On the forced Euler and Navier-Stokes equations: Linear damping and modified scattering

TL;DR

This paper investigates the long-term behavior of forced linear Euler and nonlinear Navier-Stokes equations near Couette flow, demonstrating persistence of linear inviscid damping under smooth periodic forcing and exploring stability limitations in Sobolev spaces.

Contribution

It shows that linear inviscid damping persists with smooth forcing and extends these findings to nonlinear Euler and Navier-Stokes equations, highlighting stability nuances.

Findings

Linear inviscid damping persists with smooth periodic forcing.

Stability and scattering fail in Sobolev spaces with s > -1.

Nonlinear Euler and Navier-Stokes behaviors are consistent with linear results.

Abstract

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].