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 $alpha>alpha_c$.