Stability threshold of the Couette flow in 2D Navier-Stokes Boussinesq equations on $\mathbb{T} \times [-1, 1]$ and $\mathbb{R} \times [-1, 1]$

In this paper, we study the transition threshold problem for the 2D Navier-Stokes Boussinesq equations around the Couette flow with a constant background temperature in two domains: $\mathbb{T} \times [-1, 1]$ and $\mathbb{R} \times [-1, 1]$. It was proved that if the sizes of initial perturbation