Determination of the 3D Navier-Stokes equations with damping

This paper is concerned the determination of trajectories for the three-dimensional NavierStokes equations with nonlinear damping subject to periodic boundary condition. By using the energy estimate of Galerkin approximated equation, the finite number of determining modes and asymptotic determined functionals have been shown via the Grashof numbers for the non-autonomous and autonomous damped Navier-Stokes fluid flow respectively.