Linear and nonlinear stability for the $3$D stratified Boussinesq equations with the horizontal viscosity and diffusivity

In this manuscript, we consider the $3$D Boussinesq equations for stably stratified fluids with the horizontal viscosity and thermal diffusivity and investigate the large time behavior of the solutions. Making use of the anisotropic Littlewood--Paley theory, we obtain their precise $L^1$-$L^p$ decay estimates, which provide us information on both the anisotropic and dispersive structure of the system. More precisely, we reveal that the dispersion from the skew symmetric terms of stratification makes the decay rates of some portions of the solutions faster and furthermore the third component of the velocity field exhibit the enhanced dissipative effect, which provides the additional fast decay rate.