  We consider the 2D, incompressible Navier-Stokes equations near the Couette
flow, $\omega^{(NS)} = 1 + \epsilon \omega$, set on the channel $\mathbb{T}
\times [-1, 1]$, supplemented with Navier boundary conditions on the
perturbation, $\omega|_{y = \pm 1} = 0$. We are simultaneously interested in
two asymptotic regimes that are classical in hydrodynamic stability: the long
time, $t \rightarrow \infty$, stability of background shear flows, and the
inviscid limit, $\nu \rightarrow 0$ in the presence of boundaries. Given small
($\epsilon \ll 1$, but independent of $\nu$) Gevrey 2- datum,
$\omega_0^{(\nu)}(x, y)$, that is supported away from the boundaries $y = \pm
1$, we prove the following results: \begin{align*} & \|\omega^{(\nu)}(t) -
\frac{1}{2\pi}\int \omega^{(\nu)}(t) dx \|_{L^2} \lesssim \epsilon e^{-\delta
\nu^{1/3} t}, & \text{(Enhanced Dissipation)} \\ & \langle t \rangle
\|u_1^{(\nu)}(t) - \frac{1}{2\pi} \int u_1^{(\nu)}(t) dx\|_{L^2} + \langle t
\rangle^2 \|u_2^{(\nu)}(t)\|_{L^2} \lesssim \epsilon e^{-\delta \nu^{1/3} t}, &
\text{(Inviscid Damping)} \\ &\| \omega^{(\nu)} - \omega^{(0)} \|_{L^\infty}
\lesssim \epsilon \nu t^{3+\eta}, \quad\quad t \lesssim \nu^{-1/(3+\eta)} &
\text{(Long-time Inviscid Limit)} \end{align*} This is the first nonlinear
asymptotic stability result of its type, which combines three important
physical phenomena at the nonlinear level: inviscid damping, enhanced
dissipation, and long-time inviscid limit in the presence of boundaries. The
techniques we develop represent a major departure from prior works on nonlinear
inviscid damping as physical space techniques necessarily play a central role.
In this paper, we focus on the primary nonlinear result, while tools for
handling the linearized parabolic and elliptic equations are developed in our
separate, companion work.
