On the Forced Euler and Navier–Stokes Equations: Linear Damping and Modified Scattering

Abstract

We study the asymptotic behavior of the forced linear Euler and nonlinear Navier–Stokes equations close to Couette flow on $$\mathbb {T}\times I$$ T×I. 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 $$H^{s}, s>-1$$ Hs,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 and Masmoudi (Long time instability of the Couette flow in low Gevrey spaces, 2018. arXiv:1803.01246).