We consider solutions of the Navier-Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of 3d Navier-Stokes, as well as solutions which are locally approximately self-similar. This is joint work with Pierre Germain and Benjamin Harrop-Griffiths.

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.