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.

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

DOI : https://doi.org/10.1051/cocv:2007030
Classification : 35B27,  35K57,  35P15,  74Q10
Mots clés : 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},
zbl = {1130.35307},
mrnumber = {2351401},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2007030/}
}
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. http://www.numdam.org/articles/10.1051/cocv:2007030/

[1] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | Zbl 0770.35005

[2] G. Allaire, Dispersive limits in the homogenization of the wave equation. Annales de la Faculté des Sciences de Toulouse XII (2003) 415-431. | Numdam | Zbl 1070.35006

[3] G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion. Comput. Methods Appl. Mech. Engrg. 187 (2000) 91-117. | Zbl 1126.82346

[4] G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153-208. | Zbl 0901.35005

[5] G. Allaire and F. Malige, Analyse asymptotique spectrale d'un probléme de diffusion neutronique. C. R. Acad. Sci. Paris Sér. I 324 (1997) 939-944. | Zbl 0879.35153

[6] G. Allaire and R. Orive, On the band gap structure of Hill's equation. J. Math. Anal. Appl. 306 (2005) 462-480. | Zbl 1095.34014

[7] G. Allaire and A. Piatnitski, Uniform spectral asymptotics for singularly perturbed locally periodic operator. Comm. Partial Differential Equations 27 (2002) 705-725. | Zbl 1026.35012

[8] 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. | Zbl 1072.35023

[9] 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). | MR 443383 | Zbl 0228.47001

[10] A. Benchérif-Madani and É. Pardoux, Locally periodic homogenization. Asymptot. Anal. 39 (2004) 263-279. | Zbl 1064.35017

[11] 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. | Zbl 1067.35009

[12] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978). | MR 503330 | Zbl 0404.35001

[13] Y. Capdeboscq, Homogenization of a diffusion equation with drift. C. R. Acad. Sci. Paris Sér. I 327 (1998) 807-812. | Zbl 0918.35135

[14] Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 567-594. | Zbl 1066.82530

[15] P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms. (2005) (in preparation). | MR 2233176 | Zbl 1201.35034

[16] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | Zbl 0688.35007

[17] A. Piatnitski, Asymptotic behaviour of the ground state of singularly perturbed elliptic equations. Commun. Math. Phys. 197 (1998) 527-551. | Zbl 0937.58023

[18] A. Piatnitski, Ground State Asymptotics for Singularly Perturbed Elliptic Problem with Locally Periodic Microstructure. Preprint (2006).

[19] J. Simon, Compact sets in the space ${L}^{p}\left(0,T;B\right)$. Ann. Mat. Pura Appl. 146 (1987) 65-96. | Zbl 0629.46031

[20] S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators. Asymptotic Anal. 39 (2004) 15-44. | Zbl 1072.35030

[21] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci. 90 (1981) 239-271. | Zbl 0486.35063