Stationary solutions to the critical and super-critical quasi-geostrophic equation in the scaling critical Sobolev space
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916652533153792 |
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| author | Fujii, Mikihiro |
| author_facet | Fujii, Mikihiro |
| contents | We consider the stationary problem for the quasi-geostrophic equation with the critical and super-critical dissipation and prove the unique existence of small solutions for given small external force in the scaling critical Sobolev spaces framework. Moreover, we also show that the data-to-solution map is continuous. Since the critical and super-critical case involves the derivative loss, which affects the class of the continuity of the data-to-solution map, we reveal that the map is no longer uniform continuous, in contrast to the sub-critical case, where the Lipschitz continuity holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11095 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stationary solutions to the critical and super-critical quasi-geostrophic equation in the scaling critical Sobolev space Fujii, Mikihiro Analysis of PDEs We consider the stationary problem for the quasi-geostrophic equation with the critical and super-critical dissipation and prove the unique existence of small solutions for given small external force in the scaling critical Sobolev spaces framework. Moreover, we also show that the data-to-solution map is continuous. Since the critical and super-critical case involves the derivative loss, which affects the class of the continuity of the data-to-solution map, we reveal that the map is no longer uniform continuous, in contrast to the sub-critical case, where the Lipschitz continuity holds. |
| title | Stationary solutions to the critical and super-critical quasi-geostrophic equation in the scaling critical Sobolev space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.11095 |