We review some recent advances on the problem of fully developed turbulence within the mathematical and physical points of view. From the mathematical perspective, we discuss a framework which has been developed for the rigorous treatment of the conventional statistical theory of turbulence and address some of the rigorous results which have been obtained concerning the energy cascade, the energy spectrum, the energy dissipation rate and other physical quantities of turbulent flows. Regarding the physical approach, we focus our attention on the relevance of field theoretical methods in the analysis of dimensionally reduced models (Burgers and two-dimensional incompressible turbulence), the problem of randomly advected scalars, and intermittent fluctuations in homogeneous and isotropic turbulent flows.
Navier-Stokes equations; turbulence; energy cascade; intermittency
