Skip to main content
SpringerLink
Log in

Asymptotic analysis of a multiscale parabolic problem with a rough fast oscillating interface

  • Special
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

This paper is concerned with the well posedness and homogenization for a multiscale parabolic problem in a cylinder Q of \({\mathbb {R}}^N \). A rapidly oscillating non-smooth interface inside Q separates the cylinder in two heterogeneous connected components. The interface has a periodic microstructure, and it is situated in a small neighborhood of a hyperplane which separates the two components of Q. The problem models a time-dependent heat transfer in two heterogeneous conducting materials with an imperfect contact between them. At the interface, we suppose that the flux is continuous and that the jump of the solution is proportional to the flux. On the exterior boundary, homogeneous Dirichlet boundary conditions are prescribed. We also derive a corrector result showing the accuracy of our approximation in the energy norm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

References

  1. Achdou, Y., Pironneau, O., Valentin, F.: Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147(1), 187–218 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Auriault, J.L., Ene, H.: Macroscopic modeling of heat transfer in composites with interfacial thermal barrier. Int. J. Heat Mass Transf. 37, 2885–2892 (1994)

    Article  MATH  Google Scholar 

  3. Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North Holland, Amsterdam (1978)

    MATH  Google Scholar 

  4. Blanchard, D., Carbone, L., Gaudiello, A.: Homogenization of a monotone problem in a domain with oscillating boundary. Math. Model. Numer. Anal. 33(5), 1057–1070 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bouchitte, G., Lidouh, A., Michel, J.C., Suquet, P.: Might Boundary Homogenization Help to Understand Friction?. International Centre for Theoretical Physics, SMR, Trieste (1993)

    Google Scholar 

  6. Brizzi, R., Chalot, J.-P.: Homogeneisation de Frontiere. Ph.D. Thesis, Universite de Nice, France (1978)

  7. Brizzi, R., Chalot, J.-P.: Boundary homogenization and Neumann boundary value problem. Richerche Mat. 46(2), 341–387 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Corbo Esposito, A., Donato, P., Gaudiello, A., Picard, C.: Homogenization of the p-Laplacian in a domain with oscillating boundary. Comm. Appl. Nonlinear Anal. 4, 1–23 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Carslaw, H., Jaeger, J.: Conduction of Heat in Solids. The Clarendon Press, Oxford (1947)

    MATH  Google Scholar 

  10. Chechkin, G.A., Friedman, A., Piatnitski, A.L.: The boundary-value problem in domains with very rapidly oscillating boundary. J. Math. Anal. Appl. 231(1), 213–234 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  12. Cioranescu, D., Donato, P., Roque, M.: An Introduction to Classical and Variational Partial Differential Equations. The University of the Philippines Press, Diliman, Quezon City (2012)

    MATH  Google Scholar 

  13. Donato, P.: Some corrector results for composites with imperfect interface. Rend. Math. Ser. VII 26, 189–209 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Donato, P., Jose, E.: Corrector results for a parabolic problem with a memory effect. ESAIM Math. Model. Numer. Anal. 44, 421–454 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Donato, P., Giachetti, D.: Existence and homogenization for a singular problem through rough surfaces. SIAM J. Math. Anal. 48(6), 4047–4086 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Donato, P., Piatnitski, A.: On the effective interfacial resistance through rough surfaces. Commun. Pure Appl. Anal. 9(5), 1295–1310 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Donato, P., Faella, L., Monsurró, S.: Correctors for the homogenization of a class of hyperbolic equations with imperfect interfaces. SIAM J. Math. Anal. 40, 19521978 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Donato, P., Monsurró, S.: Homogenization of two heat conductors with an interfacial contact resistance. Anal. Appl. 2, 127 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Faella, L., Monsurró, S: Memory effects arising in the homogenization of composites with inclusions. In: Topics on mathematics for smart systems. World Sci. Publ., Hackensack, USA, p. 107121 (2007)

