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-infinity,q(-1) (q > 2) is considered. In 2008, Bourgain-Pavlovic proved that the 3D-Navier-Stokes equation is ill-posted in B-infinity,infinity(-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 C-2 in the space B-infinity,q(-1) for q > 2. However he did not treat ill-posed problem in such spaces. Thus our result is an extension of these previous results. (C) 2010 Elsevier Inc. All rights reserved.