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Littlewood–Paley decomposition and Navier–Stokes equations

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. 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.