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We study the 2D Navier-Stokes equations linearized around the Couette flow$(y,0)^t$ in the periodic channel $\mathbb T \times [-1,1]$ with no-slipboundary conditions in the vanishing viscosity $\nu \to 0$ limit. We split thevorticity evolution into the free evolution (without a boundary) and a boundarycorrector that is exponentially localized to at most an $O(\nu^{1/3})$ boundarylayer. If the initial vorticity perturbation is supported away from theboundary, we show inviscid damping of both the velocity and the vorticityassociated to the boundary layer. For example, our $L^2_t L^1_y$ estimate ofthe boundary layer vorticity is independent of $\nu$, provided the initial datais $H^1$. For $L^2$ data, the loss is only logarithmic in $\nu$. Note both suchestimates are false for the vorticity in the interior. To the authors'knowledge, this inviscid decay of the boundary layer vorticity seems to be anew observation not previously isolated in the literature. Both velocity andvorticity satisfy the expected $O(\exp(-\delta\nu^{1/3}\alpha^{2/3}t))$enhanced dissipation in addition to the inviscid damping. Similar, but slightlyweaker, results are obtained also for $H^1$ data that is against the boundaryinitially. For $L^2$ data against the boundary, we at least obtain the boundarylayer localization and enhanced dissipation.

Inviscid damping and enhanced dissipation of the boundary layer for 2D Navier-Stokes linearized around Couette flow in a channel