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

It is shown that both the Camassa-Holm and Novikov equations are ill-posed in $B_{p,r}^{1+1\mathrm{/}p}(\mathbb{R})$ with $(p,r)\in [1,\mathrm{\infty }]\times (1,\mathrm{\infty }]$ in \cite{Guo2019} and well-posed in $B_{p,1}^{1+1\mathrm{/}p}(\mathbb{R})$ with $p\in [1,\mathrm{\infty })$ in \cite{Ye}. Recently, the ill-posedness for the Camassa-Holm equation in $B_{\mathrm{\infty },1}^1(\mathbb{R})$ 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 $B_{\mathrm{\infty },1}^1(\mathbb{R})$ by exhibiting the norm inflation phenomena.