GLOBAL, LOCAL AND DENSE NON-MIXING OF THE 3D EULER EQUATION - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Archive for Rational Mechanics and Analysis Année : 2020

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.
Fichier principal
Vignette du fichier
KKP_asymptotic.pdf (384.52 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-02922325 , version 1 (26-08-2020)

Identifiants

Citer

Boris Khesin, Sergei Kuksin, Daniel Peralta-Salas. GLOBAL, LOCAL AND DENSE NON-MIXING OF THE 3D EULER EQUATION. Archive for Rational Mechanics and Analysis, 2020, 238, pp.1087-1112. ⟨10.1007/s00205-020-01556-w⟩. ⟨hal-02922325⟩
16 Consultations
38 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More