Transition threshold of Couette flow for 2D Boussinesq equations

TL;DR

This paper establishes the stability threshold for 2D Boussinesq equations near Couette flow, showing conditions under which the flow remains stable, aligning with known thresholds for Navier-Stokes equations and emphasizing the sharpness of initial data regularity.

Contribution

It proves the stability threshold for 2D Boussinesq equations around Couette flow, extending understanding of flow stability in relation to viscosity and thermal diffusivity.

Findings

Stability threshold $eta o 1/3$ for the equations.

Conditions for asymptotic stability involving viscosity and diffusivity.

Consistency with optimal stability thresholds for Navier-Stokes.

Abstract

In this paper, we prove the stability threshold of $\alpha\leq \frac13$ for 2D Boussinesq equations around the Couette flow in $\mathbb{T}\times \mathbb{R}$ with Richardson number $\gamma^2>\frac14$ and different viscosity $\nu$ and thermal diffusivity $\mu$. More precisely, if $\|v_{in}-(y,0)\|_{H^{s+1/2}}+ \|\rho_{in}+\gamma^2 y-1\|_{H^{s+1/2}}\leq c(\min\{\nu,\mu\})^{1/3}$, $\frac{\nu+\mu}{2\gamma\sqrt{\nu \mu} }< 2-\varepsilon$, $s>3/2$, then the asymptotic stability holds. This stability threshold is consistent with the optimal stability threshold for the 2D Navier-Stokes equations in Sobolev space. And in the sense of inviscid damping effect, the regularity assumption of the initial data should be sharp.