Ill-posedness of the Navier–Stokes equations in a critical space in 3D

We prove that the Cauchy problem for the three-dimensional Navier–Stokes equations is ill-posed in B˙∞−1,∞ in the sense that a “norm inflation” happens in finite time. More precisely, we show that initial data in the Schwartz class S that are arbitrarily small in B˙∞−1,∞ can produce solutions arbitrarily large in B˙∞−1,∞ after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in B˙∞−1,∞ at the origin. © 2008 Elsevier Inc. All rights reserved.