Geometrical and statistical fluid dynamics: October 11 – 17, 2017

Organized by: Uriel Frisch, Konstantin Khanin and Rahul Pandit

Some of the most basic questions relating to the Euler and Navier-Stokes equations for the motion of a 3D incompressible fluid are still open. There is a strong belief that answers to these questions cannot be obtained without creative use of geometric/Lagrangian and measure-theoretic/probabilistic tools. This is the purpose of the present workshop. It will bring together two communities – mathematicians and physicists/numericists – and will establish a common language that allows them to work together on these questions.

On the mathematical side, the emphasis will be on geometrical and statistical/stochastic methods to tackle both continuous and discrete versions of hydrodynamics.

On the physical and computational side, the novelty is, to a large extent, the realization that Lagrangian methods – where one follows fluid particles – give us much more geometric and dynamic insight than, so-called, Eulerian ones.

In recent years there has been a strong renewal of interest in the Lagrangian description of flows. It was shown that the Cauchy invariants can be used to develop highly efficient Cauchy-Lagrangian numerical schemes for the Euler equation with or without boundaries. Such schemes completely bypass the usual Courant-Friedrichs-Lewy condition on the size of time steps and thus can be orders of magnitude faster than Eulerian methods.

Turbulent flow, in the limit of large Reynolds numbers, may be viewed as weak dissipative solutions of the Euler equations. There has been much progress recently regarding the Eulerian description of such flow.

There has been an important recent development in the theory of stochastic PDEs with highly singular forcing and thus very rough paths, such as the Kardar-Parisi-Zhang (KPZ) problem. It is of considerable interest to try to apply such ideas to genuine hydrodynamical problems.

Revisiting the mathematical foundations of fluid evolution models. For example, a fundamental geometrical and measure-theoretical question can now be addressed: what is the structure of arbitrary large compositions of arbitrary small action incompressible (measure-preserving) motions without any further regularity assumptions?

This workshop is also associated with the program: Geometrical and statistical fluid dynamics.