  We prove that the Cauchy problem for the three dimensional Navier-Stokes
equations is ill posed in $\dot{B}^{-1,\infty}_{\infty}$ in the sense that a
``norm inflation'' happens in finite time. More precisely, we show that initial
data in the Schwartz class $\mathcal{S}$ that are arbitrarily small in
$\dot{B}^{-1, \infty}_{\infty}$ can produce solutions arbitrarily large in
$\dot{B}^{-1, \infty}_{\infty}$ after an arbitrarily short time. Such a result
implies that the solution map itself is discontinuous in $\dot{B}^{-1,
\infty}_{\infty}$ at the origin.
