The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term
This paper studies the existence of uniform attractors for 3D micropolar equation with damping term. When $ \beta > 3 $, with initial data $ (u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2} $ and external forces $ (f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0}) $, some uni...
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AIMS Press
2024-03-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://aimspress.com/article/doi/10.3934/math.2024470?viewType=HTML |
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| author | Xue-li Song Yuan-yuan Liu Xiao-tian Xie |
| author_facet | Xue-li Song Yuan-yuan Liu Xiao-tian Xie |
| author_sort | Xue-li Song |
| collection | DOAJ |
| description | This paper studies the existence of uniform attractors for 3D micropolar equation with damping term. When $ \beta > 3 $, with initial data $ (u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2} $ and external forces $ (f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0}) $, some uniform estimates of the solution in different function spaces are given. Based on these uniform estimates, the $ ((V_{1}\times V_{2})\times(\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})), V_{1}\times V_{2}) $-continuity of the family of processes $ \{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau} $ is demonstrated. Meanwhile, the $ (V_{1}\times V_{2}, \mathbf{H}^2(\Omega)\times\mathbf{H}^2(\Omega)) $-uniform compactness of $ \{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau} $ is proved. Finally, the existence of a $ (V_{1}\times V_{2}, V_{1}\times V_{2}) $-uniform attractor and a $ (V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)) $-uniform attractor are obtained. Furthermore, the $ (V_{1}\times V_{2}, V_{1}\times V_{2}) $-uniform attractor and the $ (V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)) $-uniform attractor are verified to be the same. |
| first_indexed | 2024-04-24T21:44:18Z |
| format | Article |
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| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-24T21:44:18Z |
| publishDate | 2024-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-04ce5b8990f643feaa2be5739926b32b2024-03-21T01:23:33ZengAIMS PressAIMS Mathematics2473-69882024-03-01949608963010.3934/math.2024470The existence of uniform attractors for the 3D micropolar equations with nonlinear damping termXue-li Song 0Yuan-yuan Liu1Xiao-tian Xie2College of Science, Xi'an University of Science and Technology, Xi'an, Shaanxi, 710054, ChinaCollege of Science, Xi'an University of Science and Technology, Xi'an, Shaanxi, 710054, ChinaCollege of Science, Xi'an University of Science and Technology, Xi'an, Shaanxi, 710054, ChinaThis paper studies the existence of uniform attractors for 3D micropolar equation with damping term. When $ \beta > 3 $, with initial data $ (u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2} $ and external forces $ (f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0}) $, some uniform estimates of the solution in different function spaces are given. Based on these uniform estimates, the $ ((V_{1}\times V_{2})\times(\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})), V_{1}\times V_{2}) $-continuity of the family of processes $ \{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau} $ is demonstrated. Meanwhile, the $ (V_{1}\times V_{2}, \mathbf{H}^2(\Omega)\times\mathbf{H}^2(\Omega)) $-uniform compactness of $ \{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau} $ is proved. Finally, the existence of a $ (V_{1}\times V_{2}, V_{1}\times V_{2}) $-uniform attractor and a $ (V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)) $-uniform attractor are obtained. Furthermore, the $ (V_{1}\times V_{2}, V_{1}\times V_{2}) $-uniform attractor and the $ (V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)) $-uniform attractor are verified to be the same.https://aimspress.com/article/doi/10.3934/math.2024470?viewType=HTMLuniform attractor3d micropolar equationnonlinear damping term |
| spellingShingle | Xue-li Song Yuan-yuan Liu Xiao-tian Xie The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term AIMS Mathematics uniform attractor 3d micropolar equation nonlinear damping term |
| title | The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term |
| title_full | The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term |
| title_fullStr | The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term |
| title_full_unstemmed | The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term |
| title_short | The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term |
| title_sort | existence of uniform attractors for the 3d micropolar equations with nonlinear damping term |
| topic | uniform attractor 3d micropolar equation nonlinear damping term |
| url | https://aimspress.com/article/doi/10.3934/math.2024470?viewType=HTML |
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