Asymptotic stability for two-dimensional Boussinesq systems around the Couette flow in a finite channel

In this paper, we study the asymptotic stability for the two-dimensional Navier-Stokes Boussinesq system around the Couette flow with small viscosity ν and small thermal diffusion μ in a finite channel. In particular, we prove that if the initial velocity and initial temperature (vin,ρin) satisfies ‖vin−(y,0)‖Hx,y2≤ε0min⁡{ν,μ}12 and ‖ρin−1‖Hx1Ly2≤ε1min⁡{ν,μ}1112 for some small ε0,ε1 independent of ν,μ, then for the solution of the two-dimensional Navier-Stokes Boussinesq system, the velocity remains within O(min⁡{ν,μ}12) of the Couette flow, and approaches to Couette flow as t→∞; the temperature remains within O(min⁡{ν,μ}1112) of the constant 1, and approaches to 1 as t→∞.