Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping

The main objective of this paper is to study the existence of a finite dimensional global attractor for the three dimensional Navier-Stokes equations with nonlinear damping for r > 4. Motivated by the idea of [I], even though we can obtain the existence of a global attractor for r >= 2 by the multi-valued semiflow, it is very difficult to provide any information about its fractal dimension. Therefore, we prove the existence of a global attractor in H and provide the upper bound of its fractal dimension by the methods of l-trajectories in this paper.