Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Homogenization of a pseudo-parabolic system via a spatial-temporal decoupling: Upscaling and corrector estimates for perforated domains

1 Centre for Analysis, Scientific computing and Applications (CASA), Eindhoven University of Technology, Den Dolech 2, 5612AZ, Eindhoven, The Netherlands
2 Department of Mathematics and Computer Science, Karlstad University, Universitetsgatan 2, 651 88, Karlstad, Sweden

We determine corrector estimates quantifying the convergence speed of the upscaling of a pseudo-parabolic system containing drift terms incorporating the separation of length scales with relative size $\epsilon\ll1$. To achieve this goal, we exploit a natural spatial-temporal decomposition, which splits the pseudo-parabolic system into an elliptic partial differential equation and an ordinary differential equation coupled together. We obtain upscaled model equations, explicit formulas for effective transport coefficients, as well as corrector estimates delimitating the quality of the upscaling. Finally, for special cases we show convergence speeds for global times, i.e., $t\in\mathbf{R}_+$, by using time intervals expanding to the whole $\mathbf{R}_+$ simultaneously with passing to the homogenization limit $\epsilon\downarrow0$.
  Article Metrics

Keywords periodic homogenization; pseudo-parabolic system; mixture theory; upscaled system; corrector estimates; perforated domains

Citation: Arthur. J. Vromans, Fons van de Ven, Adrian Muntean. Homogenization of a pseudo-parabolic system via a spatial-temporal decoupling: Upscaling and corrector estimates for perforated domains. Mathematics in Engineering, 2019, 1(3): 548-582. doi: 10.3934/mine.2019.3.548


