Ill-posedness of the Euler equations and inviscid limit of the Navie-Stokes equations in Besov spaces

In this paper, we consider the Cauchy problem to the incompressible Euler and Navie-Stokes equations on the d-dimensional torus.Our aim of this paper is two fold. Firstly, we construct a new initial data and present a simple proof of the ill-posedness of the Euler equations in different senses: (1) the solution map of the Euler equations starting from $u_0$ is discontinuous at $t = 0$ in $B^s_{p,\infty}$ with $s>0$ and $1\leq p \leq \infty$, which covers the result obtained by Cheskidov and Shvydkoy in ;(2) the solution map of the Euler equations is not continuous as a map from $B^s_{p,\infty}$ to $L^\infty_T(B^s_{p,\infty})$;(3) the solution map of the Euler equations cannot be Holder continuous in time variable in Besov spaces $B^s_{p,r}$.