Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion
In this paper, $ n $-dimensional incompressible Boussinesq equations with fractional dissipation and thermal diffusion are investigated. Firstly, by applying frequency decomposition, we find that $ \Vert (u, \theta) \Vert _{L^{2}(\mathbb{R}^{n})} \rightarrow 0 $, as $ t \rightarrow \infty $. Secondl...
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2024-12-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241660 |
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| author | Xinli Wang Haiyang Yu Tianfeng Wu |
| author_facet | Xinli Wang Haiyang Yu Tianfeng Wu |
| author_sort | Xinli Wang |
| collection | DOAJ |
| description | In this paper, $ n $-dimensional incompressible Boussinesq equations with fractional dissipation and thermal diffusion are investigated. Firstly, by applying frequency decomposition, we find that $ \Vert (u, \theta) \Vert _{L^{2}(\mathbb{R}^{n})} \rightarrow 0 $, as $ t \rightarrow \infty $. Secondly, by using energy methods, we can show that if the initial data is sufficiently small in $ H^{s}(\mathbb{R}^{n}) $ with s = 1+$ \frac{n}{2}-2\alpha \, (0 < \alpha < 1) $, the global solutions are derived. Furthermore, under the assumption that the initial data $ (u_{0} $, $ \theta_{0}) $ belongs to $ L^{p}($where $ 1\le p < 2) $, using a more advanced frequency decomposition method, we establish optimal decay estimates for the solutions and their higher-order derivatives. Meanwhile, the uniqueness of the system can be obtained. In the case $ \alpha $ = 0, we obtained the regularity and decay estimate of the damped Boussinesq equation in Besov space. |
| format | Article |
| id | doaj-art-63be207d3a534e5c967a456666a080da |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
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| series | AIMS Mathematics |
| spelling | doaj-art-63be207d3a534e5c967a456666a080da2025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912348633488510.3934/math.20241660Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusionXinli Wang0Haiyang Yu1Tianfeng Wu2School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, ChinaSchool of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, ChinaSchool of Mathematics, Sichuan University, Chengdu 610065, ChinaIn this paper, $ n $-dimensional incompressible Boussinesq equations with fractional dissipation and thermal diffusion are investigated. Firstly, by applying frequency decomposition, we find that $ \Vert (u, \theta) \Vert _{L^{2}(\mathbb{R}^{n})} \rightarrow 0 $, as $ t \rightarrow \infty $. Secondly, by using energy methods, we can show that if the initial data is sufficiently small in $ H^{s}(\mathbb{R}^{n}) $ with s = 1+$ \frac{n}{2}-2\alpha \, (0 < \alpha < 1) $, the global solutions are derived. Furthermore, under the assumption that the initial data $ (u_{0} $, $ \theta_{0}) $ belongs to $ L^{p}($where $ 1\le p < 2) $, using a more advanced frequency decomposition method, we establish optimal decay estimates for the solutions and their higher-order derivatives. Meanwhile, the uniqueness of the system can be obtained. In the case $ \alpha $ = 0, we obtained the regularity and decay estimate of the damped Boussinesq equation in Besov space.https://www.aimspress.com/article/doi/10.3934/math.20241660fractional boussinesq equationsfractional dissipationthermal diffusionoptimal decayglobal well-posedness |
| spellingShingle | Xinli Wang Haiyang Yu Tianfeng Wu Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion AIMS Mathematics fractional boussinesq equations fractional dissipation thermal diffusion optimal decay global well-posedness |
| title | Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion |
| title_full | Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion |
| title_fullStr | Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion |
| title_full_unstemmed | Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion |
| title_short | Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion |
| title_sort | global well posedness and optimal decay rates for the n d incompressible boussinesq equations with fractional dissipation and thermal diffusion |
| topic | fractional boussinesq equations fractional dissipation thermal diffusion optimal decay global well-posedness |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241660 |
| work_keys_str_mv | AT xinliwang globalwellposednessandoptimaldecayratesforthendincompressibleboussinesqequationswithfractionaldissipationandthermaldiffusion AT haiyangyu globalwellposednessandoptimaldecayratesforthendincompressibleboussinesqequationswithfractionaldissipationandthermaldiffusion AT tianfengwu globalwellposednessandoptimaldecayratesforthendincompressibleboussinesqequationswithfractionaldissipationandthermaldiffusion |