Convergence of Fourier methods for the Navier-Stokes equations

Optimal orders of convergence for the Fourier-Galerkin method for two-dimensional Navier-Stokes equations in various energy norms and in Lp-norms of the velocity field are proved. Optimal orders of convergence for the Fourier-collocation method are also proved. Extensions of these results to three-dimensional Navier-Stokes equations in the original form and the rotational from are presented. Semigroup formulations and the variation of constants formula are the essential tools in the analysis.