Global, local and dense non-mixing of the 3D Euler equation

TL;DR

This paper demonstrates that in the 3D Euler equation flows, there are open and dense sets of initial conditions near certain steady states that will never evolve into or near those steady states, indicating a non-mixing property.

Contribution

It establishes a local non-mixing property for the 3D Euler flow near specific steady solutions using geometric and KAM theory techniques.

Findings

Existence of stationary solutions on $ ext{S}^3$ with nearby divergence-free fields that do not converge to steady states.

The set of initial conditions with non-converging flows is open and dense near certain steady states.

The results hold for flows on both $ ext{S}^3$ and $ ext{T}^3$ with different strengths of the non-mixing property.

Abstract

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 $\mathbb S^3$ and divergence-free vector fields $v_0$ arbitrarily close to $u_0$, whose (non-steady) evolution by the Euler flow cannot converge in the $C^k$ H\"older 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 $\mathbb T^3$. 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.