On convergence of trajectory attractors of the 3D Navier-Stokes-α model as α approaches 0

We study the relations between the long-time dynamics of the Navier-Stokes-alpha model and the exact 3D Navier-Stokes system. We prove that bounded sets of solutions of the Navier-Stokes-a model converge to the trajectory attractor u(0) of the 3D Navier-Stokes system a's the time approaches infinity and a approaches zero. In particular, we show that the trajectory attractor u(0) of the 3D Navier-Stokes-a model converges to the trajectory attractor u(0) of the 3D Navier-Stokes system as alpha -> 0+. We also construct the minimal limit u(min)( subset of u(0)) of the trajectory attractor u(alpha) as alpha -> 0+ and prove that the set u(min) is connected and strictly invariant.