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ABSTRACT. Using the dyadic decomposition of Littlewood-Paley, we find a simple condition that, when tested on an abstract Banach space X, guarantees the existence and uniqueness of a local strong solution v(t) G C([0,T); X) of the Cauchy problem for the Navier-Stokes equations in E 3 . Many examples of such Banach spaces are offered. We also prove some regularity results on the solution v(t) and we illustrate, by means of a counterexample, that the above-mentioned sufficient condition is, in general, not necessary.

We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes L~vy's midpoint displacement technique which is used to generate Brownian motion. The low-frequency terms in the expansion involve an independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections. We also obtain a way of constructing an arbitrary number of non-Gaussian continuous time processes whose second order properties are the same as those of fractional Brownian motion.Math Subject Classifications. Primary 60G18; secondary 41A58, 60F15.