A remark on the zero-filter limit for the Camassa-Holm equation in $B^s_{2,\infty}(\R)$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916626154127360 |
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| author | Qu, Guorong Lu, Jianzhong Deng, Wei |
| author_facet | Qu, Guorong Lu, Jianzhong Deng, Wei |
| contents | This paper investigates the zero-filter limit problem associated with the Camassa-Holm equation. In the work cited as \cite{C.L.L.W.L}, it was established that, under the hypothesis of initial data $u_0\in B^s_{2,r}(\R)$ with $s>\frac32$ and $1\leq r<\infty$, the solutions $\mathbf{S}_{t}^{\mathbfα}(u_0)$ of the Camassa-Holm equation exhibit convergence in the $L^\infty_T(B^s_{2,r})$ norm to the unique solution of the Burgers equation as $α\rightarrow 0$. Contrary to this result, the present study demonstrates that for initial data $u_0\in B^s_{2,\infty}(\R)$ the solutions of the Camassa-Holm equation fail to converge strongly in the $L^\infty_T(B^s_{2,\infty})$ norm to the Burgers equation as $α\rightarrow 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A remark on the zero-filter limit for the Camassa-Holm equation in $B^s_{2,\infty}(\R)$ Qu, Guorong Lu, Jianzhong Deng, Wei Analysis of PDEs 35Q35 This paper investigates the zero-filter limit problem associated with the Camassa-Holm equation. In the work cited as \cite{C.L.L.W.L}, it was established that, under the hypothesis of initial data $u_0\in B^s_{2,r}(\R)$ with $s>\frac32$ and $1\leq r<\infty$, the solutions $\mathbf{S}_{t}^{\mathbfα}(u_0)$ of the Camassa-Holm equation exhibit convergence in the $L^\infty_T(B^s_{2,r})$ norm to the unique solution of the Burgers equation as $α\rightarrow 0$. Contrary to this result, the present study demonstrates that for initial data $u_0\in B^s_{2,\infty}(\R)$ the solutions of the Camassa-Holm equation fail to converge strongly in the $L^\infty_T(B^s_{2,\infty})$ norm to the Burgers equation as $α\rightarrow 0$. |
| title | A remark on the zero-filter limit for the Camassa-Holm equation in $B^s_{2,\infty}(\R)$ |
| topic | Analysis of PDEs 35Q35 |
| url | https://arxiv.org/abs/2502.16583 |