Stability for the Boussinesq Equations with Horizontal Dissipation near the Hydrostatic Balance on ${\mathbb{R}}^2$

The hydrostatic balance is a fundamental equilibrium state in stratified fluids and plays a central role in geophysical fluid dynamics. Understanding its stability under incomplete dissipation is a longstanding challenge, since anisotropic diffusion alone is generally insufficient to control the nonlinear evolution and no robust stabilizing mechanism is known for the corresponding anisotropically dissipative Navier–Stokes equations. In this paper, we investigate the two-dimensional Boussinesq equations on ${\mathbb{R}}^2$ with only horizontal dissipation near the hydrostatic equilibrium $(U,\mathrm{\Theta })=(0,x_2)$. We show that the velocity–temperature coupling generates internal gravity waves whose dispersive decay, together with the horizontal dissipation, provides an effective stabilizing mechanism that compensates for the complete absence of vertical dissipation. This identifies a mechanism by which dispersive wave propagation restores stability in an incompletely dissipative fluid system. For sufficiently small initial perturbations in $H^k({\mathbb{R}}^2)\cap W^{3,1}({\mathbb{R}}^2)$ with $k\ge 14$, we establish the global existence and uniqueness of classical solutions together with explicit anisotropic, componentwise large-time decay rates for the velocity and temperature, including faster decay of the vertical velocity.