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