KAM theory and the 3D Euler equation

We prove that the dynamical system defined by the hydrodynamical Euler equation on any closed Riemannian 3-manifold $M$ is not mixing in the $C^k$ topology ($k > 4$ and non-integer) for any prescribed value of helicity and sufficiently large values of energy. This can be regarded as a 3D version of Nadirashvili's and Shnirelman's theorems showing the existence of wandering solutions for the 2D Euler equation. Moreover, we obtain an obstruction for the mixing under the Euler flow of $C^k$-neighborhoods of divergence-free vectorfields on $M$. On the way we construct a family of functionals on the space of divergence-free $C^1$ vectorfields on the manifold, which are integrals of motion of the 3D Euler equation. Given a vectorfield these functionals measure the part of the manifold foliated by ergodic invariant tori of fixed isotopy types. We use the KAM theory to establish some continuity properties of these functionals in the $C^k$-topology. This allows one to get a lower bound for the $C^k$-distance between a divergence-free vectorfield (in particular, a steady solution) and a trajectory of the Euler flow.