Minicourse - Laurent Chevillard

These lectures will be mostly devoted to a general introduction to the phenomenology of three-dimensional, homogeneous, isotropic, fully developed turbulent flows. I hope this material will echo with several topics proposed by lecturers of this summer school. Additionally, some key statistical ingredients of this phenomenology will be illustrated, sometimes in a schematic way, by quantifying the failures (sometimes successes!) of the solution of the stochastic heat equation. In the same vein, we will define so-called fractional Gaussian fields able to reproduce the second-order statistical structure of turbulent velocity, and their generalization to a multifractal framework that takes into account fine intermittent corrections. As a final illustration, we will show how to define these random fields as the invariant measure of some simplified partial differential equations (PDEs) forced by a smooth-in-space random term. These stochastic PDEs, linear for the most part, are designed to reproduce the phenomenon of energy cascade. This last and recent aspect is done in collaboration with G. Apolinário, G. Beck, I. Gallagher, R. Grande and J.-C. Mourrat.