Lagrangian chaos, almost sure exponential mixing, and passive scalar turbulence

In this talk we will introduce some concepts of random dynamical systems and stochastic PDEs which make it possible to study the prove that Lagrangian trajectories in a variety of stochastically forced fluid equations are chaotic and lead to exponentially fast mixing of any passive scalar transported by the flow. We will briefly discuss how to use these facts to make mathematiclaly rigorous some of the classical predictions of the statistical theory of "passive scalar turbulence" in a certain, especially simple regime. These predictions were made in the 1940s and 1950s and observed to be consistent with experimental evidence, but previously lacked any mathematical justification. The work discussed is all joint with Alex Blumenthal and Sam Punshon-Smith. 

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