Optimal orders of convergence for the Fourier-Galerkin method for two-
dimensional Navier-Stokes equations in various energy norms and in L(p
)-norms of the velocity field are proved. Optimal orders of convergenc
e for the Fourier-collocation method are also proved. Extensions of th
ese results to three-dimensional Navier-Stokes equations in the origin
al form and the rotational form are presented. Semigroup formulations
and the variation of constants formula are the essential tools in the
analysis.