Ill-posedness of the Novikov equation in the critical Besov space $B^{1}_{\infty,1}(\mathbb{R})$

TL;DR

This paper proves the ill-posedness of the Novikov equation in the critical Besov space $B^{1}_{\infty,1}(\mathbb{R})$ by demonstrating norm inflation, completing the understanding of well-posedness for this equation in critical spaces.

Contribution

It establishes the ill-posedness of the Novikov equation in the critical Besov space $B^{1}_{\infty,1}(\mathbb{R})$, filling a gap in the endpoint case analysis.

Findings

Proves ill-posedness of Novikov equation in $B^{1}_{\infty,1}(\mathbb{R})$

Demonstrates norm inflation phenomena in the critical space

Completes the endpoint case analysis for Novikov equation well-posedness

Abstract

It is shown that both the Camassa-Holm and Novikov equations are ill-posed in $B_{p,r}^{1+1/p}(\mathbb{R})$ with $(p,r)\in[1,\infty]\times(1,\infty]$ in \cite{Guo2019} and well-posed in $B_{p,1}^{1+1/p}(\mathbb{R})$ with $p\in[1,\infty)$ in \cite{Ye}. Recently, the ill-posedness for the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$ has been proved in \cite{Guo}. In this paper, we shall solve the only left an endpoint case $r=1$ for the Novikov equation. More precisely, we prove the ill-posedness for the Novikov equation in $B^{1}_{\infty,1}(\mathbb{R})$ by exhibiting the norm inflation phenomena.