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This paper establishes the nonlinear stability of the Couette flow for the 2DBoussinesq equations with only vertical dissipation. The Boussinesq equationsconcerned here model buoyancy-driven fluids such as atmospheric andoceanographic flows. Due to the presence of the buoyancy forcing, the energy ofthe standard Boussinesq equations could grow in time. It is the enhanceddissipation created by the linear non-self-adjoint operator $y\partial_x-\nu\partial_{yy}$ in the perturbation equation that makes the nonlinearstability possible. When the initial perturbation from the Couette flow $(y,0)$ is no more than the viscosity to a suitable power (in the Sobolev space$H^b$ with $b>\frac43$), we prove that the solution of the 2D Boussnesq systemwith only vertical dissipation on $\mathbb T\times \mathbb R$ remains close tothe Couette at the same order. A special consequence of this result is thestability of the Couette for the 2D Navier-Stokes equations with only verticaldissipation.

Stability of Couette flow for 2D Boussinesq system with vertical dissipation