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 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].

26 pages

References in corpus (2)

• Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow
• Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity