Zero dissipation limit of the anisotropic Boussinesq equations with Navier-slip and Neumann boundary conditions

Abstract

In this paper, we study the vanishing dissipation limit of the 2D anisotropic Boussinesq equations with the Navier-slip boundary condition for velocity field and the fixed flux boundary condition for temperature in the upper half plane. By constructing boundary layer correctors to compensate for the discrepancies between dissipative equations and non-dissipative equations at the boundary, we prove that the solutions of the anisotropic Boussinesq equations converge to the solutions of the non-dissipative Boussinesq equations in $L^2$-norm. Particularly, we find that the anisotropic dissipation coefficients only affect the rate of convergence, which is different from the phenomenon of the Dirichlet problem of the anisotropic Boussinesq equations in Wang & Xu (2021).