Ill-posedness for the periodic Camassa--Holm type equations in the end-point critical Besov space $B_{\infty ,1}^1$

Abstract

For the real-line case, it is shown that both the Camassa--Holm \cite{Guo} and Novikov equations \cite{Li-arx} are ill-posed in $B_{\infty ,1}^1$. 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 also ill-posed in $B_{\infty ,1}^1$.

arXiv admin note: substantial text overlap with arXiv:2201.02839; text overlap with arXiv:2112.14104, arXiv:2207.08190, arXiv:2104.05973