  • 1. Rendell F, Jauberthie R, Grantham M (2002) Deteriorated Concrete, London: Thomas Telford Publishing, UK.
  • 2. Sand W (2000) Microbial corrosion, In: Cahn, R.W., Haasen, P., Kramer, E.J. Editors, Materials Science and Technology: A Comprehensive Treatment, Chichester: Wiley-Vch, Vol. 1, Chap. 4.
  • 3. During EDD (1997) Corrosion Atlas: A Collection of Illustrated Case Histories, 3rd Edition., Amsterdam: Elsevier.
  • 4. Gu JD, Ford TE, Mitchell R (2011) Microbial corrosion of concrete, In: Revie, R.W. Editor, Uhlig's Corrosion Handbook, 3rd Edition., Hoboken, New Jersey: John Wiley & Sons, 451–460.
  • 5. Trethewey KR, Chamberlain J (1995) Corrosion for Science & Engineering, 2nd Edition., Harlow: Longman Group, UK.
  • 6. Elsener B (2000) Corrosion of steel in concrete, In: Cahn, R.W., Haasen, P., Kramer, E.J., Materials Science and Technology : A Comprehensive Treatment, Chichester: Wiley-Vch, Vol. 2, Chap. 8.
  • 7. Verdink Jr. ED (2011) Economics of corrosion, In: Revie, R.W., Editor, Uhlig's Corrosion Handbook, 3rd Edition., Hoboken: John Wiley & Sons, New Jersey, Chap. 3.
  • 8. Ortiz M, Popov EP (1982) Plain concrete as a composite material, Mech Mater 1: 139–150.
  • 9. Monteiro PJM (1996) Mechanical modelling of the transition zone, In: Maso, J.C. Editor, Interfacial Transition Zone in Concrete, 1st Edition., London: E & FN SPON, UK, Chap. 4. No. 11 in RILEM Report. State-of-the-Art Report prepared by RILEM Technical Committee 108-1CC, Interfaces in Cementitious Composites.
  • 10. Taylor HFW (1997) Cement Chemistry, 2nd Edition., London: Thomas Telford Publishing, UK.
  • 11. Böhm M, Devinny J, Jahani F, et al. (1998) On a moving-boundary system modeling corrosion in sewer pipes. Appl Math Comput 92: 247–269.
  • 12. Clarelli F, Fasano A, Natalini R (2008) Mathematics and monument conservation: Free boundary models of marble sulfation. SIAM J Appl Math 69: 149–168.    
  • 13. Fusi L, Farina A, Primicerio M, et al. (2014) A free boundary problem for CaCO3 neutralization of acid waters. Nonlinear Anal Real World Appl 15: 42–50.    
  • 14. Ern A, Giovangigli V (1998) The kinetic chemical equilibrium regime. Phys A 260: 49–72.    
  • 15. Giovangigli V (1999) Multicomponent Flow Modeling, Series of Modeling and Simulation in Science, Engineering and Technology, Springer Science+Business Media.
  • 16. Cowin S (2013) Continuum Mechanics of Anisotropic Materials, Berlin: Springer.
  • 17. Ciorănescu D, Saint Jean Paulin J (1998) Homogenization of Reticulated Structures, Series of Applied Mathematical Sciences, Springer-Verlag, Vol. 136.
  • 18. Sanchez-Palencia E (1980) Non-Homogeneous Media and Vibration Theory, Series of Lecture Notes in Physics, Berlin: Springer, Vol. 127.
  • 19. Vromans AJ, Muntean A, van de Ven AAF (2018) A mixture theory-based concrete corrosion model coupling chemical reactions, diffusion and mechanics. Pac J Math Ind 10: 1–21.    
  • 20. Vromans AJ (2018) A pseudoparabolic reaction-diffusion-mechanics system: Modeling, analysis and simulation, Licentiate thesis, Karlstad University.
  • 21. Vromans AJ, van de Ven AAF, Muntean A (2019) Parameter delimitation of the weak solvability for a pseudo-parabolic system coupling chemical reactions, diffusion and momentum equations. Adv Math Sci Appl 28: 273–311.
  • 22. Vromans AJ, van de Ven AAF, Muntean A (2019) Periodic homogenization of a pseudo-parabolic equation via a spatial-temporal decomposition, In: Faragó, I., Izsák, F., Simon, P. Editors, Progress in Industrial Mathematics at ECMI 2018, ECMI 2018, Mathematics in Industry, Springer, In press.
  • 23. Peszyńska M, Showalter R, Yi SY (2009) Homogenization of a pseudoparabolic system. Appl Anal 88: 1265–1282.    
  • 24. Eden M (2018) Homogenization of thermoelasticity systems describing phase transformations, PhD thesis, Universität Bremen.
  • 25. Reichelt S (2016) Error estimates for elliptic equations with not-exactly periodic coefficients. Adv Math Sci Appl 25: 117–131.
  • 26. Muntean A, Chalupecký V (2011) Homogenization Method and Multiscale Modeling, No. 34 in COE Lecture Note, Institute of Mathematics for Industry, Kyushu University, Japan.
  • 27. Marchenko VA, Krushlov E (2006) Homogenization of Partial Differential Equations, No. 46 in Progress in Mathematical Physics, Boston: Birkhäuser, Basel, Berlin. Based on the original Russian edition, translated from the original Russian by M. Goncharenko and D. Shepelsky.
  • 28. Kufner A, John O, Fučik S (1977) Function Spaces, Leyden: Noordhoff International Publishing.
  • 29. Meshkova YM, Suslina TA (2016) Homogenization of initial boundary value problems for parabolic systems with periodic coefficients. Appl Anal 95: 1736–1775. DOI 10.1080/00036811. 2015.1068300.    
  • 30. Dragomir SS (2003) Some Gronwall Type Inequalities and Applications, RGMIA Monographs, New York: Nova Science.
  • 31. Rothe F (1984) Global Solutions of Reaction-Diffusion Systems, Berlin Heidelberg: Springer-Verlag.
  • 32. Brezis H (2010) Functional Analysis, Sobolev Spaces and Partial Differential Equations, New York: Springer.
  • 33. Ciorănescu D, Donato P (1999) An Introduction to Homogenization, No. 17 in Oxford Lecture Series in Mathematics and its Applications, Oxford University Press.
  • 34. Gilbarg D, Trudinger N (1977) Elliptic Partial Differential Equations of Second Order, 1998 Edition., Berlin: Springer.
  • 35. Nguetseng G (1989) A general convergence result for a functional related to the theory of homogenization. SIAM J Math Anal 20: 608–623.    
  • 36. Lukkassen D, Nguetseng G, Wall P (2002) Two-scale convergence. Int J Pure Appl Math 2: 35–86.
  • 37. Allaire G (1992) Homogenization and two-scale convergence. SIAM J Math Anal 23: 1482–1518.    
  • 38. Pavliotis GA, Stuart AM (2008) Multiscale Methods-Averaging and Homogenization, No. 53 in Texts in Applied Mathematics, Berlin: Springer.


Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved