Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2405.17771 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910460620570624 |
|---|---|
| author | Yu, Yanghai Liu, Fang |
| author_facet | Yu, Yanghai Liu, Fang |
| contents | It is shown in \cite[Adv. Differ. Equ(2017)]{HT} that the Cauchy problem for the generalized Camassa-Holm equation is well-posed in $C^1$ and the data-to-solution map is Hölder continuous from $C^α$ to $\mathcal{C}([0,T];C^α)$ with $α\in[0,1)$. In this paper, we further show that the data-to-solution map of the generalized Camassa-Holm equation is not uniformly continuous on the initial data in $C^1$. In particular, our result also can be a complement of previous work on the classical Camassa-Holm equation in \cite[Geom. Funct. Anal(2002)]{G02}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-uniform dependence on initial data for the generalized Camassa-Holm equation in $C^1$ Yu, Yanghai Liu, Fang Analysis of PDEs It is shown in \cite[Adv. Differ. Equ(2017)]{HT} that the Cauchy problem for the generalized Camassa-Holm equation is well-posed in $C^1$ and the data-to-solution map is Hölder continuous from $C^α$ to $\mathcal{C}([0,T];C^α)$ with $α\in[0,1)$. In this paper, we further show that the data-to-solution map of the generalized Camassa-Holm equation is not uniformly continuous on the initial data in $C^1$. In particular, our result also can be a complement of previous work on the classical Camassa-Holm equation in \cite[Geom. Funct. Anal(2002)]{G02}. |
| title | Non-uniform dependence on initial data for the generalized Camassa-Holm equation in $C^1$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.17771 |