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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
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
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
Linking options:
https://www.mathnet.ru/eng/sm1144https://doi.org/10.1070/SM1991v070n01ABEH002123 https://www.mathnet.ru/eng/sm/v181/i6/p813
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Abstract page: | 430 | Russian version PDF: | 104 | English version PDF: | 15 | References: | 69 | First page: | 1 |
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