seen on the linear level seems very difficult with the current method (we are unsure about whether or not we can expect the same linear decay rate from the nonlinear solution).

Inequality is also true with the $L^2$ norm replaced by $\mathcal{G}^{\lambda^\prime;s}$ with a very similar proof.

For $\nu t^3 \ll 1$, statement (ii) essentially reduces to the main results of [@BM13].

We note that if $\omega^\nu_{R,in}$ were in $H^\sigma$ with $\sigma > 5/2$, instead of merely $L^2$, then one could take $\delta = 0$ in the hypothesis , which is consistent with the instant regularization available from the heat equation.

(150,200)(-100,0) (-100,25)[$t \ll 1$]{} (-20,5)[$t \approx {\textbf{Re}}^{1/3}$]{} (-150,80)[$t \gg {\textbf{Re}}$]{} (-140,55)[$Couette$]{} (-100,90)[$t \gg {\textbf{Re}}^{1/3}$]{} (0,140)[$Shear flows$]{}

(150,200)(-100,0) (-85,15)[$t \ll 1$]{} (-150,80)[$t \gg {\textbf{Re}}$]{} (-140,55)[$Couette$]{} (-100,90)[$t \gg 1$]{} (0,140)[$Shear flows$]{}

Schematics of the qualitative behavior predicted by Theorem \[thm:Main\] are given in Figures \[fig:Strong\] and \[fig:Weak\]. As Theorem \[thm:Main\] predicts essentially the same behavior as the main Theorem in [@BM13] before the enhanced dissipation is activated, it is natural that the proof of Theorem \[thm:Main\] also contains the entire proof of the main theorem in [@BM13]. To avoid being redundant, we simply quote without proof estimates that are proved in 