Ill-Posedness for the Cauchy Problem of the Modified Camassa-Holm Equation in \(B_{\infty ,1}^0\)

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