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On the topological size of the class of Leray solutions with algebraic decay (2307.10630v1)

Published 20 Jul 2023 in math.AP

Abstract: In 1987, Michael Wiegner in his seminal paper [17] provided an important result regarding the energy decay of Leray solutions $\boldsymbol u(\cdot,t)$ to the incompressible Navier-Stokes in $\mathbb{R}{n}$: if the associated Stokes flows had their $\hspace{-0.020cm}L{2}\hspace{-0.050cm}$ norms bounded by $O(1 + t){-\;!\alpha} $ for some $ 0 < \alpha \leq (n+2)/4 $, then the same would be true of $ |\hspace{+0.020cm} \boldsymbol u(\cdot,t) \hspace{+0.020cm} |{L{2}(\mathbb{R}{n})} $. The converse also holds, as shown by Z.Skal\'ak [15] and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier-Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two-sided bounds of the form $(1+t){-\alpha}\lesssim | \boldsymbol u(\cdot,t)|{L2(\mathbb{R}n)} \lesssim (1+t){-\alpha}$.

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