Concepts of generalized flows in Fluid Mechanics

Yann Brenier (CNRS, Centre Laurent Schwartz, Ecole Polytechnique, Palaiseau)

Since the seminal work of Euler in the 1750s, fluids have been mostly described in terms of velocity fields.

The trajectories of fluid parcels are typically recovered from their initial position by integrating the velocity field as an ordinary differential equation.

At the mathematical level, this is well understood in the framework of the Cauchy-Lipschitz theory which merely requires the velocity field to be Lipschitz continuous.

However, very small length scales are generally involved in fluid motions making the Lipschitz constant of the velocity field very large. A less demanding theory has been successively developed by DiPerna-Lions, Ambrosio, Crippa-De Lellis, which just requires the velocity field to be of bounded variation, with the interesting feature that trajectories are well defined a priori only for Lebesgue almost every initial position, which provides more flexibility in the mathematical description of fluid motions. In our lectures, we will adopt a different point of view where even the concept of velocity field is reconsidered, the basic idea being that trajectories are actually driven, through a second order differential equation, directly by the pressure field.

More generally, generalized flows will be defined as probability measures on paths (very different from the usual Wiener measure), leaving the pressure field as the only macroscopic field left in the theory.

This is perfectly consistent with the description of fluid motions in shallow domains, when vertical motions cannot be precisely observed, and also quite natural for fluid motions with strong mixing properties.