Stability threshold for 2D shear flows of the Boussinesq system near Couette

In this paper, we consider the nonlinear stability for the shear flows of the Boussinesq system in a domain T×R. We prove the nonlinear stability of the shear flow (US,ΘS)=((eνt∂yyU(y),0)⊤,αy) with U(y) close to y and α ≥ 0 in Sobolev spaces for the following two cases: (i) α ≥ 0 is small scaling with the viscosity coefficients and initial perturbation ≲min{ν,μ}1/2 and (ii) α > 0 is not small, the heat diffusion coefficient μ is fixed, and initial perturbation ≲ν1/2.