As an introduction to the workshop I will describe some basic old and new results about the Euler equation for incompressible inviscid fluids. I will focus one what can be learned from the example of the shear flow in 2d and 3d and one what are the consequences of the recent breakthroughs made first by Sheffer Shnirleman and then by De Lellis and Székelyhidi.

One of the most characteristic features of turbulence in three-dimensional incompressible fluids is enhanced energy dissipation. Most attempts to explain this phenomenon appeal to the remarkable Lagrangian properties of vorticity for inviscid flows. I review some of the ideas that have been proposed, including vortex stretching, vortex reconnection and cross-stream vortex transport, especially in light of recent work which suggests non-uniqueness of Lagrangian trajectories in the limit of zero viscosity.

I will discuss applications of harmonic analysis to the study of incompressible fluid flow. In particular, I will introduce Littlewood-Paley frequency decompositions and define Besov and related function spaces. I will then show how to construct so-called mild solutions to the Navier-Stokes equations in these spaces.