Global, Local and Dense Non-mixing of the 3D Euler Equation

We prove a non-mixing property of the flow of the 3D Euler equation which has a local nature: in any neighbourhood of a "typical" steady solution there is a generic set of initial conditions, such that the corresponding Euler flows will never enter a vicinity of the original steady one. More precisely, we establish that there exist stationary solutions u(0) of the Euler equation on S-3 and divergence-free vector fields v(0) arbitrarily close to u(0), whose (non-steady) evolution by the Euler flow cannot converge in the C-k Holder norm (k > 10 non-integer) to any stationary state in a small (but fixed a priori) C-k-neighbourhood of u(0). The set of such initial conditions v(0) is open and dense in the vicinity of u(0). A similar (but weaker) statement also holds for the Euler flow on T-3. Two essential ingredients in the proof of this result are a geometric description of all steady states near certain nondegenerate stationary solutions, and a KAM-type argument to generate knotted invariant tori from elliptic orbits.