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

In 1987, Michael Wiegner in his seminal paper (J. Lond. Math. Soc. (2), 35 (1987) 303-313) provided an important result regarding the energy decay of Leray solutions u(& BULL;,t)$ \bm{u}(\cdot ,t)$ to the incompressible Navier-Stokes in Rn$ \mathbb {R}<^>{n}$: if the associated Stokes flows had their L2$L<^>{2}$ norms bounded by O(1+t)-& alpha;$ O(1 + t)<^>{-\;\!\alpha }$ for some 0<& alpha;& LE;(n+2)/4$ 0 < \alpha \leqslant (n+2)/4$, then the same would be true of & PAR;u(& BULL;,t)& PAR;L2(Rn)$ \Vert \hspace{0.56917pt} \bm{u}(\cdot ,t) \hspace{0.56917pt} \Vert _{L<^>{2}(\mathbb {R}<^>{n})}$. The converse also holds, as shown by Skalak (J. Math. Fluid Mech. 16 (2014) 431-446) and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier-Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two-sided bounds of the form (1+t)-& alpha;& LSIM;& PAR;u(& BULL;,t)& PAR;L2(Rn)& LSIM;(1+t)-& alpha;$(1+t)<^>{-\alpha }\lesssim \Vert \bm{u}(\cdot ,t)\Vert _{L<^>2(\mathbb {R}<^>n)} \lesssim (1+t)<^>{-\alpha }$.