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 $ngeq 2$. We review the proof of strong global solvability when $alphageq alpha_{rm {tiny sc l}} (n)$, given smooth initial data. If the corresponding Euler equations for $n>2$ were to allow uncontrolled growth of the enstrophy ${1over 2} | abla u |^2_{L^2}$, 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 $alpha=alpha_{rm {tiny sc l}} (n)$.

Spontaneous symmetry breaking and finite time singularities in $d$-dimensional incompressible flow with fractional dissipation

We investigate the formation of singularities in the incompressible Navier-Stokes equations in $dgeq 2$ dimensions with a fractional Laplacian $| abla |^alpha$. We derive analytically a sufficient but not necessary condition for solutions to remain always smooth and show that finite time singularities cannot form for $alphageq alpha_c= 1+d/2$. Moreover, initial singularities become unstable for $alpha>alpha_c$.