Abstract
We investigate the formation of singularities in incompressible flows governed by Navier-Stokes equations in d⩾2 dimensions with a fractional Laplacian |∇|α. We derive analytically a sufficient but not necessary condition for the solutions to remain always smooth and show that finite-time singularities cannot form for α⩾αc=1+d/2. Moreover, initial singularities become unstable for α>αc. The scale invariance symmetry intrinsic to the Navier-Stokes system becomes spontaneously broken, except at the critical point α=αc.