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 $C^k$ 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 $\mathbb T^3$. 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.

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3D 欧拉方程的全局、局部和密集非混合

我们证明了具有局部性质的 3D 欧拉方程的流动的非混合性质：在“典型”稳定解的任何邻域中，都有一组通用的初始条件，使得相应的欧拉流动永远不会进入附近原来的稳一。更准确地说，我们确定在 $\mathbb S^3$ 上存在欧拉方程的平稳解 $u_0$ 和任意接近 $u_0$ 的无散度向量场 $v_0$，其（非稳态）演化由欧拉流不能在 $C^k$ 持有人范数（$k>10$ 非整数）中收敛到 $u_0$ 的小（但先验固定）$C^k$-邻域中的任何静止状态。一组这样的初始条件 $v_0$ 在 $u_0$ 附近是开放和密集的。类似（但较弱）的陈述也适用于 $\mathbb T^3$ 上的欧拉流。