Stability and optimal temporal decay result for the 3D Boussinesq equations with horizontal dissipation in anisotropic Sobolev spaces

In this paper, we consider 3D anisotropic Boussinesq equations with horizontal dissipation. We prove that, for the perturbed equations, the time global solution exists for small initial data in the anisotropic Sobolev spaces $H^{0,s}$ with $\frac{1}{2}<s$ and the corresponding solution of the unperturbed equations approaches the hydrostatic equilibrium. Moreover, the optimal decay result is obtained in the anisotropic Sobolev spaces $H^{0,s}$, extending the result of isotropic Sobolev spaces.