Concepts of generalized flows in Fluid Mechanics
Yann Brenier (CNRS, Centre
Laurent Schwartz, Ecole Polytechnique,
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.