Global well-posedness for the 2D Boussinesq equations with a velocity damping term

In this paper, we prove global well-posedness of smooth solutions to the two-dimensional incompressible Boussinesq equations with only a velocity damping term when the initial data is close to an nontrivial equilibrium state $(0,x_2)$. As a by-product, under this equilibrium state, our result gives a positive answer to the question proposed by [ACWX] (see P.3597).