The vanishing diffusivity limit for the 2-D Boussinesq equations with boundary effect

In this paper, we tackle the issue of the vanishing diffusivity limit of the 2-D incompressible Boussinesq equations in the half plane. Our main purpose is to study the boundary layer effect and the convergence rates as the thermal diffusion parameter ϵϵ goes to zero. Under the homogeneous Dirichlet boundary condition of velocity and the nonhomogeneous Dirichlet boundary condition in the xx-direction for temperature, we show that the boundary layer thickness is of the order O(ϵβ)O(ϵβ) with (0<β<23). In contrast with Jiang et al. (2011), the BL-thickness we got is thinner than that in Jiang et al. (2011). Moreover, we prove that as diffusivity tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion equations. We also obtain the convergence rates of the vanishing diffusivity limit.