We investigate the formation of singularities in the incompressible Navier-Stokes equations in d ≥ 2 dimensions with a fractional Laplacian |∇| α . We derive analytically a sufficient but not necessary condition for 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.Scale invariance symmetry [1,2,3] holds approximately for the nonlinear Navier-Stokes equations for incompressible flow of a Newtonian fluid [4,5]. The nonlinearity and the scale invariance together can create conditions for the energy to cascade down to increasingly finer spatial and temporal scales, e.g., turbulence [5]. In two dimensions, singularities cannot form [4]. However, more than a century since the discovery of these nonlinear parabolic partial differential equations, the question remains unanswered whether or not singularities can form in three dimensions, due to the crucial role played by scale invariance. Such fundamental problems remain the subject of ongoing investigations, due to their importance to a number of fields of physics and mathematics [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].Here we address the general question of under which conditions dissipation can overcome inertial effects to prevent the formation of singularities in finite time. We answer this question by generalizing the problem via a fractional Laplacian operator, and then analytically deriving hard inequalities based on the fact that no singularity can form provided all partial space derivatives of all orders of the velocity field remain finite for all time. In terms of the d-dimensional Fourier transformṽ(k,t) of the velocity field v(x,t), we can write (2π) d/2 ∂ n1 ∂x n1 1
