English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We prove a non-mixing property of the flow of the 3D Euler equation which hasa local nature: in any neighbourhood of a "typical" steady solution there is ageneric set of initial conditions, such that the corresponding Euler flows willnever enter a vicinity of the original steady one. More precisely, we establishthat there exist stationary solutions $u_0$ of the Euler equation on $\mathbbS^3$ 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\"oldernorm ($k>10$ non-integer) to any stationary state in a small (but fixed apriori) $C^k$-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) statementalso holds for the Euler flow on $\mathbb T^3$. Two essential ingredients inthe proof of this result are a geometric description of all steady states nearcertain nondegenerate stationary solutions, and a KAM-type argument to generateknotted invariant tori from elliptic orbits.

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