Ill-posedness and global solution for the $b$-equation
| Autor: | Guo, Yingying, Ye, Weikui |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $b\neq1.$ Comment: 10 pages |
| Databáze: | arXiv |
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