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

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.
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