Ill-posedness for the Cauchy problem of the modified Camassa-Holm equation in $B_{\mathrm{\infty },1}^0$

In this paper, we prove the norm inflation and get the ill-posedness for the modified Camassa-Holm equation in $B_{\mathrm{\infty },1}^0$. Therefore we completed all well-posedness and ill-posedness problem for the modified Camassa-Holm equation in all critical spaces $B_{p,1}^{\frac{1}{p}}$ with $p\in [1,\mathrm{\infty }]$.