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We prove that the dynamical system defined by the hydrodynamical Eulerequation 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 andsufficiently large values of energy. This can be regarded as a 3D version ofNadirashvili's and Shnirelman's theorems showing the existence of wanderingsolutions for the 2D Euler equation. Moreover, we obtain an obstruction for themixing under the Euler flow of $C^k$-neighborhoods of divergence-freevectorfields on $M$. On the way we construct a family of functionals on thespace of divergence-free $C^1$ vectorfields on the manifold, which areintegrals of motion of the 3D Euler equation. Given a vectorfield thesefunctionals measure the part of the manifold foliated by ergodic invariant toriof fixed isotopy types. We use the KAM theory to establish some continuityproperties of these functionals in the $C^k$-topology. This allows one to get alower bound for the $C^k$-distance between a divergence-free vectorfield (inparticular, a steady solution) and a trajectory of the Euler flow.

KAM theory and the 3D Euler equation