Ill-posedness issue on the Oldroyd-B model in the critical Besov spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918133732737024 |
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| author | Li, Jinlu Yu, Yanghai Zhu, Weipeng |
| author_facet | Li, Jinlu Yu, Yanghai Zhu, Weipeng |
| contents | It is proved in \cite[J. Funct. Anal., 2020]{AP} that the Cauchy problem for some Oldroyd-B model is well-posed in $\B^{d/p-1}_{p,1}(\R^d) \times \B^{d/p}_{p,1}(\R^d)$ with $1\leq p<2d$. In this paper, we prove that the Cauchy problem for the same Oldroyd-B model is ill-posed in $\B^{d/p-1}_{p,r}(\R^d) \times \B^{d/p}_{p,r}(\R^d)$ with $1\leq p\leq \infty$ and $1< r\leq\infty$ due to the lack of continuous dependence of the solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02001 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ill-posedness issue on the Oldroyd-B model in the critical Besov spaces Li, Jinlu Yu, Yanghai Zhu, Weipeng Analysis of PDEs It is proved in \cite[J. Funct. Anal., 2020]{AP} that the Cauchy problem for some Oldroyd-B model is well-posed in $\B^{d/p-1}_{p,1}(\R^d) \times \B^{d/p}_{p,1}(\R^d)$ with $1\leq p<2d$. In this paper, we prove that the Cauchy problem for the same Oldroyd-B model is ill-posed in $\B^{d/p-1}_{p,r}(\R^d) \times \B^{d/p}_{p,r}(\R^d)$ with $1\leq p\leq \infty$ and $1< r\leq\infty$ due to the lack of continuous dependence of the solution. |
| title | Ill-posedness issue on the Oldroyd-B model in the critical Besov spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.02001 |