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.
Categories: math.AP