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

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

22 pages

References in corpus (1)

• Volume preserving diffeomorphisms as Poincare maps for volume preserving flows

Cited by in corpus (4)

• Geometric Hydrodynamics in Open Problems
• Twisting in Hamiltonian Flows and Perfect Fluids
• Flexibility and rigidity of free boundary MHD equilibria
• Steady Euler flows on the 3-sphere and other Sasakian 3-manifolds