Experiments and numerical simulations show that energy dissipation in incompressible fluid turbulence tends to a positive value in the inviscid limit (infinite Reynolds number). Lars Onsager (1949) proposed an explanation for this phenomenon in terms of energy cascade for certain
singular solutions of Euler equations. We shall review current ideas on the nature of turbulent energy cascade and their status within rigorous
theory of PDE's. In particular, we shall discuss a classical picture of Geoffrey Taylor (1937) on the role of vortex line-stretching in generating
turbulent energy dissipation. Taylor's argument was based on a statistical hypothesis that material lines in a turbulent flow will tend to elongate, on average, and appealed to the Kelvin Theorem (1869) on conservation of circulations. For smooth solutions the Kelvin Theorem for all loops is equivalent to the Euler equations of motion, but we shall present rigorous results which suggest that the theorem breaks down in turbulent flow due to nonlinear effects. This turbulent "cascade of circulations" has been verified by high-Reynolds-number numerical simulations. We propose another conjecture, that circulations on material loops may be martingales of a generalized Euler flow (in the sense of Brenier and Shnirelman). We shall
show that this property has a close analogue in the "Kraichnan model" of random advection, which accounts for anomalous scalar dissipation in that model. The "Kraichnan model" is also known to probabilists as a generalized stochastic flow and its basic features have been put on a rigorous footing by Le Jan and Raimond (2002, 2004). We propose a geometric treatment of this model, formally as a diffusion process on an infinite-dimensional semi-group of volume-preserving maps.
Best constants are found for a class of multiplicative inequalities that give an estimate of the C-norm of a function in terms of the product of the L_2-norms of the powers of the Laplace operator. Special attention is given to functions defined on the sphere S^n.
This series of lectures will be concerned with the asymptotic behavior of some random dynamical system. The push-forward and pull-back approaches will be discussed. Some applications to stochastic reaction-diffusion equations and stochastic Navier-Stokes equations will be given.