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

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 $⟨D_x^{-1}⟩$ 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.