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 Mat. Sb., 2016, Volume 207, Number 4, Pages 143–172 (Mi msb8549)

Approximating the trajectory attractor of the 3D Navier-Stokes system using various $\alpha$-models of fluid dynamics

V. V. Chepyzhovab

a Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
b National Research University "Higher School of Economics" (HSE), Moscow

Abstract: We study the limit as $\alpha\to 0{+}$ of the long-time dynamics for various approximate $\alpha$-models of a viscous incompressible fluid and their connection with the trajectory attractor of the exact 3D Navier-Stokes system. The $\alpha$-models under consideration are divided into two classes depending on the orthogonality properties of the nonlinear terms of the equations generating every particular $\alpha$-model. We show that the attractors of $\alpha$-models of class I have stronger properties of attraction for their trajectories than the attractors of $\alpha$-models of class II. We prove that for both classes the bounded families of trajectories of the $\alpha$-models considered here converge in the corresponding weak topology to the trajectory attractor $\mathfrak A_0$ of the exact 3D Navier-Stokes system as time $t$ tends to infinity. Furthermore, we establish that the trajectory attractor $\mathfrak A_\alpha$ of every $\alpha$-model converges in the same topology to the attractor $\mathfrak A_0$ as $\alpha\to 0{+}$. We construct the minimal limits $\mathfrak A_{\min}\subseteq\mathfrak A_0$ of the trajectory attractors $\mathfrak A_\alpha$ for all $\alpha$-models as $\alpha\to 0{+}$. We prove that every such set $\mathfrak A_{\min}$ is a compact connected component of the trajectory attractor $\mathfrak A_0$, and all the $\mathfrak A_{\min}$ are strictly invariant under the action of the translation semigroup.
Bibliography: 39 titles.

Keywords: 3D Navier-Stokes system, $\alpha$-models of fluid dynamics, trajectory attractor.

 Funding Agency Grant Number Russian Science Foundation 14-50-00150 This research was supported by the Russian Science Foundation (project no. 14-50-00150).

DOI: https://doi.org/10.4213/sm8549

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English version:
Sbornik: Mathematics, 2016, 207:4, 610–638

Bibliographic databases:

UDC: 517.958
MSC: Primary 35Q30; Secondary 35B41, 76D05

Citation: V. V. Chepyzhov, “Approximating the trajectory attractor of the 3D Navier-Stokes system using various $\alpha$-models of fluid dynamics”, Mat. Sb., 207:4 (2016), 143–172; Sb. Math., 207:4 (2016), 610–638

Citation in format AMSBIB
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