DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 17, Number 3, March 2007 pp. 481–500 ON THE CONVERGENCE OF SOLUTIONS OF THE LERAY-α MODEL TO THE TRAJECTORY ATTRACTOR OF THE 3D NAVIER–STOKES SYSTEM Vladimir V. Chepyzhov Institute for Information Transmission Problems Russian Academy of Sciences, Bolshoy Karetniy 19 Moscow 127994, GSP-4, Russia Edriss S. Titi Department of Mathematics and Department of Mechanical and Aerospace Engineering University of California, Irvine, CA 92697-3875, USA and Department of Computer Science and Applied Mathematics, Weizmann Institute of Science Rehovot 76100, Israel Mark I. Vishik Institute for Information Transmission Problems Russian Academy of Sciences, Bolshoy Karetniy 19 Moscow 127994, GSP-4, Russia (Communicated by Roger Temam) Abstract. We study the relations between the global dynamics of the 3D Leray-α model and the 3D Navier–Stokes system. We prove that time shifts of bounded sets of solutions of the Leray-α model converges to the trajec- tory attractor of the 3D Navier–Stokes system as time tends to infinity and α approaches zero. In particular, we show that the trajectory attractor of the Leray-α model converges to the trajectory attractor of the 3D Navier–Stokes system when α → 0+ . 1. Introduction. The 3D Navier–Stokes (N.–S.) system for viscous incompressible fluids has the form  ∂ t u − ν Δu +(u ·∇)u + ∇p = g(x), ∇· u =0,x =(x 1 ,x 2 ,x 3 ) ∈ R 3 , (1) where u = ( u 1 (x,t),u 2 (x,t),u 3 (x,t) ) is the unknown velocity field of a fluid pat- tern at point x and at time t, p = p(x,t) is the unknown pressure, and g(x)= ( g 1 (x),g 2 (x),g 3 (x) ) is a given external force. The positive parameter ν is the kine- matic viscosity of the fluid. 2000 Mathematics Subject Classification. Primary: 35Q30, 37L30; Secondary: 76D03, 76F20, 76F55, 76F65. Key words and phrases. 3D Leray-α model, 3D Navier–Stokes system, trajectory attractors. 481