Document ESAIM: COCV, Vol. 13, N°4, pp. 735-749
DOI: 10.1051/cocv:2007030
Homogenization of periodic non self-adjoint problems with large drift and potential
Grégoire Allaire1 and Rafael Orive21 Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau Cedex, Paris, France; gregoire.allaire@polytechnique.fr
2 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain; rafael.orive@uam.es
(Received July 11, 2005. Revised June 29, 2006. Published online July 20, 2007.)
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
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.
Mathematics Subject Classification. 35B27, 35K57, 35P15, 74Q10
Key words: Homogenization, non self-adjoint operators, convection-diffusion, periodic medium
© EDP Sciences, SMAI 2007