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Properties of Stationary Statistical Solutions of the Three-Dimensional Navier–Stokes Equations

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Abstract

The stationary version of a modified definition of statistical solution for the three-dimensional incompressible Navier–Stokes equations introduced in a previous work is investigated. Particular types of such stationary statistical solutions and their analytical properties are addressed. Results on the support and carriers of these stationary statistical solutions are also given, showing in particular that they are supported on the weak global attractor and are carried by a more regular part of the weak global attractor containing Leray–Hopf weak solutions which are locally strong solutions. Two recurrence-type results related to these measures are also proved.

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Notes

  1. When the space average \({\bar{\mathbf {u}}}\) of \(\mathbf {u}\) does not vanish, \({\bar{\mathbf {u}}}\) is nevertheless constant in time, and the difference \(\mathbf {u}' = \mathbf {u}- {\bar{\mathbf {u}}}\) satisfies (2.1) except for the addition of lower order terms involving \({\bar{\mathbf {u}}}\). In this case, all that follows applies without any significant change. Hence, in the end, our results hold without significant change even when \({\bar{\mathbf {u}}}\ne 0\).

  2. We recall here that a regular topological space is one in which any pair of a singleton and a closed set not containing this singleton can be separated by disjoint neighborhoods, while a completely regular topological space is a regular space in which any pair of a singleton and a closed set not containing this singleton can be separated by a function, i.e. there exists a continuous real-valued function defined on the space and which vanishes at the singleton and is equal to one everywhere on the closed set. The fact that \(H_{\mathrm{w}}\) is completely regular comes from the fact that it is a topological vector space, hence it has a uniform structure [30, Section I.1.4], and is Hausdorff, and any Hausdorff topological space has a uniform structure if and only if it is completely regular [30, Section B.6].

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Ciprian Foias

  2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Ilha do Fundão, Rio de Janeiro, RJ, 21945-970, Brazil

    Ricardo M. S. Rosa

  3. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Roger M. Temam

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  1. Ciprian Foias
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  2. Ricardo M. S. Rosa
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  3. Roger M. Temam
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Correspondence to Ricardo M. S. Rosa.

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The authors dedicate this article to the memory of George Sell, with fond reminiscences of their friendly interactions with him, and with great admiration for his guidance and leadership in the theory of dynamics and dynamical systems, including the creation of the present journal.

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This work was partly supported by the National Science Foundation under the Grants NSF-DMS-1206438 and NSF-DMS-1109784, by the Research Fund of Indiana University, and by the CNPq, Brasília, Brazil, under the Grants 303654/2013-9 and 200826/2014-0 and the Cooperation Project 490124/2009-7.

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Foias, C., Rosa, R.M.S. & Temam, R.M. Properties of Stationary Statistical Solutions of the Three-Dimensional Navier–Stokes Equations. J Dyn Diff Equat 31, 1689–1741 (2019). https://doi.org/10.1007/s10884-018-9719-2

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