Ill-posedness for the Camassa–Holm equation in \(B_{p,1}^{1}\cap C^{0,1}\)

In this paper, we study the Cauchy problem for the Camassa–Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space \(B_{p,1}^{1}\cap C^{0,1}\) with \(p\in (2,\infty ]\) is discontinuous at origin. More precisely, the initial data in \(B_{p,1}^{1}\cap C^{0,1}\) can guarantee that the Camassa–Holm equation has a unique local solution in \(W^{1,p}\cap C^{0,1}\), however, this solution is instable and can have an inflation in \(B_{p,1}^{1}\cap C^{0,1}\).