Mathematics Colloquia and Seminars

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One basic assumption in the theory of turbulence is that under large scale forcing, energy is transferred throguh nonlinearity to the small scales and the system establishes a unique statistical steady state. In numerical simulations, one typically forces very few low modes when studying the direct cascade process. Statistical properties of the turbulent flow are measured or calculated through time averaging rather than ensemble averaging.

One main purpose of this talk is to rigorously establish the validity of this basic assumption. We find it convenient to study this problem in a stochastic setting.

The ergodic theory of stochastic PDEs is delicate and poorly understood (when compared with stochastic ODEs). I will describe a technique which gives a number of new results for the 2D Stochastically Forced Navier Stokes Equation. It mixes a dynamic and a statistical understanding of the equation, building on ideas from infinite dimensional dynamical systems, statistical mechanics, and Markov chain theory. One could alternatively describe the approach as coupling with infinite memory or as the lack of phase transitions when correlations decay exponentially.

Along the way I will explain how there are different mechanisms for convergence in different regimes and how this is the key to making progress. I will use a number of simple examples to illustrate these points.