  20. Gaudiello, A.: Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary. Ricerche Mat. 43(2), 239–292 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Gaudiello, A., Hadiji, R., Picard, C.: Homogenization of the Ginzburg–Landau equation in a domain with oscillating boundary. Commun. Appl. Anal. 7(2–3), 209–223 (2003)

    MathSciNet  MATH  Google Scholar 

  22. Hummel, H.-K.: Homogenization for heat transfer in polycrystals with interfacial resistances. Appl. Anal. 75(3–4), 403–424 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jose, E.: Homogenization of a parabolic problem with an imperfect interface. Rev. Roumaine Math. Pures Appl. 54(3), 189–222 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Jost, J.: Postmodern Analysis, 3rd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  25. Kohler, W., Papanicolaou, G., Varadhan, S.: Boundary and interface problems in regions with very rough boundaries. In: Chow, P., Kohler, W., Papanicolaou, G. (eds.) Multiple Scattering and Waves in Random Media, pp. 165–197. North-Holland, Amsterdam (1981)

    Google Scholar 

  26. Lancia, M.R., Mosco, U., Vivaldi, M.A.: Homogenization for conductive thin layers of pre-fractal type. J. Math. Anal. Appl. 347(1), 354–369 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Monsurró, S.: Homogenization of a two-component composite with interfacial thermal barrier. Adv. Math. Sci. Appl. 13, 43–63 (2003)

    MathSciNet  MATH  Google Scholar 

  28. Nandakumaran, A.K., Prakash, R., Raymond, J.-P.: Asymptotic analysis and error estimates for an optimal control problem with oscillating boundaries. Annali dellUniversita di Ferrara 58, 143–166 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nandakumaran, A.K., Prakash, R., Sardar, B.C.: Homogenization of an optimal control problem in a domain with highly oscillating boundary using periodic unfolding method. Math. Eng. Sci. Aerosp. 4(3), 281–303 (2013)

    MATH  Google Scholar 

  30. Neuss-Radu, M., Jaeger, W.: Effective transmission conditions for reaction–diffusion processes in domains separated by an interface. SIAM J. Math. Anal. 39(3), 687–720 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nevard, J., Keller, J.B.: Homogenization of rough boundaries and interfaces. SIAM J. Appl. Math. 57(6), 1660–1686 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sanchez-Palencia, E.: Non-homogeneous media and vibration theory. Lecture notes in physics, vol. 127. Springer, Berlin (1980)

  33. Simon, J.: Compact sets in the space \(L^p(0, T; B)\). Annali di Mat. Pura ed Appl. 146 (IV), 65–96 (1987)

    MATH  Google Scholar 

  34. Zeidler, E.: Nonlinear Functional Analysis and Its Applications II/A: Linear Monotone Operators. Springer, New York (1990)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The work of E.J. was funded by the UP System Enhanced Creative Work and Research Grant (EWCRG 2016-1-010), while that of D.O. was supported by the USAID STRIDE Visiting U.S. Professors Program. The authors were able to visit each other through the support of the following: Erasmus Mundus (IMPAKT) Staff Mobility Program (P.D. and E.J.) and Universite de Rouen and University of Houston (D.O. and E.J.).

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Université de Rouen Normandie, Avenue de l’Université, BP 12, 76801, Saint Étienne de Rouvray, France

    Patrizia Donato

  2. Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños, 4031, Los Banos, Laguna, Philippines

    Editha C. Jose

  3. Department of Mathematics, University of Houston, Houston, TX, USA

    Daniel Onofrei

Authors
  1. Patrizia Donato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Editha C. Jose
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Onofrei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Onofrei.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Donato, P., Jose, E.C. & Onofrei, D. Asymptotic analysis of a multiscale parabolic problem with a rough fast oscillating interface. Arch Appl Mech 89, 437–465 (2019). https://doi.org/10.1007/s00419-018-1415-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-018-1415-5

Keywords

Mathematics Subject Classification

Search

Navigation