\leq \epsilon\end{aligned}$$ where $$\begin{aligned}
T_{\lambda,\lambda^\prime} = \min\left\{ \frac{\lambda - \lambda'}{C_0}, 1\right\} ,
\label{eq:T:lambda}\end{aligned}$$ for some sufficiently large universal constant $C_0 >0 $.

The proof consists in fours steps:

-   We propagate the low-frequency control of the solution, which is encoded in the $L^2$ energy estimate on $u^\nu$.

-   We solve the Navier-Stokes equations perturbed about the Couette flow with initial datum $\omega^{\nu}_{S,in}$ and establish that the solution $\omega^{\nu}_{S}(t)$ lies in ${\mathcal{G}}^{\frac{4\lambda+\lambda'}{5} ;s} \cap {\mathcal{G}}^{\sqrt{\nu} t; 1}$ for all $t \leq T_2 \lesssim \nu^{1+\delta} $, with a norm less than $2 \epsilon^\prime$.

-   We solve the perturbation of around the solution $\omega^{\nu}_S$, with initial datum $\omega^{\nu}_{R,in}$ and show that the corresponding solution lies in ${\mathcal{G}}^{\sqrt{\nu} t; 1}$ for all $t \leq T_2$, with a suitable control on the norm. In view of the smallness of the initial datum $\omega^{\nu}_{R,in}$ in $L^2$ it follows that the ${\mathcal{G}}^{\frac{4\lambda+\lambda'}{5};s}$ norm of $\omega^{\nu}_{R}$ is also less than $2 \epsilon^\prime$.

-   We propagate the Gevrey-class regularity of the solution to with initial datum $\omega^{\nu}_{S}(T_2) + \omega^{\nu}_{R}(T_2) \in {\mathcal{G}}^{\frac{4\lambda+\lambda'}{5};s}$ of size $\leq 4 \epsilon^\prime$ and show that the corresponding solution $\omega^\nu$ at time $T_{\lambda,\lambda^\prime}$ obeys the bound .

For the remainder of the proof, for ease of 