Steady Euler flows on $\mathbb{R}^3$ with wild and universal dynamics

TL;DR

This paper demonstrates that steady Euler flows in three-dimensional space can exhibit any finite-dimensional conservative dynamical behavior, including complex phenomena like horseshoes and homoclinic tangencies, through the construction of universal Beltrami fields.

Contribution

The authors introduce new perturbation methods for Beltrami fields that enable the realization of arbitrary finite codimensional dynamical phenomena in steady Euler flows.

Findings

Existence of a dense set of universal steady Euler solutions in ^3.

Steady solutions can approximate any area-preserving disk diffeomorphism.

Flows exhibit complex dynamical structures like horseshoes and homoclinic tangencies.

Abstract

Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally dense set $\mathscr G$ of stationary solutions to the Euler equations in $\mathbb R^3$ such that each vector field $X\in \mathscr G$ is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincar\'e map of $X$ at some transverse section. We remark that this universality is approximate but occurs at all scales. In particular, our results establish that a steady Euler flow may exhibit any conservative finite codimensional dynamical phenomenon; this includes the existence of horseshoes accumulated by elliptic islands, increasing union of horseshoes of Hausdorff dimension $3$ or homoclinic tangencies of arbitrarily high multiplicity. The steady solutions we construct are Beltrami fields with sharp decay at infinity. To prove these results we introduce new perturbation methods in the context of Beltrami fields that allow us to import deep techniques from bifurcation theory: the Gonchenko-Shilnikov-Turaev universality theory and the Newhouse and Duarte theorems on the geometry of wild hyperbolic sets. These perturbation methods rely on two tools from linear PDEs: global approximation and Cauchy-Kovalevskaya theorems. These results imply a strong version of V.I. Arnold's vision on the complexity of Beltrami fields in Euclidean space.