Ill-posedness of the incompressible Navier-Stokes equations in ${\dot{F}}_{\mathrm{\infty }}^{-1,q}(R^3)$

In this paper, authors show the ill-posedness of 3D incompressible Navier-Stokes equations in the critical Triebel-Lizorkin spaces ${\dot{F}}_{\mathrm{\infty }}^{-1,q}({\mathbb{R}}^3)$ for any $q>2$ in the sense that arbitrarily small initial data of ${\dot{F}}_{\mathrm{\infty }}^{-1,q}({\mathbb{R}}^3)$ can lead the corresponding solution to become arbitrarily large after an arbitrarily short time. In view of the well-posedness of 3D-incompressible Navier-Stokes equations in $BM{O}^{-1}$ (i.e. the Triebel-Lizorkin space ${\dot{F}}_{\mathrm{\infty }}^{-1,2}({\mathbb{R}}^3)$) by Koch and Tataru, our work completes a dichotomy of well-posedness and ill-posedness in the Triebel-Lizorkin space framework depending on $q=2$ or $q>2$.