15:45
Euler equations for incompressible fludis describe the evolution of the divergence-free velocity of a non-viscous fluid (when viscosity is present, we have the well-known Navier-Stokes equations). V. Arnold discovered that they correspond to geodesic equations in the space of volume-preserving diffeomorphisms but several exemples show that it is not always possible to solve the corresponding variational problems inducing minimal energy displacements. A solvable relaxed version, in a non-deterministic setting (measures on the path space, with possible splitting of the particles), has been introduced by Y. Brenier who intensively studied the problem. Together with M. Bernot and A. Figalli we founded new solutions and characterization results. In the talk I'll present the most interesting features of the problem and of its solutions.