Ill-posedness of the Camassa-Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B^{1+1/p}_{p,r}$ for $pin [1,infty], rin (1,infty]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works (cite{Danchin2,Byers,HHK}).