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 (q>2) is considered. In 2008, Bourgain–Pavlović proved that the 3D-Navier–Stokes equation is ill-posed in 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 for q>2. However he did not treat ill-posed problem in such spaces. Thus our result is an extension of these previous results.