An Essay on A Statistical Theory of Turbulence

This note is based on an on-going joint work with Prof. Sakajo (Kyoto Univ) and Prof. Matsumoto (Kyoto Univ). Revisiting Kolmogorov's statistical laws (appearing in so-called Kolmogorov's Theory of 1941) and Onsager's conjecture (1949), we make an assessment of their mathematical relevance from the view point of stochastic processes. Then we need to examine the exact meaning of Kolmogorov's fundamental hypothesis, so that we introduce a \new energy dis-sipation rate", which is inspired by the Karman-Howarth-Monin relation. Our mathematical strategy viewing turbulence may not be a conventional one: we don't assume any uid equations describing the turbulence at rst, but we re-gard turbulence as an innite dimensional probability measure on an ensemble of appropriate time-dependent vector elds on the at torus T3, which describes (a part of) Kolmogorov's statistical laws. We then consider necessary prop-erties of the ensemble in which the desired probability should be constructed. Now we have a speculation that a family of incompressible Euler ows could be our candidate, according to a number of mathematical results on the Eu