On the well-posedness of the Cauchy problem for the two-component peakon system in $C^k\cap W^{k,1}$
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929656299520000 |
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| author | Karlsen, Kenneth H. Rybalko, Yan |
| author_facet | Karlsen, Kenneth H. Rybalko, Yan |
| contents | This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class $(m,n)\in C^{k}(\mathbb{R}) \cap W^{k,1}(\mathbb{R})$ with $k\in\mathbb{N}\cup\{0\}$.This system extends the celebrated Fokas-Olver-Rosenau-Qiao equation, and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: $$ \partial_t m(t,x)= \partial_x[m(t,x)(u(t,x)-\partial_xu(t,x)) (u(-t,-x)+\partial_x(u(-t,-x)))], $$ where $m(t,x)=\left(1-\partial_{x}^2\right)u(t,x)$. Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class $C^k\cap W^{k,1}$. Moreover, we derive criteria for blow-up of the local solution in this class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04723 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the well-posedness of the Cauchy problem for the two-component peakon system in $C^k\cap W^{k,1}$ Karlsen, Kenneth H. Rybalko, Yan Analysis of PDEs This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class $(m,n)\in C^{k}(\mathbb{R}) \cap W^{k,1}(\mathbb{R})$ with $k\in\mathbb{N}\cup\{0\}$.This system extends the celebrated Fokas-Olver-Rosenau-Qiao equation, and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: $$ \partial_t m(t,x)= \partial_x[m(t,x)(u(t,x)-\partial_xu(t,x)) (u(-t,-x)+\partial_x(u(-t,-x)))], $$ where $m(t,x)=\left(1-\partial_{x}^2\right)u(t,x)$. Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class $C^k\cap W^{k,1}$. Moreover, we derive criteria for blow-up of the local solution in this class. |
| title | On the well-posedness of the Cauchy problem for the two-component peakon system in $C^k\cap W^{k,1}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.04723 |