Ill-posedness of Naiver-Stokes equations and critical Besov-Morrey spaces

The blow up phenomenon in the first step of the classical Picards scheme was proved in this paper. For certain initial spaces, Bourgain-Pavlovic and Yoneda proved the ill-posedness of the Navier-Stokes equations by showing the norm inflation in certain solution spaces. But Chemin and Gallagher said the space $dot{B}^{-1,infty}_{infty}$ seems to be optimal for some solution spaces best chosen. In this paper, we consider more general initial spaces than Bourgain-Pavlovic and Yoneda did and establish ill-posedness result independent of the choice of solution space. Our result is a complement of the previous ill-posedness results on Navier-Stokes equations.