Advertisement

Model of Stationary Diffusion with Absorption In Domains With Fine-Grained Random Boundaries

  • E. Ya. Khruslov
  • L. O. KhilkovaEmail author
Article

We consider a boundary-value problem for the equation of stationary diffusion in a porous medium filled with small inclusions in the form of balls with absorbing surfaces. The process of absorption is described by the Robin nonlinear boundary condition. The locations and radii of the inclusions are randomly distributed and described by a set of finite-dimensional distribution functions. We study the asymptotic behavior of solutions to the problem when the number of balls increases and their radii decrease. We deduce a homogenized equation for the main term of the asymptotics and determine sufficient conditions for the distribution functions under which the solutions converge to the solutions of the homogenized problem in probability.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Ya. Khruslov, L. O. Khilkova, and M. V. Goncharenko, “Integral conditions for convergence of solutions of nonlinear Robin’s problem in strongly perforated domains,” J. Math. Phys., Anal., Geom., 13, No. 3, 1–31 (2017).Google Scholar
  2. 2.
    B. Cabarrubias and P. Donato, “Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary condition,” Appl. Anal., 91, No. 6, 1111–1127 (2012).MathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Cioranescu and P. Donato, “On Robin problems in perforated domains,” Math. Sci. Appl., 9, 123–135 (1997).MathSciNetzbMATHGoogle Scholar
  4. 4.
    D. Cioranescu, P. Donato, and R. Zaki, “The periodic unfolding method in perforated domains,” Port. Math., 63, No. 4, 467–496 (2006).MathSciNetzbMATHGoogle Scholar
  5. 5.
    C. Conca, J. Diaz, A. Linan, and C. Timofte, “Homogenization in chemical reactive flows,” Electron. J. Different. Equat., 40, 1–22 (2004).MathSciNetzbMATHGoogle Scholar
  6. 6.
    C. Conca, J. Diaz, A. Linan, and C. Timofte, “Homogenization results for chemical reactive flows through porous media,” in: New Trends in Continuum Mechanics (2005), pp. 99–107.Google Scholar
  7. 7.
    J. Diaz, “Two problems in homogenization of porous media,” Extracta Math., 14, 141–155 (1999).MathSciNetzbMATHGoogle Scholar
  8. 8.
    W. Jäger, O. A. Oleinik, and A. S. Shamaev, “On homogenization of solutions of boundary-value problem for the Laplace equation in partially perforated domain with the third boundary type condition on the boundary of cavities,” Trudy Mosk. Mat. Obshch., 58, 187–223 (1997).Google Scholar
  9. 9.
    T. A. Mel’nyk and O. A. Sivak, “Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain,” Ukr. Mat. Zh., 61, No. 4, 494–512 (2009); English translation: Ukr. Math. J., 61, No. 4, 592–612 (2009).Google Scholar
  10. 10.
    T. A. Mel’nyk and O. A. Sivak, “Asymptotic approximations for solutions to quasilinear and linear elliptic problems with different perturbed boundary conditions in perforated domains,” Asymptot. Anal., 75, 79–92 (2011).MathSciNetzbMATHGoogle Scholar
  11. 11.
    A. Piatnitski and V. Rybalko, “Homogenization of boundary value problems for monotone operators in perforated domains with rapidly oscillating boundary conditions of Fourier type,” J. Math. Sci., 177, No. 1, 109–140 (2011).MathSciNetCrossRefGoogle Scholar
  12. 12.
    C. Timofte, “Homogenization in nonlinear chemical reactive flows,” in: Proc. of the 9th WSEAS International Conference on Applied Mathematics (Istambul, Turkey, May 27–29, 2006), pp. 250–255.Google Scholar
  13. 13.
