Stability on 3D Boussinesq system with mixed partial dissipation
In the article, we are concerned with the three-dimensional anisotropic Boussinesq equations with the velocity dissipation in x2{x}_{2} and x3{x}_{3} directions and the thermal diffusion in only x3{x}_{3} direction. When the spatial domain is the whole space R3{{\mathbb{R}}}^{3}, the global well-pos...
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De Gruyter
2024-12-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2024-0060 |
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| author | Lin Hongxia Liu Sen Guo Xiaochuan You Ruiqi |
| author_facet | Lin Hongxia Liu Sen Guo Xiaochuan You Ruiqi |
| author_sort | Lin Hongxia |
| collection | DOAJ |
| description | In the article, we are concerned with the three-dimensional anisotropic Boussinesq equations with the velocity dissipation in x2{x}_{2} and x3{x}_{3} directions and the thermal diffusion in only x3{x}_{3} direction. When the spatial domain is the whole space R3{{\mathbb{R}}}^{3}, the global well-posedness and stability problem for the partially dissipated Boussinesq system remain the extremely challenging open problems. Attention here focuses on the periodic domain Ω=R×T2\Omega ={\mathbb{R}}\times {{\mathbb{T}}}^{2}. We aim at establishing the stability for the problem of perturbations near hydrostatic equilibrium and the large-time behavior of the perturbed solution. We first obtain the global existence of some symmetric fluids in H2(Ω){H}^{2}\left(\Omega ) for small initial data. Then the exponential decay rates for the oscillations u˜\widetilde{u} and θ\theta in H1(Ω){H}^{1}\left(\Omega ) and the homogeneous Sobolev space Hv2˙(Ω)\dot{{H}_{v}^{2}}\left(\Omega ) are also shown. The proof is based on a key observation that we can decompose the velocity uu into the average u¯\overline{u} on T2{{\mathbb{T}}}^{2} and the corresponding oscillation u˜\widetilde{u}. This enables us to establish the strong Poincaré-type inequalities on u˜\widetilde{u}, u3,θ{u}_{3},\theta and some anisotropic inequalities, which ensure the establishment of the closed priori estimates. In addition, we also prove the oscillations in one direction u˜(2),u˜(3){\widetilde{u}}^{\left(2)},{\widetilde{u}}^{\left(3)} in H1(Ω){H}^{1}\left(\Omega ) decay to zero exponentially. |
| format | Article |
| id | doaj-art-83682d86ddca4f31aec46c33822dcb22 |
| institution | Kabale University |
| issn | 2191-950X |
| language | English |
| publishDate | 2024-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj-art-83682d86ddca4f31aec46c33822dcb222025-01-20T11:08:04ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2024-12-011311893191710.1515/anona-2024-0060Stability on 3D Boussinesq system with mixed partial dissipationLin Hongxia0Liu Sen1Guo Xiaochuan2You Ruiqi3Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, P.R. ChinaSchool of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, People’s Republic of ChinaGeomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, P.R. ChinaGeomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, P.R. ChinaIn the article, we are concerned with the three-dimensional anisotropic Boussinesq equations with the velocity dissipation in x2{x}_{2} and x3{x}_{3} directions and the thermal diffusion in only x3{x}_{3} direction. When the spatial domain is the whole space R3{{\mathbb{R}}}^{3}, the global well-posedness and stability problem for the partially dissipated Boussinesq system remain the extremely challenging open problems. Attention here focuses on the periodic domain Ω=R×T2\Omega ={\mathbb{R}}\times {{\mathbb{T}}}^{2}. We aim at establishing the stability for the problem of perturbations near hydrostatic equilibrium and the large-time behavior of the perturbed solution. We first obtain the global existence of some symmetric fluids in H2(Ω){H}^{2}\left(\Omega ) for small initial data. Then the exponential decay rates for the oscillations u˜\widetilde{u} and θ\theta in H1(Ω){H}^{1}\left(\Omega ) and the homogeneous Sobolev space Hv2˙(Ω)\dot{{H}_{v}^{2}}\left(\Omega ) are also shown. The proof is based on a key observation that we can decompose the velocity uu into the average u¯\overline{u} on T2{{\mathbb{T}}}^{2} and the corresponding oscillation u˜\widetilde{u}. This enables us to establish the strong Poincaré-type inequalities on u˜\widetilde{u}, u3,θ{u}_{3},\theta and some anisotropic inequalities, which ensure the establishment of the closed priori estimates. In addition, we also prove the oscillations in one direction u˜(2),u˜(3){\widetilde{u}}^{\left(2)},{\widetilde{u}}^{\left(3)} in H1(Ω){H}^{1}\left(\Omega ) decay to zero exponentially.https://doi.org/10.1515/anona-2024-00603d boussinesq equationspartial dissipationstabilitydecay rates35a0535q3576d03 |
| spellingShingle | Lin Hongxia Liu Sen Guo Xiaochuan You Ruiqi Stability on 3D Boussinesq system with mixed partial dissipation Advances in Nonlinear Analysis 3d boussinesq equations partial dissipation stability decay rates 35a05 35q35 76d03 |
| title | Stability on 3D Boussinesq system with mixed partial dissipation |
| title_full | Stability on 3D Boussinesq system with mixed partial dissipation |
| title_fullStr | Stability on 3D Boussinesq system with mixed partial dissipation |
| title_full_unstemmed | Stability on 3D Boussinesq system with mixed partial dissipation |
| title_short | Stability on 3D Boussinesq system with mixed partial dissipation |
| title_sort | stability on 3d boussinesq system with mixed partial dissipation |
| topic | 3d boussinesq equations partial dissipation stability decay rates 35a05 35q35 76d03 |
| url | https://doi.org/10.1515/anona-2024-0060 |
| work_keys_str_mv | AT linhongxia stabilityon3dboussinesqsystemwithmixedpartialdissipation AT liusen stabilityon3dboussinesqsystemwithmixedpartialdissipation AT guoxiaochuan stabilityon3dboussinesqsystemwithmixedpartialdissipation AT youruiqi stabilityon3dboussinesqsystemwithmixedpartialdissipation |