slightly restricted subset as the *resonant intervals* $$\begin{aligned}
\mathbf I_{k,\eta} = \left\{
\begin{array}{lr} 
I_{k,\eta} & 2\sqrt{{\left\vert\eta\right\vert}} \leq t_{k,\eta}, \\ 
\emptyset & otherwise.
\end{array} 
\right. \end{aligned}$$ Note this is the same as putting a slightly more stringent requirement on $k$: $k \leq \frac{1}{2}\sqrt{{\left\vert\eta\right\vert}}$.

Proof of Theorem \[thm:Main\] {#sec:Proof}
=============================

Instant regularization and O(1) times
-------------------------------------

A physical aspect of our work is to verify that the instant regularization due to the viscosity replaces the qualitative regularity requirements in [@BM13] with the more physically natural quantitative regularity requirement . Indeed, we have the following result:

\[prop:InstantReg\] For all $\epsilon > 0$, $\lambda > \lambda^\prime > 0$, $\delta>0$, and $s \in (1/2,1)$, there exists an $\epsilon^\prime = \epsilon'(s,\lambda,\lambda^\prime,\delta)$, a $K_0 = K_0(s,\lambda,\lambda^\prime,\delta)$, and a $\nu_0 = \nu_0(s,\lambda,\lambda^\prime,\delta)$ such that if $\omega_{in}^\nu$ is mean-zero and can be written as the sum of a smooth part and a rough part $\omega_{in}^{\nu} = \omega^\nu_{S,in} + \omega^\nu_{R,in}$ with $$\begin{aligned}
{\left\lVert |\nabla|^{-1} \omega^\nu_{in} \right\rVert}_{2} + {\left\lVert \omega^\nu_{S,in} \right\rVert}_{{\mathcal{G}}^{\lambda;s}}  + e^{K_0\nu^{-\frac{(3+\delta)s}{2(1-s)}}}   {\left\lVert \omega^\nu_{R,in} \right\rVert}_{2}  \leq \epsilon^\prime,  
\label{ineq:Instant}\end{aligned}$$ then for all $\nu \leq \nu_0$ the solution $\omega^\nu$ to 2D NSE near Couette flow , with initial datum $\omega_{in}^\nu$ satisfies $$\begin{aligned}
\label{eq:analytic:to:do}
{\left\lVert u^\nu(T_{\lambda,\lambda^\prime}) \right\rVert}_2 + {\left\lVert  \omega^\nu(T_{\lambda,\lambda^\prime})  \right\rVert}_{{\mathcal{G}}^{\frac{3\lambda}{4} + \frac{\lambda^\prime}{4};s}} 