# Stability of the Couette flow under the 2D steady Navier-Stokes flow

## TL;DR

This paper analyzes the stability of shear flows, specifically Couette and Poiseuille flows, under the 2D stationary Navier-Stokes equations, revealing stability in certain function spaces and instability in others.

## Contribution

It establishes the stability and instability conditions of Couette and Poiseuille flows in various function spaces, highlighting the role of anisotropic cut-off functions.

## Key findings

- Couette flow is stable in ^{1,q} for 1<q<
- Couette flow is unstable in ^{1,}
- Poiseuille flow is stable in ^{1,q} for 4/3<q

## Abstract

In this note, we investigate the stability property of shear flows under the 2D stationary Navier-Stokes equations, and we obtain that the Couette flow $(y,0)$ is stable under the space of $\mathcal{D}^{1,q}(\mathbb{R}^2)$ for any $1<q<\infty$ and unstable in the space of $\mathcal{D}^{1,\infty}(\mathbb{R}^2)$. A key observation is the anisotropic cut-off function. We also consider the Poiseuille flow $(y^2,0)$, which is stable in $\mathcal{D}^{1,q}(\mathbb{R}^2)$ with $\frac43<q\leq4.$