Non-conservation of a generalized helicity in the Euler equations
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917192372584448 |
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| author | Giri, Vikram Kwon, Hyunju Novack, Matthew |
| author_facet | Giri, Vikram Kwon, Hyunju Novack, Matthew |
| contents | For a $C^1_{t,x}$ solution $u$ to the incompressible 3D Euler equations, the helicity $H(u(t))=\int_{\mathbb{T}^3} u \cdot \textrm{curl}\, u$ is constant in time. For general low-regularity weak solutions, it is not always clear how to define the helicity, or whether it must be constant in time in the case that there is a clear definition. In this paper, we define a generalized helicity which extends the classical definitions and construct weak solutions of Euler of almost Onsager-critical regularity in $L^3$ with prescribed generalized helicity and kinetic energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05869 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-conservation of a generalized helicity in the Euler equations Giri, Vikram Kwon, Hyunju Novack, Matthew Analysis of PDEs For a $C^1_{t,x}$ solution $u$ to the incompressible 3D Euler equations, the helicity $H(u(t))=\int_{\mathbb{T}^3} u \cdot \textrm{curl}\, u$ is constant in time. For general low-regularity weak solutions, it is not always clear how to define the helicity, or whether it must be constant in time in the case that there is a clear definition. In this paper, we define a generalized helicity which extends the classical definitions and construct weak solutions of Euler of almost Onsager-critical regularity in $L^3$ with prescribed generalized helicity and kinetic energy. |
| title | Non-conservation of a generalized helicity in the Euler equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.05869 |