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

Abstract

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 $B_{p,\infty }^s$ solution for the Euler equations. We furthermore prove that the solution map of the Euler equation is not continuous in the Besov spaces from $B_{p,\infty }^s$ to $L_T^{\infty }{B}_{p,\infty }^s$ for $s>1+d/p$ with $1\le p\le \infty$ and in the Hölder 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.