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Russian Mathematical Surveys, 2014, Volume 69, Issue 2, Pages 359–381
DOI: https://doi.org/10.1070/RM2014v069n02ABEH004891
(Mi rm9583)
 

This article is cited in 31 scientific papers (total in 31 papers)

A unified approach to determining forms for the 2D Navier–Stokes equations — the general interpolants case

C. Foiasa, M. S. Jollyb, R. Kravchenkoc, E. S. Titide

a Texas A&M University, College Station, USA
b Indiana University, Bloomington, USA
c University of Chicago, Chicago, USA
d Weizmann Institute of Science, Rehovot, Israel
e University of California, Irvine, USA
References:
Abstract: It is shown that the long-time dynamics (the global attractor) of the 2D Navier–Stokes system is embedded in the long-time dynamics of an ordinary differential equation, called a determining form, in a space of trajectories which is isomorphic to $C^1_b(\mathbb{R};\mathbb{R}^N)$ for sufficiently large $N$ depending on the physical parameters of the Navier–Stokes equations. A unified approach is presented, based on interpolant operators constructed from various determining parameters for the Navier–Stokes equations, namely, determining nodal values, Fourier modes, finite volume elements, finite elements, and so on. There are two immediate and interesting consequences of this unified approach. The first is that the constructed determining form has a Lyapunov function, and thus its solutions converge to the set of steady states of the determining form as the time goes to infinity. The second is that these steady states of the determining form can be uniquely identified with the trajectories in the global attractor of the Navier–Stokes system. It should be added that this unified approach is general enough that it applies, in an almost straightforward manner, to a whole class of dissipative dynamical systems.
Bibliography: 23 titles.
Keywords: Navier–Stokes equation, inertial manifold, determining forms, determining modes, dissipative dynamical systems.
Funding agency Grant number
National Science Foundation DMS-1109784
DMS-1008661
DMS-1109638
DMS-1009950
DMS-1109640
DMS-1109645
Minerva Stiftung
The work of the first author was supported by NSF (grant no. DMS-1109784), that of the second author by NSF (grant nos. DMS-1008661 and DMS-1109638), and that of the fourth author by NSF (grant nos. DMS-1009950, DMS-1109640, and DMS-1109645), as well as the Minerva Stiftung/Foundation.
Received: 27.10.2013
Russian version:
Uspekhi Matematicheskikh Nauk, 2014, Volume 69, Issue 2(416), Pages 177–200
DOI: https://doi.org/10.4213/rm9583
Bibliographic databases:
Document Type: Article
UDC: 517.954+517.957
MSC: Primary 35Q30; Secondary 76D05
Language: English
Original paper language: Russian
Citation: C. Foias, M. S. Jolly, R. Kravchenko, E. S. Titi, “A unified approach to determining forms for the 2D Navier–Stokes equations — the general interpolants case”, Uspekhi Mat. Nauk, 69:2(416) (2014), 177–200; Russian Math. Surveys, 69:2 (2014), 359–381
Citation in format AMSBIB
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  • This publication is cited in the following 31 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Успехи математических наук Russian Mathematical Surveys
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    Russian version PDF:194
    English version PDF:20
    References:65
    First page:18
     
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