# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021068
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## Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations

 Department of Applied Mathematics, Chengdu University of Technology, Chengdu 610059, P.R. China

* Corresponding author: mllpzh@126.com (Liangliang Ma)

Received  September 2020 Revised  November 2020 Early access March 2021

Fund Project: The research of L. Ma was supported by the National Natural Science Foundation of China (No. 11571243, 11971331), China Scholarship Council (No. 202008515084), Opening Fund of Geomathematics Key Laboratory of Sichuan Province (No. scsxdz2020zd02), and Teacher's development Scientific Research Staring Foundation of Chengdu University of Technology (No. 10912-KYQD2019_07717)

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 $t\rightarrow\infty$.

Citation: Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021068
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