Homogenization of periodic non self-adjoint problems with large drift and potential
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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 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.

DOI: 10.1051/cocv:2007030
Classification: 35B27,  35K57,  35P15,  74Q10
Keywords: 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/}
}
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
PY  - 2007
DA  - 2007///
SP  - 735
EP  - 749
VL  - 13
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2007030/
UR  - https://zbmath.org/?q=an%3A1130.35307
UR  - https://www.ams.org/mathscinet-getitem?mr=2351401
UR  - https://doi.org/10.1051/cocv:2007030
DO  - 10.1051/cocv:2007030
LA  - en
ID  - COCV_2007__13_4_735_0
ER  - 
%0 Journal Article
%A Allaire, Grégoire
%A Orive, Rafael
%T Homogenization of periodic non self-adjoint problems with large drift and potential
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2007
%P 735-749
%V 13
%N 4
%I EDP-Sciences
%U https://doi.org/10.1051/cocv:2007030
%R 10.1051/cocv:2007030
%G en
%F COCV_2007__13_4_735_0
Allaire, Grégoire; Orive, Rafael. Homogenization of periodic non self-adjoint problems with large drift and potential. ESAIM: Control, Optimisation and Calculus of Variations, Volume 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

[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

[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

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

[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

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

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

[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

[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 | Zbl

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

[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

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

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

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

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

[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

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

[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 (0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | Zbl

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

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

Cited by Sources: