数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (4): 1466-1486.doi: 10.1007/s10473-024-0415-5

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THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE

Saiguo Xu1, Zhong Tan2,3,*   

  1. 1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
    2. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China;
    3. Shenzhen Research Institute of Xiamen University, Shenzhen 518057, China
  • 收稿日期:2022-12-05 修回日期:2023-04-16 出版日期:2024-08-25 发布日期:2024-08-30

THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE

Saiguo Xu1, Zhong Tan2,3,*   

  1. 1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
    2. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China;
    3. Shenzhen Research Institute of Xiamen University, Shenzhen 518057, China
  • Received:2022-12-05 Revised:2023-04-16 Online:2024-08-25 Published:2024-08-30
  • Contact: *E-mail: tan85@xmu.edu.cn
  • About author:E-mail: xsgsxx@126.com
  • Supported by:
    This work was supported by National Natural Science Foundation of China (12071391, 12231016) and the Guangdong Basic and Applied Basic Research Foundation (2022A1515010860).

摘要: This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena. The Boussinesq system focused on here is anisotropic, and involves only horizontal dissipation and thermal damping. In the 2D case $\mathbb{R}^2$, due to the lack of vertical dissipation, the stability and large-time behavior problems have remained open in a Sobolev setting. For the spatial domain $\mathbb{T}\times\mathbb{R}$, this paper solves the stability problem and gives the precise large-time behavior of the perturbation. By decomposing the velocity $u$ and temperature $\theta$ into the horizontal average $(\bar{u},\bar{\theta})$ and the corresponding oscillation $(\tilde{u},\tilde{\theta})$, we can derive the global stability in $H^2$ and the exponential decay of $(\tilde{u},\tilde{\theta})$ to zero in $H^1$. Moreover, we also obtain that $(\bar{u}_2,\bar{\theta})$ decays exponentially to zero in $H^1$, and that $\bar{u}_1$ decays exponentially to $\bar{u}_1(\infty)$ in $H^1$ as well; this reflects a strongly stratified phenomenon of buoyancy-driven fluids. In addition, we establish the global stability in $H^3$ for the 3D case $\mathbb{R}^3$.

关键词: Boussinesq equations, partial dissipation, stability, decay

Abstract: This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena. The Boussinesq system focused on here is anisotropic, and involves only horizontal dissipation and thermal damping. In the 2D case $\mathbb{R}^2$, due to the lack of vertical dissipation, the stability and large-time behavior problems have remained open in a Sobolev setting. For the spatial domain $\mathbb{T}\times\mathbb{R}$, this paper solves the stability problem and gives the precise large-time behavior of the perturbation. By decomposing the velocity $u$ and temperature $\theta$ into the horizontal average $(\bar{u},\bar{\theta})$ and the corresponding oscillation $(\tilde{u},\tilde{\theta})$, we can derive the global stability in $H^2$ and the exponential decay of $(\tilde{u},\tilde{\theta})$ to zero in $H^1$. Moreover, we also obtain that $(\bar{u}_2,\bar{\theta})$ decays exponentially to zero in $H^1$, and that $\bar{u}_1$ decays exponentially to $\bar{u}_1(\infty)$ in $H^1$ as well; this reflects a strongly stratified phenomenon of buoyancy-driven fluids. In addition, we establish the global stability in $H^3$ for the 3D case $\mathbb{R}^3$.

Key words: Boussinesq equations, partial dissipation, stability, decay

中图分类号: 

  • 35Q35