Enhanced dissipation and stability of Poiseuille flow for two-dimensional Boussinesq system

We investigate the nonlinear stability problem for the two-dimensional Boussinesq system around the Poiseuille flow in a finite channel. The system has the characteristic of Navier-slip boundary condition for the velocity and Dirichlet boundary condition for the temperature, with a small viscosity ν and small thermal diffusion μ, respectively. More precisely, we prove that if the initial velocity and initial temperature satisfies||u₀-(1-y²,0) ||_H7/2+≤ c₀min μ,ν 2/3 and ||θ₀||H¹+|||Dₓ|1/8θ₀||H¹≤ c₁min μ,ν ^31/24 for some small constants c₀ and c₁ which are both independent of μ,ν, then we can reach the conclusion that the velocity remains within O( min μ,ν 2/3) of the Poiseuille flow; the temperature remains O( min μ,ν 31/24) of the constant 0, and approaches to 0 as t→∞.