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 x R. We prove the nonlinear stability of the shear flow (U-S, Theta(S)) = ((e(nu t partial derivative yy) U(y), 0)(?), alpha y) with U(y) close to y and alpha >= 0 in Sobolev spaces for the following two cases: (i) alpha >_ 0 is small scaling with the viscosity coefficients and initial perturbation < min{nu,mu}(1/2) and (ii) alpha > 0 is not small, the heat diffusion coefficient mu is fixed, and initial perturbation <= nu(1/2). Published under an exclusive license by AIP Publishing.