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 (v_in,ρ_in) satisfies [Formula presented] and [Formula presented] 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 [Formula presented] of the Couette flow, and approaches to Couette flow as t→∞; the temperature remains within [Formula presented] of the constant 1, and approaches to 1 as t→∞.