Ill-posedness in the critical Sobolev space for the $b$-Novikov equation

TL;DR

This paper demonstrates norm inflation in the critical Sobolev space for the $b$-Novikov equation, establishing ill-posedness at the critical regularity level.

Contribution

It completes the well-posedness theory by proving ill-posedness at the critical Sobolev space $H^{3/2}( )$ for the $b$-Novikov equation.

Findings

Norm inflation occurs in $H^{3/2}( )$ for the $b$-Novikov equation.

The result confirms ill-posedness at the critical Sobolev space.

The well-posedness is only known for $s>3/2$, now extended to show ill-posedness at $s=3/2$.

Abstract

This article proves norm inflation in the critical Sobolev space $H^{3/2}(\mathbb{R})$ for the $b$-Novikov equation, which is a $1$-parameter family of Camassa-Holm-type equations with cubic nonlinearities. This result completes the well-posedness theory for this equation, which was previously known to be locally well-posed in $H^{s}(\mathbb{R})$ for $s>3/2$ and ill-posed in $H^{s}(\mathbb{R})$ for $s<3/2$.