Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations

Stability problem on perturbations near the hydrostatic balance is one of the important issues for Boussinesq equations. This paper focuses on the asymptotic stability and large-time behavior problem of perturbations of the 2D fractional Boussinesq equations with only fractional velocity dissipation or fractional thermal diffusivity. Since the linear portion of the Boussinesq equations plays a crucial role in the stability properties, we firstly study the linearized fractional Boussinesq equations with only fractional velocity dissipation or fractional thermal diffusivity and complete the following work: 1) assessing the stability and obtaining the precise large-time asymptotic behavior for solutions to the linearized system satisfied the perturbation; 2) understanding the spectral property of the linearization; 3) showing the $ H^2 $-stability for the linearized system, and prove that the $ L^2 $-norm of $ \nabla{u} $ and $ \Delta{u} $ (or $ \nabla\theta $ and $ \Delta\theta $), the $ L^\varrho $-norm $ (2<\varrho<\infty) $ of $ u $ and $ \nabla{u} $ (or $ \theta $ and $ \nabla\theta $) are all approaching to zero as $ t\rightarrow\infty $ when $ \alpha = 1 $ and $ \eta = 0 $ (or $ \nu = 0 $ and $ \beta = 1 $). Secondly, we obtain the $ H^1 $-stability for the full nonlinear system and prove the $ L^\varrho $-norm $ (2<\varrho<\infty) $ of $ \theta $ and the $ L^2 $-norm of $ \nabla\theta $ approaching to zero as $ t\rightarrow\infty $.