  We study the asymptotic behavior of the forced linear Euler and nonlinear
Navier-Stokes equations close to Couette flow in a periodic channel. As our
main result we show that for smooth time-periodic forcing linear inviscid
damping persists, i.e. the velocity field (weakly) asymptotically converges.
However, stability and scattering to the transport problem fail in $H^{s},
s>-1$. We further show that this behavior is consistent with the nonlinear
Euler equations and that a similar result also holds for the nonlinear
Navier-Stokes equations. Hence, these results provide an indication that
nonlinear inviscid damping may still hold in Sobolev regularity in the above
sense despite the Gevrey regularity instability results of [Deng-Masmoudi
2018].
