Singularities of the equations of fluid motion

We explore some implications of the observed multifractal nature of the turbulent energy-dissipation field and of velocity derivatives of increasing order on the near-singularities of the Navier-Stokes equations and the singularities of Euler equations. Although these singularities occur on fractal sets of dimension close to (and only marginally less than) 3, it is shown that most of the energy dissipation is concentrated on a subset of fractal dimension about 2.87 and volume zero. Similar statements can be made with respect to velocity derivatives. In particular, it is shown that the higher the order of the velocity derivative, the less space filling the corresponding singularities become.