Ill-posedness for the Euler equations in Besov spaces

In the paper, we consider the Cauchy problem to the Euler equations in ${\mathbb{R}}^d$ with $d\ge 2$. We construct an initial data $u_0\in B^\sigma_{p,\infty}$ showing that the corresponding solution map of the Euler equations starting from $u_0$ is discontinuous at $t=0$ in the metric of $B_{p,\mathrm{\infty }}^{\sigma }$, which implies the ill-posedness for this equation in $B_{p,\mathrm{\infty }}^{\sigma }$. We generalize the periodic result of Cheskidov and Shvydkoy \cite{Cheskidov}.