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Homogenization of a parabolic equation in perforated domain with Neumann boundary condition

  • A. K. Nandakumaran
  • M. RajeshEmail author
Article

Abstract

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains
$$\begin{gathered} \partial _t b(\tfrac{x}{\varepsilon },u_\varepsilon ) - diva(\tfrac{x}{\varepsilon },u_\varepsilon ,\nabla u_\varepsilon ) = f(x,t) in \Omega _\varepsilon \times (0,T), \hfill \\ a(\tfrac{x}{\varepsilon },u_\varepsilon ,\nabla u_\varepsilon ) \cdot v_\varepsilon = 0 on \partial S_\varepsilon \times (0,T), \hfill \\ u_\varepsilon = 0 on \partial \Omega \times (0,T), \hfill \\ u_\varepsilon (x,0) = u_0 (x) in \Omega _\varepsilon \hfill \\ \end{gathered} $$
. Here, ΩɛS ɛ is a periodically perforated domain. We obtain the homogenized equation and corrector results. The homogenization of the equations on a fixed domain was studied by the authors [15]. The homogenization for a fixed domain and\(b(\tfrac{x}{\varepsilon },u_\varepsilon ) \equiv b(u_\varepsilon )\) has been done by Jian [11].

Keywords

Homogenization perforated domain two-scale convergence correctors 

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References

  1. [1]
    Allaire G, Homogenization and two-scale convergence,SIAM J. Math. Anal. 23 (1992) 1482–1518.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Allaire G, Murat M and Nandakumar AK, Appendix of ‘Homogenization of the Neumann problem with nonisolated holes’,Asymptotic Anal. 7(2) (1993) 81–95zbMATHMathSciNetGoogle Scholar
  3. [3]
    Alt H W and Luckhaus S, Quasilinear elliptic-parabolic differential equations,Math. Z. 183(1983) 311–341zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Avellaneda M and Lin F H, Compactness methods in the theory of homogenization,Comm. Pure Appl. Math. 40 (1987) 803–847zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Bensoussan B, Lions J L and Papanicolaou G, Asymptotic Analysis of Periodic Structures (Amsterdam: North Holland) (1978)Google Scholar
  6. [6]
    Brahim-Omsmane S, Francfort G A and Murat F, Correctors for the homogenization of wave and heat equations,J. Math. Pures Appl. 71 (1992) 197–231MathSciNetGoogle Scholar
  7. [7]
    Clark G W and Showalter R E, Two-scale convergence of a model for flow in a partially fissured medium,Electron. J. Differ. Equ. 1999(2) (1999) 1–20MathSciNetGoogle Scholar
  8. [8]
    Douglas Jr. J, Peszyńska M and Showalter R E, Single phase flow in partially fissured media,Trans. Porous Media 28 (1995) 285–306CrossRefGoogle Scholar
  9. [9]
    Fusco N and Moscariello G, On the homogenization of quasilinear divergence structure operators,Ann. Mat. Pura Appl. 146(4) (1987) 1–13zbMATHMathSciNetGoogle Scholar
  10. [10]
    Hornung U, Applications of the homogenization method to flow and transport in porous media, Notes of Tshingua Summer School on Math. Modelling of Flow and Transport in Porous Media (ed.) Xiao Shutie (Singapore: World Scientific) (1992) pp. 167–222Google Scholar
  11. [11]
    Jian H, On the homogenization of degenerate parabolic equations,Acta Math. Appl. Sinica 16(1) (2000) 100–110zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Jikov V V, Kozlov S M and Oleinik O A, Homogeniztion of Differential Operators and Integral Functionals (Berlin: Springer-Verlag) (1994)Google Scholar
  13. [13]
    Ladyzenskaya O A, Solonnikov V and Ural’tzeva N N, Linear and quasilinear equations of parabolic type,Am. Math. Soc. Transl. Mono.? 23, (1968)Google Scholar
  14. [14]
    Nguetseng G, A general convergence result of a functional related to the theory of homogenization,SIAM J. Math. Anal. 20 (1989) 608–623zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Nandakumaran A K and Rajesh M, Homogenization of a nonlinear degenerate parabolic differential equation,Electron. J. Differ. Equ. 2001(17) (2001) 1–19MathSciNetGoogle Scholar
  16. [16]
    Oleinik O A, Kozlov S M and Zhikov V V, On G-convergence of parabolic operators,Ouspekhi Math. Naut. 36(1) (1981) 11–58MathSciNetGoogle Scholar

Copyright information

© Printed in India 2002

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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