Rearrangement, Convection and Competition
Rearrangement theory is about reorganizing a given function (or map) in some specific order (monotonicity, cycle monotonicity etc...). This is somewhat similar to the convection phenomenon in fluid mechanics, where fluid parcels are continuously reorganized in a stabler way (heavy fluid at bottom and light fluid at top). This can also be related to some competition models in economy, where agents act according to their rank. In our talk, we make these analogies more precise by analysing the Navier-Stokes equations with Boussinesq approximation of the buoyancy force and two distinct approximations of this model (Darcy and Boussinesq). We will see how these approximations are related to the concept, well known in optimal transport theory, of rearrangement of maps as gradient of convex functions.
Yann Brenier, University of Nice, France