On the long-time behavior of scale-invariant solutions to the 2d Euler equation and applications
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866908585296920576 |
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| author | Elgindi, Tarek. M. Murray, Ryan. W. Said, Ayman. R. |
| author_facet | Elgindi, Tarek. M. Murray, Ryan. W. Said, Ayman. R. |
| contents | We study the long-time behavior of scale-invariant solutions of the 2d Euler equation satisfying a discrete symmetry. We show that all scale-invariant solutions with bounded variation on $\mathbb{S}^1$ relax to states that are piece-wise constant with finitely many jumps.
All continuous scale-invariant solutions become singular and homogenize in infinite time. On $\mathbb{R}^2$, this corresponds to generic infinite-time spiral and cusp formation. The main tool in our analysis is the discovery of a monotone quantity that measures the number of particles that are moving away from the origin.
This monotonicity also applies locally to solutions of the 2d Euler equation that are $m$-fold symmetric ($m\geq 4$) and have radial limits at the point of symmetry.
Our results are also applicable to the Euler equation on a large class of surfaces of revolution (like $\mathbb{S}^2$ and $\mathbb{T}^2$). Our analysis then gives generic spiraling of trajectories and infinite-time loss of regularity for globally smooth solutions on any such smooth surface, under a discrete symmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_08418 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the long-time behavior of scale-invariant solutions to the 2d Euler equation and applications Elgindi, Tarek. M. Murray, Ryan. W. Said, Ayman. R. Analysis of PDEs Mathematical Physics 35Qxx We study the long-time behavior of scale-invariant solutions of the 2d Euler equation satisfying a discrete symmetry. We show that all scale-invariant solutions with bounded variation on $\mathbb{S}^1$ relax to states that are piece-wise constant with finitely many jumps. All continuous scale-invariant solutions become singular and homogenize in infinite time. On $\mathbb{R}^2$, this corresponds to generic infinite-time spiral and cusp formation. The main tool in our analysis is the discovery of a monotone quantity that measures the number of particles that are moving away from the origin. This monotonicity also applies locally to solutions of the 2d Euler equation that are $m$-fold symmetric ($m\geq 4$) and have radial limits at the point of symmetry. Our results are also applicable to the Euler equation on a large class of surfaces of revolution (like $\mathbb{S}^2$ and $\mathbb{T}^2$). Our analysis then gives generic spiraling of trajectories and infinite-time loss of regularity for globally smooth solutions on any such smooth surface, under a discrete symmetry. |
| title | On the long-time behavior of scale-invariant solutions to the 2d Euler equation and applications |
| topic | Analysis of PDEs Mathematical Physics 35Qxx |
| url | https://arxiv.org/abs/2211.08418 |