Corrector results for a parabolic problem with a memory effect

Patrizia Donato; Editha C. Jose

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 3, page 421-454
  • ISSN: 0764-583X

Abstract

top
The aim of this paper is to provide the correctors associated to the homogenization of a parabolic problem describing the heat transfer. The results here complete the earlier study in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] on the asymptotic behaviour of a problem in a domain with two components separated by an ε-periodic interface. The physical model established in [Carslaw and Jaeger, The Clarendon Press, Oxford (1947)] prescribes on the interface the condition that the flux of the temperature is proportional to the jump of the temperature field, by a factor of order εγ. We suppose that -1 < γ ≤ 1. As far as the energies of the homogenized problems are concerned, we consider the cases -1 < γ < 1 and γ = 1 separately. To obtain the convergence of the energies, it is necessary to impose stronger assumptions on the data. As seen in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] and [Faella and Monsurrò, Topics on Mathematics for Smart Systems, World Sci. Publ., Hackensack, USA (2007) 107–121] (also in [Donato et al., J. Math. Pures Appl.87 (2007) 119–143]), the case γ = 1 is more interesting because of the presence of a memory effect in the homogenized problem.

How to cite

top

Donato, Patrizia, and Jose, Editha C.. "Corrector results for a parabolic problem with a memory effect." ESAIM: Mathematical Modelling and Numerical Analysis 44.3 (2010): 421-454. <http://eudml.org/doc/250829>.

@article{Donato2010,
abstract = { The aim of this paper is to provide the correctors associated to the homogenization of a parabolic problem describing the heat transfer. The results here complete the earlier study in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] on the asymptotic behaviour of a problem in a domain with two components separated by an ε-periodic interface. The physical model established in [Carslaw and Jaeger, The Clarendon Press, Oxford (1947)] prescribes on the interface the condition that the flux of the temperature is proportional to the jump of the temperature field, by a factor of order εγ. We suppose that -1 < γ ≤ 1. As far as the energies of the homogenized problems are concerned, we consider the cases -1 < γ < 1 and γ = 1 separately. To obtain the convergence of the energies, it is necessary to impose stronger assumptions on the data. As seen in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] and [Faella and Monsurrò, Topics on Mathematics for Smart Systems, World Sci. Publ., Hackensack, USA (2007) 107–121] (also in [Donato et al., J. Math. Pures Appl.87 (2007) 119–143]), the case γ = 1 is more interesting because of the presence of a memory effect in the homogenized problem. },
author = {Donato, Patrizia, Jose, Editha C.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Periodic homogenization; correctors; heat equation; interface problems; periodic homogenization; interface problem; homogeneous Dirichlet boundary condition},
language = {eng},
month = {4},
number = {3},
pages = {421-454},
publisher = {EDP Sciences},
title = {Corrector results for a parabolic problem with a memory effect},
url = {http://eudml.org/doc/250829},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Donato, Patrizia
AU - Jose, Editha C.
TI - Corrector results for a parabolic problem with a memory effect
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/4//
PB - EDP Sciences
VL - 44
IS - 3
SP - 421
EP - 454
AB - The aim of this paper is to provide the correctors associated to the homogenization of a parabolic problem describing the heat transfer. The results here complete the earlier study in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] on the asymptotic behaviour of a problem in a domain with two components separated by an ε-periodic interface. The physical model established in [Carslaw and Jaeger, The Clarendon Press, Oxford (1947)] prescribes on the interface the condition that the flux of the temperature is proportional to the jump of the temperature field, by a factor of order εγ. We suppose that -1 < γ ≤ 1. As far as the energies of the homogenized problems are concerned, we consider the cases -1 < γ < 1 and γ = 1 separately. To obtain the convergence of the energies, it is necessary to impose stronger assumptions on the data. As seen in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] and [Faella and Monsurrò, Topics on Mathematics for Smart Systems, World Sci. Publ., Hackensack, USA (2007) 107–121] (also in [Donato et al., J. Math. Pures Appl.87 (2007) 119–143]), the case γ = 1 is more interesting because of the presence of a memory effect in the homogenized problem.
LA - eng
KW - Periodic homogenization; correctors; heat equation; interface problems; periodic homogenization; interface problem; homogeneous Dirichlet boundary condition
UR - http://eudml.org/doc/250829
ER -

