  In this paper, we prove the norm inflation and get the ill-posedness for the
modified Camassa-Holm equation in $B_{\infty,1}^0$. 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 $p\in[1,\infty]$.

Analysis of PDEs