notation we drop the $\nu$ upper index for $\omega$ and $u$. Without loss of generality we may assume the “rough” part of the initial datum $\omega_{R,in}$ is supported on frequencies away from the set $\{ k=0, |\eta|\leq 1\}$. Further, we define $u_S(t) = {\nabla}^{\perp}\Delta^{-1}\omega_S(t)$ and $u_R(t) = {\nabla}^{\perp}\Delta^{-1}\omega_R(t)$, the velocities associated with the smooth and rough solutions (both in $L^2$ initially due to the assumptions on the initial data and the frequency support of $\omega_{R,in}$).

[*Proof of (a).*]{} The standard $L^2$ energy estimate for $u_S$ obtained from yields $$\frac 12 \frac{d}{dt} {\left\lVert u_S \right\rVert}_2^2 +  \nu {\left\lVert \omega_S \right\rVert}_2^2 \leq  \left| \int u^{x}_S u^{y}_S \right| \leq  {\left\lVert P_{\neq 0} u_S \right\rVert}_2^2$$ and therefore $$\begin{aligned}
{\left\lVert u_S(t) \right\rVert}_2 \leq 3 {\left\lVert u_{S,in} \right\rVert}_2 \leq 3 ( {\left\lVert |\nabla|^{-1} \omega_{in} \right\rVert}_2 + {\left\lVert \omega_{S,in} \right\rVert}_2) \leq 6 \epsilon^\prime 
\label{eq:u:S:control}\end{aligned}$$ for all $t \leq 1$. Note that the same bound for the full solution $u$ holds on $[0,1]$.\
[*Proof of (b).*]{} Consider the analytic Fourier multiplier $$\begin{aligned}
M_A(t,k,\eta) &= {\left\langle k,\eta \right\rangle}^\sigma e^{\sqrt{\nu}\; t\; |k,\eta| }\end{aligned}$$ for some $\sigma > 5/2$. As in [@FoiasTemam89], the solution $\omega_S$ of with initial datum $\omega_{S,in}$ obeys the a priori estimate $$\begin{aligned}
 &\frac{1}{2} \frac{d}{dt} {\left\lVert M_A \omega_S 