Ill-posedness of basic equations of fluid dynamics in Besov spaces

We give a construction of a divergence-free vector field $u_0 \in H^s \cap B^{-1}_{\infty,\infty}$,forall$s<1/2$, such that any Leray-Hopf solution to the Navier-Stokes equation starting from $u_0$ is discontinuous at $t=0$ in the metric of $B_{\mathrm{\infty },\mathrm{\infty }}^{-1}$. For the Euler equation a similar result is proved in all Besov spaces $B_{r,\mathrm{\infty }}^s$ where $s>0$ if $r>2$, and $s>n(2\mathrm{/}r-1)$ if $1\le r\le 2$.