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Mathematics of the USSR-Sbornik, 1991, Volume 70, Issue 1, Pages 241–261
DOI: https://doi.org/10.1070/SM1991v070n01ABEH002123
(Mi sm1144)
 

This article is cited in 7 scientific papers (total in 7 papers)

Averaging on a background of vanishing viscosity

S. M. Kozlova, A. L. Piatnitskib

a Moscow Engineering Building Institute
b P. N. Lebedev Physical Institute, Russian Academy of Sciences
References:
Abstract: Elliptic equations of the form
\begin{gather*} \biggl(\mu a_{ij}\biggl (\frac x\varepsilon\biggr)\frac\partial{\partial x_i} \frac\partial{\partial x_j}+\varepsilon^{-1}v_i\biggl (\frac x\varepsilon\biggr) \frac\partial{\partial x_i}\biggr)u^{\mu,\varepsilon}(x)=0, \\ u^{\mu,\varepsilon}\big|_{\partial\Omega}=\varphi(x) \end{gather*}
with periodic coefficients are considered; $\mu$ and $\varepsilon$ are small parameters. For potential fields $v(y)$ and constants $a_{ij}=\delta_{ij}$, the asymptotic behavior as $\mu\to 0$ of the coefficients of the averaged operator (which is customarily also called the effective diffusion) is studied. It is shown that as $\mu\to 0$ the effective diffusion $\sigma(\mu)=\sigma_{ij}(\mu)$ decays exponentially, and the limit $\lim\limits_{\mu\to 0}\mu\ln\sigma(\mu)$ is found. Sufficient conditions are found for the existence of a limit operator as $\mu$ and $\varepsilon$ tend to 0 simultaneously. The structure of this operator depends on the symmetry reserve of the coefficients $a_{ij}(y)$ and $v_i(y)$; in particular, it may decompose into independent operators in subspaces of lower dimension.
Received: 09.03.1989
Russian version:
Matematicheskii Sbornik, 1990, Volume 181, Number 6, Pages 813–832
Bibliographic databases:
UDC: 517.9
MSC: 35J25
Language: English
Original paper language: Russian
Citation: S. M. Kozlov, A. L. Piatnitski, “Averaging on a background of vanishing viscosity”, Mat. Sb., 181:6 (1990), 813–832; Math. USSR-Sb., 70:1 (1991), 241–261
Citation in format AMSBIB
\Bibitem{KozPia90}
\by S.~M.~Kozlov, A.~L.~Piatnitski
\paper Averaging on a~background of vanishing viscosity
\jour Mat. Sb.
\yr 1990
\vol 181
\issue 6
\pages 813--832
\mathnet{http://mi.mathnet.ru/sm1144}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1072299}
\zmath{https://zbmath.org/?q=an:0709.35007|0732.35006}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?1991SbMat..70..241K}
\transl
\jour Math. USSR-Sb.
\yr 1991
\vol 70
\issue 1
\pages 241--261
\crossref{https://doi.org/10.1070/SM1991v070n01ABEH002123}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1991GG78300015}
Linking options:
  • https://www.mathnet.ru/eng/sm1144
  • https://doi.org/10.1070/SM1991v070n01ABEH002123
  • https://www.mathnet.ru/eng/sm/v181/i6/p813
  • This publication is cited in the following 7 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник - 1989–1990 Sbornik: Mathematics
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    Abstract page:430
    Russian version PDF:104
    English version PDF:15
    References:69
    First page:1
     
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