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.
Categories: math.AP