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

Abstract

In 1987, Michael Wiegner in his seminal paper [17] provided an important result regarding the energy decay of Leray solutions $\mathbf{u}(\cdot ,t)$ to the incompressible Navier-Stokes in ${\mathbb{R}}^n$: if the associated Stokes flows had their $L^2$ norms bounded by $O(1+t{)}^{-\alpha }$ for some $0<\alpha \le (n+2)/4$, then the same would be true of $\Vert \mathbf{u}(\cdot ,t){\Vert }_{L^2({\mathbb{R}}^n)}$. The converse also holds, as shown by Z.Skalák [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 $(1+t{)}^{-\alpha }\lesssim \Vert \mathbf{u}(\cdot ,t){\Vert }_{L^2({\mathbb{R}}^n)}\lesssim (1+t{)}^{-\alpha }$.

Bulletin of the London Mathematical Society, In press