Some new Liouville type theorems for 3D steady tropical climate model
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916043807522816 |
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| author | Dong, Yanyan Fang, Yan Zhang, Zhibing |
| author_facet | Dong, Yanyan Fang, Yan Zhang, Zhibing |
| contents | In this paper, we establish two major classes of Liouville type results for the three-dimensional stationary tropical climate model. The first class is obtained under the assumptions imposed on $u,v,θ$ whereas the second one relies on the assumptions imposed on $u,v,\nablaθ$. Using the energy method and an iteration argument, we obtain Liouville type theorems under the condition that Lebesgue norms of the smooth solutions on the annulus satisfy some power-law growth conditions. As a consequence, we show that a smooth solution is trivial provided that it belongs to some Lebesgue spaces or satisfies some decay conditions at infinity. Furthermore, with the aid of a contradiction argument and by developing a systematic framework to handle the energy function associated with the non-trivial solutions, we obtain a logarithmic improvement of our Liouville type theorems. Our new framework is very effective for establishing logarithmic improvements of Liouville type theorems for coupled fluid equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09423 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some new Liouville type theorems for 3D steady tropical climate model Dong, Yanyan Fang, Yan Zhang, Zhibing Analysis of PDEs 35B53, 35Q35, 35A02 In this paper, we establish two major classes of Liouville type results for the three-dimensional stationary tropical climate model. The first class is obtained under the assumptions imposed on $u,v,θ$ whereas the second one relies on the assumptions imposed on $u,v,\nablaθ$. Using the energy method and an iteration argument, we obtain Liouville type theorems under the condition that Lebesgue norms of the smooth solutions on the annulus satisfy some power-law growth conditions. As a consequence, we show that a smooth solution is trivial provided that it belongs to some Lebesgue spaces or satisfies some decay conditions at infinity. Furthermore, with the aid of a contradiction argument and by developing a systematic framework to handle the energy function associated with the non-trivial solutions, we obtain a logarithmic improvement of our Liouville type theorems. Our new framework is very effective for establishing logarithmic improvements of Liouville type theorems for coupled fluid equations. |
| title | Some new Liouville type theorems for 3D steady tropical climate model |
| topic | Analysis of PDEs 35B53, 35Q35, 35A02 |
| url | https://arxiv.org/abs/2504.09423 |