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Sbornik: Mathematics, 2007, Volume 198, Issue 12, Pages 1703–1736
DOI: https://doi.org/10.1070/SM2007v198n12ABEH003902
(Mi sm3832)
 

This article is cited in 33 scientific papers (total in 33 papers)

On convergence of trajectory attractors of the 3D Navier–Stokes-$\alpha$ model as $\alpha$ approaches 0

M. I. Vishika, E. S. Titibc, V. V. Chepyzhova

a Institute for Information Transmission Problems, Russian Academy of Sciences
b Weizmann Institute of Science
c University of California, Irvine
References:
Abstract: We study the relations between the long-time dynamics of the Navier–Stokes-$\alpha$ model and the exact 3D Navier–Stokes system. We prove that bounded sets of solutions of the Navier–Stokes-$\alpha$ model converge to the trajectory attractor $\mathfrak A_0$ of the 3D Navier–Stokes system as the time approaches infinity and $\alpha$ approaches zero. In particular, we show that the trajectory attractor $\mathfrak A_\alpha$ of the Navier–Stokes-$\alpha$ model converges to the trajectory attractor $\mathfrak A_0$ of the 3D Navier–Stokes system as $\alpha\to0+$. We also construct the minimal limit $\mathfrak A_{\min}(\subseteq\!\mathfrak A_0)$ of the trajectory attractor $\mathfrak A_\alpha$ as $\alpha\to0+$ and prove that the set $\mathfrak A_{\min}$ is connected and strictly invariant.
Bibliography: 35 titles.
Received: 23.01.2007
Russian version:
Matematicheskii Sbornik, 2007, Volume 198, Number 12, Pages 3–36
DOI: https://doi.org/10.4213/sm3832
Bibliographic databases:
UDC: 517.958
MSC: Primary 35Q30, 35B41; Secondary 76D05
Language: English
Original paper language: Russian
Citation: M. I. Vishik, E. S. Titi, V. V. Chepyzhov, “On convergence of trajectory attractors of the 3D Navier–Stokes-$\alpha$ model as $\alpha$ approaches 0”, Mat. Sb., 198:12 (2007), 3–36; Sb. Math., 198:12 (2007), 1703–1736
Citation in format AMSBIB
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\by M.~I.~Vishik, E.~S.~Titi, V.~V.~Chepyzhov
\paper On convergence of trajectory attractors of the 3D~Navier--Stokes-$\alpha$
model as $\alpha$ approaches~0
\jour Mat. Sb.
\yr 2007
\vol 198
\issue 12
\pages 3--36
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\crossref{https://doi.org/10.4213/sm3832}
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\transl
\jour Sb. Math.
\yr 2007
\vol 198
\issue 12
\pages 1703--1736
\crossref{https://doi.org/10.1070/SM2007v198n12ABEH003902}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-40749129222}
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  • https://doi.org/10.1070/SM2007v198n12ABEH003902
  • https://www.mathnet.ru/eng/sm/v198/i12/p3
  • This publication is cited in the following 33 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Russian version PDF:276
    English version PDF:31
    References:115
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