Non-uniform Dependence on Initial Data for the Camassa–Holm Equation in the Critical Besov Space

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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.