Steady 3d Euler flows via a topology-preserving convex integration scheme
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909464405213184 |
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| author | Enciso, Alberto Peñafiel-Tomás, Javier Peralta-Salas, Daniel |
| author_facet | Enciso, Alberto Peñafiel-Tomás, Javier Peralta-Salas, Daniel |
| contents | Given any smooth solenoidal vector field $v_0$ on $\mathbf T^3$, we show the existence of infinitely many Hölder-continuous steady Euler flows $v$ with the same topology as $v_0$, in certain weak sense. In particular, we show that $v$ possesses a unique flow of the highest Hölder regularity, which is conjugate to the flow of $v_0$ via a volume-preserving Hölder homeomorphism of $\mathbf T^3$. This result extends to the case of Euler equations on toroidal domains, which has applications to the study of plasmas. The proof relies on a novel convex integration scheme incorporating the key idea that the velocity field of the subsolutions must remain diffeomorphic to $v_0$ at each iteration step. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13632 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Steady 3d Euler flows via a topology-preserving convex integration scheme Enciso, Alberto Peñafiel-Tomás, Javier Peralta-Salas, Daniel Analysis of PDEs Given any smooth solenoidal vector field $v_0$ on $\mathbf T^3$, we show the existence of infinitely many Hölder-continuous steady Euler flows $v$ with the same topology as $v_0$, in certain weak sense. In particular, we show that $v$ possesses a unique flow of the highest Hölder regularity, which is conjugate to the flow of $v_0$ via a volume-preserving Hölder homeomorphism of $\mathbf T^3$. This result extends to the case of Euler equations on toroidal domains, which has applications to the study of plasmas. The proof relies on a novel convex integration scheme incorporating the key idea that the velocity field of the subsolutions must remain diffeomorphic to $v_0$ at each iteration step. |
| title | Steady 3d Euler flows via a topology-preserving convex integration scheme |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.13632 |