### Article

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

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 0 of the exact 3D Navier-Stokes system as time t tends to infinity. Furthermore, we establish that the trajectory attractor α of every α-model converges in the same topology to the attractor 0 as α → 0+. We construct the minimal limits min ⊆ 0 of the trajectory attractors α for all α-models as α → 0+. We prove that every such set min is a compact connected component of the trajectory attractor 0, and all the min are strictly invariant under the action of the translation semigroup. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.