Chaos in the incompressible Euler equation on manifolds of high dimension

TL;DR

This paper demonstrates the existence of complex, chaotic, and invariant structures in solutions to the incompressible Euler equations on high-dimensional manifolds, revealing rich dynamical behaviors.

Contribution

It constructs finite-dimensional families of solutions exhibiting chaos and invariant structures, linking Euler dynamics to arbitrary finite-dimensional vector fields on manifolds.

Findings

Existence of strange attractors and chaos in Euler solutions

Construction of invariant manifolds of arbitrary topology

Reduction of Euler equations to finite-dimensional ODEs approximating any vector field

Abstract

We construct finite dimensional families of non-steady solutions to the Euler equations, existing for all time, and exhibiting all kinds of qualitative dynamics in the phase space, for example: strange attractors and chaos, invariant manifolds of arbitrary topology, and quasiperiodic invariant tori of any dimension. The main theorem of the paper, from which these families of solutions are obtained, states that for any given vector field $X$ on a closed manifold $N$, there is a Riemannian manifold $M$ on which the following holds: $N$ is diffeomorphic to a finite dimensional manifold in the phase space of fluid velocities (the space of divergence-free vector fields on $M$) that is invariant under the Euler evolution, and on which the Euler equation reduces to a finite dimensional ODE that is given by an arbitrarily small perturbation of the vector field $X$ on $N$.