[en] We study the limit as $\mathrm{\alpha <!--\; ??\; -->}\to <!--\; ???\; -->0+$ of the long-time dynamics for various approximate $\mathrm{\alpha <!--\; ??\; -->}$-models of a viscous incompressible fluid and their connection with the trajectory attractor of the exact 3D Navier-Stokes system. The $\mathrm{\alpha <!--\; ??\; -->}$-models under consideration are divided into two classes depending on the orthogonality properties of the nonlinear terms of the equations generating every particular $\mathrm{\alpha <!--\; ??\; -->}$-model. We show that the attractors of $\mathrm{\alpha <!--\; ??\; -->}$-models of class I have stronger properties of attraction for their trajectories than the attractors of $\mathrm{\alpha <!--\; ??\; -->}$-models of class II. We prove that for both classes the bounded families of trajectories of the $\mathrm{\alpha <!--\; ??\; -->}$-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 ${\mathfrak{A}}_{\mathrm{\alpha <!--\; ??\; -->}}$ of every $\mathrm{\alpha <!--\; ??\; -->}$-model converges in the same topology to the attractor ${\mathfrak{A}}_0$ as $\mathrm{\alpha <!--\; ??\; -->}\to <!--\; ???\; -->0+$. We construct the minimal limits ${\mathfrak{A}}_{min}\subseteq <!--\; ???\; -->{\mathfrak{A}}_0$ of the trajectory attractors ${\mathfrak{A}}_{\mathrm{\alpha <!--\; ??\; -->}}$ for all $\mathrm{\alpha <!--\; ??\; -->}$-models as $\mathrm{\alpha <!--\; ??\; -->}\to <!--\; ???\; -->0+$. 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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