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On the initial value problem for the two-coupled Camassa–Holm system in Besov spaces

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Abstract

Considered herein is the Cauchy problem for the two-coupled Camassa–Holm system. Based on the local well-posedness results for this problem, it is shown that the solution map \(z_{0}\mapsto z(t)\) of this problem in the periodic case is not uniformly continuous in Besov spaces \(B^{s}_{p,r}(\mathbb {T})\times B^{s}_{p,r}(\mathbb {T}) \) with \(s>\max \{3/2,1+1/p\}, 1\le p,r\le \infty \) by using the method of approximate solutions. In the non-periodic case, the non-uniform continuity of this solution map in Besov spaces \(B^{s}_{2,r}(\mathbb {R})\times B^{s}_{2,r}(\mathbb {R}) \) with \(s>3/2, 2\le r\le \infty \) is established. Finally, the Hölder continuity of the solution map in Besov spaces is proved.

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Acknowledgements

The work of Wang is supported by National Natural Science Foundation Grant-11471259 and the National Science Basic Research Program of Shaanxi(Program No. 2019JM-007).

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  1. School of Mathematics, Northwest University, Xi’an, 710069, People’s Republic of China

    Haiquan Wang & Gezi Chong

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  1. Haiquan Wang
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  2. Gezi Chong
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Correspondence to Haiquan Wang.

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Communicated by Adrian Constantin.

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Wang, H., Chong, G. On the initial value problem for the two-coupled Camassa–Holm system in Besov spaces. Monatsh Math 193, 479–505 (2020). https://doi.org/10.1007/s00605-020-01385-9

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  • DOI: https://doi.org/10.1007/s00605-020-01385-9

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