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

TL;DR

This paper analyzes the linearized 2D Navier-Stokes equations around Couette flow in a channel, demonstrating inviscid damping and enhanced dissipation effects for boundary layer vorticity, with results valid in the vanishing viscosity limit.

Contribution

It introduces new results on inviscid damping of boundary layer vorticity and quantifies enhanced dissipation effects in the linearized setting, especially for boundary-supported initial data.

Findings

Inviscid damping of boundary layer vorticity is established.

Enhanced dissipation occurs at an exponential rate depending on viscosity and frequency.

Boundary layer vorticity estimates are independent of viscosity for certain initial data.

Abstract

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-slip boundary conditions in the vanishing viscosity $\nu \to 0$ limit. We split the vorticity evolution into the free evolution (without a boundary) and a boundary corrector that is exponentially localized to at most an $O(\nu^{1/3})$ boundary layer. If the initial vorticity perturbation is supported away from the boundary, we show inviscid damping of both the velocity and the vorticity associated to the boundary layer. For example, our $L^2_t L^1_y$ estimate of the boundary layer vorticity is independent of $\nu$, provided the initial data is $H^1$. For $L^2$ data, the loss is only logarithmic in $\nu$. Note both such estimates are false for the vorticity in the interior. To the authors' knowledge, this inviscid decay of the boundary layer vorticity seems to be a new observation not previously isolated in the literature. Both velocity and vorticity satisfy the expected $O(\exp(-\delta\nu^{1/3}\alpha^{2/3}t))$ enhanced dissipation in addition to the inviscid damping. Similar, but slightly weaker, results are obtained also for $H^1$ data that is against the boundary initially. For $L^2$ data against the boundary, we at least obtain the boundary layer localization and enhanced dissipation.