    V. A. Marchenko and E.Ya. Khuslov, “Boundary-value problems with fine-grained boundaries,” Mat. Sb., 65, 458–472 (1964).MathSciNetGoogle Scholar
  14. 14.
    S. Kaizu, “The Poisson equation with semilinear boundary conditions in domains with many tiny holes,” J. Fac. Sci. Univ. Tokyo. Sect. IA. Math., 36, 43–86 (1989).MathSciNetzbMATHGoogle Scholar
  15. 15.
    M. Goncharenko, “The asymptotic behavior of the third boundary-value problem solutions in domains with fine-grained boundaries,” Math. Sci. Appl., 9, 203–213 (1997).MathSciNetzbMATHGoogle Scholar
  16. 16.
    A. Brillard, D. Gómez, M. Lobo, E. Pérez, and T. A. Shaposhnikova, “Boundary homogenization in perforated domains for adsorption problems with an advection term,” Appl. Anal., 1–17 (2016).Google Scholar
  17. 17.
    J. Diaz, D. Gómez-Castro, and C. Timofte, “The effectiveness factor of reaction-diffusion equations: homogenization and existence of optimal pellet shapes,” J. Elliptic Parabol. Equat., 2, 119–129 (2016).MathSciNetCrossRefGoogle Scholar
  18. 18.
    J. Diaz, D. Gómez-Castro, T. A. Shaposhnikova, and M. N. Zubova, “The effectiveness factor of reaction-diffusion equations: homogenization and existence of optimal pellet shapes,” Electron. J. Different. Equat., 178, 1–25 (2017).Google Scholar
  19. 19.
    D. Gómez, E. Pérez, and T. A. Shaposhnikova, “On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems,” Asymptot. Anal., 80, 289–322 (2012).MathSciNetzbMATHGoogle Scholar
  20. 20.
    W. Jäger, M. Neuss-Radu, and T. A. Shaposhnikova, “Homogenization limit for the diffusion equation with nonlinear flux condition on the boundary of very thin holes periodically distributed in a domain, in case of a critical size,” Dokl. Mat., 82, No. 2, 736–740 (2010).Google Scholar
  21. 21.
    M. E. Pérez, M. N. Zubova, and T. A. Shaposhnikova, “A homogenization problem in a domain perforated by tiny isoperimetric holes with nonlinear Robin-type boundary conditions,” Dokl. Mat., 90, No. 1, 489–494 (2014).CrossRefGoogle Scholar
  22. 22.
    M. N. Zubova and T. A. Shaposhnikova, “Homogenization of boundary-value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem,” Different. Equat., 47, No. 1, 78–90 (2011).MathSciNetCrossRefGoogle Scholar
  23. 23.
    M. N. Zubova and T. A. Shaposhnikova, “Homogenization of the boundary-value problem for the Laplace operator in a perforated domain with a rapidly oscillating nonhomogeneous Robin-type condition on the boundary of holes in the critical case,” Dokl. Mat., 96, No. 1, 344–347 (2017).CrossRefGoogle Scholar
  24. 24.
    L. A. Khilkova, “Averaging of the diffusion equation in domains with fine-grained boundary with Robin-type nonlinear boundary condition,” Visn. Kharkiv. Nats. Univ., Ser. Mat. Prykl. Mat. Mekh., 84, 93–111 (2016).Google Scholar
  25. 25.
    E. Ya. Khuslov and L. A. Khilkova, “Robin nonlinear problem in domains with fine-grained random boundaries,” Dop. Nats. Akad. Nauk Ukr., No. 9, 3–8 (2017).Google Scholar
  26. 26.
    A. N. Shiryaev, Probability-1 [in Russian], MTSNMO, Moscow (2004).Google Scholar
  27. 27.
    N. N. Bogolyubov, Selected Works [in Russian], Vol. 2, Naukova Dumka, Kiev (2004).Google Scholar
  28. 28.
    I. I. Gikhman, A.V. Skorokhod, and M. I. Yadrenko, Probability Theory and Mathematical Statistics [in Russian], Vyshcha Shkola, Kiev (1973).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for Low Temperature Physics and Engineering, Ukrainian National Academy of SciencesKharkovUkraine

Personalised recommendations