The asymptotic behavior and the Euler time discretization analysis are presented for the two-dimensional non-stationary Navier-Stokes problem. If the data ν and lim t→∞ f (t) satisfy a uniqueness condition corresponding to the stationary Navier-Stokes problem, we then obtain the convergence of the non-stationary Navier-Stokes problem to the stationary Navier-Stokes problem and the uniform boundedness, stability and error estimates of the Euler time discretization for the non-stationary Navier-Stokes problem.