Stability and algebra decay for 2D Boussinesq system with partial horizontal dissipation and horizontal diffusion

This paper is concerned with two dimensional Boussinesq equations involving the horizontal dissipation in the first component of the velocity and horizontal temperature diffusion. Due to the lack of so much dissipation, the stability issue becomes more challenging than that in [10]. When the spatial domain is $ \Omega = T\times R $ with $ T = [0,1] $ being a 1D periodic box, we establish the stability and build the precise large-time behavior of perturbations near the hydrostatic equilibrium. We further prove that the oscillation parts of the velocity and the temperature only share the decay rate as $ (1+t)^{-\frac12} $, which is a different phenomenon from the corresponding results in [10].