effect), however, the high regularity requirement will come as a quantitative requirement that the data be $L^2$ close to Gevrey class (see Theorem \[thm:Main\] below). Specifically, Gevrey-$\frac{1}{s}$ with [$s \in(1/2,1)$]{} data with a distance in $L^2$ that diminishes as a function of Reynolds number, eventually collapsing to a neighborhood in Gevrey-$\frac{1}{s}$ in the inviscid limit (consistent with [@BM13]). In view of the regularity requirement needed to obtain inviscid damping in 2D Euler, the type of initial data we consider is natural as the $L^2$ piece must be small enough to have negligible influence above the dissipative length scale (although it is not clear whether or not the diameter of the $L^2$ neighborhood in is optimal). The Gevrey-$\frac{1}{s}$ norm is given by $${\left\lVert f \right\rVert}_{{\mathcal{G}}^{\lambda;s}}^2 = \sum_{k}\int {\left\vert\hat{f}_k(\eta)\right\vert}^2 e^{2\lambda{\left\vertk,\eta\right\vert}^s} d\eta$$ where $\hat{f}_k(\eta)$ is the Fourier transform of $f$ at frequency $(k,\eta)$.

Statement of Theorem {#sec:Statement}
--------------------

We first state the main result, and then follow the statement by remarks, a discussion and a brief summary of the proof.

\[thm:Main\] For all $s \in (1/2,1)$, $\lambda > \lambda^\prime > 0$, $\delta > 0$ and all integers $\alpha \geq 1$ there exists $\epsilon_0 = \epsilon_0(\alpha,s,\lambda,\lambda^\prime,\delta)$ and $K_0 = K_0(s,\lambda,\lambda^\prime,\delta)$ such that if $\omega_{in}^\nu$ 