Ill-posedness of the 3D-Navier–Stokes equations in a generalized Besov space near BMO−1

The ill-posedness of the 3D-Navier–Stokes equations in a generalized Besov space which is smaller than B−1 ∞,q (q > 2) is considered. In 2008, Bourgain–Pavlovi´c proved that the 3D-Navier–Stokes equation is ill-posed in B−1 ∞,∞ by showing norm inflation phenomena of the solution for some initial data. On the other hand, in 2008, Germain proved that the flow map is not C2 in the space B−1 ∞,q for q >2. However he did not treat ill-posed problem in such spaces. Thus our result is an extension of these previous results. © 2010 Elsevier Inc. All rights reserved