Ill-posedness of the 3D-Navier–Stokes equations in a generalized Besov space near [formula omitted

Abstract

The ill-posedness of the 3D-Navier–Stokes equations in a generalized Besov space which is smaller than B ∞ , q − 1 ( q > 2 ) is considered. In 2008, Bourgain–Pavlović 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 C 2 in the space B ∞ , 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.