The statistical approach to 2D incompressible fluid dynamics

Grotto Francesco, Università di Pisa

The statistical approach to turbulence, that is the use of random fields in the mathematical modeling of turbulent flows, has its roots in the classical works of the first half of the XX century, by renowned scholars such as Taylor, Kolmogorov, Onsager, Batchelor. Later developments by Kraichnan and many others have established the subject as a fundamental topic in fluid dynamics, both from the physical and mathematical point of view. Incompressible fluids in 2D are an idealization particularly suited to describe flows such as those common in geophysics. In this context the main efforts involving the statistical approach have been devoted to the description of characteristic 2D phenomena, the most notable example being the inverse cascade of energy in turbulent flows. I will give a brief review on this wide topic, focusing on relevant research directions in mathematics such as: Onsager's negative temperature ensembles, non-uniqueness and instability in 2D Euler's equations, negative eddy viscosity phenomena and the formation of coherent structures.

Area: IS11 - Stochastic Fluid Dynamics (Francesco Grotto)

Keywords: 2D Euler equations, 2D Navier-Stokes equations, Point Vortex models