Knots and links in steady solutions of the Euler equation

TL;DR

This paper proves that any locally finite link can be realized as stream lines of a steady Euler flow through a near-identity smooth diffeomorphism, linking topology with fluid dynamics.

Contribution

It establishes the existence of steady Euler solutions with prescribed knotted and linked stream lines via smooth diffeomorphisms close to the identity.

Findings

Any locally finite link can be realized as stream lines of a steady Euler flow.

The diffeomorphism can be made arbitrarily close to the identity in any $C^r$ norm.

Connects topological link structures with solutions of the Euler equations.

Abstract

Given any possibly unbounded, locally finite link, we show that there exists a smooth diffeomorphism transforming this link into a set of stream (or vortex) lines of a vector field that solves the steady incompressible Euler equation in $\mathbb{R}^3$. Furthermore, the diffeomorphism can be chosen arbitrarily close to the identity in any $C^r$ norm.