GLOBAL, LOCAL AND DENSE NON-MIXING OF THE 3D EULER EQUATION
Résumé
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 Hölder 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.
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