Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913975276404736 |
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| author | Dolce, Michele Mescolini, Giulia |
| author_facet | Dolce, Michele Mescolini, Giulia |
| contents | Building on an approach introduced by Golovkin in the '60s, we show that nonuniqueness in some forced PDEs is a direct consequence of the existence of a self-similar linearly unstable eigenvalue: the key point is a clever choice of the forcing term removing complicated nonlinear interactions. We use this method to give a short and self-contained proof of nonuniqueness in 2D perfect fluids, first obtained in Vishik's groundbreaking result. In particular, we present a direct construction of a forced self-similar unstable vortex, where we treat perturbatively the self-similar operator in a new and more quantitative way. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_18452 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations Dolce, Michele Mescolini, Giulia Analysis of PDEs 35Q31, 35Q35 Building on an approach introduced by Golovkin in the '60s, we show that nonuniqueness in some forced PDEs is a direct consequence of the existence of a self-similar linearly unstable eigenvalue: the key point is a clever choice of the forcing term removing complicated nonlinear interactions. We use this method to give a short and self-contained proof of nonuniqueness in 2D perfect fluids, first obtained in Vishik's groundbreaking result. In particular, we present a direct construction of a forced self-similar unstable vortex, where we treat perturbatively the self-similar operator in a new and more quantitative way. |
| title | Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations |
| topic | Analysis of PDEs 35Q31, 35Q35 |
| url | https://arxiv.org/abs/2411.18452 |