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

TL;DR

This paper demonstrates that several shallow water wave equations, including the Camassa-Holm equation, are ill-posed in critical Sobolev and Besov spaces, indicating norm inflation and instability in these function spaces.

Contribution

It proves ill-posedness for a class of shallow water equations in critical function spaces, resolving an open problem from prior research.

Findings

Norm inflation occurs in critical Sobolev space $H^{3/2}$.

Ill-posedness extends to Besov spaces $B^{1+1/p}_{p,r}$.

Results apply to both real-line and torus cases.

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 $H^{3/2}$ and even in the Besov space $B^{1+1/p}_{p,r}$ for $p\in [1,\infty], r\in (1,\infty]$. 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}).