The ill-posedness for the rotation Camassa-Holm equation in Besov space $B^{1}_{\infty,1}(\mathbb{R})$

TL;DR

This paper demonstrates local ill-posedness of the rotation Camassa-Holm equation in a specific Besov space by constructing initial data that leads to norm inflation, highlighting the equation's instability in this setting.

Contribution

It provides the first proof of ill-posedness for the rotation Camassa-Holm equation in the Besov space $B^{1}_{ abla,1}$, using a novel construction of initial data.

Findings

Norm inflation in $B^{1}_{ abla,1}$ space

Ill-posedness of the rotation Camassa-Holm equation

Instability of solutions in the specified Besov space

Abstract

In this paper, we present a construction of $u_{0}\in B^{1}_{\infty,1}$ and get the local ill-posedness for the rotation Camassa-Holm equation modelling the equatorial water waves with the weak Coriolis effect by proving the norm inflation.