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

In this paper, 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. When the initial perturbation from the Poiseuille flow (1 − y2, 0) is no more than the viscosity to a suitable power, we prove that the solution of the 2D Boussinesq system on T×R remains close to the Poiseuille flow at the same order.