DESCRIPTIONS, DISCRETIZATIONS, AND COMPARISONS OF TIME/SPACE COLORED AND WHITE NOISE FORCINGS OF THE NAVIER-STOKES EQUATIONS

We systematically consider the stochastic Navier-Stokes equations with temporal-spatial correlated and uncorrelated noises for given autocovariances. In particular, for temporal colored noise, random processes are used to characterize statistical properties instead of Karhunen-Loeve expansions in a temporal aspect. For each kind of temporal-spatial noises, we present detailed definitions and discussions of the noises and their properties. The proposed techniques are useful for general settings of partial differential equations with colored or white forcing. We then apply the discretized colored/white forcings to facilitate numerical experiments in the context of finite element discretizations and compare the efficiency and regularity features of the system resulting from the experiments.