Stability threshold of the Couette flow for Navier-Stokes Boussinesq system with large Richardson number $\gamma^2>\frac{1}{4}$

TL;DR

This paper establishes the nonlinear asymptotic stability of the Couette flow in a stratified fluid with a large Richardson number, under small initial perturbations, extending understanding of flow stability in stratified Navier-Stokes Boussinesq systems.

Contribution

The paper proves nonlinear stability of Couette flow for Richardson number greater than 1/4, with explicit conditions on initial perturbations in a stratified Navier-Stokes Boussinesq system.

Findings

Proves nonlinear asymptotic stability for large Richardson number.

Identifies initial perturbation bounds ensuring stability.

Extends stability analysis to stratified flows with strong stratification.

Abstract

In this paper, we study the nonlinear asymptotic stability of the Couette flow in the stably stratified regime, namely the Richardson number $\gamma^2>\frac{1}{4}$. Precisely, we prove that if the initial perturbation $(u_{in},\vartheta_{in})$ of the Couette flow $v_s=(y,0)$ and the linear temperature $\rho_s=-\gamma^2y+1$ satisfies $\|u_{in}\|_{H^{s+1}}+\|\vartheta_{in}\|_{H^{s+2}}\leq \epsilon_0\nu^{\frac{1}{2}}$, then the asymptotic stability holds.