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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.23857 |
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| _version_ | 1866908304153772032 |
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| author | Wang, Guodong |
| author_facet | Wang, Guodong |
| contents | In this paper, we establish two stability theorems for steady or traveling solutions of the two-dimensional incompressible Euler equation in a finite periodic channel, extending Arnold's classical work from the 1960s. Compared to Arnold's approach, we employ a compactness argument rather than relying on the negative definiteness of the energy-Casimir functional. The isovortical property of the Euler equation and Burton's rearrangement theory play an essential role in our analysis. As a corollary, we prove for the first time the existence of a class of stable non-shear flows when the ratio of the channel's height to its length is less than or equal to $\sqrt{3}/2.$ Two rigidity results are also obtained as byproducts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23857 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear stability of plane ideal flows in a periodic channel Wang, Guodong Analysis of PDEs In this paper, we establish two stability theorems for steady or traveling solutions of the two-dimensional incompressible Euler equation in a finite periodic channel, extending Arnold's classical work from the 1960s. Compared to Arnold's approach, we employ a compactness argument rather than relying on the negative definiteness of the energy-Casimir functional. The isovortical property of the Euler equation and Burton's rearrangement theory play an essential role in our analysis. As a corollary, we prove for the first time the existence of a class of stable non-shear flows when the ratio of the channel's height to its length is less than or equal to $\sqrt{3}/2.$ Two rigidity results are also obtained as byproducts. |
| title | Nonlinear stability of plane ideal flows in a periodic channel |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.23857 |