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 Mat. Sb., 1990, Volume 181, Number 6, Pages 813–832 (Mi msb1144)

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

Abstract: Elliptic equations of the form
\begin{gather*} (\mu a_{ij}(\frac x\varepsilon)\frac\partial{\partial x_i} \frac\partial{\partial x_j}+\varepsilon^{-1}v_i(\frac x\varepsilon) \frac\partial{\partial x_i})u^{\mu,\varepsilon}(x)=0,
u^{\mu,\varepsilon}|_{\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.

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English version:
Mathematics of the USSR-Sbornik, 1991, 70:1, 241–261

Bibliographic databases:

UDC: 517.9
MSC: 35J25

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/msb1144} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1072299} \zmath{https://zbmath.org/?q=an:0709.35007|0732.35006} \adsnasa{http://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{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1991GG78300015} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bensoussan A., Kozlov S., “Effective Diffusion in a Periodic-Flow”, Comptes Rendus Acad. Sci. Ser. I-Math., 315:7 (1992), 765–768
2. Kozlov S., Piatnitski A., “Effective Diffusion for a Parabolic Operator with Periodic Potential”, SIAM J. Appl. Math., 53:2 (1993), 401–418
3. Mark I. Freidlin, Richard B. Sowers, “A comparison of homogenization and large deviations, with applications to wavefront propagation”, Stochastic Processes and their Applications, 82:1 (1999), 23
4. M. L. Kleptsyna, A. L. Piatnitski, “Homogenization of a random non-stationary convection-diffusion problem”, Russian Math. Surveys, 57:4 (2002), 729–751
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