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Mat. Sb., 2003, Volume 194, Number 1, Pages 121–146 (Mi msb709)  

This article is cited in 19 scientific papers (total in 19 papers)

Asymptotic analysis of a double porosity model with thin fissures

L. S. Pankratov, V. A. Rybalko

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine

Abstract: An initial-boundary-value problem is considered for the parabolic equation
$$ \Phi^\varepsilon(x)u^\varepsilon_t-\operatorname{div}(A^\varepsilon(x) \nabla u^\varepsilon)=f^\varepsilon(x), \qquad x\in\Omega, \quad t>0, $$
with discontinuous diffusion tensor $A^\varepsilon(x)$. This tensor is assumed to degenerate as $\varepsilon\to0$ in the whole of the domain $\Omega$ except on a set ${\mathscr F}^{(\varepsilon)}$ of asymptotically small measure. It is shown that the behaviour of the solutions $u^\varepsilon$ as $\varepsilon\to0$ is described by a homogenized model with memory.

DOI: https://doi.org/10.4213/sm709

Full text: PDF file (421 kB)
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English version:
Sbornik: Mathematics, 2003, 194:1, 123–150

Bibliographic databases:

UDC: 517.946
MSC: 35K20, 35B27, 35R05
Received: 18.12.2001 and 14.08.2002

Citation: L. S. Pankratov, V. A. Rybalko, “Asymptotic analysis of a double porosity model with thin fissures”, Mat. Sb., 194:1 (2003), 121–146; Sb. Math., 194:1 (2003), 123–150

Citation in format AMSBIB
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    1. Amaziane B., Bourgeat A., Goncharenko M., Pankratov L., “Characterization of the flow for a single fluid in an excavation damaged zone”, C. R. Méc., Acad. Sci. Paris, 332:1 (2004), 79–84  crossref  zmath  isi  elib
    2. Amaziane B., Goncharenko M., Pankratov L., “$\Gamma_D$-convergence for a class of quasilinear elliptic equations in thin structures”, Math. Methods Appl. Sci., 28:15 (2005), 1847–1865  crossref  mathscinet  zmath  isi  elib
    3. Amaziane B., Goncharenko M., Pankratov L., “Homogenization of a degenerate triple porosity model with thin fissures”, European J. Appl. Math., 16:3 (2005), 335–359  crossref  mathscinet  zmath  isi  elib
    4. Amaziane B., Pankratov L., Piatnitski A., “Homogenization of a class of quasilinear elliptic equations in high-contrast fissured media”, Proc. Roy. Soc. Edinburgh Sect. A, 136:6 (2006), 1131–1155  crossref  mathscinet  zmath  isi  elib
    5. Amaziane B., Pankratov L., “On the homogenization of some linear problems in domains weakly connected by a system of traps”, Math. Methods Appl. Sci., 30:15 (2007), 1855–1883  crossref  mathscinet  zmath  isi  elib
    6. Amaziane B., Pankratov L., Piatnitski A, “Homogenization of a single phase flow through a porous medium in a thin layer”, Math. Models Methods Appl. Sci., 17:9 (2007), 1317–1349  crossref  mathscinet  zmath  isi  elib
    7. Braides A., Briane M., “Homogenization of non-linear variational problems with thin low-conducting layers”, Appl. Math. Optim., 55:1 (2007), 1–29  crossref  mathscinet  zmath  isi  elib
    8. Khrabustovskyi A., Stephan H., “Positivity and time behavior of a linear reaction-diffusion system, non-local in space and time”, Math. Methods Appl. Sci., 31:15 (2008), 1809–1834  crossref  mathscinet  zmath  isi  elib
    9. Zhao Hongxing, Yao Zheng-an, “Homogenization of a non-stationary Stokes flow in porous medium including a layer”, J. Math. Anal. Appl., 342:1 (2008), 108–124  crossref  mathscinet  zmath  isi
    10. Amaziane B., Pankratov L., Rybalko V., “On the homogenization of some double-porosity models with periodic thin structures”, Appl. Anal., 88:10-11 (2009), 1469–1492  crossref  mathscinet  zmath  isi  elib
    11. Amaziane B., Pankratov L., Piatnitski A., “Nonlinear flow through double porosity media in variable exponent Sobolev spaces”, Nonlinear Anal. Real World Appl., 10:4 (2009), 2521–2530  crossref  mathscinet  zmath  isi  elib
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    13. Sili A., “A diffusion equation through a highly heterogeneous medium”, Applicable Analysis, 89:6 (2010), 893–904  crossref  mathscinet  zmath  isi  elib
    14. Zhao H., Yao Zh.-a., “Effective models of the Navier–Stokes flow in porous media with a thin fissure”, J Math Anal Appl, 387:2 (2012), 542–555  crossref  mathscinet  zmath  isi
    15. Mladen Jurak, Leonid Pankratov, Anja Vrbaški, “A fully homogenized model for incompressible two-phase flow in double porosity media”, Applicable Analysis, 2015, 1  crossref
    16. Braides A. Piat V.Ch. Solci M., “Discrete Double-Porosity Models For Spin Systems”, Math. Mech. Complex Syst., 4:1 (2016), 79–102  crossref  mathscinet  zmath  isi  elib  scopus
    17. Amaziane B., Jurak M., Pankratov L., Vrbaski A., “Some Remarks on the Homogenization of Immisciblein Compressible Two-Phase Flow in Double Porosity Media”, Discrete Contin. Dyn. Syst.-Ser. B, 23:2 (2018), 629–665  crossref  mathscinet  isi
    18. Konyukhov A., Pankratov L., Voloshin A., “The Homogenized Kondaurov Type Non-Equilibrium Model of Two-Phase Flow in Multiscale Non-Homogeneous Media”, Phys. Scr., 94:5 (2019), 054002  crossref  isi
    19. Voloshin A., Pankratov L., Konyukhov A., “Homogenization of Kondaurov'S Non-Equilibrium Two-Phase Flow in Double Porosity Media”, Appl. Anal., 98:8 (2019), 1429–1450  crossref  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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