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

Abstract

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 [1], even though we can obtain the existence of a global attractor for $r≥ 2$ by the multi-valued semi-flow, 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 $\ell$-trajectories in this paper.