On Arnold-type stability theorems for the Euler equation on a sphere
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| Format: | Preprint |
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2024
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| _version_ | 1866909248435257344 |
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| author | Cao, Daomin Wang, Guodong |
| author_facet | Cao, Daomin Wang, Guodong |
| contents | In this paper, we establish three Arnold-type stability theorems for steady or rotating solutions of the incompressible Euler equation on a sphere. Specifically, we prove that if the stream function of a flow solves a semilinear elliptic equation with a monotone nonlinearity, then, under appropriate conditions, the flow is stable or orbitally stable in the Lyapunov sense. In particular, our theorems apply to degree-2 Rossby-Haurwitz waves. These results are achieved via a variational approach, with the key ingredient being to show that the flows under consideration satisfy the conditions of two Burton-type stability criteria which are established in this paper. As byproducts, we obtain some sharp rigidity results for solutions of semilinear elliptic equations on a sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_06752 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Arnold-type stability theorems for the Euler equation on a sphere Cao, Daomin Wang, Guodong Analysis of PDEs In this paper, we establish three Arnold-type stability theorems for steady or rotating solutions of the incompressible Euler equation on a sphere. Specifically, we prove that if the stream function of a flow solves a semilinear elliptic equation with a monotone nonlinearity, then, under appropriate conditions, the flow is stable or orbitally stable in the Lyapunov sense. In particular, our theorems apply to degree-2 Rossby-Haurwitz waves. These results are achieved via a variational approach, with the key ingredient being to show that the flows under consideration satisfy the conditions of two Burton-type stability criteria which are established in this paper. As byproducts, we obtain some sharp rigidity results for solutions of semilinear elliptic equations on a sphere. |
| title | On Arnold-type stability theorems for the Euler equation on a sphere |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.06752 |