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 and small thermal diffusion, respectively. More precisely, we prove that if the initial velocity and initial temperature satisfies||u₀- (1-y², 0) ||₇^₇{₂+} c₀, ^2{3} and ||₀||₇℉+|||Dₓ|^1{8}₀||₇℉ c₁, ^{31{24}} for some small constants c₀ and c₁ which are both independent of, , then we can reach the conclusion that the velocity remains within O (, ^2{3}) of the Poiseuille flow; the temperature remains O (, ^31{24}) of the constant 0, and approaches to 0 as t.