# 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

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topAllaire, 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

top- G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal.23 (1992) 1482–1518.
- G. Allaire, Dispersive limits in the homogenization of the wave equation. Annales de la Faculté des Sciences de ToulouseXII (2003) 415–431.
- G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion. Comput. Methods Appl. Mech. Engrg.187 (2000) 91–117.
- G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl.77 (1998) 153–208.
- G. Allaire and F. Malige, Analyse asymptotique spectrale d'un probléme de diffusion neutronique. C. R. Acad. Sci. Paris Sér. I324 (1997) 939–944.
- G. Allaire and R. Orive, On the band gap structure of Hill's equation. J. Math. Anal. Appl.306 (2005) 462–480.
- G. Allaire and A. Piatnitski, Uniform spectral asymptotics for singularly perturbed locally periodic operator. Comm. Partial Differential Equations27 (2002) 705–725.
- G. Allaire, Y. Capdeboscq, A. Piatnitski, V. Siess and M. Vanninathan, Homogenization of periodic systems with large potentials. Arch. Rational Mech. Anal.174 (2004) 179–220.
- P.H. Anselone, Collectively compact operator approximation theory and applications to integral equations. Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1971).
- A. Benchérif-Madani and É. Pardoux, Locally periodic homogenization. Asymptot. Anal.39 (2004) 263–279.
- A. Benchérif-Madani and É. Pardoux, Homogenization of a diffusion with locally periodic coefficients. Séminaire de Probabilités XXXVIII Lect. Notes Math.1857 (2005) 363–392.
- A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).
- Y. Capdeboscq, Homogenization of a diffusion equation with drift. C. R. Acad. Sci. Paris Sér. I327 (1998) 807–812.
- Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift. Proc. Roy. Soc. Edinburgh Sect. A132 (2002) 567–594.
- P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms. (2005) (in preparation).
- G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20 (1989) 608–623.
- A. Piatnitski, Asymptotic behaviour of the ground state of singularly perturbed elliptic equations. Commun. Math. Phys.197 (1998) 527–551.
- A. Piatnitski, Ground State Asymptotics for Singularly Perturbed Elliptic Problem with Locally Periodic Microstructure. Preprint (2006).
- J. Simon, Compact sets in the space ${L}^{p}(0,T;B)$. Ann. Mat. Pura Appl.146 (1987) 65–96.
- S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators. Asymptotic Anal.39 (2004) 15–44.
- M. Vanninathan, Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci.90 (1981) 239–271.

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