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.
The existence of singular solutions of the incompressible Navier-Stokes system with singular external forces, the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions (including stationary ones), and a pointwise loss of smoothness for solutions are proved in the same function space of pseudomeasure type. r
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