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 -periodic functions modulated by a macroscopic variable, where is a small parameter. The mean free path of the particles is also of order . We assume that the leading eigenvalue of the periodicity cell problem admits a unique minimum in the domain at a point where its hessian matrix is positive definite. This assumption yields a concentration phenomenon around , as goes to , at a new scale of the order of which is superimposed with the usual 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 , i.e., of the form , where 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.
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 = {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/
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