Ill-posedness for the periodic Camassa–Holm type equations in critical Besov spaces

Abstract

In this paper, by presenting a new construction of initial data which leads to the norm inflation phenomena, we prove that both the periodic Camassa–Holm and Novikov equations are ill-posed in \(B_{\infty ,1}^{1}\). We also construct a new example for initial data and present a short proof of the ill-posedness for the periodic Camassa–Holm equation in \(B^{1+1/p}_{p,r}\) with \((p,r)\in (1,\infty ]\times (1,\infty ]\).