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

Abstract

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 H3/2 and even in the Besov space Bp,r1+1/p for p∈[1,∞],r∈(1,∞]. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5,14,16]).