The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation
This paper studies a special 2D anisotropic incompressible Boussinesq equation in $ {\mathbb{T}}^2 $ with $ \mathbb{T} = [-\frac{1}{2}, \frac{1}{2}] $ being a 1D periodic box. The system concerned here possesses vertical dissipation only in the vertical component of the velocity and vertical heat di...
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AIMS Press
2025-02-01
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| Series: | Communications in Analysis and Mechanics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/cam.2025005 |
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| author | Hongxia Lin Sabana Qing Sun Ruiqi You Xiaochuan Guo |
| author_facet | Hongxia Lin Sabana Qing Sun Ruiqi You Xiaochuan Guo |
| author_sort | Hongxia Lin |
| collection | DOAJ |
| description | This paper studies a special 2D anisotropic incompressible Boussinesq equation in $ {\mathbb{T}}^2 $ with $ \mathbb{T} = [-\frac{1}{2}, \frac{1}{2}] $ being a 1D periodic box. The system concerned here possesses vertical dissipation only in the vertical component of the velocity and vertical heat diffusion. When the buoyancy forcing is not present, the 2D Boussinesq equation is a 2D Navier-Stokes equation with vertical dissipation only in the vertical component. The stability and large-time behavior problem on the solutions to the 2D Navier-Stokes equation with only vertical or horizontal dissipation remains unknown. When coupled with the temperature, the global regularity to the system with vertical dissipation and vertical diffusion in $ {\mathbb{R}}^2 $ has been solved by Cao and Wu (Arch. Ration. Mech. Anal., 208(2013), 985-1004). The stability with horizontal dissipation and horizontal diffusion in the periodic domain $ \mathbb{T} \times \mathbb{R} $ has also been established by Dong, Wu, Xu, and Zhu (Calc. Var. Partial Differential Equations, 60(2021)) recently. Now whether the solution of the 2D system remains stable has yet to be solved when the velocity has vertical dissipation only in the $ u_2 $ equation. This paper aims to solve the problem and investigates the stability and large-time behavior of the solution to the special 2D Boussinesq equations on perturbations near the hydrostatic equilibrium. The basic idea here is to decompose the physical quantity $ f $ into its horizontal average, vertical average, and their corresponding oscillations. By establishing the strong Poincaré-type inequalities and several anisotropic inequalities related to the oscillations, we are able to obtain $ H^2 $-stability of the solution under the assumptions that the initial data is sufficiently small and obeys some symmetries. Furthermore, the exponential decay rates for the oscillation parts in $ H^1 $ are also established. |
| format | Article |
| id | doaj-art-6f7afe407e074668bd9fcaf8fe73f12b |
| institution | DOAJ |
| issn | 2836-3310 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Communications in Analysis and Mechanics |
| spelling | doaj-art-6f7afe407e074668bd9fcaf8fe73f12b2025-08-20T03:17:43ZengAIMS PressCommunications in Analysis and Mechanics2836-33102025-02-0117110012710.3934/cam.2025005The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipationHongxia Lin0Sabana1Qing Sun2Ruiqi You3Xiaochuan Guo4School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, P.R. ChinaSchool of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, P.R. ChinaSchool of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, P.R. ChinaSchool of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, P.R. ChinaSchool of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, P.R. ChinaThis paper studies a special 2D anisotropic incompressible Boussinesq equation in $ {\mathbb{T}}^2 $ with $ \mathbb{T} = [-\frac{1}{2}, \frac{1}{2}] $ being a 1D periodic box. The system concerned here possesses vertical dissipation only in the vertical component of the velocity and vertical heat diffusion. When the buoyancy forcing is not present, the 2D Boussinesq equation is a 2D Navier-Stokes equation with vertical dissipation only in the vertical component. The stability and large-time behavior problem on the solutions to the 2D Navier-Stokes equation with only vertical or horizontal dissipation remains unknown. When coupled with the temperature, the global regularity to the system with vertical dissipation and vertical diffusion in $ {\mathbb{R}}^2 $ has been solved by Cao and Wu (Arch. Ration. Mech. Anal., 208(2013), 985-1004). The stability with horizontal dissipation and horizontal diffusion in the periodic domain $ \mathbb{T} \times \mathbb{R} $ has also been established by Dong, Wu, Xu, and Zhu (Calc. Var. Partial Differential Equations, 60(2021)) recently. Now whether the solution of the 2D system remains stable has yet to be solved when the velocity has vertical dissipation only in the $ u_2 $ equation. This paper aims to solve the problem and investigates the stability and large-time behavior of the solution to the special 2D Boussinesq equations on perturbations near the hydrostatic equilibrium. The basic idea here is to decompose the physical quantity $ f $ into its horizontal average, vertical average, and their corresponding oscillations. By establishing the strong Poincaré-type inequalities and several anisotropic inequalities related to the oscillations, we are able to obtain $ H^2 $-stability of the solution under the assumptions that the initial data is sufficiently small and obeys some symmetries. Furthermore, the exponential decay rates for the oscillation parts in $ H^1 $ are also established.https://www.aimspress.com/article/doi/10.3934/cam.20250052d anisotropic boussinesq equationspartial dissipationstabilityvertical thermal diffusion |
| spellingShingle | Hongxia Lin Sabana Qing Sun Ruiqi You Xiaochuan Guo The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation Communications in Analysis and Mechanics 2d anisotropic boussinesq equations partial dissipation stability vertical thermal diffusion |
| title | The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation |
| title_full | The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation |
| title_fullStr | The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation |
| title_full_unstemmed | The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation |
| title_short | The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation |
| title_sort | stability and decay of 2d incompressible boussinesq equation with partial vertical dissipation |
| topic | 2d anisotropic boussinesq equations partial dissipation stability vertical thermal diffusion |
| url | https://www.aimspress.com/article/doi/10.3934/cam.2025005 |
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