Nonlinear stability of 3-D plane Couette flow for Boussinesq system in a finite channel

In this paper, we study the hydrodynamics stability of the 3-D plane Couette flow $v_s=(y,0,0)$ with constant temperature state ${\rho }_s$ for the Boussinesq system in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$ with non-slip boundary condition in $y$ direction. We prove that if the initial velocity field $v_{in}$ and initial temperature ${\rho }_{in}$ satisfy \begin{align*} |v_{in}-(y,0,0)|{H^2} \leq c_0 ν, \ \ |ρ{in}-ρ_s|_{H^2} \leq c_0 ν^2, \end{align*} for some $c_0>0$ independent of the viscosity and thermal diffusion $\nu$, then the solutions to 3D Boussinesq system do not transit from the Couette flow and constant temperature state. To our best knowledge, this is the first result about the nonlinear stability of plane Couette flow for 3-D Boussinesq system in the domain with non-slip boundary condition. This result, together with [18], implies that the strong boundary layer effect does not lead to strong instability.