Stability threshold of Couette flow for Boussinesq equations in R²

This paper establishes the asymptotic stability threshold for the Couette flow (y, 0) under the 2D Boussinesq system in R². It was proved that for initial perturbations in Sobolev spaces with controlled low horizontal frequencies, the stability threshold is at most \1{3+, 23+\}, extending the known threshold results from the periodic case Tₓ Rᵧ to the whole space. The core innovations are twofold: First, the Dₓ^-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 M₃ that effectively absorbs the |Dₓ|^1/3 derivative structure induced by the temperature equation while handling nonlinear echo cascades.