---
author:
- 'Jacob Bedrossian[^1] and Nader Masmoudi[^2] and Vlad Vicol[^3]'
bibliography:
- 'eulereqns\_vlad.bib'
title: 'Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow'
---

Abstract {#abstract .unnumbered}
========

In this work we study the long time, inviscid limit of the 2D Navier-Stokes equations near the periodic Couette flow, and in particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin’s 1887 linear analysis. At high Reynolds number $\textbf{Re}$, we prove that the solution behaves qualitatively like 2D Euler for times $t \lesssim \textbf{Re}^{1/3}$, and in particular exhibits inviscid damping (e.g. the vorticity weakly approaches a shear flow). For times $t \gtrsim \textbf{Re}^{1/3}$, [which is sooner than the natural dissipative time scale $O(\textbf{Re})$]{}, the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by an enhanced dissipation effect. Afterward, the remaining shear flow decays on very long time scales $t \gtrsim \textbf{Re}$ back to the Couette flow. When properly defined, the dissipative length-scale in this setting is $\ell_D \sim \textbf{Re}^{-1/3}$, larger than the scale $\ell_D \sim \textbf{Re}^{-1/2}$ predicted in classical Batchelor-Kraichnan 2D turbulence theory. The class of initial data we study is the sum 