of a sufficiently smooth function and a small (with respect to ${\textbf{Re}}^{-1}$) $L^2$ function.

Introduction
============

Consider the 2D incompressible Navier-Stokes equations in the vicinity of the Couette flow: $$\label{def:NSE}
\left\{
\begin{array}{l}
  \omega_t + y\partial_x\omega + U \cdot {\nabla}\omega = \nu\Delta \omega, \\ 
  U  = {\nabla}^{\perp}\Delta^{-1} \omega,  \\\omega(t=0) =\omega_{in}. 
\end{array}
\right.$$ Here, $t\in {\mathbb R}_+$, $(x,y) \in \mathbb {\mathbb T}\times {\mathbb R}$, ${\nabla}^\perp = (-\partial_y,\partial_x)$, $U= (U^x,U^y)$ is the velocity, and $\omega$ is the vorticity of the [*perturbation*]{} to the Couette flow. The physical velocity is $(y,0) + U$ and the total vorticity of the flow is $-1 + \omega$. We denote the streamfunction of the perturbation by $\psi = \Delta^{-1}\omega$. We have already non-dimensionalized by normalizing the Couette flow to $(y,0)$ and the side-length of the torus to $2\pi$, which together with the kinematic viscosity of the fluid, sets the Reynolds number ${\textbf{Re}}$. We use the notation $\nu = \textbf{Re}^{-1}$. If $\nu = 0$, equivalently $\textbf{Re} = \infty$, the fluid is called *inviscid* and corresponds to the incompressible 2D Euler equations. The velocity satisfies the momentum equation: $$\label{def:NSEMomentum}
\left\{
\begin{array}{l}
  U_t + y\partial_x U + (U^y,0) + U \cdot {\nabla}U = -{\nabla}P + \nu \Delta U, \\ 
  {\nabla}\cdot U 