Advertisement

Applications of Mathematics

, Volume 42, Issue 5, pp 321–343 | Cite as

Homogenization of parabolic equations an alternative approach and some corrector-type results

  • Anders Holmbom
Article

Abstract

We extend and complete some quite recent results by Nguetseng [Ngu1] and Allaire [All3] concerning two-scale convergence. In particular, a compactness result for a certain class of parameterdependent functions is proved and applied to perform an alternative homogenization procedure for linear parabolic equations with coefficients oscillating in both their space and time variables. For different speeds of oscillation in the time variable, this results in three cases. Further, we prove some corrector-type results and benefit from some interpolation properties of Sobolev spaces to identify regularity assumptions strong enough for such results to hold.

partial differential equations homogenization two-scale convergence linear parabolic equations oscillating coefficients in space and time variable dissimilar speeds of oscillation admissible test functions corrector results compactness result interpolation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ada]
    R. A. Adams: Sobolev Spaces. Academic Press, New York. 1975.Google Scholar
  2. [All1]
    G. Allaire: Two-scale convergence and homogenization of periodic structures. School on homogenization, ICTP, Trieste, September 6–17. 1993.Google Scholar
  3. [All2]
    G. Allaire: Homogenization of the unsteady Stokes equation in porous media. Progress in pdes: calculus of variation, applications, Pitman Research notes in mathematics Series 267 (C. Bandle et al., eds.). Longman Higher Education, New York, 1992.Google Scholar
  4. [All3]
    G. Allaire: Homogenization and two-scale convergence. SIAM Journal of Mathematical Analysis, 23 (1992), no. 6, 1482–1518.Google Scholar
  5. [Alt]
    H. W. Alt: Lineare Funktionalanalysis. Springer-Verlag, 1985.Google Scholar
  6. [Att]
    H. Attouch: Variational Convergence of Functions and Operators. Pitman Publishing Limited, 1984.Google Scholar
  7. [Bens]
    A. Bensoussan, J. L. Lions, G. Papanicolau: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, North-Holland, 1978.Google Scholar
  8. [BeLö]
    J. Bergh, J. Löfström: Interpolation Spaces. An Introduction. Grundlehren der mathematischen Wissenschaft, Springer-Verlag, 1976.Google Scholar
  9. [BraOts]
    S. Brahim-Otsmane, G. Francfort, F. Murat: Correctors for the homogenization of the wave and heat equation. J. Math. Pures Appl 9 (1992).Google Scholar
  10. [CoFo]
    P. Constantin, C. Foiaş: Navier-Stokes equations. The University of Chicago Press, Chicago, 1989.Google Scholar
  11. [DM]
    G. Dal Maso: An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications, Volume 8, Birkhäuser Boston. 1993.Google Scholar
  12. [Defr]
    A. Defranceschi: An introduction to homogenization and G-convergence. School on homogenization, ICTP, Trieste, September 6–17, 1993.Google Scholar
  13. [Edw]
    R. E. Edwards: Functional Analysis. Holt, Rinehart and Winston, New York, 1965.Google Scholar
  14. [HolWel]
    A. Holmbom, N. Wellander: Some results for periodic and non-periodic two-scale convergence. Working paper No. 33 University of Gävle/Sandviken, 1996.Google Scholar
  15. [Kuf]
    A. Kufner: Function Spaces. Nordhoff International, Leyden, 1977.Google Scholar
  16. [LiMa]
    J. L. Lions, E. Magenes: Non Homogeneous Boundary Value problems and Applications II. Springer-Verlag, Berlin, 1972.Google Scholar
  17. [Nand]
    A. K. Nandakumaran: Steady and evolution Stokes equations in a porous media with Non-homogeneous boundary data. A homogenization process. Differential and Integral Equations 5 (1992), no. 1, 73–93.Google Scholar
  18. [Ngul]
    G. Nguetseng: A general convergence result for a functional related to the theory of homogenization. SIAM Journal of Mathematical Analysis 20 (1989), no. 3, 608–623.Google Scholar
  19. [Ngu2]
    G. Nguetseng: Thèse d'Etat. Université Paris 6, 1984.Google Scholar
  20. [Per]
    L. E. Persson, L. Persson, J. Wyller, N. Svanstedt: The Homogenization Method—An Introduction. Studentlitteratur Publishing, 1993.Google Scholar
  21. [SaPa]
    E. Sanchez-Palencia: Non-Homogeneous Media and Vibration Theory. Springer Verlag, 1980.Google Scholar
  22. [Tem]
    R. Temam: Navier Stokes Equation. North-Holland, 1984.Google Scholar
  23. [Zei]
    E. Zeidler: Nonlinear Functional Analysis and its Applications II. Springer Verlag, 1990.Google Scholar
  24. [Ziem]
    W. Ziemer: Weakly Differentiable Functions. Springer Verlag, 1989.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 1997

Authors and Affiliations

  • Anders Holmbom
    • 1
    • 2
  1. 1.Resource and Design OptimizationÖstersundSweden, and
  2. 2.Department of MathematicsLuleå University of TechnologyLuleåSweden

Personalised recommendations