On the topological size of the class of Leray solutions with algebraic decay

TL;DR

This paper explores the algebraic decay rates of Leray solutions to the Navier-Stokes equations, establishing equivalences and generic bounds for their energy decay in the context of fluid dynamics.

Contribution

It provides a simplified proof of the equivalence between decay rates of Stokes flows and Leray solutions, and demonstrates that algebraic decay solutions typically satisfy two-sided bounds.

Findings

Equivalence between decay rates of Stokes flows and Leray solutions.

Leray solutions with algebraic decay generally satisfy two-sided bounds.

The results clarify the genericity of algebraic decay estimates for Navier-Stokes solutions.

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 $(1+t)^{-\alpha}\lesssim \| \boldsymbol u(\cdot,t)\|_{L^2(\mathbb{R}^n)} \lesssim (1+t)^{-\alpha}$.