Stability threshold of Couette flow for Boussinesq equations in $\mathbb{R}^2$

TL;DR

This paper determines the stability threshold for 2D Boussinesq Couette flow in the whole space, extending previous periodic results, through novel frequency control and multiplier techniques.

Contribution

It introduces new methods for controlling horizontal frequencies and a modified multiplier to analyze stability thresholds in unbounded domains.

Findings

Stability threshold is at most 1/3+ and 2/3+ for initial data in Sobolev spaces.

Extended known periodic results to the whole space $\\mathbb{R}^2$.

Developed new analytical tools for frequency control and nonlinear analysis.

Abstract

This paper establishes the asymptotic stability threshold for the Couette flow $(y,0)$ under the 2D Boussinesq system in $\mathbb{R}^2$. It was proved that for initial perturbations in Sobolev spaces with controlled low horizontal frequencies, the stability threshold is at most $\left\{\frac{1}{3}+, \frac{2}{3}+\right\}$, extending the known threshold results from the periodic case $\mathbb{T}_x \times \mathbb{R}_y$ to the whole space. The core innovations are twofold: First, the $\langle D_x^{-1} \rangle$ control on the initial data simultaneously resolves horizontal frequency singularities and optimizes integral indices when applying Young's convolution inequality. Second, we develop a modified multiplier $\mathcal{M}_3$ that effectively absorbs the $|D_x|^{1/3}$ derivative structure induced by the temperature equation while handling nonlinear echo cascades.