Stability and Exponential Decay for the 2D Anisotropic Navier–Stokes Equations with Horizontal Dissipation

The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near Boussinesq systems helps gain insight into certain weather phenomena. 2D system focused here anisotropic involves only horizontal dissipation thermal diffusion. Due to lack vertical dissipation, precise large-time behavior problem difficult. When spatial domain $\mathbb R^2$, Sobolev setting remains open. T\times \mathbb R$, this paper solves specifies perturbation. By decomposing velocity $u$ temperature $\theta$ average $(\bar u, \bar\theta)$ corresponding oscillation $(\widetilde \widetilde \theta)$, deriving various inequalities, we are able establish global space $H^2$. In addition, prove that \theta)$ decays exponentially zero $H^1$ $(u, converges \bar\theta)$. This result reflects stratification phenomenon buoyancy-driven fluids.