On convergence of trajectory attractors of 3D

We study the relations between the long-time dynamics of the Navier–Stokes-α model and the exact 3D Navier–Stokes system. We prove that bounded sets of solutions of the Navier–Stokes-α model converge to the trajectory attractor A0 of the 3D Navier–Stokes system as time tends to infinity and α approaches zero. In particular, we show that the trajectory attractor Aα of the Navier–Stokes-α model converges to the trajectory attractor A0 of the 3D Navier–Stokes system when α→ 0+. We also construct the minimal limit Amin (⊆ A0) of the trajectory attractor Aα as α→ 0+ and we prove that the set Amin is connected and strictly invariant. Introduction (Date: January 17, 2007) In this paper, we study the connection between the long-time dynamics of solutions of the Lagrange averaged Navier–Stokes-α model (N.–S.-α model) and the exact 3D Navier–Stokes system (3D N.–S. system) with periodic boundary conditions. The Navier–Stokes-α model (also known as the viscous 3D Camassa–Holm system) under the consideration was introduced in the works [1] – [6] (see also [7] and the references