Stability threshold of the Poiseuille flow in 2D Navier-Stokes Boussinesq equations

We study the transition threshold problem for the 2D Navier–Stokes Boussinesq equations around the Poiseuille flow on three domains: $\mathbb{T}\times \mathbb{R}$, $\mathbb{T}\times [0,\mathrm{\infty })$, and $\mathbb{T}\times [-1,1]$. More precisely, we prove that if the initial perturbations of the vorticity and the temperature are assumed size $O(\min \{{\nu }^{\text{fr}34},{\mu }^{\text{fr}34},{\nu }^{-\text{fr}14}\mu \})$ and $O(\min \{{\nu }^{\text{fr}94},{\mu }^{\text{fr}94}\})$ in an anisotropic Sobolev space respectively, then for the solution to this system, its size remains so for all time, and the nonlinear enhanced dissipation persists with a rate proportional to $O(\min \{{\nu }^{\text{fr}12},{\mu }^{\text{fr}12}\})$.