SeMA Journal

, Volume 58, Issue 1, pp 53–95 | Cite as

Homogenization of a nonstationary convection-diffusion equation in a thin rod and in a layer

  • G. Allaire
  • I. Pankratova
  • A. Piatnitski


The paper deals with the homogenization of a non-stationary convection-diffusion equation defined in a thin rod or in a layer with Dirichlet boundary condition. Under the assumption that the convection term is large, we describe the evolution of the solution’s profile and determine the rate of its decay. The main feature of our analysis is that we make no assumption on the support of the initial data which may touch the domain’s boundary. This requires the construction of boundary layer correctors in the homogenization process which, surprisingly, play a crucial role in the definition of the leading order term at the limit. Therefore we have to restrict our attention to simple geometries like a rod or a layer for which the definition of boundary layers is easy and explicit.

Key words

Homogenization convection-diffusion localization thin cylinder layer 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    (1445458) I. A. Aleksandrova, The spectral method in asymptotic diffusion problems with drift. Math. Notes 59 (1996), no. 5–6, 554–556.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    (1450451) G. Allaire, F. Malige, Analyse asymptotique spectrale d’un problème de diffusion neutronique, C.R. Acad. Sci. Paris, Série I, t.324, 939–944 (1997).MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    (2324490) G. Allaire, A.-L. Raphael, Homogenization of a convection-diffusion model with reaction in a porous medium. C. R. Math. Acad. Sci. Paris 344 (2007), no. 8, 523–528.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    (2852263) G. Allaire, I. Pankratova, A. Piatnitski, Homogenization and concentration for a diffusion equation with large convection in a bounded domain, Journal of Functional Analysis, 262, no. l, pp. 300–330 (2012).MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    (0435594) D.G. Aronson, Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607–694.MathSciNetMATHGoogle Scholar
  6. [6]
    (0985952) M. Avellaneda, F.-H. Lin, Homogenization of Poisson’s kernel and applications to boundary control. J. Math. Pures Appl. (9) 68 (1989), no. 1, 1–29.MathSciNetGoogle Scholar
  7. [7]
    (0797571) N.S. Bakhvalov, G.P. Panasenko, Homogenization: Averaging processes in periodic media, Nauka, Moscow, 1984(Russian); English transl.,Kluwer, Dordrecht/Boston/London, 1989.Google Scholar
  8. [8]
    (0503330) A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structure, Studies in Mathematics and its Applications, 5. North-Holland Publishing Co., Amsterdam-New York, 1978.Google Scholar
  9. [9]
    (1663726) Y. Capdeboscq, Homogenization of a diffusion equation with drift. C. R. Acad. Sci. Paris Ser. I Math. 327 (1998), no. 9, 807–812.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    (1796243) Y. Capdeboscq, Homogenization of a neutronic multigroup evolution model. Asymptot. Anal. 24 (2000), no. 2, 143–165.MathSciNetMATHGoogle Scholar
  11. [11]
    (1912416) Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 3, 567–594.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    (2233176) Donato, P., Piatnitski, A., Averaging of nonstationary parabolic operators with large lower order terms. Multi Scale Problems and Asymptotic Analysis, GAKUTO Internat. Ser. Math. Sci. Appl., 24, 153–165 (2005).MathSciNetGoogle Scholar
  13. [13]
    (2825170) D. Gérard-Varet, N. Masmoudi, Homogenization in polygonal domains, J. Eur. Math. Soc.. 13, 1477–1503 (2011).MATHCrossRefGoogle Scholar
  14. [14]
    D. Gérard-Varet, N. Masmoudi, Homogenization and boundary layer, to appear in Acta Mathematica.Google Scholar
  15. [15]
    (1814364) D. Gilbarg and N.S. Trudinger, “Elliptic Partial Differential Equations of Second Order,” Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.Google Scholar
  16. [16]
    (0737902) Kozlov, S. M. Reducibility of quasiperiodic differential operators and averaging. (Russian) Trudy Moskov. Mat. Obshch. 46 (1983), 99–123.MathSciNetGoogle Scholar
  17. [17]
    (0241822) Ladyzenskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N.. Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1967.Google Scholar
  18. [18]
    (2156660) E. Marušić-Paloka, A. Piatnitski, Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection, Journal of London Math. Soc., Vol. 72 (2005), No. 2, p. 391–409.MATHCrossRefGoogle Scholar
  19. [19]
    (2133084) G. Panasenko, Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht, 2005.Google Scholar
  20. [20]
    (2505654) I. Pankratova, A. Piatnitski, On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder, DCDS-B, 11(4) (2009).Google Scholar
  21. [21]
    (1366209) G.C. Papanicolaou, Diffusion in random media, Surveys in applied mathematics, Vol. 1, 205–253, Plenum, New York, 1995.Google Scholar
  22. [22]
    (0699735) A. Piatnitski, Averaging of a singularly perturbed equation with rapidly oscillating coefficients in a layer Math. USSR-Sb. 49 (1984), no. 1, 19–40.CrossRefGoogle Scholar
  23. [23]
    (0635561) M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90:239–271 (1981).MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    (1329546) V.V. Zhikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994.MATHGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2012

Authors and Affiliations

  1. 1.Ecole PolytechniquePalaiseau CedexFrance
  2. 2.Narvik University CollegeNarvikNorway
  3. 3.Ecole PolytechniquePalaiseau CedexFrance
  4. 4.Lebedev Physical Institute RASMoscowRussia

Personalised recommendations