On convergence of trajectory attractors of the 3D Navier–Stokes-$\alpha$ model as $\alpha$ 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-$\alpha$ model converge to the trajectory attractor $\mathfrak A_0$ of the 3D Navier–Stokes system as the time approaches infinity and $\alpha$ approaches zero. In particular, we show that the trajectory attractor $\mathfrak A_\alpha$ of the Navier–Stokes-$\alpha$ model converges to the trajectory attractor $\mathfrak A_0$ of the 3D Navier–Stokes system as $\alpha\to0+$. We also construct the minimal limit $\mathfrak A_{\min}(\subseteq\!\mathfrak A_0)$ of the trajectory attractor $\mathfrak A_\alpha$ as $\alpha\to0+$ and prove that the set $\mathfrak A_{\min}$ is connected and strictly invariant.
Bibliography: 35 titles.