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

We study the limit as alpha -> 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 u(0) of the exact 3D Navier-Stokes system as time t tends to infinity. Furthermore, we establish that the trajectory attractor u(alpha) of every alpha-model converges in the same topology to the attractor u(0) as alpha -> 0+. We construct the minimal limits u(min) subset of u(0) of the trajectory attractors u(alpha) for all alpha-models as alpha -> 0+. We prove that every such set u(min) is a compact connected component of the trajectory attractor u(0), and all the u(min) are strictly invariant under the action of the translation semigroup.