A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| author | Castro, Ángel Faraco, Daniel Mengual, Francisco Solera, Marcos |
| author_facet | Castro, Ángel Faraco, Daniel Mengual, Francisco Solera, Marcos |
| contents | We give a simpler proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation in the vorticity class $L^1\cap L^p$ with $2<p<\infty$. The main simplification is an alternative construction of a smooth and compactly supported unstable vortex, which is split into two steps: Firstly, we construct a piecewise constant unstable vortex, and secondly, we find a regularization through a fixed point argument. This simpler structure of the unstable vortex yields a simplification of the other parts of Vishik's proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation Castro, Ángel Faraco, Daniel Mengual, Francisco Solera, Marcos Analysis of PDEs We give a simpler proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation in the vorticity class $L^1\cap L^p$ with $2<p<\infty$. The main simplification is an alternative construction of a smooth and compactly supported unstable vortex, which is split into two steps: Firstly, we construct a piecewise constant unstable vortex, and secondly, we find a regularization through a fixed point argument. This simpler structure of the unstable vortex yields a simplification of the other parts of Vishik's proof. |
| title | A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.15995 |