On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system

Abstract

We study the relations between the global dynamics of the 3DLeray-$\alpha $ model and the 3D Navier-Stokes system. We provethat time shifts of bounded sets of solutions of the Leray-$\alpha$ model converges to the trajectoryattractor of the 3D Navier-Stokes system as time tends to infinity and $\alpha $ approaches zero. In particular, we show that thetrajectory attractor of the Leray-$\alpha $ model converges to thetrajectory attractor of the 3D Navier-Stokes system when $\alpha\rightarrow 0\+.$