On the Navier-Stokes equations in scaling-invariant spaces in any dimension

We study the Navier-Stokes equations with a dissipative term that is generalized through a fractional Laplacian in any dimension higher than two. We extend the horizontal Biot-Savart law beyond dimension three. Using the anisotropic Littlewood-Paley theory with which we distinguish the first two directions from the rest, we obtain a blow-up criteria for its solution in norms which are invariant under the rescaling of these equations. The proof goes through for the classical Navier-Stokes equations if dimension is three, four or five. We also give heuristics and partial results toward further improvement.