Georgia Institute of Technology College of Sciences

Consider the 2-d ideal incompressible fluid moving inside a bounded domain (say 2-d torus). It is described by 2-d Euler equations which have unique global solution; thus, we have a dynamical system in the space of sufficiently regular incompressible vector fields. The global properties of this system are poorly studied, and, as much as we know, paradoxical. It turns out that there exists a global attractor (in the energy norm), i.e. a set in the phase space attracting all trajectories (in spite the fact that the system is conservative). This apparent contradiction leads to some deep questions of non-equilibrium statistical mechanics.