Abstract
In this paper we establish the local Lyapunov property of certain $\mathrm{L}^{p}$ and Besov norms of the vorticity fields. We have resolved in part, a certain open problem posed by Tosio Kato for the three-dimensional Navier-Stokes equation by studying the vorticity equation. The local dissipativity of the sum of linear and non-linear operators of the vorticity equation is established. One of the main techniques used here is Littlewood-Paley analysis.
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