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

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