KAM theory and the 3d Euler equation

Consider the Euler equation on a closed 3d manifold (e.g. on the 3-torus or on the 3-sphere). Let $u(t,x)$ be its solution, and $\omega(t,x)$ – the vorticity of this solution. By the Kelvin theorem the vorticity-vectorfields, calculated for different values of time, are conjugated by means of volume-preserving diffeomorphisms. Therefore the quantity $\kappa(\omega)$, equal to the volume of the set which is the union of all two-dimensional invariant tori of $\omega$ is time-independent. We use KAM and the Arnold theorem on the structure of steady-states of the 3d Euler to prove that $\kappa$ is continuous at $\omega$'s which are non-degenerate steady-states of the equation, and use this integral of motion to study qualitative properties of the dynamical system, defined by the equation in the space of sufficiently smooth vector-fields. Namely, to study its non-ergodicity and study the problem “does the manifold of steady-states of the equation attracts (in a suitable sense) all trajectories which start from its vicinity?”.
This is a joint work with B. Khesin and D. Peralta-Salas.