Stability threshold of Couette flow for 2D Boussinesq equations in Sobolev spaces

Abstract

Consider the nonlinear stability of the Couette flow in the Boussinesq equations with vertical dissipation on T×R. We prove that if the initial perturbations uin and θin to the Couette flow vs=(y,0)⊤ and θs=1, respectively, satisfy ‖uin‖HN+1+ν−12‖θin‖HN+ν−13‖|∂x|13θ‖HN≪ν13, N>7, then the resulting solution remains close to the Couette flow in L2 at the same order for all time.