Improved stability threshold of the two-dimensional Couette flow for Navier-Stokes-Boussinesq systems via quasi-linearization

In this paper, we improve the size requirement of the perturbations for the asymptotic stability of the Couette flow in stratified fluids governed by the two-dimensional Navier-Stokes-Boussinesq system. More precisely, the size of perturbed temperature is improved to $ \nu^{2/3} $ from $ \nu^{5/6} $ in the paper of Zhang and Zi [J. Math. Pure. Anal. 179:123-182 (2023)]. The idea is the quasi-linearization. The main system is decomposed into two or more equations: a good equation (might be linear) that carries the regularity and size of the initial data, and some quasi-linear and nonlinear equations that contain the nonlinear part, which start from zero initial data.