Analytical structure of fluid flows (joint work with Aleksander Danielski)

Consider the motion of an ideal (i.e. inviscid) incompressible fluid inside a 2-dimensional bounded domain. It is described by the Euler equations for the velocity field, and, equivalently, by the Lagrange equations for the fluid particle trajectories. A surprising property of the flows was found recently: for any sufficiently regular flow, the particle trajectories are real analytic curves, while the velocity field may be of finite regularity (Uriel Frisch has expressed this fact as a "very smooth ride in the stormy sea"). In particular, the flow lines of any stationary (time independent) flow are real-analytic curves. The further development of this result is the theorem stating that the set of stationary flows can be endowed with a structure of an analytic Banach manifold. The general hypothesis says that the phase space of the fluid contains a (very strange) attractor whose every component is an analytic Banach manifold, and the set of stationary flows is one of its components. In this talk I'll tell about the nontrivial history of this discovery, explain the ideas of the proof, and tell about the conjectures on the global structure of the fluid as a dynamical system. No preliminary knowledge of the fluid dynamics is assumed.