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

We give a construction of a divergence-free vector field $u_0 \in H^s \cap B^{-1}_{\infty,\infty}$, for all $s<1/2$, with arbitrarily small norm $\Vert u_0\Vert _{B^{-1}_{\infty,\infty}}$ such that any Leray-Hopf solution to the Navier-Stokes equation starting from $u_0$ is discontinuous at $t=0$ in the metric of $B^{-1}_{\infty,\infty}$. For the Euler equation a similar result is proved in all Besov spaces $B^s_{r,\infty}$ where $s>0$ if $r>2$, and $s>n(2/r-1)$ if $1 \leq r \leq 2$. This includes the space $B^{1/3}_{3,\infty}$, which is known to be critical for the energy conservation in ideal fluids.