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

Journal Article · Sat Apr 30 04:00:00 EDT 2016 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM8549· OSTI ID:22875622

Chepyzhov, V V ^[1]

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1. Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow (Russian Federation)

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

• OSTI ID: 22875622
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 4 Vol. 207; ISSN 1064-5616
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
ATTRACTORS
FLUID MECHANICS
NAVIER-STOKES EQUATIONS
NONLINEAR PROBLEMS
TOPOLOGY
TRAJECTORIES