Ill-posedness in $B^s_{p,\infty}$ of the Euler equations: Non-continuous dependence

In this paper, we solve an open problem left in the monographs \cite[Bahouri-Chemin-Danchin, (2011)]{BCD}. Precisely speaking, it was obtained in \cite[Theorem 7.1 on pp293, (2011)]{BCD} the existence and uniqueness of $B^s_{p,\infty}$ solution for the Euler equations. We furthermore prove that the solution map of the Euler equation is not continuous in the Besov spaces from $B^s_{p,\infty}$ to $L_T^\infty B^s_{p,\infty}$ for $s>1+d/p$ with $1\leq p\leq \infty$ and in the H\"{o}lder spaces from $C^{k,\alpha}$ to $L_T^\infty C^{k,\alpha}$ with $k\in \mathbb{N}^+$ and $\alpha\in(0,1)$, which later covers particularly the ill-posedness of $C^{1,\alpha}$ solution in \cite[Trans. Amer. Math. Soc., (2018)]{MYtams}. Beyond purely technical aspects on the choice of initial data, a remarkable novelty of the proof is the construction of an approximate solution to the Burgers equation.