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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
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”, Sb. Math., 198:12 (2007), 1703–1736
Citation in format AMSBIB
\Bibitem{VisTitChe07}
\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 Sb. Math.
\yr 2007
\vol 198
\issue 12
\pages 1703--1736
\mathnet{http://mi.mathnet.ru//eng/sm3832}
\crossref{https://doi.org/10.1070/SM2007v198n12ABEH003902}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2380803}
\zmath{https://zbmath.org/?q=an:1137.37037}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000253636300008}
\elib{https://elibrary.ru/item.asp?id=9602054}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-40749129222}
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  • https://www.mathnet.ru/eng/sm3832
  • 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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    Abstract page:1013
    Russian version PDF:277
    English version PDF:33
    References:116
    First page:11
     
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