Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866917563937587200 |
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| author | Lafleche, Laurent Vasseur, Alexis F. Vishik, Misha |
| author_facet | Lafleche, Laurent Vasseur, Alexis F. Vishik, Misha |
| contents | It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In [{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it exists, such a solution becomes linearly unstable close to the blow-up time. In this paper, we show that the same phenomenon holds even in the more rigid axisymmetric case. To obtain this result, we first prove a blow-up criterion involving only the toroidal component of the vorticity. The instability of blow-up profiles is also investigated. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2009_12603 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations Lafleche, Laurent Vasseur, Alexis F. Vishik, Misha Analysis of PDEs 76B03, 35B35, 35B44 It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In [{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it exists, such a solution becomes linearly unstable close to the blow-up time. In this paper, we show that the same phenomenon holds even in the more rigid axisymmetric case. To obtain this result, we first prove a blow-up criterion involving only the toroidal component of the vorticity. The instability of blow-up profiles is also investigated. |
| title | Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations |
| topic | Analysis of PDEs 76B03, 35B35, 35B44 |
| url | https://arxiv.org/abs/2009.12603 |