A note on the Cauchy problem for the two-component Novikov system

Abstract

Considered herein is the initial value problem for the two-component Novikov system with peakons. 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 \(s>\max \{5/2,2+1/p\}, 1\le p,r\le \infty \) and \(B^{5/2}_{2,1}({\mathbb {T}})\times B^{5/2}_{2,1}({\mathbb {T}})\) through the method of approximate solutions. Furthermore, it is in the non-periodic case that the non-uniform continuity of this solution map in Besov spaces \(B^{s}_{p,r}({\mathbb {R}})\times B^{s}_{p,r}({\mathbb {R}})\) with \(s>\max \{5/2,2+1/p\}, 1\le p,r\le \infty \) is discussed by constructing new subtle initial data. Finally, the Hölder continuity of the solution map in Besov spaces is proved.

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Acknowledgements

The work is supported by National Natural Science Foundation Grant-11471259 and the National Science Basic Research Program of Shaanxi (Program Nos. 2019JM-007, 2020JC-37).

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Correspondence to Haiquan Wang.

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Wang, H., Chong, G. & Wu, L. A note on the Cauchy problem for the two-component Novikov system. J. Evol. Equ. (2021). https://doi.org/10.1007/s00028-020-00657-z

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Keywords

  • The two-component Novikov system
  • Non-uniform dependence
  • Hölder continuity
  • Besov spaces

Mathematics Subject Classification

  • 35B30
  • 35G25