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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2306.04193 |
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| _version_ | 1866915031783833600 |
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| author | Kiselev, Alexander Luo, Xiaoyutao |
| author_facet | Kiselev, Alexander Luo, Xiaoyutao |
| contents | We consider the patch problem for the $α$-SQG system with the values $α=0$ and $α= \frac{1}{2}$ being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint $C^{k,β}$ Hölder spaces, as well as in $W^{2,p},$ $1<p<\infty$ spaces. In stark contrast to the Euler case, we prove that for $0<α< \frac{1}{2}$, the $α$-SQG patch problem is strongly illposed in \emph{every} $C^{2,β} $ Hölder space with $β<1$. Moreover, in a suitable range of regularity, the same strong illposedness holds for \emph{every} $W^{2,p}$ Sobolev space unless $p=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_04193 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The $α$-SQG patch problem is illposed in $C^{2,β}$ and $W^{2,p}$ Kiselev, Alexander Luo, Xiaoyutao Analysis of PDEs We consider the patch problem for the $α$-SQG system with the values $α=0$ and $α= \frac{1}{2}$ being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint $C^{k,β}$ Hölder spaces, as well as in $W^{2,p},$ $1<p<\infty$ spaces. In stark contrast to the Euler case, we prove that for $0<α< \frac{1}{2}$, the $α$-SQG patch problem is strongly illposed in \emph{every} $C^{2,β} $ Hölder space with $β<1$. Moreover, in a suitable range of regularity, the same strong illposedness holds for \emph{every} $W^{2,p}$ Sobolev space unless $p=2$. |
| title | The $α$-SQG patch problem is illposed in $C^{2,β}$ and $W^{2,p}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2306.04193 |