Fractal geometry of turbulence - Navier-Stokes equations and the Hausdorff dimension

Two theorems on the role of the Hausdorff dimension in the flow of viscous incompressible fluids governed by Navier-Stokes equations are stated. The first theorem states that the set of moments when the liquid is turbulent in the Leray sense has a Hausdorff dimension equal at most to one half. The second theorem states that at any moment the fluid is turbulent, its singularities are encompassed by a set S whose dimension is at most a little greater than one.