The mathematics of turbulent fluids

The main motivations which nowadays guide the development of the mathematical Theory of Turbulence find their germs in the groundbreaking works of Kolmogorov and Onsager from the last century. After a brief description of those physical theories, I will review on the current state of the art in their mathematical counterpart in which the "Convex Integration" methods have been playing a major role in the last 15 years. However, the theory of turbulent flows remains vastly incomplete in view of several ubiquitous experimental phenomena which still lack their mathematical validation. A clear example is the emergence of "Intermittency". In this context, I will present some recent works that try to get a better understanding about intermittent scalings. The approach suggested here is more geometric/measure-theoretic than analytic. The presentation will be kept quite light in terms of proofs and technical details that will be replaced by some easy toy problems in which all the main concepts are apparent.