A remark on ill-posedness

Abstract

Norm inflation implies certain discontinuous dependence of the solution on the initial value. The well-posedness of the mild solution means the existence and uniqueness of the fixed points of the corresponding integral equation. For ${\rm BMO}^{-1}$, Auscher-Dubois-Tchamitchian proved that Koch-Tataru's solution is stable. In this paper, we construct a non-Gauss flow function to show that, for classic Navier-Stokes equations, wellposedness and norm inflation may have no conflict and stability may have meaning different to $L^{\infty}(({\rm BMO}^{-1})^{n})$.