In the early 1940s, Kolmogorov found explicit estimates for some
important physical quantities associated with very turbulent fluid
flows. These estimates were obtained by an elementary dimensional argument
subject to a few pertinent physical assumptions.
During the past 15 years several groups of researchers have attempted to
derive rigorously Kolmogorov's estimates from the properties of the
Navier-Stokes equations. In this lecture we will discuss both Kolmogorov's
theory and its rigorous partial confirmation arrived at in collaboration
with O.Manley, R.Temam and associates.