Ill-posedness for the Cauchy problem of the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$

For the famous Camassa-Holm equation, the well-posedness in $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $ p\in [1,\infty)$ and the ill-posedness in $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $ p\in [1,\infty],\ r\in (1,\infty]$ had been studied in \cite{d1,d2,glmy,yyg}, that is to say, it only left an open problem in the critical case $B^{1}_{\infty,1}(\mathbb{R})$ proposed by Danchin in \cite{d1,d2}. In this paper, we solve this problem by proving the norm inflation and hence the ill-posedness for the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$. Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critial Besov spaces $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $ p\in [1,\infty]$ have been completed. Finally, since the norm inflation occurs by choosing an special initial data $u_0\in B^{1}_{\infty,1}(\mathbb{R})$ but $u^2_{0x}\notin B^{0}_{\infty,1}(\mathbb{R})$ (an example implies $B^{0}_{\infty,1}(\mathbb{R})$ is not a Banach algebra), we then prove that this condition is necessary. That is, if $u^2_{0x}\in B^{0}_{\infty,1}(\mathbb{R})$ holds, then the Camassa-Holm equation has a unique solution $u(t,x)\in \mathcal{C}_T(B^{1}_{\infty,1}(\mathbb{R}))\cap \mathcal{C}^{1}_T(B^{0}_{\infty,1}(\mathbb{R}))$ and the norm inflation will not occur.