A Solvability criterion for Navier-Stokes equations in high dimensions

We define the Ladyzhenskaya-Lions exponent $\alpha_{\rm {\tiny \sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-\Delta)^{\alpha}$ in ${\Bbb R}^n$, for all $n\geq 2$. We review the proof of strong global solvability when $\alpha\geq \alpha_{\rm {\tiny \sc l}} (n)$, given smooth initial data. If the corresponding Euler equations for $n>2$ were to allow uncontrolled growth of the enstrophy ${1\over 2} \|\nabla u \|^2_{L^2}$, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for $\alpha<\alpha_{\rm {\tiny \sc l}} (n)$. The energy is critical under scale transformations only for $\alpha=\alpha_{\rm {\tiny \sc l}} (n)$.