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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  1. G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal.23 (1992) 1482–1518.  Zbl0770.35005
  2. G. Allaire, Dispersive limits in the homogenization of the wave equation. Annales de la Faculté des Sciences de ToulouseXII (2003) 415–431.  Zbl1070.35006
  3. G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion. Comput. Methods Appl. Mech. Engrg.187 (2000) 91–117.  Zbl1126.82346
  4. G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl.77 (1998) 153–208.  Zbl0901.35005
  5. G. Allaire and F. Malige, Analyse asymptotique spectrale d'un probléme de diffusion neutronique. C. R. Acad. Sci. Paris Sér. I324 (1997) 939–944.  Zbl0879.35153
  6. G. Allaire and R. Orive, On the band gap structure of Hill's equation. J. Math. Anal. Appl.306 (2005) 462–480.  Zbl1095.34014
  7. G. Allaire and A. Piatnitski, Uniform spectral asymptotics for singularly perturbed locally periodic operator. Comm. Partial Differential Equations27 (2002) 705–725.  Zbl1026.35012
  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.  Zbl1072.35023
  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).  Zbl0228.47001
  10. A. Benchérif-Madani and É. Pardoux, Locally periodic homogenization. Asymptot. Anal.39 (2004) 263–279.  Zbl1064.35017
  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.  Zbl1067.35009
  12. A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).  Zbl0404.35001
  13. Y. Capdeboscq, Homogenization of a diffusion equation with drift. C. R. Acad. Sci. Paris Sér. I327 (1998) 807–812.  Zbl0918.35135
  14. Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift. Proc. Roy. Soc. Edinburgh Sect. A132 (2002) 567–594.  Zbl1066.82530
  15. P. Donato and A. Piatnitski, Averaging of nonstationary parabolic operators with large lower order terms. (2005) (in preparation).  Zbl1201.35034
  16. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20 (1989) 608–623.  Zbl0688.35007
  17. A. Piatnitski, Asymptotic behaviour of the ground state of singularly perturbed elliptic equations. Commun. Math. Phys.197 (1998) 527–551.  Zbl0937.58023
  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 ( 0 , T ; B ) . Ann. Mat. Pura Appl.146 (1987) 65–96.  Zbl0629.46031
  20. S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators. Asymptotic Anal.39 (2004) 15–44.  Zbl1072.35030
  21. M. Vanninathan, Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci.90 (1981) 239–271.  Zbl0486.35063

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