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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2402.15128 |
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| _version_ | 1866909118215749632 |
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| author | Guo, Yingying Ye, Weikui |
| author_facet | Guo, Yingying Ye, Weikui |
| contents | In this paper, we consider the Cauchy problem for the $b$-equation. Firstly, for $s>\frac32,$ if $u_{0}(x)\in H^{s}(\mathbb{R})$ and $m_{0}(x)=u_{0}(x)-u_{0xx}(x)\in L^{1}(\mathbb{R}),$ the global solutions of the $b$-equation is established when $b\geq1$ or $b\leq1.$ It's worth noting that our global result is a new result which doesn't need the condition that $m_{0}(x)$ keeps its sign. For $s<\frac32,$ it is shown (see [13]) that the Cauchy problem of the $b$-equation is ill-posed in Sobolev space $H^{s}(\mathbb{R})$ when $b>1$ or $b<1.$ In the present paper, for $s=\frac32,$ we prove that the Cauchy problem of the $b$-equation is also ill-posed in $H^{\frac32}(\mathbb{R})$ in the sense of norm inflation by constructing a class of special initial data when $b\neq1.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15128 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ill-posedness and global solution for the $b$-equation Guo, Yingying Ye, Weikui Analysis of PDEs 35Q53, 35G25, 35D30 In this paper, we consider the Cauchy problem for the $b$-equation. Firstly, for $s>\frac32,$ if $u_{0}(x)\in H^{s}(\mathbb{R})$ and $m_{0}(x)=u_{0}(x)-u_{0xx}(x)\in L^{1}(\mathbb{R}),$ the global solutions of the $b$-equation is established when $b\geq1$ or $b\leq1.$ It's worth noting that our global result is a new result which doesn't need the condition that $m_{0}(x)$ keeps its sign. For $s<\frac32,$ it is shown (see [13]) that the Cauchy problem of the $b$-equation is ill-posed in Sobolev space $H^{s}(\mathbb{R})$ when $b>1$ or $b<1.$ In the present paper, for $s=\frac32,$ we prove that the Cauchy problem of the $b$-equation is also ill-posed in $H^{\frac32}(\mathbb{R})$ in the sense of norm inflation by constructing a class of special initial data when $b\neq1.$ |
| title | Ill-posedness and global solution for the $b$-equation |
| topic | Analysis of PDEs 35Q53, 35G25, 35D30 |
| url | https://arxiv.org/abs/2402.15128 |