References

top
  1. J.L. Auriault and H. Ene, Macroscopic modelling of heat transfer in composites with interfacial thermal barrier. International J. Heat Mass Transfer37 (1994) 2885–2892.  Zbl0900.73453
  2. A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).  Zbl0404.35001
  3. S. Brahim-Otsman, G.A. Francfort and F. Murat, Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl.8 (1992) 197–231.  Zbl0837.35016
  4. M. Briane, A. Damlamian and P. Donato, H-convergence in Perforated Domains, in Nonlinear Partial Differential Equations and Their Applications – Collège de France SeminarXIII, D. Cioranescu and J.L. Lions Eds., Pitman Research Notes in Mathematics Series391, Longman, New York, USA (1998) 62–100.  Zbl0943.35005
  5. H.S. Carslaw and J.C. Jaeger, Conduction of heat in solids. The Clarendon Press, Oxford, UK (1947).  Zbl0029.37801
  6. D. Cioranescu and P. Donato, Homogénéisation du problème de Neumann non homogène dans des ouverts perforés. Asymptot. Anal.1 (1988) 115–138.  Zbl0683.35026
  7. D. Cioranescu and P. Donato, Exact internal controllability in perforated domains. J. Math. Pures Appl.68 (1989) 185–213.  Zbl0627.35057
  8. D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications17. Oxford Univ. Press, New York, USA (1999).  
  9. D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl.71 (1979) 590–607.  Zbl0427.35073
  10. D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Springer-Verlag, New York (1999).  Zbl0929.35002
  11. D. Cioranescu, P. Donato, F. Murat and E. Zuazua, Homogenization and corrector for the wave equation in domains with small holes. Ann. Scuola Norm. Sup. Pisa Cl. Sci.2 (1999) 251–293.  Zbl0807.35077
  12. P. Donato, Some corrector results for composites with imperfect interface. Rend. Math. Ser. VII26 (2006) 189–209.  Zbl1129.35008
  13. P. Donato and S. Monsurrò, Homogenization of two heat conductors with an interfacial contact resistance. Anal. Appl.2 (2004) 1–27.  Zbl1083.35014
  14. P. Donato and A. Nabil, Approximate controllability of linear parabolic equations in perforated domains. ESAIM: COCV6 (2001) 21–38.  Zbl0964.35015
  15. P. Donato and A. Nabil, Homogenization and correctors for the heat equation in perforated domains. Chin. Ann. Math. B25 (2004) 143–156.  Zbl1085.35022
  16. P. Donato, A. Gaudiello and L. Sgambati, Homogenization of bounded solutions of elliptic equations with quadratic growth in periodically perforated domains. Asymptot. Anal.16 (1998) 223–243.  Zbl0944.35009
  17. P. Donato, L. Faella and S. Monsurrò, Homogenization of the wave equation in composites with imperfect interface: a memory effect. J. Math. Pures Appl.87 (2007) 119–143.  Zbl1112.35017
  18. P. Donato, L. Faella and S. Monsurrò, Correctors for the homogenization of a class of hyperbolic equations with imperfect interfaces. SIAM J. Math. Anal.40 (2009) 1952–1978.  Zbl1197.35029
  19. L. Faella and S. Monsurrò, Memory Effects Arising in the Homogenization of Composites with Inclusions, Topics on Mathematics for Smart Systems. World Sci. Publ., Hackensack, USA (2007) 107–121.  Zbl1114.74048
  20. H.K. Hummel, Homogenization for heat transfer in polycrystals with interfacial resistances. Appl. Anal.75 (2000) 403–424.  Zbl1024.80005
  21. E. Jose, Homogenization of a parabolic problem with an imperfect interface. Rev. Roumaine Math. Pures Appl.54 (2009) 189–222.  Zbl1199.35015
  22. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Volume 1. Dunod, Paris, France (1968).  Zbl0165.10801
  23. R. Lipton, Heat conduction in fine scale mixtures with interfacial contact resistance . SIAM J. Appl. Math.58 (1998) 55–72.  Zbl0913.35010
  24. R. Lipton and B. Vernescu, Composite with imperfect interface. Proc. Soc. Lond. A452 (1996) 329–358.  Zbl0872.73033
  25. M.L. Mascarenhas, Linear homogenization problem with time dependent coefficient. Trans. Amer. Math. Soc.281 (1984) 179–195.  Zbl0536.45003
  26. S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier. Adv. Math. Sci. Appl.13 (2003) 43–63.  Zbl1052.35022
  27. S. Monsurrò, Erratum for the paper “Homogenization of a two-component composite with interfacial thermal barrier” (in Vol. 13, pp. 43–63, 2003). Adv. Math. Sci. Appl.14 (2004) 375–377.  Zbl1052.35022
  28. S.E. Pastukhova, Homogenization of nonstationary problems in the theory of elasticity on thin periodic structures from the standpoint of the convergence of hyperbolic semigroups in a variable Hilbert space. Sovrem. Mat. Prilozh.16, Differ. Uravn. Chast. Proizvod. (2004) 64–97 (Russian). Translation in J. Math. Sci. (N. Y.)133 (2006) 949–998.  
  29. R.E. Showalter, Distributed microstructure models of porous media, in Flow in porous media (Oberwolfach (1992)), J. Douglas and U. Hornung Eds., Internat. Ser. Numer. Math.114, Birkhäuser, Basel, Switzerland (1993) 155–163.  Zbl0805.76082
  30. L. Tartar, Cours Peccot. Collège de France, France, unpublished (1977).  
  31. L. Tartar, Quelques remarques sur l'homogénéisation, in Functional Analysis and Numerical Analysis, Proc. Japan-France Seminar 1976, Japanese Society for the Promotion of Science (1978) 468–482.  
  32. L. Tartar, Memory effects and homogenization. Arch. Rational Mech. Anal.3 (1990) 121–133.  Zbl0725.45012

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.