Homogenization of periodic non self-adjoint problems with large drift and potential

Grégoire Allaire; Rafael Orive

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 4, page 735-749
  • ISSN: 1292-8119

Abstract

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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 ε the period, the potential or zero-order term is scaled as ε - 2 and the drift or first-order term is scaled as ε - 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.

How to cite

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Allaire, Grégoire, and Orive, Rafael. "Homogenization of periodic non self-adjoint problems with large drift and potential." ESAIM: Control, Optimisation and Calculus of Variations 13.4 (2007): 735-749. <http://eudml.org/doc/250000>.

@article{Allaire2007,
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. },
author = {Allaire, Grégoire, Orive, Rafael},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Homogenization; non self-adjoint operators; convection-diffusion; periodic medium},
language = {eng},
month = {7},
number = {4},
pages = {735-749},
publisher = {EDP Sciences},
title = {Homogenization of periodic non self-adjoint problems with large drift and potential},
url = {http://eudml.org/doc/250000},
volume = {13},
year = {2007},
}

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
DA - 2007/7//
PB - EDP Sciences
VL - 13
IS - 4
SP - 735
EP - 749
AB - 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.
LA - eng
KW - Homogenization; non self-adjoint operators; convection-diffusion; periodic medium
UR - http://eudml.org/doc/250000
ER -

References

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