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Homogenization of periodic non self-adjoint problemswith large drift and potential

Published online by Cambridge University Press:  20 July 2007

Grégoire Allaire  and
Rafael Orive
Grégoire Allaire
Affiliation:
Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau Cedex, Paris, France; gregoire.allaire@polytechnique.fr
Rafael Orive
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain; rafael.orive@uam.es
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Abstract

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 ε the period, the potential or zero-order term is scaled as $\varepsilon^{-2}$ and the drift or first-order term is scaled as $\varepsilon^{-1}$. 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

Homogenizationnon self-adjoint operatorsconvection-diffusionperiodic medium
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 4 , October 2007 , pp. 735 - 749
Copyright
© EDP Sciences, SMAI, 2007

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