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: T times R, T times [0, infty), and T times [-1, 1]. More precisely, we prove that if the initial perturbations of the vorticity and the temperature are assumed size O(min{ν^{fr34},μ^{fr34},ν^{-fr14}μ}) and O(min{ν^{fr94},μ^{fr94}}) 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{ν^{fr12},μ^{fr12}}).