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We consider the homogenization of both the parabolic and eigenvalue problems for a singularly perturbed convection-diffusion equation in a periodic medium. All coefficients of the equation may vary both on the macroscopic scale and on the periodic microscopic scale. Denoting by the period, the potential or zero-order term is scaled as and the drift or first-order term is scaled as . Under a structural hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the domain with non-degenerate quadratic behavior, we prove an exponential localization at this minimum point. The homogenized problem features a diffusion equation with quadratic potential in the whole space.
Keywords: homogenization, non self-adjoint operators, convection-diffusion, periodic medium
@article{COCV_2007__13_4_735_0,
author = {Allaire, Gr\'egoire and Orive, Rafael},
title = {Homogenization of periodic non self-adjoint problems with large drift and potential},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {735--749},
publisher = {EDP-Sciences},
volume = {13},
number = {4},
year = {2007},
doi = {10.1051/cocv:2007030},
mrnumber = {2351401},
zbl = {1130.35307},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007030/}
}
TY - JOUR AU - Allaire, Grégoire AU - Orive, Rafael TI - Homogenization of periodic non self-adjoint problems with large drift and potential JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 SP - 735 EP - 749 VL - 13 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007030/ DO - 10.1051/cocv:2007030 LA - en ID - COCV_2007__13_4_735_0 ER -
%0 Journal Article %A Allaire, Grégoire %A Orive, Rafael %T Homogenization of periodic non self-adjoint problems with large drift and potential %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 735-749 %V 13 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007030/ %R 10.1051/cocv:2007030 %G en %F COCV_2007__13_4_735_0
Allaire, Grégoire; Orive, Rafael. Homogenization of periodic non self-adjoint problems with large drift and potential. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 735-749. doi: 10.1051/cocv:2007030
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