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Contact type solutions and non-mixing of the 3D Euler equations

Robert Cardona, Francisco Torres de Lizaur

Abstract

We prove that on any closed Riemannian three-manifold $(M,g)$ the time-dependent Euler equations are non-mixing on the space of smooth volume-preserving vector fields endowed with the $C^1$-topology, for any fixed helicity and large enough energy, solving a problem posed by Khesin, Misiolek, and Shnirelman. To prove this, we introduce a new framework that assigns contact/symplectic geometry invariants to large sets of time-dependent solutions to the Euler equations on any 3-manifold with an arbitrary fixed metric. This greatly broadens the scope of contact topological methods in hydrodynamics, which so far have had applications only for stationary solutions and without fixing the ambient metric. We further use this framework to prove that spectral invariants obtained from Floer theory, concretely embedded contact homology, define new non-trivial continuous first integrals of the Euler equations in certain regions of the phase space endowed with the $C^{1,s}$-topology, producing countably many disjoint invariant open sets.

Contact type solutions and non-mixing of the 3D Euler equations

Abstract

We prove that on any closed Riemannian three-manifold the time-dependent Euler equations are non-mixing on the space of smooth volume-preserving vector fields endowed with the -topology, for any fixed helicity and large enough energy, solving a problem posed by Khesin, Misiolek, and Shnirelman. To prove this, we introduce a new framework that assigns contact/symplectic geometry invariants to large sets of time-dependent solutions to the Euler equations on any 3-manifold with an arbitrary fixed metric. This greatly broadens the scope of contact topological methods in hydrodynamics, which so far have had applications only for stationary solutions and without fixing the ambient metric. We further use this framework to prove that spectral invariants obtained from Floer theory, concretely embedded contact homology, define new non-trivial continuous first integrals of the Euler equations in certain regions of the phase space endowed with the -topology, producing countably many disjoint invariant open sets.
Paper Structure (19 sections, 24 theorems, 103 equations, 1 figure)

This paper contains 19 sections, 24 theorems, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Divergence-free vector fields and classical invariants
  3. Divergence-free fields on Riemannian manifolds
  4. Primitive bounds of exact two-forms
  5. The classical invariants of the Euler equations: energy, helicity, homotopy
  6. Open properties of vorticities
  7. Contact type vorticities
  8. Contact type solutions of the Euler equation
  9. Cieliebak's criterion
  10. Initial conditions with prescribed data
  11. Vorticity fields with prescribed helicity
  12. Local vortex stretching
  13. Applications to ideal hydrodynamics
  14. Non $C^1$-mixing of the Euler equations
  15. Stationary solutions in the three-sphere
  16. ...and 4 more sections

Key Result

Theorem 1

Let $(M,g)$ be a closed Riemannian three-manifold. For any helicity $h\neq 0$, any homotopy class of non-vanishing vector fields $a$, and any energy $e \geq e_{0}$ (where $e_{0}$ depends on $h$ and $a$), there exist two $C^1$-open sets $C_{a,h,e}$ and $N_{a,h,e}$ of $\mathcal{V}_{a,h,e}$ such that for each $t$ where the local flow defined by the Euler equation $\varphi_t:\mathfrak{X}_\mu(M) \long

Figures (1)

  • Figure 1: Flowlines of $X$ and of $(F_K)_*(X)$.

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • Remark 7
  • Theorem 8: Gray Gray
  • Definition 9
  • ...and 42 more