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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.01504 |
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| _version_ | 1866916693438103552 |
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| author | Karlsen, Kenneth H. Rybalko, Yan |
| author_facet | Karlsen, Kenneth H. Rybalko, Yan |
| contents | We study the Cauchy problem for the two-component Novikov system with initial data $u_0, v_0$ in $H^1(\mathbb{R})$ such that the product $(\partial_x u_0)\partial_x v_0$ belongs to $L^2(\mathbb{R})$. We construct a global semigroup of conservative weak solutions. We also discuss the potential concentration phenomena of $(\partial_x u)^2dx$, $(\partial_x v)^2dx$, and $\left((\partial_x u)^2(\partial_x v)^2\right)dx$, which contribute to wave-breaking and may occur for a set of time with nonzero measure. Finally, we establish the continuity of the data-to-solution map in the uniform norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01504 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global semigroup of conservative weak solutions of the two-component Novikov equation Karlsen, Kenneth H. Rybalko, Yan Analysis of PDEs We study the Cauchy problem for the two-component Novikov system with initial data $u_0, v_0$ in $H^1(\mathbb{R})$ such that the product $(\partial_x u_0)\partial_x v_0$ belongs to $L^2(\mathbb{R})$. We construct a global semigroup of conservative weak solutions. We also discuss the potential concentration phenomena of $(\partial_x u)^2dx$, $(\partial_x v)^2dx$, and $\left((\partial_x u)^2(\partial_x v)^2\right)dx$, which contribute to wave-breaking and may occur for a set of time with nonzero measure. Finally, we establish the continuity of the data-to-solution map in the uniform norm. |
| title | Global semigroup of conservative weak solutions of the two-component Novikov equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.01504 |