Suppression of Fluid Echoes and Sobolev Stability Threshold for 2D Dissipative Fluid Equations Around Couette Flow

TL;DR

This paper investigates the stability of 2D dissipative fluid equations near Couette flow, introducing bounds that connect nonlinear stability to linear analysis, and improves stability thresholds for several fluid models.

Contribution

It provides a unified approach to relate nonlinear stability to linear stability analysis for various fluid equations around Couette flow.

Findings

Sobolev stability threshold for Boussinesq equations improved to 1/3.

Stability threshold for MHD equations improved to slightly above 1/3.

Bound for nonlinear interactions reduces stability proof to linear analysis.

Abstract

We study the Sobolev stability thresholds of 2d dissipative fluid equations around Couette flow on the domain $\mathbb T\times \mathbb R$. We prove a bound for general nonlinear interactions, which, for several fluid equations, reduces the proof of nonlinear stability to a linear stability analysis. We apply this approach to the examples of Navier-Stokes, Boussinesq and magnetohydrodynamic equations around Couette flow. This improves the Sobolev stability threshold for the Boussinesq equations around Couette flow and large affine temperature to $1/3$ and for the MHD equations around Couette flow and constant magnetic field to $1/3^+$.