Hydrodynamics and the geometry of the diffeomorphisms group on the torus
by Ana Bela Cruzeiro (GFMUL / Departamento de Matemática do IST, Portugal)
In the spirit of Arnold's approach to hydrodynamics, where Euler equation can be regarded as a geodesic on the group of diffeomorphisms (in this case of the torus), we study the geometry of this group.
- We prove that for the 2-dimensional case the Ricci curvature is positive. Nevertheless probabilistic arguments show that the Euler flow is not ergodic (based on a joint work with P. Malliavin).
- By stochastically perturbing Arnold's framework we derive Navier-Stokes equation as a generalized geodesic (work in progress)