The celebrated Kolmogorov's K41 theory of fully developed turbulence
attempts to explain and quantify "wild, but typical" behaviors of
"chaotic" fluids, most notably the lack of conservation of the total
energy. The loss of energy is not due to friction between fluid
molecules, but rather to the limited regularity of the flow.
Kolmogorov's theory is numerically and experimentally validated to a
very large extent, however, very little is known in rigorous
mathematical terms. In my lecture, I will present some aspects of
Kolmogorov's theory, and illustrate recent results (in collaboration
with M. Colombo and M. Sorella) on a related question for the linear
advection equation.