Approximations Of The Stochastic 3d Navier-Stokes Equations With Damping

The stochastic three-dimensional Navier-Stokes equation with damping is considered in this paper. We show that solutions of three-dimensional stochastic Navier-Stokes equation with damping driven by Brownian motion can be approximated by three-dimensional stochastic Navier-Stokes equation with damping driven by pure jump noise/random kicks on the spaces D([0, T], V) and D([0, T], H) for 3 < beta < 5 with any alpha > 0 and alpha >= 1/4 as beta = 3.