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Non-uniform continuity on initial data for the two-component b-family system in Besov space

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Abstract

This paper studies a two-component b-family system, which includes the two-component Camassa-Holm system and the two-component Degasperis-Procesi system as special case. It is shown that the solution map of this system is not uniformly continuous on the initial data in Besov spaces \(B_{p, r}^{s-1}({\mathbb {R}})\times B_{p, r}^s({\mathbb {R}})\) with \(s>\max \{1+\frac{1}{p}, \frac{3}{2}\}\), \(1\le p, r< \infty \). Our result covers and extends the previous non-uniform continuity in Sobolev spaces \(H^{s-1}({\mathbb {R}})\times H^s({\mathbb {R}})\) for \(s>\frac{5}{2}\) to Besov spaces (Nonlinear Anal., 2014, 111: 1-14). Compared with the generalized rotation b-family system considered by Holmes et al. (Z. Angew. Math. Mech., 2021), our non-uniform continuity is established in a broader range of Besov spaces.

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Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No.12001163) and Natural Science Foundation of Henan Province (Grant No.212300410164).

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Authors and Affiliations

  1. College of Information and Management Science, Henan Agricultural University, Zhengzhou, 450002, Henan, China

    Xing Wu & Jie Cao

  2. School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450000, Henan, China

    Cui Li

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  1. Xing Wu
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Wu, X., Li, C. & Cao, J. Non-uniform continuity on initial data for the two-component b-family system in Besov space. Monatsh Math 201, 547–563 (2023). https://doi.org/10.1007/s00605-022-01790-2

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