Zbl0937.58023
- J.E. Potter, Matrix quadratic solutions, J. SIAM Appl. Math.14 (1966) 496-501. Zbl0144.02001
- D.L. Russel, Mathematics of finite-dimensional control systems, theory and design. Lecture Notes in Pure Appl. Math.43 (1979).
- R. Sentis, Study of the corrector of the eigenvalue of a transport operator. SIAM J. Math. Anal. (1985) 151-166. Zbl0609.45002

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