Abstract
Whether or not the data-to-solution map of the Cauchy problem for the Camassa–Holm equation and Novikov equation in the critical Besov space \(B_{2,1}^{3/2}({\mathbb {R}})\) is uniformly continuous remains open. In the paper, we aim at solving the open question left in the previous works (Li et al. in J Differ Equ 269:8686–8700, 2020a; J Math Fluid Mech 22:50, 2020b) and giving a negative answer to this problem.
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Acknowledgements
The authors want to thank the referees for their constructive comments and helpful suggestions which greatly improved the presentation of this paper. J. Li is supported by the National Natural Science Foundation of China (Grant No. 11801090) and Postdoctoral Science Foundation of China (2020T130129 and 2020M672565). Y. Yu is supported by the Natural Science Foundation of Anhui Province (No. 1908085QA05) and the PhD Scientific Research Start-up Foundation of Anhui Normal University. W. Zhu is partially supported by the National Natural Science Foundation of China (Grant No. 11901092) and Natural Science Foundation of Guangdong Province (No. 2017A030310634).
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Li, J., Wu, X., Yu, Y. et al. Non-uniform Dependence on Initial Data for the Camassa–Holm Equation in the Critical Besov Space. J. Math. Fluid Mech. 23, 36 (2021). https://doi.org/10.1007/s00021-021-00571-5
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DOI: https://doi.org/10.1007/s00021-021-00571-5